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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

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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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My Reflection in Mathematics in the Modern World

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Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

• Updated on  
• Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

Table of Contents

• 1 Essay on Importance of Mathematics in our Daily life in 100 words 
• 2 Essay on Importance of Mathematics in our Daily life in 200 words
• 3 Essay on Importance of Mathematics in our Daily Life in 350 words

Essay on Importance of Mathematics in our Daily life in 100 words 

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Essay on Importance of Mathematics in our Daily life in 200 words

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

Also Read:-   My Aim in Life

Also Read: How to Prepare for UPSC in 6 Months?

Essay on Importance of Mathematics in our Daily Life in 350 words

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.  From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

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Reflections: Students in Math Class

Published by patrick honner on June 14, 2012 June 14, 2012

At the end of the term I ask students to write simple reflections on their experiences from the year:  what they learned about math, about the world, about themselves.  It’s one of the many ways I get students writing in math class .

It’s a great way to model reflection as part of the learning process, and it’s also a good way for me to get feedback about the student experience.

Mostly, it’s fun!  I love sharing and discussing the reflections with students, and it always results in great end-of-year conversations.

Here are some of my favorites.

After learning a little more about math, I think math is created rather than discovered.  This makes mathematicians and scientists the creators, not merely the seekers.

I learned a lot of things from my classmates that I wouldn’t have learned if I were to just study on my own.

I have learned that I still have very much to learn about myself.

Mathematics is magical; it can lead you to a dead end, but then it can miraculously open up an exit.

Learning how to think of things in three dimensions completely changed the way I saw math.

By seeing algebraic and geometric interpretations, I learned how to communicate math in more ways.

The process which turns a difficult problem into a relatively easy problem is the beauty of math.

One of the best parts of reflection is how much it gets you thinking about the future.  Plenty of food for thought here.

For more resources, see my Writing in Math Class  page.

Related Posts

• Writing in Math Class
• Writing in Math Class: Peer Review
• Why Write in Math Class?

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patrick honner

Math teacher in Brooklyn, New York

Hilary · August 7, 2012 at 3:39 pm

These are great!

admin · August 8, 2012 at 1:18 am

Yeah, inspiring and thoughtful stuff. It’s a great way to make kids conscious of the role of reflection in learning while getting some practical teaching advice, too.

The key is to get the students writing and reflecting on a regular basis. By the end of the year, the students will have great things to say plus the tools and motivaiton to say them.

Annette · June 17, 2018 at 5:09 pm

I know this is an old post, but this is truly inspiring and I hope you encourage students to continue doing reflections!

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Resources Teaching Writing

Math that connects where we’re going to where we’ve been — quanta magazine.

My latest column for Quanta Magazine is about the power of creative thinking in mathematics, and how understanding problems from different perspectives can lead us to surprising new conclusions. It starts with one of my Read more…

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People tell me my job is easy. You get summers off. You only work nine months of the year. You’re done at 3 pm. You get paid to babysit. Students at that school won’t succeed Read more…

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Home — Essay Samples — Science — Mathematics in Everyday Life — Mathematics In Everyday Life: Most Vital Discipline

Mathematics in Everyday Life: Most Vital Discipline

• Categories: Mathematics in Everyday Life

About this sample

Words: 795 |

Published: Mar 14, 2019

Words: 795 | Pages: 2 | 4 min read

Works Cited

• Benacerraf, P. (1991). Mathematics as an object of knowledge. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings (pp. 1-13). Cambridge University Press.
• EdReady. (n.d.). Home. Retrieved from https://www.edready.org/
• Puttaswamy, T. K. (2012). Engineering mathematics. Dorling Kindersley (India) Pvt. Ltd.
• Steen, L. A. (Ed.). (2001). Mathematics today: Twelve informal essays. Springer Science & Business Media.
• Suter, B. W. (2012). Mathematics education: A critical introduction. Bloomsbury Academic.
• Tucker, A. W. (2006). Applied combinatorics. John Wiley & Sons.
• Vakil, R. (2017). A mathematical mosaic: Patterns & problem solving. Princeton University Press.
• Wolfram MathWorld. (n.d.). MathWorld--The web's most extensive mathematics resource. Retrieved from http://mathworld.wolfram.com/
• Wu, H. H. (2011). The mis-education of mathematics teachers. Educational Studies in Mathematics, 77(1), 1-20.
• Ziegler, G. M., & Aigner, M. (2012). Proofs from THE BOOK. Springer Science & Business Media.

