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Reflecting on My Own Math Experiences

Hi thank you so much for being here..

Welcome! I am so glad you have come across this post! My name is Julia Park and I am a senior at Millersville University! I am an Early Childhood Education major and I have learned so much so far! If you have a moment, feel free to check out my previous blog posts!

In my last post, I shared information about learning centers in math class! In this post, I will be reflecting on my mathematical journey. My experiences in math have really shaped the way I teach my students.

My Early Math Memories

I believe that early math experiences can really shape a child’s mindset towards mathematics. It has definitely shaped mine. Unfortunately, it has been a long journey of growing my interest in math, and I am still working on it! 

When I was in elementary school, even up until my time at Millersville, math has been a huge struggle for me. I have grown up with the incredibly damaging misconception that you have to be a “math person” to excel in math.  A lot of my peers had the same mindset, which made it even harder to let go of those limiting thoughts. 

I discussed this in my growth mindset blog post , but “math people” do not exist! I have my own reasons as to why I thought there were math people, but children’s experiences often vary. I think my fixed mindset was formed from experiences with not-so-nice teachers, the pressure of time limits and the need for accuracy in class, and a lack of hands-on learning. Those are just a few ideas of why I think I have had a tough time with math and I will be discussing more ideas later in this post!

Although it was hard to get through math class sometimes, I am really grateful that I have had these experiences because I can learn from them and relate to my own students. I want my students to feel comfortable with asking for help and to know that it is possible to learn and grow in many ways!

What I Have Learned From Past Teachers

Through my time as a student in math class, I have had many different experiences with a variety of teachers. I want to share the good and the bad of what I have gone through because I think it is beneficial for teachers to reflect on all experiences related to learning. We can take what we learn to inform our own teaching practices. 

Positive approaches I have learned from teachers:

• Providing assistance outside of class
• Using a hands-on learning approach
• Giving time to practice skills in class
• Utilizing interactive math games
• Facilitating class discussions 
• Being kind and encouraging when a student is struggling 

Approaches of teachers that were difficult for me: 

• Focusing on accuracy only and not effort
• Putting pressure on students to turn in extensive assignments with a limited amount of time
• Teaching new concepts too fast
• Using too many lectures and PowerPoint presentations
• Not having time to reflect on concepts in class
• Being intimidating when a student is struggling 

Every student learns differently. These experiences are unique to me and not everyone will be able to relate to what I have taken from my past math classes. However, I think it is important to recognize that although one strategy might work for one student, it might not work for another student.  This notion emphasizes the need for differentiation. I will be discussing differentiation more in the next section. 

Strategies I Want to Use to Teach Math

As I finish this semester at Millersville University, I am leaving with so many new ways of teaching math that I was not even aware of previously. I have a new passion for making math class fun and interesting for my students. The following are some examples of strategies I would love to incorporate in my future math class: 

• My math instruction will be differentiated based on my students’ needs. I will monitor their progress through various assessments and observations to modify or individualize my instruction when needed.
• Hands-on learning will be included to increase the engagement and participation of my students. I want to make math fun and exciting!!
• Class discussions will be a huge part of my mathematics instruction. Discussions in math class promote a deeper understanding of mathematical concepts in children.  
• I would love to try to use interactive notebooks to organize my students’ learning and create engaging experiences. I had not heard of these notebooks until this year and I love them!
• Technology , manipulatives , and children’s literature are just a few tools I plan on using to enhance mathematics instruction for my students. 
• Parent involvement is very important for a child’s education and I will consistently keep in contact with families to increase this involvement. 
• I am very passionate about modeling a growth mindset for my students. I want my students to believe in themselves and in their ability to grow.
• I will strive to create a safe and welcoming environment for my students. I want them to be comfortable with sharing their ideas and to not fear making mistakes. To do this, I will value effort just as much as accuracy. 

Mistakes Are Learning Opportunities!

One of the biggest lessons I have learned throughout my time at Millersville is that making mistakes is okay. I used to put so much pressure on myself to be perfect and know everything, but that is not healthy. Teachers are not robots made to feed information to students. Instead, we have a purpose to learn alongside our students and to welcome mistakes as learning opportunities.

I am much more comfortable now being honest with my students in moments of uncertainty. I would rather figure something out with them than provide them with the wrong information. It’s really fun to explore ideas with students and work together toward a common goal. These experiences with students are valuable and strengthen the student-teacher relationship. When children trust their teachers, they are more engaged, motivated, and feel an increased amount of comfort when reaching out for help and sharing their thoughts with others. 

Check out my blog post about growth mindset to learn more about the importance of making mistakes and the value of having a positive mindset in math class!

Thank you so much for reading!

I had a blast sharing my mathematical experiences with you all! I have grown so much through the years and I can’t wait to keep growing as I gain more experience. I hope you learned about some ways you can teach mathematics in your own classroom! Thank you for reading. I sincerely appreciate it!

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Hi! I am Julia Park and I'm a junior at Millersville University. I am currently studying Early Childhood Education. I am so excited to share my journey through my new blog! View all posts by Julia Park

Student Experience Research Network Blog

January 22, 2020

Exploring Student Experience in Mathematics Learning

Chloe stroman.

Mathematics, especially when compared to subjects in the arts and humanities, is often mischaracterized as a static set of facts and rules, insusceptible to bias, and disconnected from other disciplines and issues in the world.

But in mathematics contexts, students have the same psychological needs that they have in other academic settings to belong, feel a sense of relevance and purpose, and believe that their abilities can grow. And, as in all subjects, students’ experience in mathematics is shaped by their identity and the norms and narratives that comprise their home and classroom culture.

At a recent panel, MSN Director of Research and Senior Fellow Shanette Porter moderated a discussion about how designing mathematics contexts with a focus on culture, curiosity, vulnerability, and structures for failure and redemption can lead to better mathematics learning and achievement.

The video of the session featuring Nathan Alexander (Morehouse College), Elham Kazemi (University of Washington), Dan Meyer (Desmos), and Na’ilah Suad Nasir (Spencer Foundation) is now available online.

Nathan Alexander and Na’ilah Suad Nasir are mentors in our Inclusive Mathematics Environments Early Career Fellowship , along with several members of the MSN. In the past year, many researchers in our community and their colleagues have furthered our understanding of how learning mindsets, classroom practices, relationships between educators and students, and inequitable access to mathematics courses influence students’ experiences and outcomes.

• Jo Boaler shared an excerpt of her new book Mathematical Mindsets that outlines research on, and strategies for, teaching mathematics in a conceptual rather than procedural way in order to combat mathematics anxiety and build students’ ability to think flexibly about mathematics.
• In a sample of German fourth graders, Andrei Cimpian found that teachers’ belief that success in mathematics requires innate ability may be an obstacle to fostering engagement and learning among students receiving lower grades.
• Chris Hulleman and Inclusive Mathematics Environments Early Career Fellow Stacy Priniski developed and tested three utility value interventions in online high school mathematics courses , illustrating the costs and benefits of different implementation choices (paywall) . Stephanie Wormington and Chris Hulleman found that community college students in online developmental mathematics courses received lower grades than face-to-face students , with online adult learners receiving particularly low grades. In a third study co-authored by Chris Hulleman, a utility value intervention administered in an intermediate mathematics course at a two-year college had a positive effect on men’s passing rates (paywall) .
• Yasmiyn Irizarry examined teachers’ ratings of academic ability across different racialized groups, finding that black first-graders received lower ratings in language and literacy and Asian first-graders received higher ratings in mathematics (paywall).
• Michal Kurlaender co-authored a report on mathematics course-taking among high school seniors in California , finding patterns of inequitable access to advanced mathematics pathways based on students’ race/ethnicity and school characteristics, which may in turn have limited students’ postsecondary opportunity.
• Jamaal Matthews wrote about inclusive pedagogies that can be used to create mathematics classrooms that support black girls’ learning and well-being. In another study, he examined teachers’ formative experiences as mathematics students and how those experiences relate to the teachers’ classroom care practices and student outcomes. Both papers are available on the THREADS website .
• Barbara Schneider and National Study of Learning Mindsets Early Career Fellow Soobin Kim found positive effects of a statewide college-preparatory policy , called the Michigan Merit Curriculum, on high school mathematics course-taking and college enrollment (paywall).
• A study by Tanner LeBaron Wallace revealed that adolescents’ perceptions of how well their mathematics teacher knew them were related to their self-reported value of mathematics and classroom experience , particularly for students of color (paywall) .
• Among female college students in virtual classrooms, Gregory Walton found that students who experienced sexist behavior from their instructor performed worse , regardless of the gender of the student’s avatar representing them in the classroom.

Several recently published snapshots from our National Study of Learning Mindsets Early Career Fellowship and our Mindsets and the Learning Environment research portfolio also investigate topics in mathematics:

• The formation of learning profiles in context: Mathematics anxiety, achievement, and interest in adolescents  by Michael Broda
• Estimating the impact of growth mindset on high school mathematics performance and course-taking  by Soobin Kim
• Can a growth mindset program overcome persistent messages about the stability of intelligence?  by Alison Koenka
• Adolescents’ fixed mindset and stereotype concerns in mathematics: Their relations to anxiety, challenge avoidance, and achievement  by Eunjin Seo
• Relations among students’ motivation, mathematics anxiety, and mathematics achievement  by Nicole Sorhagen
• The relationship among growth mindset classroom climate, trust and respect, and student performance in mathematics by Chloe Stroman

We are excited to continue learning, as a network, about what it takes to create mathematics environments that support positive learning experiences for all students.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License .

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What Students Are Saying About the Value of Math

We asked teenagers: Do you see the point in learning math? The answer from many was “yes.”

By The Learning Network

“Mathematics, I now see, is important because it expands the world,” Alec Wilkinson writes in a recent guest essay . “It is a point of entry into larger concerns. It teaches reverence. It insists one be receptive to wonder. It requires that a person pay close attention.”

In our writing prompt “ Do You See the Point in Learning Math? ” we wanted to know if students agreed. Basic arithmetic, sure, but is there value in learning higher-level math, such as algebra, geometry and calculus? Do we appreciate math enough?

The answer from many students — those who love and those who “detest” the subject alike — was yes. Of course math helps us balance checkbooks and work up budgets, they said, but it also helps us learn how to follow a formula, appreciate music, draw, shoot three-pointers and even skateboard. It gives us different perspectives, helps us organize our chaotic thoughts, makes us more creative, and shows us how to think rationally.

Not all were convinced that young people should have to take higher-level math classes all through high school, but, as one student said, “I can see myself understanding even more how important it is and appreciating it more as I get older.”

Thank you to all the teenagers who joined the conversation on our writing prompts this week, including students from Bentonville West High School in Centerton, Ark, ; Harvard-Westlake School in Los Angeles ; and North High School in North St. Paul, Minn.

Please note: Student comments have been lightly edited for length, but otherwise appear as they were originally submitted.

“Math is a valuable tool and function of the world.”

As a musician, math is intrinsically related to my passion. As a sailor, math is intertwined with the workings of my boat. As a human, math is the building block for all that functions. When I was a child, I could very much relate to wanting a reason behind math. I soon learned that math IS the reason behind all of the world’s workings. Besides the benefits that math provides to one’s intellect, it becomes obvious later in life that math is a valuable tool and function of the world. In music for example, “adolescent mathematics” are used to portray functions of audio engineering. For example, phase shifting a sine wave to better project sound or understanding waves emitted by electricity and how they affect audio signals. To better understand music, math is a recurring pattern of intervals between generating pitches that are all mathematically related. The frets on a guitar are measured precisely to provide intervals based on a tuning system surrounding 440Hz, which is the mathematically calculated middle of the pitches humans can perceive and a string can effectively generate. The difference between intervals in making a chord are not all uniform, so guitar frets are placed in a way where all chords can sound equally consonant and not favor any chord. The power of mathematics! I am fascinated by the way that math creeps its way into all that I do, despite my plentiful efforts to keep it at a safe distance …

— Renan, Miami Country Day School

“Math isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life.”

I believe learning mathematics is both crucial to the learning and development of 21st century students and yet also not to be imposed upon learners too heavily. Aside from the rise in career opportunity in fields centered around mathematics, the skills gained while learning math are able to be translated to many facets of life after a student’s education. Learning mathematics develops problem solving skills which combine logic and reasoning in students as they grow. The average calculus student may complain of learning how to take derivatives, arguing that they will never have to use this after high school, and in that, they may be right. Many students in these math classes will become writers, musicians, or historians and may never take a derivative in their life after high school, and thus deem the skill to do so useless. However, learning mathematics isn’t about taking derivatives or solving for x, it’s about having the skills to do so and putting them to use elsewhere in life. A student who excels at calculus may never use it again, but with the skills of creativity and rational thinking presented by this course, learning mathematics will have had a profound effect on their life.

— Cam, Glenbard West

“Just stop and consider your hobbies and pastimes … all of it needs math.”

Math is timing, it’s logic, it’s precision, it’s structure, and it’s the way most of the physical world works. I love math — especially algebra and geometry — as it all follows a formula, and if you set it up just right, you can create almost anything you want in at least two different ways. Just stop and consider your hobbies and pastimes. You could be into skateboarding, basketball, or skiing. You could be like me, and sit at home for hours on end grinding out solves on a Rubik’s cube. Or you could be into sketching. Did you know that a proper drawing of the human face places the eyes exactly halfway down from the top of the head? All of it needs math. Author Alec Wilkinson, when sharing his high school doubting view on mathematics, laments “If I had understood how deeply mathematics is embedded in the world …” You can’t draw a face without proportions. You can’t stop with your skis at just any angle. You can’t get three points without shooting at least 22 feet away from the basket, and get this: you can’t even ride a skateboard if you can’t create four congruent wheels to put on it.