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Engaging Maths

Promoting student reflection to improve mathematics learning.

Critical reflection is a skill that doesn’t come naturally for many students, yet it is one of the most important elements of the learning process. As teachers, not only should we practice what we preach by engaging in critical reflection of our practice, we also need to be modelling critical reflection skills to our students so they know what it looks like, sounds like, and feels like (in fact, a Y chart is a great reflection tool).

How often do you provide opportunities for your students to engage in deep reflection of their learning? Consider Carol Dweck’s research on growth mindset. If we want to convince our students that our brains have the capability of growing from making mistakes and learning from those mistakes, then critical reflection must be part of the learning process and must be included in every mathematics lesson.

What does reflection look like within a mathematics lesson, and when should it happen?Reflection can take many forms, and is often dependent on the age and abilities of your students. For example, young students may not be able to write fluently, so verbal reflection is more appropriate and can save time. Verbal reflections, regardless of the age of the student, can be captured on video and used as evidence of learning. Video reflections can also be used to demonstrate learning during parent/teacher conferences. Another reflection strategy for young students could be through the use of drawings. Older students could keep a mathematics journal, which is a great way of promoting non-threatening, teacher and student dialogue. Reflection can also occur amongst pairs or small groups of students.

How do you promote quality reflection? The use of reflection prompts is important. This has two benefits: first, they focus students’ thinking and encourage depth of reflection; and second, they provide information about student misconceptions that can be used to determine the content of the following lessons. Sometimes teachers fall into the trap of having a set of generic reflection prompts. For example, prompts such as “What did you learn today?”, “What was challenging?” and “What did you do well?” do have some value, however if they are over-used, students will tend to provide generic responses. Consider asking prompts that relate directly to the task or mathematical content.

An example of powerful reflection prompts is the REAL Framework, from Munns and Woodward (2006). Although not specifically written for mathematics, these reflection prompts can be adapted. One great benefit of the prompts is that they fit into the three dimensions of engagement: operative, affective, and cognitive. The following table represents reflection prompts from one of four dimensions identified by Munns and Woodward: conceptual, relational, multidimensional and unidimensional.

Finally, student reflection can be used to promote and assess the proficiencies (Working Mathematically in NSW) from the Australian Curriculum: Mathematics as well as mathematical concepts. It can be an opportunity for students to communicate mathematically, use reasoning, and show evidence of understanding. It can also help students make generalisations and consider how the mathematics can be applied elsewhere.

How will you incorporate reflection into your mathematics lessons? Reflection can occur at any time throughout the lesson, and can occur more than once per lesson. For example, when students are involved in a task and you notice they are struggling or perhaps not providing appropriate responses, a short, sharp verbal reflection would provide opportunity to change direction and address misconceptions. Reflection at the conclusion of a lesson consolidates learning, and also assists students in recognising the learning that has occurred. They are more likely to remember their learning when they’ve had to articulate it either verbally or in writing.

And to conclude, some reflection prompts for teachers (adapted from the REAL Framework):

• How have you encouraged your students to think differently about their learning of mathematics?
• What changes to your pedagogy are you considering to enhance the way you teach mathematics?
• Explain how your thinking about mathematics teaching and learning is different today from yesterday, and from what it could be tomorrow?

Munns, G., & Woodward, H. (2006). Student engagement and student self-assessment: the REAL framework. Assessment in Education, 13 (2), 193-213.

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The Complete IB Extended Essay Guide: Examples, Topics, and Ideas

International Baccalaureate (IB)

IB students around the globe fear writing the Extended Essay, but it doesn't have to be a source of stress! In this article, I'll get you excited about writing your Extended Essay and provide you with the resources you need to get an A on it.

If you're reading this article, I'm going to assume you're an IB student getting ready to write your Extended Essay. If you're looking at this as a potential future IB student, I recommend reading our introductory IB articles first, including our guide to what the IB program is and our full coverage of the IB curriculum .

IB Extended Essay: Why Should You Trust My Advice?

I myself am a recipient of an IB Diploma, and I happened to receive an A on my IB Extended Essay. Don't believe me? The proof is in the IBO pudding:

If you're confused by what this report means, EE is short for Extended Essay , and English A1 is the subject that my Extended Essay topic coordinated with. In layman's terms, my IB Diploma was graded in May 2010, I wrote my Extended Essay in the English A1 category, and I received an A grade on it.

What Is the Extended Essay in the IB Diploma Programme?

The IB Extended Essay, or EE , is a mini-thesis you write under the supervision of an IB advisor (an IB teacher at your school), which counts toward your IB Diploma (learn more about the major IB Diploma requirements in our guide) . I will explain exactly how the EE affects your Diploma later in this article.

For the Extended Essay, you will choose a research question as a topic, conduct the research independently, then write an essay on your findings . The essay itself is a long one—although there's a cap of 4,000 words, most successful essays get very close to this limit.

Keep in mind that the IB requires this essay to be a "formal piece of academic writing," meaning you'll have to do outside research and cite additional sources.

The IB Extended Essay must include the following:

• A title page
• Contents page
• Introduction
• Body of the essay
• References and bibliography

Additionally, your research topic must fall into one of the six approved DP categories , or IB subject groups, which are as follows:

• Group 1: Studies in Language and Literature
• Group 2: Language Acquisition
• Group 3: Individuals and Societies
• Group 4: Sciences
• Group 5: Mathematics
• Group 6: The Arts

Once you figure out your category and have identified a potential research topic, it's time to pick your advisor, who is normally an IB teacher at your school (though you can also find one online ). This person will help direct your research, and they'll conduct the reflection sessions you'll have to do as part of your Extended Essay.

As of 2018, the IB requires a "reflection process" as part of your EE supervision process. To fulfill this requirement, you have to meet at least three times with your supervisor in what the IB calls "reflection sessions." These meetings are not only mandatory but are also part of the formal assessment of the EE and your research methods.

According to the IB, the purpose of these meetings is to "provide an opportunity for students to reflect on their engagement with the research process." Basically, these meetings give your supervisor the opportunity to offer feedback, push you to think differently, and encourage you to evaluate your research process.

The final reflection session is called the viva voce, and it's a short 10- to 15-minute interview between you and your advisor. This happens at the very end of the EE process, and it's designed to help your advisor write their report, which factors into your EE grade.

Here are the topics covered in your viva voce :

• A check on plagiarism and malpractice
• Your reflection on your project's successes and difficulties
• Your reflection on what you've learned during the EE process

Your completed Extended Essay, along with your supervisor's report, will then be sent to the IB to be graded. We'll cover the assessment criteria in just a moment.

We'll help you learn how to have those "lightbulb" moments...even on test day!  

What Should You Write About in Your IB Extended Essay?

You can technically write about anything, so long as it falls within one of the approved categories listed above.

It's best to choose a topic that matches one of the IB courses , (such as Theatre, Film, Spanish, French, Math, Biology, etc.), which shouldn't be difficult because there are so many class subjects.

Here is a range of sample topics with the attached extended essay:

• Biology: The Effect of Age and Gender on the Photoreceptor Cells in the Human Retina
• Chemistry: How Does Reflux Time Affect the Yield and Purity of Ethyl Aminobenzoate (Benzocaine), and How Effective is Recrystallisation as a Purification Technique for This Compound?
• English: An Exploration of Jane Austen's Use of the Outdoors in Emma
• Geography: The Effect of Location on the Educational Attainment of Indigenous Secondary Students in Queensland, Australia
• Math: Alhazen's Billiard Problem
• Visual Arts: Can Luc Tuymans Be Classified as a Political Painter?

You can see from how varied the topics are that you have a lot of freedom when it comes to picking a topic . So how do you pick when the options are limitless?

How to Write a Stellar IB Extended Essay: 6 Essential Tips

Below are six key tips to keep in mind as you work on your Extended Essay for the IB DP. Follow these and you're sure to get an A!

#1: Write About Something You Enjoy

You can't expect to write a compelling essay if you're not a fan of the topic on which you're writing. For example, I just love British theatre and ended up writing my Extended Essay on a revolution in post-WWII British theatre. (Yes, I'm definitely a #TheatreNerd.)

I really encourage anyone who pursues an IB Diploma to take the Extended Essay seriously. I was fortunate enough to receive a full-tuition merit scholarship to USC's School of Dramatic Arts program. In my interview for the scholarship, I spoke passionately about my Extended Essay; thus, I genuinely think my Extended Essay helped me get my scholarship.

But how do you find a topic you're passionate about? Start by thinking about which classes you enjoy the most and why . Do you like math classes because you like to solve problems? Or do you enjoy English because you like to analyze literary texts?