— Marshall, Union High School, Vancouver, WA

“Math gives us a different perspective on everyday activities.”

Even though the question “why do we even do math?” is asked all the time, there is a deeper meaning to the values it shares. Math gives us a different perspective on everyday activities, even if those activities in our routine have absolutely nothing to do with mathematical concepts itself. Geometry, for instance, allows us to think on a different level than simply achieving accuracy maintains. It trains our mind to look at something from various viewpoints as well as teaching us to think before acting and organizing chaotic thoughts. The build up of learning math can allow someone to mature beyond the point where if they didn’t learn math and thought through everything. It paves a way where we develop certain characteristics and traits that are favorable when assisting someone with difficult tasks in the future.

— Linden, Harvard-Westlake High School, CA

“Math teaches us how to think.”

As explained in the article, math is all around us. Shapes, numbers, statistics, you can find math in almost anything and everything. But is it important for all students to learn? I would say so. Math in elementary school years is very important because it teaches how to do simple calculations that can be used in your everyday life; however middle and high school math isn’t used as directly. Math teaches us how to think. It’s far different from any other subject in school, and truly understanding it can be very rewarding. There are also many career paths that are based around math, such as engineering, statistics, or computer programming, for example. These careers are all crucial for society to function, and many pay well. Without a solid background in math, these careers wouldn’t be possible. While math is a very important subject, I also feel it should become optional at some point, perhaps part way through high school. Upper level math classes often lose their educational value if the student isn’t genuinely interested in learning it. I would encourage all students to learn math, but not require it.

— Grey, Cary High School

“Math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful.”

Although I loved math as a kid, as I got older it felt more like a chore; all the kids would say “when am I ever going to use this in real life?” and even I, who had loved math, couldn’t figure out how it benefits me either. This was until I started asking my dad for help with my homework. He would go on and on about how he used the math I was learning everyday at work and even started giving me examples of when and where I could use it, which changed my perspective completely. Ultimately, I believe that math is a valuable tool for everyone to learn, but students need better influences to show them why it’s useful and where they can use it outside of class.

— Lilly, Union High School

“At the roots of math, it teaches people how to follow a process.”

I do believe that the math outside of arithmetic, percentages, and fractions are the only math skills truly needed for everyone, with all other concepts being only used for certain careers. However, at the same time, I can’t help but want to still learn it. I believe that at the roots of math, it teaches people how to follow a process. All mathematics is about following a formula and then getting the result of it as accurately as possible. It teaches us that in order to get the results needed, all the work must be put and no shortcuts or guesses can be made. Every equation, number, and symbol in math all interconnect with each other, to create formulas that if followed correctly gives us the answer needed. Everything is essential to getting the results needed, and skipping a step will lead to a wrong answer. Although I do understand why many would see no reason to learn math outside of arithmetic, I also see lessons of work ethics and understanding the process that can be applied to many real world scenarios.

— Takuma, Irvine High School

“I see now that math not only works through logic but also creativity.”

A story that will never finish resembling the universe constantly expanding, this is what math is. I detest math, but I love a never-ending tale of mystery and suspense. If we were to see math as an adventure it would make it more enjoyable. I have often had a closed mindset on math, however, viewing it from this perspective, I find it much more appealing. Teachers urge students to try on math and though it seems daunting and useless, once you get to higher math it is still important. I see now that math not only works through logic but also creativity and as the author emphasizes, it is “a fundamental part of the world’s design.” This view on math will help students succeed and have a more open mindset toward math. How is this never-ending story of suspense going to affect YOU?

— Audrey, Vancouver, WA union high school

“In some word problems, I encounter problems that thoroughly interest me.”

I believe math is a crucial thing to learn as you grow up. Math is easily my favorite subject and I wish more people would share my enthusiasm. As Alec Wilkinson writes, “Mathematics, I now see, is important because it expands the world.” I have always enjoyed math, but until the past year, I have not seen a point in higher-level math. In some of the word problems I deal with in these classes, I encounter problems that thoroughly interest me. The problems that I am working on in math involve the speed of a plane being affected by wind. I know this is not riveting to everyone, but I thoroughly wonder about things like this on a daily basis. The type of math used in the plane problems is similar to what Alec is learning — trigonometry. It may not serve the most use to me now, but I believe a thorough understanding of the world is a big part of living a meaningful life.

— Rehan, Cary High School

“Without high school classes, fewer people get that spark of wonder about math.”

I think that math should be required through high school because math is a use-it-or-lose-it subject. If we stop teaching math in high school and just teach it up to middle school, not only will many people lose their ability to do basic math, but we will have fewer and fewer people get that spark of wonder about math that the author had when taking math for a second time; after having that spark myself, I realized that people start getting the spark once they are in harder math classes. At first, I thought that if math stopped being required in high school, and was offered as an elective, then only people with the spark would continue with it, and everything would be okay. After thinking about the consequences of the idea, I realized that technology requires knowing the seemingly unneeded math. There is already a shortage of IT professionals, and stopping math earlier will only worsen that shortage. Math is tricky. If you try your best to understand it, it isn’t too hard. However, the problem is people had bad math teachers when they were younger, which made them hate math. I have learned that the key to learning math is to have an open mind.

— Andrew, Cary High School

“I think math is a waste of my time because I don’t think I will ever get it.”

In the article Mr. Wilkinson writes, “When I thought about mathematics at all as a boy it was to speculate about why I was being made to learn it, since it seemed plainly obvious that there was no need for it in adult life.” His experience as a boy resonates with my experience now. I feel like math is extremely difficult at some points and it is not my strongest subject. Whenever I am having a hard time with something I get a little upset with myself because I feel like I need to get everything perfect. So therefore, I think it is a waste of my time because I don’t think I will ever get it. At the age of 65 Mr. Wilkinson decided to see if he could learn more/relearn algebra, geometry and calculus and I can’t imagine myself doing this but I can see myself understanding even more how important it is and appreciating it more as I get older. When my dad was young he hated history but, as he got older he learned to appreciate it and see how we can learn from our past mistakes and he now loves learning new things about history.

— Kate, Cary High School

“Not all children need to learn higher level math.”

The higher levels of math like calculus, algebra, and geometry have shaped the world we live in today. Just designing a house relates to math. To be in many professions you have to know algebra, geometry, and calculus such as being an economist, engineer, and architect. Although higher-level math isn’t useful to some people. If you want to do something that pertains to math, you should be able to do so and learn those high levels of math. Many things children learn in math they will never use again, so learning those skills isn’t very helpful … Children went through so much stress and anxiety to learn these skills that they will never see again in their lives. In school, children are using their time learning calculus when they could be learning something more meaningful that can prepare them for life.

— Julyssa, Hanover Horton High School

“Once you understand the basics, more math classes should be a choice.”

I believe that once you get to the point where you have a great understanding of the basics of math, you should be able to take more useful classes that will prepare you for the future better, rather than memorizing equations after equations about weird shapes that will be irrelevant to anything in my future. Yes, all math levels can be useful to others’ futures depending on what career path they choose, but for the ones like me who know they are not planning on encountering extremely high level math equations on the daily, we should not have to take math after a certain point.

— Tessa, Glenbard West High School

“Math could shape the world if it were taught differently.”

If we learned how to balance checkbooks and learn about actual life situations, math could be more helpful. Instead of learning about rare situations that probably won’t come up in our lives, we should be learning how to live on a budget and succeed money-wise. Since it is a required class, learning this would save more people from going into debt and overspending. In schools today, we have to take a specific class that doesn’t sound appealing to the average teenager to learn how to save and spend money responsibly. If it was required in math to learn about that instead of how far Sally has to walk then we would be a more successful nation as a whole. Math could shape the world differently but the way it is taught in schools does not have much impact on everyday life.

— Becca, Bentonville West High School

“To be honest, I don’t see the point in learning all of the complicated math.”

In a realistic point of view, I need to know how to cut a cake or a piece of pie or know how to divide 25,000 dollars into 10 paychecks. On the other hand, I don’t need to know the arc and angle. I need to throw a piece of paper into a trash can. I say this because, in all reality and I know a lot of people say this but it’s true, when are we actually going to need this in our real world lives? Learning complicated math is a waste of precious learning time unless you desire to have a career that requires these studies like becoming an engineer, or a math professor. I think that the fact that schools are still requiring us to learn these types of mathematics is just ignorance from the past generations. I believe that if we have the technology to complete these problems in a few seconds then we should use this technology, but the past generations are salty because they didn’t have these resources so they want to do the same thing they did when they were learning math. So to be honest, I don’t see the point in learning all of the complicated math but I do think it’s necessary to know the basic math.

— Shai, Julia R Masterman, Philadelphia, PA

Learn more about Current Events Conversation here and find all of our posts in this column .

Why I Center Student Experiences in My Math Class

Fourth grade teacher Neven Holland describes how equitable mathematics and quality content help to ensure all students have multiple avenues to access and approach math learning.

By Neven Holland Fourth Grade Teacher, Memphis-Shelby County Schools

If my younger self met me now, he would be very surprised that I ended up becoming a math teacher. He would probably remember how hard it was to feel connected to mathematics because the content rarely felt relevant to any personal experiences. 

I can’t recall many (if any) teachers who conveyed why math mattered in the real world or the role it could play in my life. The instructional materials I had access to were mostly pages of problems and numbers and felt entirely unrelated to me, my family, or the community where I lived . I never had math teachers who looked like me or offered examples of how I could succeed in mathematics or who highlighted why my experiences were valuable and applicable to what we were learning. 

"Throughout my education, math remained a subject meant for others, for those who could solve problems exactly like the teacher did and who could see themselves in the materials."

Because I struggled to connect with the content as well as my teachers, I simply felt like I wasn’t good at math—I wasn’t a “math person.” Throughout my education, math remained a subject meant for others, for those who could solve problems exactly like the teacher did and who could see themselves in the materials. 

So how did I end up as a fourth grade math teacher in Memphis, Tennessee? I was on a path to a career in forensic psychology but quickly realized this was not the career for me. I was searching for a real calling which ultimately led me to a teacher residency program. At the time, I honestly wasn’t sure about teaching, but I took a leap of faith and enrolled. It turned out to be the best decision I ever made.

During my residency I experienced the beauty and power of mathematics for the first time. The class was so different from any I had taken before. The instructor incorporated our experiences, asked us questions to probe our thinking, and pushed us to share our ideas. She encouraged mistakes and multiple ways of arriving at an answer. I began to see mathematics as a way to understand the world and creatively solve problems. 

Not only did my time in teacher training help me to realize how much I loved math, I also came to understand that for all students to see math the way I now did, accessibility was key. I promised myself that I would teach math in a way that celebrated and incorporated my students’ experiences through high-quality, engaging, and culturally relevant content. 

"I promised myself that I would teach math in a way that celebrated and incorporated my students' experiences through high-quality, engaging, and culturally relevant content."

There are so many misunderstandings right now about what a culturally relevant and equitable curriculum means in the classroom. For me, it’s quite simple: creating a learning environment that honors student needs and experiences. And when teachers are supported with high-quality materials, they’re not only provided with a foundation of grade-level content, they can also leverage built in scaffolds and embedded assessments to ensure targeted supports for each student. 

3 Ways to Center Student Experiences in Mathematics

When I talk about equitable mathematics, I’m talking about classrooms where students are centered in the learning while working toward the mastery of college and career-ready standards. Equitable mathematics ensures that all students have the same opportunity to engage in the content, that every voice is being heard, and that students have multiple avenues to access math learning.

"To be clear: quality materials offer a strong standards- aligned foundation, but it's up to skilled teachers to bring the content to life and to connect that content to students' actual lives."

This type of classroom can’t exist for students when teachers are not supported with high-quality materials or the mindset and environment to incorporate and celebrate student experiences. To be clear: quality materials offer a strong standards-aligned foundation, but it’s up to skilled teachers to bring the content to life and to connect that content to students’ actual lives. 

I believe wholeheartedly that math can be a vehicle to understand the world, but only if students can connect to the skills and concepts and only if we show them how mathematics relates to their world. Here are three ways to get started today:

1. Bring student experiences and their local surroundings into problem solving.

In my first year in the classroom I showed my students a picture of the Golden Gate Bridge as part of a lesson and was met with blank stares. Most of my Tennessean students had never been to San Francisco. Then I switched it up and brought in the Memphis Pyramid and FedEx Forum—those little shifts made a big difference! My students were immediately more engaged in the lesson.  Quality materials are flexible and adaptable to local environments and student backgrounds. They offer suggestions and guidance on how to incorporate local context into the current skills students are building. Mixing in the familiar when introducing new concepts is one way to make math relevant for your students and help them feel connected to the learning. 

2 . Find ways to engage all students.

We know as teachers that it’s not enough to reach just some students. Without engagement, it’s unlikely kids are hearing a word you are saying. High-quality materials help teachers to honor every student’s contribution. For example, no matter what a student’s home language is, materials should be designed in a way that allows every student to access the content and learning. Strong materials also include built in scaffolds, differentiation guidance, and embedded assessments to help target the needs of each student.  When math materials reflect the lived experiences of students, they are able to make sense of the content and become more confident about what they are learning.