Keep in mind that there's no right or wrong answer when it comes to choosing your Extended Essay topic. You're not more likely to get high marks because you're writing about science, just like you're not doomed to failure because you've chosen to tackle the social sciences. The quality of what you produce—not the field you choose to research within—will determine your grade.

Once you've figured out your category, you should brainstorm more specific topics by putting pen to paper . What was your favorite chapter you learned in that class? Was it astrophysics or mechanics? What did you like about that specific chapter? Is there something you want to learn more about? I recommend spending a few hours on this type of brainstorming.

One last note: if you're truly stumped on what to research, pick a topic that will help you in your future major or career . That way you can use your Extended Essay as a talking point in your college essays (and it will prepare you for your studies to come too!).

#2: Select a Topic That Is Neither Too Broad nor Too Narrow

There's a fine line between broad and narrow. You need to write about something specific, but not so specific that you can't write 4,000 words on it.

You can't write about WWII because that would be a book's worth of material. You also don't want to write about what type of soup prisoners of war received behind enemy lines, because you probably won’t be able to come up with 4,000 words of material about it. However, you could possibly write about how the conditions in German POW camps—and the rations provided—were directly affected by the Nazis' successes and failures on the front, including the use of captured factories and prison labor in Eastern Europe to increase production. WWII military history might be a little overdone, but you get my point.

If you're really stuck trying to pinpoint a not-too-broad-or-too-narrow topic, I suggest trying to brainstorm a topic that uses a comparison. Once you begin looking through the list of sample essays below, you'll notice that many use comparisons to formulate their main arguments.

I also used a comparison in my EE, contrasting Harold Pinter's Party Time with John Osborne's Look Back in Anger in order to show a transition in British theatre. Topics with comparisons of two to three plays, books, and so on tend to be the sweet spot. You can analyze each item and then compare them with one another after doing some in-depth analysis of each individually. The ways these items compare and contrast will end up forming the thesis of your essay!

When choosing a comparative topic, the key is that the comparison should be significant. I compared two plays to illustrate the transition in British theatre, but you could compare the ways different regional dialects affect people's job prospects or how different temperatures may or may not affect the mating patterns of lightning bugs. The point here is that comparisons not only help you limit your topic, but they also help you build your argument.

Comparisons are not the only way to get a grade-A EE, though. If after brainstorming, you pick a non-comparison-based topic and are still unsure whether your topic is too broad or narrow, spend about 30 minutes doing some basic research and see how much material is out there.

If there are more than 1,000 books, articles, or documentaries out there on that exact topic, it may be too broad. But if there are only two books that have any connection to your topic, it may be too narrow. If you're still unsure, ask your advisor—it's what they're there for! Speaking of advisors...

Don't get stuck with a narrow topic!

#3: Choose an Advisor Who Is Familiar With Your Topic

If you're not certain of who you would like to be your advisor, create a list of your top three choices. Next, write down the pros and cons of each possibility (I know this sounds tedious, but it really helps!).

For example, Mr. Green is my favorite teacher and we get along really well, but he teaches English. For my EE, I want to conduct an experiment that compares the efficiency of American electric cars with foreign electric cars.

I had Ms. White a year ago. She teaches physics and enjoyed having me in her class. Unlike Mr. Green, Ms. White could help me design my experiment.

Based on my topic and what I need from my advisor, Ms. White would be a better fit for me than would Mr. Green (even though I like him a lot).

The moral of my story is this: do not just ask your favorite teacher to be your advisor . They might be a hindrance to you if they teach another subject. For example, I would not recommend asking your biology teacher to guide you in writing an English literature-based EE.

There can, of course, be exceptions to this rule. If you have a teacher who's passionate and knowledgeable about your topic (as my English teacher was about my theatre topic), you could ask that instructor. Consider all your options before you do this. There was no theatre teacher at my high school, so I couldn't find a theatre-specific advisor, but I chose the next best thing.

Before you approach a teacher to serve as your advisor, check with your high school to see what requirements they have for this process. Some IB high schools require your IB Extended Essay advisor to sign an Agreement Form , for instance.

Make sure that you ask your IB coordinator whether there is any required paperwork to fill out. If your school needs a specific form signed, bring it with you when you ask your teacher to be your EE advisor.

#4: Pick an Advisor Who Will Push You to Be Your Best

Some teachers might just take on students because they have to and aren't very passionate about reading drafts, only giving you minimal feedback. Choose a teacher who will take the time to read several drafts of your essay and give you extensive notes. I would not have gotten my A without being pushed to make my Extended Essay draft better.