3. Encourage students to ask questions, talk with each other, and engage in math practices .

A key innovation from college and career-ready math standards is the concept of mathematics modeling . This practice utilizes real-world situations, and students determine how to best solve the questions in front of them. Not only does modeling ensure that lessons can be taught with multiple representations, but students are often excited to be leading their own learning. Mathematics can be about discovery rather than a predetermined answer they have to get right. Anyone who remembers their own time as a student will not be surprised to hear that when kids feel that they belong—and when their experiences are visible and celebrated as part of learning— they perform better. (And building these strong relationships with students is good for our practice as teachers as well). Investing in equitable math practices and high-quality materials are key actions that school systems can take to help every student become problem solvers who can think creatively to meet challenges inside and outside of the classroom. Instead of the world being divided into math people and non-math people, we can show students that we are all math people. 

Neven Holland is a fourth-grade teacher for Memphis-Shelby County Schools (MSCS) in Tennessee. Neven is a 2022 Tennessee state finalist for the Presidential Award for Excellence in Mathematics and Science Teaching. He is an EdReports mathematics reviewer and a Klawe Fellow.

Related Resources

Confessions of a Lifelong Math Nerd

My Journey Toward Loving Math

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Active Learning in Mathematics, Part IV: Personal Reflections

By Benjamin Braun, Editor-in-Chief , University of Kentucky; Priscilla Bremser, Contributing Editor , Middlebury College; Art Duval,  Contributing Editor , University of Texas at El Paso; Elise Lockwood,  Contributing Editor , Oregon State University; and Diana White,  Contributing Editor , University of Colorado Denver.

Editor’s note: This is the fourth article in a series devoted to active learning in mathematics courses.  The other articles in the series can be found here .

In contrast to our first three articles in this series on active learning, in this article we take a more personal approach to the subject.  Below, the contributing editors for this blog share aspects of our journeys into active learning, including the fundamental reasons we began using active learning methods, why we have persisted in using them, and some of our most visceral responses to our own experiences with these methods, both positive and negative.  As is clear from these reflections, mathematicians begin using active learning techniques for many different reasons, from personal experiences as students (both good and bad) to the influence of colleagues, conferences, and workshops.  The path to active learning is not always a smooth one, and is almost always a winding road.

Because of this, we believe it is important for mathematics teachers to share their own experiences, both positive and negative, in the search for more meaningful student engagement and learning.   We invite all our readers to share their own stories in the comments at the end of this post.  We also recognize that many other mathematicians have shared their experiences in other venues, so at the end of this article we provide a collection of links to essays, blog posts, and book chapters that we have found inspirational.

There is one more implicit message contained in the reflections below that we want to highlight.  All mathematics teachers, even those using the most ambitious student-centered methods, use a range of teaching techniques combined in different ways.  In our next post, we will dig deeper into the idea of instructor “telling” to gain a better understanding of how an effective balance can be found between the process of student discovery and the act of faculty sharing their expertise and experience.

Priscilla Bremser:

I began using active learning methods for several reasons, but two interconnected ones come to mind.  First, Middlebury College requires all departments to contribute to the First-Year Seminar program, which places every incoming student into a small writing-intensive class. The topic is chosen by the instructor, while guidelines for writing instruction apply to all seminars.  As I have developed and taught my seminars over the years, I’ve become convinced that students learn better when they are required to express themselves clearly and precisely, rather than simply listening or reading.  At some point it became obvious that the same principle applies in my other courses as well, and hence I was ready to try some of the active learning approaches I’d been hearing about at American Mathematical Society meetings and reading about in journals .

Second, I got a few student comments on course evaluations, especially for Calculus courses, that suggested I was more helpful in office hours than in lecture.  Thinking it through, I realized that in office hours, I routinely and repeatedly ask students about their own thinking, whereas in lecture, I was constantly making assumptions about student thinking, and relying on their responses to “Any questions?” for guidance, which didn’t elicit enough information to address the misunderstandings around the room. One way to make class more like office hours is to put students into small groups. I then set ground rules for participation and ask for a single set of problem solutions from each group. This encourages everyone to speak some mathematics in each class session, and to ask for clarity and precision from classmates.  Because I’m joining each conversation for a while, I get a more accurate perception of students’ comprehension levels.

This semester I’m teaching Mathematics for Teachers, using an IBL textbook by Matthew Jones . I’ve already seen several students throw fists up in the air, saying “I get it now!  That’s so cool!” How well I remember having that response to my first Number Theory course; it’s why I went into teaching at this level in the first place.  On the other hand, a Linear Algebra student who insists that  “I learn better from reading a traditional textbook” leaves me feeling rather deflated. It seems that I’ve failed to convey why I direct the course the way that I do, or at least I haven’t yet succeeded.  The truth is, though, that I used to feel the same way.  I regarded mathematics as a solitary pursuit, in which checking in with classmates was a sign of weakness.  Had I been required to discuss my thinking regularly during class and encouraged to do so between sessions, I would have developed a more solid foundation for my later learning. Remembering this inspires me to be intentional with students, and explain repeatedly why I direct my courses the way that I do.  Most of them come around eventually.

Elise Lockwood:

I have a strong memory of being an undergraduate in a discrete mathematics course, trying desperately to understand the formulas for permutations, combinations, and the differences between the two. The instructor had presented the material, perhaps providing an example or two, but she had not provided an opportunity for us to actively explore and understand why the formulas might make sense. By the time I was working on homework, I simply tried (and often failed) to apply the formulas I had been given. I strongly disliked and feared counting problems for years after that experience. It wasn’t until much later that I took a combinatorics course as a master’s student. Here, the counting material was brought to life as we were given opportunities to work through problems during class, to unpack formulas, and to come to understand the subtlety and wonder of counting. The teacher did not simply present a formula and move on, assuming we understood it. Rather, he persisted by challenging us to make sense of what was going on in the problems we solved.

For example, we once were discussing a counting problem in class (I can’t recall if it was an in-class problem or a problem that had been assigned for homework). During this discussion, it became clear that students had answered the problem in two different ways — both of them seemed to make sense logically, but they did not yield the same numerical result. The instructor did not just tell us which answer was right, but he used the opportunity to have us consider both answers, facilitating a (friendly) debate among the class about which approach was correct. We had to defend whichever answer we thought was correct and critique the one we thought was incorrect. This had the effect not only of engaging us and piquing our curiosity about a correct solution, but it made us think more carefully and deeply about the subtleties of the problem.

Now, studying how students solve counting problems is the primary focus of my research in mathematics education. My passion for the teaching and learning of counting was probably in large part formed by the frustrations I felt as an undergraduate and the elation I later experienced when I actually understood some of the fundamental ideas.

When I have been given the opportunity to teach counting over the years (in discrete mathematics or combinatorics classes, or in courses for pre-service teachers), I have tried my hardest to facilitate my students’ active engagement with the material during class. This has not taken an inordinate amount of time or effort: instead of just giving students the formulas off the bat, I give them a series of counting problems that both introduce counting as a problem solving activity and motivate (and build up to) some key counting formulas. For example, students are given problems in which they list some outcomes and appreciate the difference between permutations and combinations firsthand. I have found that a number of important issues and ideas (concerns about order, errors of overcounting, key binomial identities) can emerge on their own through the students’ activity, making any subsequent discussion or lecture much more meaningful for students. When I incorporate these kinds of activities for my students, I am consistently impressed at the meaning they are able to make of complex and notoriously tricky ideas.

More broadly, these pedagogical decisions I make are also based on my belief about the nature of mathematics and the nature of what it means to learn mathematics. Through my own experiences as a student, a teacher, and a researcher, I have become convinced that providing students with opportunities to actively engage with and think about mathematical concepts — during class, and not just on their own time — is a beneficial practice. My experience with the topic of counting (something near and dear to my heart) is but one example of the powerful ways in which student engagement can be leverage for deep and meaningful mathematical understanding.

Diana White:

What stands out most to me as I reflect upon my journey into active learning is not so much how or why I got involved, but the struggles that I faced during my first few years as a tenure-track faculty member as I tried to switch from being a good “lecturer” to all out inquiry-based learning.  I was enthusiastic and ambitious, but lacking in the skills to genuinely teach in the manner in which I wanted.

As a junior faculty member, I was already sold on the value of inquiry-based learning and student-centered teaching.  I had worked in various ways with teachers as a graduate student at the University of Nebraska and as a post-doc at the University of South Carolina, including teaching math content courses for elementary teachers and assisting with summer professional development courses for teachers.  Then, the summer before I started my current position, I attended both the annual Legacy of R.L. Moore conference and a weeklong workshop on teaching number theory with IBL through the MAA PREP program.  The enthusiasm and passion at both of these was contagious.  

However, upon starting my tenure track position, I jumped straight in, with extremely ambitious goals for my courses and my students, ones for which I did not have the skills to implement yet.  In hindsight, it was too much for me to try to both switch from being a good “lecturer” to doing full out IBL and running an intensely student centered classroom, all while teaching new courses in a new place.  I tried to do way too much too soon, and in many ways that was not healthy for either me or the students, as evidenced by low student evaluations and frustrations on both sides.

Figuring out specifically what was going wrong was a challenge, though.  Those who came to observe, both from my department and our Center for Faculty Development, did not find anything specific that was major, and student comments were somewhat generic – frustration that they felt the class was disorganized and that they were having to teach themselves the material.  

I thus backtracked to more in the center of the spectrum, using an interactive lecture  Things smoothed out and students became happier.  What I am not at all convinced of, though, is that this decision was best for student learning.  Despite the unhappiness on both our ends when I was at the far end of the active learning spectrum, I had ample evidence (both from assessments and from direct observation of their thought processes in class) that students were both learning how to think mathematically and building a sense of community outside the classroom.  To this day, I feel torn, like I made a decision that was best for student satisfaction, as well as for how my colleagues within my department perceive me.  Yet I remain convinced that my students are now learning less, and that there are students who are not passing my classes who would have passed had I taught using more active learning. (It was impossible to “hide” with my earlier classes, due to the natural accountability built into the process, so struggling students had to confront their weaknesses much sooner.)

It is hard for me to look back with regrets, as the lessons learned have been quite powerful and no doubt shaped who I am today.  However, I would offer some thoughts, aimed primarily at junior faculty.  

Don’t be afraid to start slow.  Even if it’s not where you want to end up, just getting started is still an important first step.  Negative perceptions from students and colleagues are incredibly hard to overcome.

Don’t underestimate the importance of student buy-in, or of faculty buy-in.  I found many faculty feel like coverage and exposure are essential, and believe strongly that performance on traditional exams is an indicator of depth of knowledge or ability to think mathematically.

Don’t be afraid to politely request to decline teaching assignments.  When I was asked to teach the history of mathematics, a course for which I had no knowledge of or background in, I wasn’t comfortable asking to teach something else instead.  While it has proved really beneficial to my career (I’m now part of an NSF grant related to the use of primary source projects in the undergraduate mathematics classroom), I was in no way qualified to take that on as a first course at a new university.

I have personally gained a tremendous amount from my participation in the IBL community, perhaps most importantly a sense of community with others who believe strongly in active learning.  

My first experience with active learning in mathematics was as a student at the Hampshire College Summer Studies in Mathematics program during high school.  Although I’d had good math teachers in junior high and high school, this was nothing like I’d seen before: The first day of class, we spent several hours discussing one problem (the number of regions formed in 3-dimensional space by drawing \(n\) planes), drawing pictures and making conjectures; the rest of the summer was similar.  The six-week experience made such an impression on me, that (as I realized some years later) most of the educational innovations I have tried as a teacher have been an attempt to recreate that experience in some way for my own students.

When I was an undergraduate, I noticed that classes where all I did was furiously take notes to try to keep up with the instructor were not nearly as successful for me as those where I had to do something.  Early in my teaching career, I got a big push towards using active learning course structures from teaching “ reform calculus ” and courses for future elementary school teachers.  In each case, this was greatly facilitated by my sitting in on another instructor’s section that already incorporated these structures.  Later I learned, through my participation in a K-16 mathematics alignment initiative , the importance of conceptual understanding among the levels of cognitive demand , and this helped me find the language to describe what I was trying to achieve.

Over time, I noticed that students in my courses with more active learning seemed to stay after class more often to discuss mathematics with me or with their peers, and to provide me with more feedback about the course.  This sort of engagement, in addition to being good for the students, is very addictive to me.  My end-of-semester course ratings didn’t seem to be noticeably different, but the written comments students submitted were more in-depth, and indicated the course was more rewarding in fundamental ways.  As with many habits, after I’d done this for a while, it became hard not to incorporate at least little bits of interactivity (think-pair-share, student presentation of homework problems), even in courses where external forces keep me from incorporating more radical active learning structures.

Of course, there are always challenges to overcome.  The biggest difficulty I face with including any sort of active learning is how much more time it takes to get students to realize something than it takes to simply tell them.  I also still find it hard to figure out the right sort of scaffolding to help students see their way to a new concept or the solution to a problem.  Still, I keep including as much active learning as I can in each course.  The parts of classes I took as a student (going back to junior high school) that I remember most vividly, and the lessons I learned most thoroughly, whether in mathematics or in other subjects, were the activities, not the lectures.  Along the same lines, I occasionally run into former students who took my courses many years ago, and it’s the students who took the courses with extensive active learning, much more than those who took more traditional courses, who still remember all these years later details of the course and how much they learned from it.

Other Essays and Reflections:

Benjamin Braun, The Secret Question (Are We Actually Good at Math?), http://blogs.ams.org/matheducation/2015/09/01/the-secret-question-are-we-actually-good-at-math/

David Bressoud, Personal Thoughts on Mature Teaching, in How to Teach Mathematics, 2nd Edition , by Steven Krantz, American Mathematical Society, 1999.   Google books preview

Jerry Dwyer, Transformation of a Math Professor’s Teaching, http://blogs.ams.org/matheducation/2014/06/01/transformation-of-a-math-professors-teaching/

Oscar E. Fernandez, Helping All Students Experience the Magic of Mathematics, http://blogs.ams.org/matheducation/2014/10/10/helping-all-students-experience-the-magic-of-mathematics/

Ellie Kennedy, A First-timer’s Experience With IBL, http://maamathedmatters.blogspot.com/2014/09/a-first-timers-experience-with-ibl.html

Bob Klein, Knowing What to Do is not Doing, http://maamathedmatters.blogspot.com/2015/07/knowing-what-to-do-is-not-doing.html

Evelyn Lamb, Blogs for an IBL Novice, http://blogs.ams.org/blogonmathblogs/2015/09/21/blogs-for-an-ibl-novice/

Carl Lee, The Place of Mathematics and the Mathematics of Place, http://blogs.ams.org/matheducation/2014/10/01/the-place-of-mathematics-and-the-mathematics-of-place/

Steven Strogatz, Teaching Through Inquiry: A Beginner’s Perspectives, Parts I and II,  http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-1,  http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-2

Francis Su, The Lesson of Grace in Teaching, http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html

2 Responses to Active Learning in Mathematics, Part IV: Personal Reflections

In response to Priscilla Bremser, I feel as though it is almost elementary that students who are able to precisely express themselves are better to understand the information conceptually. What I mean by this is that the students who are able to interact with the information will get a better idea of what that information means conceptually rather than the students who simply listen to lecturing.

In regards to your second point, I also find this point to be important, even though it may seem obvious. Similarly to your first point, students who get more personal interaction with the instructor will probably be more likely to understand the information that is being presented. Since I am still in school, we have been discussing the best ways to prompt questions from students. Asking “are there any questions” is not a good way to do this. Breaking up into groups is a good way to see where the students are at conceptually.

However, this may prove to be tricky at the college level because of class size. One way to battle this is to ask for thumbs (either up, down, or in the middle) as to whether they understand the information being presented. This practice will give you a good idea at where the class is as a whole in a quick snapshot and students will be less likely to feel as though they are being singled out.

A few points in this post resonated with me particularly well. First, when Priscilla said that she was more helpful in office hours than in lecture because she asked students about their own thinking in the former, I agreed with it from a student’s perspective. Making class feel more like office hours, with more one-on-one time, helps students feel more like individual learners in the classroom. By suggesting small group work in order to facilitate more participation and allow for more analysis of each student’s performance, I feel that Bremser is acknowledging the ineffectiveness of using the phrase “Any questions”, which is something I try not to use, and hate to hear in my college classes. I also can relate to what Diana White says about trying to switch teaching styles as you would flip a switch. Not having the skills necessary to be at the level you want will be frustrating, and I know that as a future teacher, I will want to be successful right out of the gate. I know that this is unreasonable, and largely impossible, but this is more of a personality flaw that I will have to suppress. When it comes to being evaluated by others, I will have to recognize that many of my evaluators were once young teachers themselves, with the same aspirations, the same experience, and probably the same results as me. I will have to be patient, and use their feedback (and my own) to improve my teaching over time, rather than overnight. I wonder if this is a good assessment of what I should expect of myself when I begin teaching.

Comments are closed.

Opinions expressed on these pages were the views of the writers and did not necessarily reflect the views and opinions of the American Mathematical Society.

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Mathematics Experience in Our Life Research Paper

Introduction.

Mathematics is applied in every aspect of this life be it in our personal or professional lives. Mathematics is used to budget money and time, these aspects do not require formulas. People do not even realize they are applying mathematical concepts when budgeting. As a child I used to enjoy counting the number of stairs, this is whenever I went up the stairs. Mathematics was interesting by then; it involved simple concepts that did not require complex formulas. As I grew up, I realized that mathematics was not as simple as I thought. My high school teacher could introduce new concepts now and then; with time I realized that with mathematics a lot of practice was required. Mathematics during my childhood was interesting; the teacher could make us practice until it became almost automated without us realizing it. Allen (2001) says that teachers use this tactic at elementary level to make mathematics interesting.

I compare mathematics to climbing a wall, sometimes it is very easy while it can prove to be very hard sometimes. One of my mathematics teachers used to describe mathematics as ‘a voyage of discovery. I am opposed to this, this is because sometimes we attempt to find solutions to mathematical problems but do not get to arrive at the correct answer, instead, we get more and more confused. At one point, I remember our elementary teacher telling us that one divided by zero equals zero, this concept however changed when I joined high school, we were told that the answer is infinite. Could be that the elementary teacher did not want to go into detail about this since it would have been hard for us to understand.

In his book Overcoming math anxiety Tobias (2002), states that most people enjoy learning mathematics when they fully understand the concept. This statement is agreeable, it is common to find students doing calculations and arriving at an answer but if asked questions, they cannot infer anything from the answer they have gotten. I have experienced this severally, at some point; I could completely get absorbed when solving some mathematics problems that I even forget the time factor. This is when I can comfortably state what the answer means. When it becomes hard to understand a mathematical concept, I spend less time trying to understand the concept since the efforts are not in most cases rewarding. A topic such as ordinary differential and linear algebra gave me a hard time. It is interesting to note that most students will have varying ideas of a topic; this just shows that the topic has not been well presented or is too hard for the students to understand (Tobias, 2002).

One thing that I love about mathematics is that the formulas do not change. Formulas that were introduced by Plato are still being used today (Allen, 2001). This field of study cannot be compared to other fields such as business and arts where ideas keep changing with time. Some simple formulas that were applied in elementary levels are still used in higher levels of learning. I have realized that with a good mastery of most of the formulas, it can be easy to solve most of the problems. To some extent I love mathematics. The main reason is that when it becomes too easy concepts, I get absorbed and enjoy doing the calculations.

Mathematics was very enjoyable during my childhood years. The teacher was very dedicated when it came to teaching this subject. I had good mastery of the simple concepts taught and one thing that I enjoyed doing most is counting the number of stairs as I climbed up. Today, mathematics has ceased to be my favorite subject; this does not however mean that I hate it. The thing is that I only enjoy dealing with simple concepts. Hard concepts give me a hard time. I must admit that I was not born a mathematician like some people. The little concepts that I have are enough since they enable me to go about my budgeting without any problems.

Allen, D. S. (2001 ). Mathematics experience: contributing factors to the math anxiety and avoidance behaviors of female elementary school pre-service teachers. Texas: Texas Tech University Press.

Tobias, S. (2002). Overcoming math anxiety . New York: W.W. Norton Press.

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Article Contents

1 background to the study, 2 theoretical perspectives, 3 methodology, 5 discussion, 6 conclusion, a. a. sample tasks from the writing intervention, b b. student questionnaire, students’ perceptions of mathematics writing and its impact on their enjoyment and self-confidence.

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Tandeep Kaur, Mark Prendergast, Students’ perceptions of mathematics writing and its impact on their enjoyment and self-confidence, Teaching Mathematics and its Applications: An International Journal of the IMA , Volume 41, Issue 1, March 2022, Pages 1–21, https://doi.org/10.1093/teamat/hrab008

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There have been universal endorsements of the benefits of writing as an effective medium of communicating mathematically. Writing and learning are seen as isomorphic to each other and writing can facilitate the comprehension of mathematical thinking through intrapersonal communication. Through a short writing intervention, this study investigates students’ perceptions on the use of writing in the mathematics classroom and explores the impact of writing on students’ affective domains of self-confidence and enjoyment levels in mathematics. A mixed-methods approach was employed using a pre-test, intervention, post-test design for the study. Quantitative data were collected through a questionnaire adapted from the Attitudes Towards Mathematics Inventory ( Tapia & Marsh, 2004 ), which was administered before and after the intervention. An analysis of the quantitative data revealed a significant increase in students’ mean scores for both enjoyment and self-confidence. Qualitative data collected in the form of students’ reflections of the writing intervention indicated that, overall, students had a positive perception of writing as a means of communicating in the mathematics classroom.

In India, the National Curriculum Framework (2005) describes mathematical communication as an important feature of any mathematical undertaking. It is recommended that mathematical communication which uses unambiguous and precise language is crucial for developing an appreciation of the subject. Matsuura et al. (2013) determined that such endeavours can be effective in the development of mathematical thinking and habits of mind. ‘These habits are not about particular definitions, theorems, or algorithms that one might find in a textbook; instead, they are about the thinking, mental habits, and research techniques that mathematicians employ to develop such definitions, theorems, or algorithms’ ( Matsuura et al. , 2013 , p. 736). Despite such recognized importance, there has been no research to date investigating the issue of mathematical writing in India. This present study is the first of its kind to be carried out in the Indian context.

Mathematics has evolved significantly over the past number of decades and so have the teaching practices associated with it. Various reforms have been implemented worldwide from time to time, with a focus on differing strategies to enhance deeper mathematical understanding and engagement among students. One feature of mathematical reform that has been under continuous scrutiny is ‘mathematical communication’. It has been suggested by the educational policies of almost all countries that it is necessary for students to be able to communicate mathematically and that mathematical communication should be at the heart of mathematical teaching. For example, in the USA, the National Council of Teachers of Mathematics (1989 , 2000) called for mathematical literacy for all, stressing the need for mathematical communication. Similarly, India’s National Curriculum Framework (2005) states that ‘Children see mathematics as something to talk about, to communicate through’ (p. 43). In Ireland, a recent reform of lower secondary level education has identified communication as one of the key skills of the entire curriculum. More specifically, communication also underpins a unifying strand across the Irish mathematics syllabus. The syllabus states that ‘Students should be able to communicate mathematics effectively in verbal and written form’ ( NCCA, 2018 , p. 9).

With specific reference to writing as the mode of communication, there has been increasing interest in recent years towards its role in mathematics classrooms. While participation in discourses and debates of mathematics for improved mathematical learning has been frequently emphasized ( Burton & Morgan, 2000 ), formal recommendations for mathematical writing are less explicitly available ( Casa et al. , 2016 ). Despite this, studies and interventions have consistently reported the benefits of writing in learning mathematics ( Fry & Villagomez, 2012 ; Knox, 2017 ; Kostos & Shin, 2010 ; Kuzle, 2013 ; Pugalee, 2001 ).

1.1 Mathematical writing—the perceived benefits

Bangert-Drowns et al. (2004) determine that writing and learning are isomorphic to each other. In all aspects of life, writing serves as a psychologically powerful instrument, providing a vent for thoughts through deep reflection and understanding ( Hacker et al. , 2009 ). In the classroom, writing assists students in constructing new knowledge through activities such as exploration, representation, investigation and justification ( Countryman, 1992 ). Being a planned and conscious process, it strengthens current knowledge while building new connections at the same time ( Kenney et al. , 2013 ; Kuzle, 2013 ). Pugalee (2001 ) notes that in order to express one’s thoughts through writing, inner speech has to be compressed maximally by connecting current knowledge to the new knowledge gained. With specific reference to mathematics, various studies have focused on the effects of mathematical writing in the mathematics classroom. Such research has revealed enhanced metacognitive thinking, self-confidence and enjoyment levels, which leads to increased mathematical achievement ( Knox, 2017 ; Kostos & Shin, 2010 ; Kuzle, 2013 ; Pugalee, 2001 ).

1.2 Writing and the affective domains

Emotions are fundamental to learning and can affect students’ thought processes as well as memory ( Hinton et al. , 2008 ; Westen, 1999 ). For example, threatening situations, such as peer competition, parental pressure, exam stress, image in front of teacher, etc., can affect learning in a negative way ( Wolfe & Brandt, 1998 ). Writing aids in dealing with such issues and may help bring positive changes in the affective constructs such as self-confidence and enjoyment ( Countryman, 1992 ).

Several studies have revealed that mathematics learning is highly influenced by learners’ mathematics-related beliefs, especially self-confidence ( Hannula & Malmivuori, 1997 ; Hannula et al. , 2004 ). In fact, self-confidence has been observed as the greatest non-cognitive predictor for academic achievement among other self-belief measures such as self-efficacy and self-concept ( Stankov et al. , 2014 ). On the other hand, a lack of self-confidence may negatively impact students’ motivation to learn ( Boekaerts & Rozendaal, 2010 ). Engaging students to communicate their mathematical ideas through writing may instil in them a higher level of self-confidence and critical thinking skills ( Quitadamo & Kurtz, 2007 ).

Another important construct is students’ enjoyment while learning. Many studies have evidenced a positive relationship between the level of enjoyment while performing a task and students’ attitudes and self-efficacy beliefs ( Ahmed et al. , 2010 ; Lorsbach & Jinks, 1999 ; Sakiz et al. , 2012 ). These effects are greatly enhanced if students enjoy the tasks they are working on ( Bramlett & Herron, 2009 ). Mathematical writing is one such task that may prove to be an enjoyable class activity, while also adhering to formal curricular demands.

However, little research has been conducted to examine the role of writing in relation to these affective domains ( Miller & Meece, 1997 ). With such a dearth of research in mind, this paper investigates students’ attitudes towards mathematics with regards to their enjoyment and self-confidence after a short writing intervention.