Ask a teacher that you have experience with through class or an extracurricular activity. Do not ask a teacher that you have absolutely no connection to. If a teacher already knows you, that means they already know your strengths and weaknesses, so they know what to look for, where you need to improve, and how to encourage your best work.

Also, don't forget that your supervisor's assessment is part of your overall EE score . If you're meeting with someone who pushes you to do better—and you actually take their advice—they'll have more impressive things to say about you than a supervisor who doesn't know you well and isn't heavily involved in your research process.

Be aware that the IB only allows advisors to make suggestions and give constructive criticism. Your teacher cannot actually help you write your EE. The IB recommends that the supervisor spends approximately two to three hours in total with the candidate discussing the EE.

#5: Make Sure Your Essay Has a Clear Structure and Flow

The IB likes structure. Your EE needs a clear introduction (which should be one to two double-spaced pages), research question/focus (i.e., what you're investigating), a body, and a conclusion (about one double-spaced page). An essay with unclear organization will be graded poorly.

The body of your EE should make up the bulk of the essay. It should be about eight to 18 pages long (again, depending on your topic). Your body can be split into multiple parts. For example, if you were doing a comparison, you might have one third of your body as Novel A Analysis, another third as Novel B Analysis, and the final third as your comparison of Novels A and B.

If you're conducting an experiment or analyzing data, such as in this EE , your EE body should have a clear structure that aligns with the scientific method ; you should state the research question, discuss your method, present the data, analyze the data, explain any uncertainties, and draw a conclusion and/or evaluate the success of the experiment.

#6: Start Writing Sooner Rather Than Later!

You will not be able to crank out a 4,000-word essay in just a week and get an A on it. You'll be reading many, many articles (and, depending on your topic, possibly books and plays as well!). As such, it's imperative that you start your research as soon as possible.

Each school has a slightly different deadline for the Extended Essay. Some schools want them as soon as November of your senior year; others will take them as late as February. Your school will tell you what your deadline is. If they haven't mentioned it by February of your junior year, ask your IB coordinator about it.

Some high schools will provide you with a timeline of when you need to come up with a topic, when you need to meet with your advisor, and when certain drafts are due. Not all schools do this. Ask your IB coordinator if you are unsure whether you are on a specific timeline.

Below is my recommended EE timeline. While it's earlier than most schools, it'll save you a ton of heartache (trust me, I remember how hard this process was!):

• January/February of Junior Year: Come up with your final research topic (or at least your top three options).
• February of Junior Year: Approach a teacher about being your EE advisor. If they decline, keep asking others until you find one. See my notes above on how to pick an EE advisor.
• April/May of Junior Year: Submit an outline of your EE and a bibliography of potential research sources (I recommend at least seven to 10) to your EE advisor. Meet with your EE advisor to discuss your outline.
• Summer Between Junior and Senior Year: Complete your first full draft over the summer between your junior and senior year. I know, I know—no one wants to work during the summer, but trust me—this will save you so much stress come fall when you are busy with college applications and other internal assessments for your IB classes. You will want to have this first full draft done because you will want to complete a couple of draft cycles as you likely won't be able to get everything you want to say into 4,000 articulate words on the first attempt. Try to get this first draft into the best possible shape so you don't have to work on too many revisions during the school year on top of your homework, college applications, and extracurriculars.
• August/September of Senior Year: Turn in your first draft of your EE to your advisor and receive feedback. Work on incorporating their feedback into your essay. If they have a lot of suggestions for improvement, ask if they will read one more draft before the final draft.
• September/October of Senior Year: Submit the second draft of your EE to your advisor (if necessary) and look at their feedback. Work on creating the best possible final draft.
• November-February of Senior Year: Schedule your viva voce. Submit two copies of your final draft to your school to be sent off to the IB. You likely will not get your grade until after you graduate.

Remember that in the middle of these milestones, you'll need to schedule two other reflection sessions with your advisor . (Your teachers will actually take notes on these sessions on a form like this one , which then gets submitted to the IB.)

I recommend doing them when you get feedback on your drafts, but these meetings will ultimately be up to your supervisor. Just don't forget to do them!

The early bird DOES get the worm!

How Is the IB Extended Essay Graded?

Extended Essays are graded by examiners appointed by the IB on a scale of 0 to 34 . You'll be graded on five criteria, each with its own set of points. You can learn more about how EE scoring works by reading the IB guide to extended essays .