This study also aimed to capture students’ experiences of the writing intervention. It has been argued in the literature that learners’ perceptions of classroom events strongly influence their classroom behaviour and response to teaching approaches ( Marx, 1983 ; Roese & Sherman, 2007 ; Struyven et al. , 2005 ). Therefore, capturing students’ perceptions may provide useful insights for successful reform efforts. However, at the moment, such insights are not given due consideration ( Levin, 2000 ). This paper thus explores participants’ written reflections of the intervention and investigates their perceptions of it.

Keeping up with the goal of promoting mathematical communication, the Elementary Writing Mathematical Task Force from the University of Connecticut in the USA proposed the following four types of mathematical writing, with their purposes described as follows:

Exploratory (to personally make sense of a problem, situation or one’s own ideas)

Explanatory/informative (to describe, to explain)

Argumentative (to construct or critique an argument)

Mathematically creative (to document original ideas, problems and/or solutions, to convey fluency and flexibility in thinking, to elaborate on ideas)

( Casa et al. , 2016 , p. 4)

It has been suggested by the Task Force that all students should have exposure to all types of writing. However, there is an ambiguity in comprehending the context of writing in relation to mathematics ( Bossé & Faulconer, 2008 ). The Task Force highlighted two categories of writing that take place in mathematics classrooms—‘writing about mathematics’ and ‘mathematical writing’. Writing about mathematics stresses on the learning of literacy skills e.g. a mathematics autobiography, while mathematical writing emphasizes the use of mathematical symbols and vocabulary with an aim to develop mathematical reasoning.

The present study focused on the ‘explanatory’ type of mathematical writing where the purpose is to describe or explain ideas for mathematical reasoning. The study also explored the perceptions of participants for its use in mathematics class. The study aims to address the following research questions:

(1) What effect, if any, does a short mathematical writing intervention have on students’ self-confidence and enjoyment levels in mathematics?

(2) What are students’ perceptions about the use of mathematical writing in the classroom?

The study used a pre-test, intervention, post-test design to explore students’ perceptions and investigated the impact of a short writing intervention on students’ enjoyment and self-confidence in mathematics. A convergent mixed-methods approach was employed to look into the research phenomena from different viewpoints ( Creswell & Plano Clark, 2011 ), with both quantitative and qualitative data being collected. The instruments used for addressing the research questions were questionnaires and participants’ reflections. The analysis and triangulation of this data enabled a ‘complete, holistic and contextual portrayal’ of the research ( Clark & Creswell, 2008 , p. 109).

3.1 Study sample

The study was conducted in a secondary co-educational school in New Delhi, India, which is affiliated with the country’s Central Board of Secondary Education. This school caters for approximately 1000 students from Kindergarten to Grade 10. It was selected through a purposive sampling method as its location was in close proximity to one of the researchers and they had existing contacts in the school. Students from a Grade 7 and Grade 8 (aged 12–15 years old) class group were invited to partake in the study and 55 students agreed to participate. The sample was made up of 38 males (69%) and 17 females (31%), with a mean age of 13 years. For all the students, Hindi was their first language and English was their second language. However, the language of instruction for all students in the school was English.

3.2 The intervention

Realizing that students are the primary stakeholders in the field of education and that any educational reform affects them foremost, an intervention was designed to measure the impact of writing on students’ attitudes, especially the affective domains of enjoyment and self-confidence. Six sessions were administered for the intervention, in addition to an introductory session which detailed the purpose of the study. The methodology of the study was also shared at this opening session, ensuring that each student understood the research instruments that would be used. Each session was conducted for a duration of 40 min. These sessions took place during regular school study hours and in the periods allocated for extra-curricular activities. This meant that participants’ formal studies were not interrupted by any activities related to this research.

Tasks for the intervention were selected from the Trends in International Mathematics and Science Study (TIMSS) 2011 grade 8 mathematics assessment items. These items manifest various ways of measuring students’ understanding in several content and cognitive domains (see Appendix A for sample tasks). Six tasks were selected, one for each session of the intervention. As this study focused particularly on the ‘explanatory’ genre of mathematical writing, tasks based on cognitive domains of reasoning were specifically selected. The criteria for selection of tasks were based on students’ previous knowledge and their current class group curriculum. Two tasks were based on identification of patterns, one was a geometry problem and the remaining three tasks were multiple choice based (see Appendix A). Each session started with class discussion of a specific task wherein students came up with different ways of solving the problem at hand. This was then followed by students’ explanation of their understanding and thought processes through writing. The researcher was present throughout each session to offer any assistance if required. No formal assessment was done for the submitted writings. Oral feedback was provided, and suggestions given to improve their writing for the subsequent sessions. One of the tasks from the intervention is specified in Fig. 1 .

Here is a pattern:

What will the next line in the pattern be? (Item number M042186, Cognitive Domain—Reasoning).

Example of a multiple choice-based task (SOURCE: TIMSS 2011 Assessment. Copyright © 2013 International Association for the Evaluation of Educational Achievement (IEA). Publisher: TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College, Chestnut Hill, MA and International Association for the Evaluation of Educational Achievement (IEA), IEA Secretariat, Amsterdam, the Netherlands.)

3.3 Data collection

Quantitative data were collected in the form of questionnaires which were administered both before and after the intervention. The questionnaires were adopted from the Attitudes Towards Mathematics Inventory (ATMI; Tapia & Marsh, 2004 ). The subscales of enjoyment and self-confidence were selected for the present study. The enjoyment subscale consisted of a total of 10 statements and the self-confidence subscale comprised of 15 statements (see Appendix B). Each statement had alternative response options which were based on a 5-point Likert-type scale. Every positive statement in the questionnaires was scored from 5 to 1, ranging from 5 = ‘strongly agree’ to 1 = ‘strongly disagree’. Negatively worded items were scored in the reverse direction ranging from 1 = ‘strongly agree’ to 5 = ‘strongly disagree’. The maximum score for enjoyment subscale was 50 and for the self-confidence subscale was 75, giving an overall total score of 125.

After the final session of the intervention, qualitative data were collected in the form of participants’ written reflections. This enabled participants to share their experience regarding the use of mathematical writing as a tool for learning. Participants were prompted to reflect and write about whether they liked the intervention and to include their reasoning. Varied opinions emerged as they expressed their perceptions about the use of writing in the mathematics classroom.

3.4 Data analysis

The quantitative data for the study were analysed using SPSS (Statistical Package for the Social Sciences) software. Mean scores for the pre- and post-intervention tests were compared by descriptive analysis. In addition, t -tests were conducted to determine the statistical significance of the findings.

In order to ensure the internal consistency of the quantitative scales, Cronbach alpha coefficients were calculated. The Cronbach alpha coefficients were recorded as 0.83 for the self-confidence subscale and 0.84 for the enjoyment subscale. The alpha coefficient for the overall scale was 0.95. High values of Cronbach alpha coefficients (suggested values greater than 0.70— Nunnally and Bernstein, 1994 ) indicate good internal consistency and ensure content validity for the data.

The qualitative data were analysed thematically by one of the authors. Following this, a convergence model of triangulation design ( Creswell et al. , 2007 ) was used to interpret the results from both sets (qualitative and quantitative) of analysis. This enabled the authors to examine the convergence, consistency or inconsistency of the overall data ( Ary et al. , 2010 ). Findings for both types of data are detailed in subsequent sections.

As discussed previously, both quantitative and qualitative data were collected in order to address the research questions. Findings from the quantitative data will be presented first followed by those from the qualitative data.

4.1 Quantitative findings

Descriptive statistics revealed an increase in the mean scores for both the enjoyment and self-confidence scales, from ‘pre-intervention’ to ‘post-intervention’. Figure 2 illustrates a comparison of mean scores on the enjoyment (EN) and self-confidence (SC) subscales.

The results of a long jump competition were reported as follows:

Average length

Team A 3.6 m

Team B 4.8 m

There was the same number of students in each team. Which statement about the competition MUST be true?

Comparison of mean scores on enjoyment and self-confidence.

As illustrated from Fig. 2 , the mean enjoyment scores increased from 35 (before the intervention) to 40 (after the intervention) from a total score of 50. The mean self-confidence scores rose from 49 (before the intervention) to 54 (after the intervention) from a total score of 75.

In addition, a paired-samples t -test was conducted to assess the statistical significance of the findings. The differences in enjoyment scores from pre-intervention ( M  = 35, SD = 7) to post-intervention ( M  = 40, SD = 5), with t (54) = −10.8, p  < 0.05 (two-tailed), were statistically significant. The mean increase in enjoyment scores was recorded as 5 with a 95% confidence interval ranging from 5.9 to 4.1. For the effect size, eta square was calculated and found to be 0.6. As suggested by Cohen (1988) , values greater than 0.1 indicate a large effect size. Therefore, it can be concluded that there was a large effect size with a significant increase in enjoyment mean scores recorded from before and after the intervention.

Similarly, for self-confidence, the increase in mean scores from the pre-intervention test ( M  = 49, SD = 9) to the post-intervention test ( M  = 54, SD = 9), with t (54) = −9.4, p  < 0.05 (two-tailed), was statistically significant. The 95% confidence interval ranged from 5.9 to 3.8 with an increase of 5 in the self-confidence mean scores. The eta square statistic (0.6) also indicated a large effect size, implying that participants felt greater levels of self-confidence in learning mathematics aided by the writing intervention.

4.1.1 Further analysis

A statement-wise analysis of the two subscales for both the pre- and post-intervention revealed some noteworthy findings. Six out of 10 statements on the enjoyment subscale showed noticeable changes in the responses marked by the participants at both testing points. As an example, for the statement ‘I really like mathematics’, 20 out of 55 students (36% of the participants) recorded their response as ‘strongly agree’ in the post-test as compared to 8 students (14%) in the pre-test.

In the self-confidence questionnaire analysis, 5 out of 15 statements showed clear changes in terms of responses obtained from participants. For example, for the statement ‘I am always under a terrible strain in a mathematics class’, 23 students (42% of the participants) recorded the response ‘strongly disagree’ in the post-test as compared to 9 participants (16%) in the pre-test. Interestingly, for the statement ‘When I hear the word mathematics, I have a feeling of dislike’, there were no responses in favour of ‘strongly agree’ or ‘agree’ in the post-test questionnaire as compared to 9 responses (2 for ‘strongly agree’ and 7 for ‘agree’) in the pre-test.

4.1.2 Gender-based findings

A descriptive analysis was performed to explore for any gender-based differences in the mean scores. Table 1 compares the mean scores of male and female participants in both the pre- and post-intervention tests and these are further illustrated by the line graph in Fig. 3 .

Comparison of mean scores gender-wise

Gender-based differences.

A general look at the line graph determines that females scored slightly higher on both the EN and SC subscales for the pre-intervention tests. While this trend continued in the post-intervention tests, the scores between both groups were closer.

To test the statistical significance of these gender-based differences, an independent-samples t -test was conducted on the mean scores of both males and females for each of the subscales, both before and after the intervention.

Regarding the enjoyment subscale, the tests recorded t (53) = −0.5, p  > 0.05 (two-tailed) for the pre-intervention and t (53) = −0.3, p  > 0.05 (two-tailed) for the post-intervention. For the self-confidence subscale, the tests showed the values as t (53) = −0.9, p  > 0.05 (two-tailed) for pre-intervention and t (53) = −0.4, p  > 0.05 (two-tailed) for the post-intervention. These findings revealed that differences between the scores obtained by the two gender groups were not statistically significant.

4.2 Qualitative findings

An analysis of participants’ reflections provided an interesting glimpse of their thoughts and opinions about writing. Based on responses that emerged from the qualitative data, participants’ reflections were coded into the six themes which are outlined in Table 2 . This table also shows the percentage of responses under each theme in the data sample.

% Distribution of themes for feedback responses

As evident from Table 2 , 29% of the participants related the writing activity to increased content knowledge and a greater understanding of the mathematics (either directly or indirectly). Many of these responses signalled the importance of connections to prior knowledge using phrases such as ‘ I need to know first what is average ’ and ‘ you have to remember what you have done before ’ e.g. a student wrote the following: ‘ for solving, I just need to remember the formula but for writing I have to remember the work of [the] previous class also and how it comes ’.

Support for non-gradation of the writing assignments ranked second in the list of themes that emerged from the data and was noted in 20% of the respondents’ reflections. Furthermore, 15% of participants expressed that it was easier to explain through writing as opposed to an oral explanation. For example, one student expressed ‘ I like it. I know all [the] answers, but I am shy to speak it to all my class. When [the] teacher tells me to explain it, I cannot do it. But by writing, I can explain. So, I like it ’. Moreover, 11% of participants also noted an ‘ improved efficiency in writing ’ in general.

However, the lengthy nature of writing activities was a cause of concern for 16% of participants, as could be found in their responses for feedback which included phrases like ‘ lengthy activity ’, ‘ It takes a long time to write ’ and ‘solving is quicker than writing ’.

Finally, 9% of the participants found the task of writing challenging due to a lack of vocabulary to express themselves. They mentioned that they were more comfortable in solving questions mathematically as opposed to writing explanations using the English language.

Sample responses from participants’ reflections under the above-mentioned themes are presented in Table 3 .

Sample responses for participants’ attitude towards writing

There were also some responses where the participants liked the activity provided some conditions were met. Their responses marked the presence of ‘If’, ‘but’, etc. For example,

‘…. if teacher is there to guide me how to write and explain’

‘… but not for big questions. You have to write a lot’

‘……only when it is for activity, not for exam’

A full excerpt from a student’s similar response is cited here for reference:

‘I think writing helps to understand more by explaining and also it is a good brain exercise when we have to remember what we done before in the previous class. But in the [examination] paper, we have to solve only and then we get full marks. So, we should not waste time in writing because in this time we can solve many questions if we remember the formula’.

Although this student acknowledged the benefit of writing in terms of understanding more, they preferred giving a procedural solution or stating the answer directly. This student recognized that memorizing formulae is enough for getting good marks in an examination (owing to the particular marking scheme adopted). This raises the question of whether current assessment systems and education policies are really in favour of developing students’ understanding or are merely ranking procedures that reward rote learning.

A detailed discussion of these findings with illustrative examples and links to the research questions and relevant literature is included in the next section.

In this section, each research question is addressed, and findings discussed with relevant supporting references from the literature.

5.1 Research question 1: what effects, if any, does a short mathematical writing intervention have on students’ self-confidence and enjoyment levels in mathematics?

5.1.1 impact of the intervention on students’ enjoyment.

The analysis of the quantitative data revealed that participants’ mean enjoyment scores increased from 34.79 (prior to the intervention) to 39.82 (after the intervention). Results of a paired-samples t -test confirmed the statistical significance of this increase. In addition, these findings were supported through participants’ reflections of their perceptions of the writing intervention. Responses such as ‘ I like it ’, ‘ It was enjoyable ’, ‘… I like my maths class as this ’ indicate participants’ perceived enjoyment and positive attitude towards the writing intervention. A few students even expressed a desire for writing to be a regular feature of their mathematics lessons. For instance,

‘...Can we have it in our daily class also?’

‘…I think it should be weekly or on alternate days’

‘…We should have it in our daily class also’

This enjoyment was facilitated in a number of ways. For example, the role of a comfortable and interactive classroom environment was emphasized in the literature ( Firmender et al. , 2017 ; Hidi, 2000 ) and this was given due consideration throughout this study. The aim was to make students feel stress-free while writing their solution strategies.

5.1.2 Impact of the intervention on students’ self-confidence

Writing can develop greater confidence in mathematics by providing students with opportunities to grapple with mathematical ideas ( Powell, 1997 ). This assertion is particularly relevant to the present study, given the results of the self-confidence questionnaire. The increase in participants’ mean scores for self-confidence from 49.48 (pre-test) to 54.40 (post-test) is statistically significant as confirmed by the paired-samples t -test. The results are also supported by participants’ reflections of the writing intervention.

Participants felt an increase in confidence for a variety of reasons, for example, a greater understanding, improved efficiency in writing, clarity of thoughts, etc. Some excerpts from participants’ reflections are provided in Fig. 4 to highlight the findings specifically in relation to confidence.

Sample examples relating to confidence.

Sample of participants’ work.

Sample examples relating to non-gradation.

The use of phrases such as ‘ was sure’ , ‘ understood more’ and ‘ can improve’ in these excerpts are indicative of students’ increased confidence through writing and help to reaffirm the quantitative findings.

The reflection process which occurs during writing provides learners with an opportunity to look at their own thoughts and refine them in accordance with the information to be conveyed. ‘ Such acquisition of control and monitoring capabilities engenders in students’ feelings of accomplishment...students develop faith in themselves as learners who are capable of doing and understanding mathematics ’ ( Powell, 1997 , p. 23). According to Bandura (1977 ), performance accomplishments are the most powerful contributors to one’s self-efficacy beliefs. The confidence gained through these small accomplishments leads to motivation which brings further enjoyment for the task at hand.

5.2 Research question 2: what are students’ perceptions about the use of mathematical writing in the classroom?

A variety of student perceptions about the use of mathematical writing emerged from participants’ reflections. As detailed in Section 4.2, these responses were categorized into themes which will now be discussed considering similar studies from the literature.

5.2.1 Increase in content knowledge/understanding

As mentioned previously, 29% of participants noted an increase in their content knowledge and/or understanding. One participant wrote the following:

‘I don’t want to do maths for marks. This activity I enjoyed because you are doing it for your understanding, not as a paper (exam). I understood more when I solved my answers by writing’.

Participants’ feedback included statements such as ‘able to understand more by writing’, ‘writing helps to understand in a good way’, ‘by writing, there is less confusion for the answer’ and that ‘it will also clear any doubts that you had since you have to provide reasons on why your answer is correct’. Figure 5 presents a students’ work from the six sessions of the intervention, along with the feedback for how the student felt about the writing intervention.

These findings resonate with the results from many other studies which have reported the instrumental role of writing in a greater acquisition of content knowledge through a deeper engagement with the subject (for example, Borasi & Rose, 1989 ; Craig, 2016 ; Porter & Masingila, 2000 ; Pugalee, 2004 ).

A noteworthy observation was the reflection of thoughts through writing, as shown by the following response.

‘ When we write, we can revise it many times and we can know if we get a right or wrong answer. If I see that I am thinking wrong, I can start again with some other method but if we are just explaining orally, you have said all [the] words, and you cannot go back and change your answer. So, I like this part of writing that you can see what you are thinking and change it any time before submitting [the] final answer ’.

This excerpt is indicative of the importance of the reflection process that occurs while writing. Even though students may not recognize this on-going process, it is one of the many potential benefits of writing ( Craig, 2011 ; Ray Parsons, 2011 ). Effective learning occurs while resolving the cognitive conflicts in writers’ minds and results in metacognitive development ( Kuzle, 2013 ; Pugalee, 2004 ).

Sample examples relating to preference over oral explanations.

Sample examples relating to written efficiency.

5.2.2 Support for non-gradation of the writing assignments

In the qualitative data, many participants (20%) attributed the non-grading criterion of the written tasks as one of the reasons that they were comfortable with writing. They commented that while writing, they did not experience fear or anxiety of mathematics and enjoyed the activity without any stress as ‘it was not an exam’. Some participants’ reflections relevant to this are provided in Fig. 6 .

The literature provides evidence that anxiety regarding grading and assessment not only disrupts students’ capability to reason and understand but also causes a disliking for the subject ( Wells, 1994 ). In other words, the fear of being assessed may obstruct learning from taking place naturally and may hold back students from even attempting various mathematical tasks. This raises concerns regarding the adequacy of current assessment systems which often fail to assess the process and are more focused on the product of learning ( Little et al. , 2017 ). Another point of importance here is that participants were more comfortable because their wrong answers or mistakes were not highlighted. This is noteworthy, especially in the domain of mathematics where making mistakes can be an integral part of the learning experience. The fear of making mistakes may inhibit the brain’s growth and capacity to learn and understand ( Boaler & Dweck, 2016 ).

5.2.3 Ease of written explanation as opposed to oral

Among the various benefits of writing is its efficacy to reach out to diverse learners ( Bakewell, 2008 ). This assertion proved to be particularly true for the current study where approximately 15% of participants expressed being more comfortable with written explanations as opposed to oral. There were varied reasons for this response with many signalling a lack of confidence for class interactions. Some excerpts are provided in Fig. 7 to illustrate this.

These findings, as well as others from the literature, highlight that writing is a useful medium for empowering students who feel too shy to take part in class discussions. For example, a year-long study by Fry and Villagomez (2012) in the USA showed that writing helped introverts who seldom took part in class interactions. Students who participated in the writing-to-learn activities of that study showed an increased engagement with the course content. Furthermore, other research (for example, Pugalee, 2004 ) also notes the benefits of written explanations over oral, thus providing a rationale for writing to be an important vehicle for learning.

5.2.4 Improved efficiency in writing

Another feature perceived by participants in favour of writing was improved competence in their writing skills. Some students (11%) commented that they felt an improvement as they learned new words and gained confidence for writing. A few participants also expressed that they expect to improve further if they keep practicing. Figure 8 provides some of the participants’ reflections under this theme.

These findings are supported by existing literature in the field. It has been reported that an improved use of vocabulary (both in terms of formal mathematical vocabulary as well as the usage of complete sentences and linking words) is an associated advantage of providing reasoning in mathematical writing ( Cohen et al. , 2015 ). Rubenstein (2007 ) contends that in order to communicate mathematically, students must learn how to use correct mathematical language and this learning is supported through writing. Although participants in this study did not report an improvement in mathematics vocabulary, in particular, it could be expected that enjoyment and confidence gained through an improved comprehension in general may enthuse them with a liking for mastering discipline-specific language.

5.2.5 More time consuming

There were mixed responses from respondents in relation to the time-factor. Nine out of 55 students commented that writing takes a longer time and that they preferred giving a direct answer to the problem. Some of these contended that the aim of solving a mathematical task is getting the correct answer and thus viewed writing as a ‘waste of time’. A smaller number also felt that even though writing results in improved learning and is a good brain exercise, the lengthy nature of this activity trumps its benefits and thus, it may be ‘good for some problems but not for all’. Excerpts from some participants’ responses that fall into this category are presented in Fig. 9 .

The time-consuming nature of writing, as reported by the participants of this study, has also been reported by many others. In fact, the constraint of time acts as a potential drawback to the implementation of writing in regular teaching ( Baxter et al., 2005 ; McIntosh & Draper, 2001 ). On the contrary, Porter and Masingila (2000 ) assert that the success of writing in promoting a deeper mathematical understating might be primarily due to the increased time that is spent on writing for a given task. In fact, they consider whether the primary contributor in the process is the time spent on the task or the writing itself.

Sample examples relating to time consumption.

Sample example related to language.

5.2.6 Linguistic barriers

Although English was the language of instruction in the school where this study was carried out, 5 out of 55 students manifested a difficulty in using the English language for their explanations. For these students, a lack of language proficiency hindered their ability to explain their reasoning. For example, one student wrote the following: ‘… I know the maths of answer but not English words ’. Findings from other studies (for example, Craig, 2011 ; Porter & Masingila, 2000 ) confirm the prevalence of such linguistic difficulties for students.

The following reflection ( Fig. 10 ) is noteworthy and is worth mentioning with respect to this theme.

In this instance, although there were no rewards or incentives for the participants, this child wanted to be a good writer. She started learning new words to be more able to express herself. This one excerpt sets an example of how writing may instil a desire to learn more and implies that writing is a beneficial medium for inter-disciplinary learning.

This study sought to examine students’ perceptions and explore the impact of mathematical writing on students’ affective constructs of enjoyment and self-confidence. An analysis of the quantitative data revealed an increase in the mean scores for both enjoyment and self-confidence. Results of t -tests confirmed that these increases were statistically significant. A further analysis revealed there to be no gender-related differences.

A thematic analysis of participants’ reflections of the writing intervention also signalled a positive perception towards such activities. Overall, participants gave a positive response towards the intervention and reported an increase in content knowledge/understanding as the main reason. As well as supporting the non-grading of the tasks, some students also noted the ease of written explanations as opposed to oral and an improved efficiency in writing. At the same time, the time-consuming nature of the activity and a lack of proficiency in English language emerged as the factors of concern for a few participants.

In conclusion, findings from this study indicated a progressive shift in students’ attitude post-intervention. Hence, although this was a short intervention with a relatively small cohort, it can be inferred that mathematical writing has the potential to increase students’ enjoyment and self-confidence in mathematics and has a positive impact on their learning. In contrast with the traditional methods of teaching, writing activities in mathematics may serve as an effective medium for transforming students’ mindsets and fostering positive attitudes towards the subject.

However, the benefits of writing are contingent on a host of factors such as the nature of the writing tasks allotted to students, the intensity of intervention by instructors, the students’ ability to exploit its benefits, etc. It is necessary to keep these factors in mind as these may neutralize the positives that can be gained from writing. Furthermore, students and teachers may hold different views about mathematical writing which may affect the quality and nature of writing in a mathematics classroom. In addition, time-bound learning also constrains the integration of writing into classrooms. Future research might gauge the effect of other contextual factors, for example, classroom environment, motivational and constructive feedback, etc. that may aid in bringing out positive changes in students’ affective domains while learning. Additional research is also required to investigate how writing, if incorporated into the regular curriculum, may change learners’ as well as teachers’ beliefs about the nature of mathematics.

It is important to keep in mind that the absence of a comparison or control group may affect the generalizability of the results of this study and that the positive results may not be solely due to the intervention. Several other factors such as the non-routine nature of the mathematics tasks, the activity-based sessions, non-grading of work, absence of teacher, etc., could have contributed to the findings. Nonetheless, the results of the study point to the potential of mathematical writing to be used as an effective scaffolding tool for students’ mathematics learning.

Additionally, the effect of various other contextual factors cannot be ignored. There is a chance that participants’ enhanced enjoyment and self-confidence in this study could have been affected by the nature of the research context itself. For example, participants may be keen to impress and please the researcher which might also have acted as a restraining factor affecting the results of the study in a positive or negative manner. Activity-based intervention with no assessment involved may also have contributed to the heightened enjoyment and reduced levels of anxiety in participants.

Finally, the study has emphasized a domain of mathematics education where there is a dearth of research. It has thus added value to the existing body of research and is particularly enriching from an Indian context. In terms of future educational policy, it may prove to be helpful and act as a starting point for further research in India and indeed in other education systems around the world.