• Criterion A: Focus and Method (6 points maximum)
• Criterion B: Knowledge and Understanding (6 points maximum)
• Criterion C: Critical Thinking (12 points maximum)
• Criterion D: Presentation (4 points maximum)
• Criterion E: Engagement (6 points maximum)

How well you do on each of these criteria will determine the final letter grade you get for your EE. You must earn at least a D to be eligible to receive your IB Diploma.

Although each criterion has a point value, the IB explicitly states that graders are not converting point totals into grades; instead, they're using qualitative grade descriptors to determine the final grade of your Extended Essay . Grade descriptors are on pages 102-103 of this document .

Here's a rough estimate of how these different point values translate to letter grades based on previous scoring methods for the EE. This is just an estimate —you should read and understand the grade descriptors so you know exactly what the scorers are looking for.

Here is the breakdown of EE scores (from the May 2021 bulletin):

How Does the Extended Essay Grade Affect Your IB Diploma?

The Extended Essay grade is combined with your TOK (Theory of Knowledge) grade to determine how many points you get toward your IB Diploma.

To learn about Theory of Knowledge or how many points you need to receive an IB Diploma, read our complete guide to the IB program and our guide to the IB Diploma requirements .

This diagram shows how the two scores are combined to determine how many points you receive for your IB diploma (3 being the most, 0 being the least). In order to get your IB Diploma, you have to earn 24 points across both categories (the TOK and EE). The highest score anyone can earn is 45 points.

Let's say you get an A on your EE and a B on TOK. You will get 3 points toward your Diploma. As of 2014, a student who scores an E on either the extended essay or TOK essay will not be eligible to receive an IB Diploma .

Prior to the class of 2010, a Diploma candidate could receive a failing grade in either the Extended Essay or Theory of Knowledge and still be awarded a Diploma, but this is no longer true.

Figuring out how you're assessed can be a little tricky. Luckily, the IB breaks everything down here in this document . (The assessment information begins on page 219.)

40+ Sample Extended Essays for the IB Diploma Programme

In case you want a little more guidance on how to get an A on your EE, here are over 40 excellent (grade A) sample extended essays for your reading pleasure. Essays are grouped by IB subject.

• Business Management 1
• Chemistry 1
• Chemistry 2
• Chemistry 3
• Chemistry 4
• Chemistry 5
• Chemistry 6
• Chemistry 7
• Computer Science 1
• Economics 1
• Design Technology 1
• Design Technology 2
• Environmental Systems and Societies 1
• Geography 1
• Geography 2
• Geography 3
• Geography 4
• Geography 5
• Geography 6
• Literature and Performance 1
• Mathematics 1
• Mathematics 2
• Mathematics 3
• Mathematics 4
• Mathematics 5
• Philosophy 1
• Philosophy 2
• Philosophy 3
• Philosophy 4
• Philosophy 5
• Psychology 1
• Psychology 2
• Psychology 3
• Psychology 4
• Psychology 5
• Social and Cultural Anthropology 1
• Social and Cultural Anthropology 2
• Social and Cultural Anthropology 3
• Sports, Exercise and Health Science 1
• Sports, Exercise and Health Science 2
• Visual Arts 1
• Visual Arts 2
• Visual Arts 3
• Visual Arts 4
• Visual Arts 5
• World Religion 1
• World Religion 2
• World Religion 3

What's Next?

Trying to figure out what extracurriculars you should do? Learn more about participating in the Science Olympiad , starting a club , doing volunteer work , and joining Student Government .

Studying for the SAT? Check out our expert study guide to the SAT . Taking the SAT in a month or so? Learn how to cram effectively for this important test .

Not sure where you want to go to college? Read our guide to finding your target school . Also, determine your target SAT score or target ACT score .

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How Do We Support Students in Reflecting on Mathematics?

by Cynthia Garland Dore | Dec 4, 2017 | 0 comments

When students engage in math experiences that include time to reflect on their reasoning and the thinking of others they are more likely to become self-reflective. They become better at thinking about thinking and making connections to other mathematical concepts and contexts. But what does it look and sound like when students reflect on mathematical ideas?

Consider the following discussion from a grade 1 classroom, where students are sharing their strategies for solving a problem about two groups of toy cars (6+7). After recording two strategies in which the students counted all of the cars, the teacher asks, “Can anybody see a way that Hunter’s strategy is similar to Danny’s? … What’s the same about them?”