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Tandeep Kaur is a research associate and a doctoral student at the Institute of Education, Dublin City University. She has extensive experience in teaching mathematics at the secondary and senior secondary level. Her research interests lie in exploration of best practices in mathematics education, initial teacher education and students’ mental health and well-being. E-mail: [email protected]

Mark Prendergast is a senior lecturer in Education in the School of Education at University College Cork. His teaching and research interests include mathematics education, teacher education and working with non-traditional students. E-mail: [email protected]

(All tasks were selected from TIMSS (2011) 8th-Grade Mathematics Concepts and Mathematics Items SOURCE: TIMSS 2011 Assessment. Copyright © 2013 International Association for the Evaluation of Educational Achievement (IEA). Publisher: TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College, Chestnut Hill, MA and International Association for the Evaluation of Educational Achievement (IEA), IEA Secretariat, Amsterdam, the Netherlands.)

Each student in team B jumped farther than any student in team A.

After every student in team A jumped, there was a student in team B who jumped farther.

As a group, team B jumped farther than team A.

Some students in team A jumped farther than some students in team B.

(Item number M042269, Cognitive Domain—Reasoning)

EN1. I get a great deal of satisfaction out of solving a mathematics problem.

EN2. I have usually enjoyed studying mathematics in school.

EN3. Mathematics is dull and boring.

EN4. I like to solve new problems in mathematics.

EN5. I would prefer to do an assignment in mathematics than to write an essay.

EN6. I really like mathematics.

EN7. I am happier in a mathematics class than in any other class.

EN8. Mathematics is a very interesting subject.

EN9. I am comfortable expressing my own ideas on how to look for solutions to a difficult problem in mathematics.

EN10. I am comfortable answering questions in mathematics class.

SC1. Mathematics is one of my most dreaded subjects.

SC2. My mind goes blank and I am unable to think clearly when working with mathematics.

SC3. Studying mathematics makes me feel nervous.

SC4. Mathematics makes me feel uncomfortable.

SC5. I am always under a terrible strain in a mathematics class.

SC6. When I hear the word mathematics, I have a feeling of dislike.

SC7. It makes me nervous to even think about having to do a mathematics problem.

SC8. Mathematics does not scare me at all.

SC9. I expect to do fairly well in any mathematics class I take.

SC10. I am always confused in my mathematics class.

SC11. I have a lot of self-confidence when it comes to mathematics.

SC12. I am able to solve mathematics problems without too much difficulty.

SC13. I feel a sense of insecurity when attempting mathematics.

SC14. I learn mathematics easily.

SC15. I believe I am good at solving mathematics problems.

(ATMI; Tapia & Marsh, 2004 ).

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My Reflection in Mathematics in the Modern World

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Home | Science | Mathematics

My Experience with Mathematics

• Updated July 25, 2023
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Many people perceive mathematics as boring and all about numbers, letters, and equations. But Mathematics plays a very vital role in everyday life. It is found everywhere in life and work. There are many applications of math in the society. But before appreciating those applications and use of mathematics we should go to school and learn all of those. It is not easy studying mathematics nor fun but Mathematics is a subject given since grade 1, I think, that requires a high amount of paying attention and listening to the teachers. And based on my experience it is indeed a difficult subject to take.

Math is a big part of my life not just mine but all of us because of course it is used all the time. However some people, including me at the past, have anxiety when I think about Mathematics because I don’t like math and we are tired of solving for “x”. In the past, I literally don’t like and understand why it was part and given to us because I don’t even get any knowledge every after the class. My first impression of it was good, somehow, but when junior high school happened, I came to dislike it, especially Grade 9-10 but time goes by I came to enjoy it again and again. Seeing the nature and what is really mathematics is all about makes me feel amazed and adore math all the time. I think the teachers of Mathematics have a great impact in learning and loving Mathematics. I observed how It was explained by the teacher makes me like and love Math again and again.

My experience with math was initially good, because of the teachers I had. Teachers make all the difference when it comes to presenting math and I think social media has a great influence also in learning and appreciating math. Thinking back about all my experiences when it came to learning math, I didn’t have the best experiences because I struggled a lot to understand the different concepts, formulas, techniques and tools that are used when solving different equations but yet time goes by and at present I appreciate it and it was fun and the best.

• Math is Fun
• Khan Academy
• Wolfram MathWorld
• IXL Learning
• Math Central – University of Regina
• The On-Line Encyclopedia of Integer Sequences (OEIS)

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25 Learning Experiences Examples

A learning experience is any experience a student has in which they learn something. They can be both intentional and unintentional and both in and outside of schools.

Learning experiences can be structured, or unstructured, direct, or vicarious. They can come from listening to a lecture, reading, participating in an activity, or through observation.

For teachers in structured learning situations, it is often suggested that learning experiences be:

• Tied to educational objectives
• Meaningful and engaging
• Age appropriate
• Match the learning style of students
• Connected to real-life situations
• Varied and dynamic
• Culturally aware
• Designed to allow students options

Types of Learning Experiences

• Structured – These are experiences that have a clear procedure to ensure that the learning occurs. Example: A chemistry professor uses direct instruction to explain how atoms are held together by chemical bonds to form molecules.
• Unstructured – These are experiences that have learning as a goal, but allow students to find their own way to learn the lesson. Example: Students are given a set of materials such as paper towel tubes, tape, scissors, and cardboard to make their own marble runs (see also: unstructured play ).  
• Experiential – These situations involve students ‘experiencing’ what it’s like to be in a certain situation. Example: business students engage in a simulation that involves assuming different roles in a labor-contract negotiation.
• Collaborative – This involves learning that occurs alongside and with other learners. Example: Students must work in teams to develop a customer satisfaction survey, collect data, analyze the data and graph the results.
• Observational – This involves passive learning, where students come to a realization simply through watching something occur. Example: A young child observes their father cracking eggs to make breakfast and then tries to mimic the actions.
• Reading – Written text can be used as a form of instruction and therefore can facilitate learning. Example: Students are assigned to read the next chapter for homework and take the sample test at the end.
• Independent – This learning experience doesn’t involve a formal teacher, but is instigated and pursued by the learner themselves. Example: A college student takes computer programming courses online during their summer vacation.
• Blended – This type of learning involves a mix of teacher instruction and student-led inquiry. Example: The assignment involves students listening to their professor’s lecture in the classroom, supplemented with material from an online seminar.  
• Project-Based – The student is provided a project, and learning occurs through the process of completing the project. Example: Fifth graders make a poster on volcanoes that includes text, diagrams, and photos.  
• Sensory Based – This learning takes place when our senses (touch, feel, taste, smell, sight) give us stimuli that help us comprehend our world. Example: A 12-month-old grasps an unfamiliar object, examines it visually, tries to pull it apart, smells it and then puts it in its mouth to determine if it is edible.

Learning Experience Examples

• Internship: During internships, we get to learn what it’s like to do a certain job and whether we’d enjoy it.
• Apprenticeship: During apprenticeships, we learn on the job, which helps to develop practical rather than theoretical skills.
• Reading a good book: Reading books isn’t just fun. A good book also teaches us moral and life lessons.
• Project-based learning tasks: In this learning experience, students are given projects to complete. In the process of doing the project, learning naturally occurs.
• Inquiry-based learning task : Students are given a puzzle to investigate, and, through investigation, they gain deep knowledge.
• Lightbulb moment: A lightbulb moment is any moment where you finally reach a realization about something (see also: threshold concepts)
• Take your kid to work day: Attending a parent’s workplace is often highly educational to a child who can get an insight into what it means to work in a certain profession.
• Losing: Losing in a sporting game helps you learn to be gracious in defeat.
• Failing: In failure, we often get a chance to reflect and figure out what we did wrong so we succeed next time.
• Success: Failure often teaches more than success, but if we reflect on our successes, we can also learn what we did right to ensure we succeed again in the future.
• Observation: Observing another child getting in trouble and deciding it’s best not to misbehave.
• Conducting primary research : Research studies, such as dissertations at university, are designed to learn something new (often that no one knew before).
• Experimentation: Experiments allow us to test hypotheses that lead to new insights on a topic.
• Self-reflection : Through self-reflection, we explore how our personal experiences have educational value.
• Vicarious punishment : People see the negative consequences of someone else’s actions, so they decide not to participate in those behaviors themselves ( see also: vicarious reinforcement ).
• Writing an essay: The process of constructing an essay involves conducting research and figuring out how to structure an argument, which helps you to develop your knowledge.
• Teamwork tasks: Students are often set teamwork tasks not only to complete a curriculum outcome, but also to learn how to get along with others to reach a common goal.
• Attending a presentation: Presentations from teachers, colleagues, or mentors are designed to help structure educational information into a clear and simple learning experience.
• Professional development days: Professional development days can involve learning about the newest innovations in an industry so we can remain relevant and skilled practitioners.
• Seminars: In university, seminars are small group learning experiences that facilitate conversation between peers.
• One-to-one coaching: One to one coaching is a valuable learning experience because it’s catered directly to the student’s needs, unlike whole group instruction.
• Embarrassment: Embarrassment is often very confronting and leaves a big impression, which teaches us to avoid certain behaviors in the future.
• Play: Both children and adults engage in play to learn about ourself, our bodies, how to develop social skills , and so much more (see also: play based learning ).
• Conversation: Through one-to-one conversations, we can gather other people’s perspectives on issues, which can help us to learn more about the issues.
• Teachable moments : Teachable moments are everyday instances that can help elucidate an important lesson for students.
• Disciplinary scenarios: Good disciplinary techniques should teach a clear lesson, such as “this behavior is unacceptable” or “this action leads to this negative consequence”.

Learning Experiences Case Studies

1. inquiry-based learning experience.

According to Lee et al. (2004), inquiry-based learning is an “array of classroom practices that promote student learning through guided and, increasingly, independent investigation of complex questions and problems, often for which there is no single answer” (p. 9).

For example, in a traditional anthropology course, a professor will lecture while students diligently take notes that are committed to rote memory and later regurgitated on an exam.

However, in an inquiry-based lesson, instead of telling students about cultural artifacts, their relevance to a specific culture, and what they were used for, the lesson would be reversed.

The professor gives the artifacts to the students and then says nothing.

The students then set out to examine the objects and conduct their own research. They try to identify what the objects are, what they are used for, which culture they belong to, and whatever else they can uncover.

This is a type of learning experience that is far more engaging to students. The information is processed at a much deeper level and their interest and motivation is far higher than what would occur in a traditional lecture format.  

2. Service-Oriented Learning Experience

Service-oriented learning refers to when students apply academic concepts to help address community or societal needs. This type of learning experience contains several elements of other types of learning. It is often experiential, collaborative, and project-based.

For example, the Growing Voters report by Tufts University provides institutions with a valuable framework for facilitating participation of the next generation of U. S. voters.

The framework identifies ways that educators and community leaders can “…close voting gaps, expand the electorate, and support a more equitable and representative American democracy” .

This is the type of learning experience for students that also addresses a societal need. However, it’s more than just volunteering: “…service-learning applies equal focus to both learning and the service goals . It requires an academic context and is designed so that that the service and learning goals are mutually reinforcing” (Starting Point, n.d.).

3. Performance-Based Learning Experience

Performance-based learning involves students developing specific skills related to the subject being studied. It helps them see the connection between abstract academic concepts and how those concepts manifest themselves in the real world.

For example, this group of math teachers spent a tremendous amount of time designing a learning experience called Mission Relief . The students play different roles in a simulated emergency scenario involving an airplane.

By applying mathematical formulas and various aeronautical concepts, the students are tasked with guiding the plane to safety.

Performance-based learning is far more interesting to students than traditional formats. It completely transforms the learning experience.

Students process the information more deeply and learn about the subtle nuances of a subject that can only be appreciated through experience.

4. Internships

An internship is when a student works in an organization for several months, for free. That organization could be a small business, large corporation, or non-profit organization. Internships are great ways for students to gain practical experience.

Majoring in a subject domain involves processing a lot of abstract information, taking a lot of notes, writing papers, and studying for exams.

But, there is no way for a student to know if they would actually like to have a career in that line of work without having any experience actually doing the job.

So, an internship is a valuable opportunity for students to dip their toe in the water and find out what the profession is really like.

The results can be quite surprising. Many times, a student will discover that the daily job responsibilities are completely unlike what they envisioned. In other cases, students’ career interests are affirmed, even strengthened, as they discover the job is even more exciting than they imagined.

5. Study Abroad

In the era of globalization, it has never been more important to attain some cross-cultural experience. Many occupations today and in the future will involve collaborating with people that are located in foreign lands. That’s why studying abroad is so valuable.

Many universities offer students the unique opportunity to study in a foreign country. These programs can be for as short as a few weeks or as long as an entire academic year.

Students can stay with a host family that has been carefully chosen, or live in a campus dormitory.

In addition, the credits they receive for the college courses they take transfer to their home institution.

The benefits are numerous: cultural enlightenment, development of a global perspective, forming new friendships, even becoming proficient in a second language.

Of course, not every aspect of studying abroad is super fantastic. Beware of the infamous culture shock .

There are many types of learning experiences . Students today have many options that were once never even imagined.

Educational practices have evolved to be inclusive and dynamic. Teachers and professors take into account the characteristics of their students, their learning styles, and their motivation levels.

Students can learn in the classroom, in the real world, in the virtual world, or in another country. Today, the options are limitless. 

Furco, A. and Billig, S.H., (2002) Service-Learning: The Essence of the Pedagogy . Greenwich, CT: Information Age Publishing.

Lee, V. S., Greene, D. B., Odom, J., Schechter, E., & Slatta, R. W. (2004). What is inquiry guided learning. In V. S. Lee (Ed.), Teaching and learning through inquiry: A guidebook for institutions and instructors (pp. 3-15). Sterling, VA: Stylus Publishing.

Starting Point. (n.d.). What is service learning? Retrieved November 2, 2022, from https://serc.carleton.edu/introgeo/service/what.html

Wirkala, C., & Kuhn, D. (2011). Problem-based learning in K–12 education: Is it effective and how does it achieve its effects? American Educational Research Journal, 48 (5), 1157–1186. https://doi.org/10.3102/0002831211419491

Dave Cornell (PhD)

Dr. Cornell has worked in education for more than 20 years. His work has involved designing teacher certification for Trinity College in London and in-service training for state governments in the United States. He has trained kindergarten teachers in 8 countries and helped businessmen and women open baby centers and kindergartens in 3 countries.

• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ 25 Positive Punishment Examples
• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ 25 Dissociation Examples (Psychology)
• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ 15 Zone of Proximal Development Examples
• Dave Cornell (PhD) https://helpfulprofessor.com/author/dave-cornell-phd/ Perception Checking: 15 Examples and Definition

Chris Drew (PhD)

This article was peer-reviewed and edited by Chris Drew (PhD). The review process on Helpful Professor involves having a PhD level expert fact check, edit, and contribute to articles. Reviewers ensure all content reflects expert academic consensus and is backed up with reference to academic studies. Dr. Drew has published over 20 academic articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education and holds a PhD in Education from ACU.

• Chris Drew (PhD) #molongui-disabled-link 25 Positive Punishment Examples
• Chris Drew (PhD) #molongui-disabled-link 25 Dissociation Examples (Psychology)
• Chris Drew (PhD) #molongui-disabled-link 15 Zone of Proximal Development Examples
• Chris Drew (PhD) #molongui-disabled-link Perception Checking: 15 Examples and Definition

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Contents of Volume 61, Number 2 HTML articles powered by AMS MathViewer View front and back matter from the print issue

Facility for Rare Isotope Beams

At michigan state university, frib professorial assistant earns goldwater scholarship.

Aaron Philip, a professorial assistant at FRIB, has earned a  Barry M. Goldwater Scholarship , becoming Michigan State University’s fifty-fifth Goldwater Scholar.

The Goldwater Foundation seeks sophomores and juniors committed to a research career in STEM fields with the potential for significant future contribution in their chosen field. The award provides $7,500 per year in funding for 51 students for undergraduate tuition and living expenses.

For the 2024 Goldwater Scholarship competition, 1,353 undergraduates were nominated by 446 institutions. Philip was among 438 scholars selected. The funding for the award is a collaboration between the U.S. Congress and the Department of Defense’s National Defense Education Program.

Philip is a second-year Michigan State University student from Los Alamos, New Mexico studying Physics and Advanced Mathematics in the College of Natural Science. He is also a member of the Honors College.

“I am honored and humbled to join the ranks of Spartan Goldwater Scholars. I share this recognition with my professors, research mentors, fellow students, and family who have all supported me and cultivated my passion to pursue a career in physics research,” Philip said. “Specifically, I would like to thank my research mentors over the past few years for their guidance, encouragement, and mentorship: Drs. Pablo Giuliani, Kyle Godbey, Witek Nazarewicz, Odelia Schwartz, Jianliang Qian, and Benjamin Nebgen.”

Philip is passionate about pursuing a career in research addressing micro-scale physics problems using analytic approaches, high performance computing, and AI. He has contributed to diverse research projects through his roles as a professorial assistant at FRIB, a Discovering America researcher with MSU’s Math Department, a student intern at the Theoretical Division of Los Alamos National Laboratory (LANL), and as a Computer Science Research Experience for Undergraduates (REU) student at the University of Miami.

“Aaron joined our nuclear theory research group at the Facility for Rare Isotope Beams at Michigan State University in August 2022 as an undergraduate research assistant. An incoming first-year undergraduate student, he came extremely well prepared to directly work in forefront research and quickly managed to get acquainted with the necessary tools and background knowledge,” Kyle Godbey, a research assistant professor at FRIB, and Witold Nazarewicz, John A. Hannah Distinguished Professor of Physics and chief scientist at FRIB, said.

“During the course of his work, Aaron was able to reach a level of mastery of theoretical and computational methods on par with the current experts in the field. We consider ourselves to be incredibly lucky to have Aaron as a member of our research group and we have no doubt that he will go on to have a successful research career,” Godbey and Nazarewicz said.

“Aaron’s research at the Facility for Rare Isotope Beams has been exemplary, and his mentorship activities embody the values of care and support that empower excellence at the MSU Honors College. We congratulate Aaron on being named a Goldwater Scholar,” Long said.

Philip has written two papers and presented at various conferences, including MSU’s Mathematics and Data Science Conferences, the University of Miami’s Computer Science REU Poster Presentation, and a LANL Lab Directed Research and Development Review. He also serves as a student tutor through the Mathematics Learning Center and at East Lansing High School.

“Congratulations to Aaron on this esteemed achievement,” said FRIB Laboratory Director Thomas Glasmacher. “Being named a Goldwater Scholar is a testament to Aaron’s dedication and outstanding efforts. We are so proud he is furthering his research pursuits at FRIB and honored to be part of his journey as he prepares to become a leader in our field.”

Read the original article on the  MSUToday website .

Michigan State University operates the Facility for Rare Isotope Beams as a user facility for the U.S. Department of Energy Office of Science (DOE-SC), supporting the mission of the DOE-SC Office of Nuclear Physics.

The End of Foreign-Language Education

Thanks to AI, people may no longer feel the need to learn a second language.

Listen to this article

Produced by ElevenLabs and News Over Audio (NOA) using AI narration.

A few days ago, I watched a video of myself talking in perfect Chinese. I’ve been studying the language on and off for only a few years, and I’m far from fluent. But there I was, pronouncing each character flawlessly in the correct tone, just as a native speaker would. Gone were my grammar mistakes and awkward pauses, replaced by a smooth and slightly alien-sounding voice. “My favorite food is sushi,” I said— wo zui xihuan de shiwu shi shousi —with no hint of excitement or joy.

I’d created the video using software from a Los Angeles–based artificial-intelligence start-up called HeyGen. It allows users to generate deepfake videos of real people “saying” almost anything based on a single picture of their face and a script, which is paired with a synthetic voice and can be translated into more than 40 languages. By merely uploading a selfie taken on my iPhone, I was able to glimpse a level of Mandarin fluency that may elude me for the rest of my life.

HeyGen’s visuals are flawed—the way it animates selfies almost reminded me of the animatronics in Disney’s It’s a Small World ride—but its language technology is good enough to make me question whether learning Mandarin is a wasted effort. Neural networks, the machine-learning systems that power generative-AI programs such as ChatGPT, have rapidly improved the quality of automatic translation over the past several years, making even older tools like Google Translate far more accurate.

At the same time, the number of students studying foreign languages in the U.S. and other countries is shrinking. Total enrollment in language courses other than English at American colleges decreased 29.3 percent from 2009 to 2021, according to the latest data from the Modern Language Association, better known as the MLA. In Australia, only 8.6 percent of high-school seniors were studying a foreign language in 2021—a historic low. In South Korea and New Zealand , universities are closing their French, German, and Italian departments. One recent study from the education company EF Education First found that English proficiency is decreasing among young people in some places.

Many factors could help explain the downward trend, including pandemic-related school disruptions, growing isolationism, and funding cuts to humanities programs. But whether the cause of the shift is political, cultural, or some mix of things, it’s clear that people are turning away from language learning just as automatic translation becomes ubiquitous across the internet.

Read: High-school English needed a makeover before ChatGPT

Within a few years, AI translation may become so commonplace and frictionless that billions of people take for granted the fact that the emails they receive, videos they watch, and albums they listen to were originally produced in a language other than their native one. Something enormous will be lost in exchange for that convenience. Studies have suggested that language shapes the way people interpret reality. Learning a different way to speak, read, and write helps people discover new ways to see the world—experts I spoke with likened it to discovering a new way to think. No machine can replace such a profoundly human experience. Yet tech companies are weaving automatic translation into more and more products. As the technology becomes normalized, we may find that we’ve allowed deep human connections to be replaced by communication that’s technically proficient but ultimately hollow.

AI language tools are now in social-media apps, messaging platforms, and streaming sites. Spotify is experimenting with using a voice-generation tool from the ChatGPT maker OpenAI to translate podcasts in the host’s own voice, while Samsung is touting that its new Galaxy S24 smartphone can translate phone calls as they’re occurring . Roblox, meanwhile, claimed last month that its AI translation tool is so fast and accurate , its English-speaking users might not realize that their conversation partner “is actually in Korea.” The technology—which works especially well for “ high-resource languages ” such as English and Chinese, and less so for languages such as Swahili and Urdu—is being used in much more high-stakes situations as well, such as translating the testimony of asylum seekers and firsthand accounts from conflict zones. Musicians are already using it to translate songs , and at least one couple credited it with helping them to fall in love.

One of the most telling use cases comes from a start-up called Jumpspeak, which makes a language-learning app similar to Duolingo and Babbel. Instead of hiring actual bilingual actors, Jumpspeak appears to have used AI-generated “people” reading AI-translated scripts in at least four ads on Instagram and Facebook. At least some of the personas shown in the ads appear to be default characters available on HeyGen’s platform. “I struggled to learn languages my whole life. Then I learned Spanish in six months, I got a job opportunity in France, and I learned French. I learned Mandarin before visiting China,” a synthetic avatar says in one of the ads, while switching between all three languages. Even a language-learning app is surrendering to the allure of AI, at least in its marketing.

Alexandru Voica, a communications professional who works for another video-generating AI service, told me he came across Jumpspeak’s ads while looking for a program to teach his children Romanian, the language spoken by their grandparents. He argued that the ads demonstrated how deepfakes and automated-translation software could be used to mislead or deceive people. “I'm worried that some in the industry are currently in a race to the bottom on AI safety,” he told me in an email. (The ads were taken down after I started reporting this story, but it’s not clear if Meta or Jumpspeak removed them; neither company returned requests for comment. HeyGen also did not immediately respond to a request for comment about its product being used in Jumpspeak’s marketing.)

The world is already seeing how all of this can go wrong. Earlier this month, a far-right conspiracy theorist shared several AI-generated clips on X of Adolf Hitler giving a 1939 speech in English instead of the original German. The videos, which were purportedly produced using software from a company called ElevenLabs, featured a re-creation of Hitler’s own voice. It was a strange experience, hearing Hitler speak in English, and some people left comments suggesting that they found him easy to empathize with: “It sounds like these people cared about their country above all else,” one X user reportedly wrote in response to the videos. ElevenLabs did not immediately respond to a request for comment. ( The Atlantic uses ElevenLabs’ AI voice generator to narrate some articles.)

Read: The last frontier of machine translation

Gabriel Nicholas, a research fellow at the nonprofit Center for Democracy and Technology, told me that part of the problem with machine-translation programs is that they’re often falsely perceived as being neutral, rather than “bringing their own perspective upon how to move text from one language to another.” The truth is that there is no single right or correct way to transpose a sentence from French to Russian or any other language—it’s an art rather than a science. “Students will ask, ‘How do you say this in Spanish?’ and I’ll say, ‘You just don’t say it the same way in Spanish; the way you would approach it is different,’” Deborah Cohn, a Spanish- and Portuguese-language professor at Indiana University Bloomington who has written about the importance of language learning for bolstering U.S. national security , told me.

I recently came across a beautiful and particularly illustrative example of this fact in an article written by a translator in China named Anne. “Building a ladder between widely different languages, such as Chinese and English, is sometimes as difficult as a doctor building a bridge in a patient's heart,” she wrote. The metaphor initially struck me as slightly odd, but thankfully I wasn’t relying on ChatGPT to translate Anne’s words from their original Mandarin. I was reading a human translation by a professor named Jeffrey Ding, who helpfully noted that Anne may have been referring to a type of heart surgery that has recently become common in China. It's a small detail, but understanding that context brought me much closer to the true meaning of what Anne was trying to say.

Read: The college essay is dead

But most students will likely never achieve anything close to the fluency required to tell whether a translation rings close enough to the original or not. If professors accept that automated technology will far outpace the technical skills of the average Russian or Arabic major, their focus would ideally shift from grammar drills to developing cultural competency , or understanding the beliefs and practices of people from different backgrounds. Instead of cutting language courses in response to AI, schools should “stress more than ever the intercultural components of language learning that tremendously benefit the students taking these classes,” Jen William, the head of the School of Languages and Cultures at Purdue University and a member of the executive committee of the Association of Language Departments, told me.

Paula Krebs, the executive director of the MLA, referenced a beloved 1991 episode of Star Trek: The Next Generation to make a similar point. In “Darmok,” the crew aboard the starship Enterprise struggles to communicate with aliens living on a planet called El-Adrel IV. They have access to a “universal translator” that allows them to understand the basic syntax and semantics of what the Tamarians are saying, but the greater meaning of their utterances remains a mystery.

It later becomes clear that their language revolves around allegories rooted in the Tamarians’ unique history and practices. Even though Captain Picard was translating all the words they were saying, he “couldn’t understand the metaphors of their culture,” Krebs told me. More than 30 years later, something like a universal translator is now being developed on Earth. But it similarly doesn’t have the power to bridge cultural divides the way that humans can.