Jamie : That um he both put numbers.

Teacher : So they both labeled with numbers? That is one thing that they did the same. Does anyone see another way that these are similar strategies?

Luke : Um well they were counting all of them after they put ‘em up? …  They just counted all of them.

Teacher : So in both cases, Luke, you’re right. They drew or used a finger to count every single car to get the total of how many.

Asking students to compare strategies and representations encourages them to reflect on their own thinking and the thinking of others. This teacher focuses students’ attention on how the two counting all strategies are similar. Students must first make sense of each strategy, both of which represent someone else’s thinking. Then, they must step back and reflect on the ways in which those strategies are the same.

Later in the discussion, the teacher focuses the discussion on a different strategy, which involves using known facts to solve the problem.

Elizabeth knows that 6+6=12, and uses it to solve 6+7=13. She is able to reflect on how she solved this problem in order to share it with the class, no small feat for a 6-year old. In addition, Derek sees a relationship between thinking about this problem as 6+6+1 and 7+7-1. The teacher supports such reflection with her comments and questions, which name and explicate the strategy and encourage students to make sense of it and compare it to their own.

How does one build such a classroom culture, where reflection is an ongoing, natural part of doing mathematics?  Teachers help such a culture develop when they:

• anticipate and plan for potential reflection opportunities based on the mathematics at hand
• actively listen to and observe students at work, and use the information gathered to inform discussions and reflection opportunities
• I wonder what would happen if ____?
• I noticed ______ and ______ are the (same/different) because…
• As you’re working on _____ look for _____.
• Compare your thinking with ____.
• What are you noticing?
• comparing and contrasting several different strategies
• making sense of a strategy that’s different from one’s own
• using different representations and contexts to explore one strategy
• investigating an error or misunderstanding

Building a classroom culture where students make connections to the thinking of others takes time. As teachers encourage and promote student reflection about math, thinking about their thinking becomes a natural part of how they learn and make sense of what they’ve learned.

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2 Ways to Encourage Reflection on Math Concepts

Open-ended questions guide students to participate and to think mathematically, which cements their learning.

For many students, math is a subject where every question has one (and only one) correct answer. If a student is asked, “What is two plus two?” the only acceptable response is “Four.”

What if students were also asked, “Why does two plus two equal four?” Reflection questions like this, which are purposely open-ended, do not have a single correct answer. Instead, these questions remove the fear of being wrong and encourage mathematical thinking, participation, and growth.

“Reflection questions are important for students and help move the focus from performance to learning,” says Stanford professor Jo Boaler , who believes that “assessment plays a key role in the messages given to students about their potential, and many classrooms need to realign their assessment approach in order to encourage growth instead of fixed mindsets among students.”

In addition to performance-focused questions and assessments (“What is the total sum of the interior angles of a triangle?”), you can ask open-ended reflection questions that encourage mathematical thinking and participation (“Why do you think the total sum of the interior angles of a triangle always equals 180 degrees?”). The second question shifts the focus from performance toward thinking, learning, and engaging with mathematics without the fear of being wrong.

How can you incorporate reflection questions into your math lessons? Try these two useful strategies.

Which One Doesn’t Belong?

If you grew up watching Sesame Street , you probably remember the “One of These Things Is Not Like the Others” segment, where viewers had to identify one object out of a set of four that did not belong. This simple activity helps children to identify similarities and differences, and this type of thinking can be extended to learning math.

Which One Doesn’t Belong? (WODB) math activities present students with four different visual graphics that are all similar and different from each other in some way. This four-quadrant activity is my go-to for getting whole-class participation, as each option can be argued as the correct answer.

Observe the photo above of a WODB activity showing the numbers 22, 33, 44, and 50, and identify which choice does not belong and explain why. Since the graphics are purposely ambiguous and have overlapping similarities and differences, there is no single correct answer. One student might conclude that 50 doesn’t belong because it is the only number not divisible by 11. Another student may also believe that 50 doesn’t belong but for a different reason, namely that it is the only number with two different digits. A third student might conclude that 33 doesn’t belong because it is the only odd number. With this one graphic, you can easily spark a deep mathematical discussion where all students are eager to participate and share their thinking without any fear of being wrong.

WODB activities can be used for any math topic and can include images, numbers, charts, and graphs. They can also be used as formative assessments where students write their responses on sticky notes and stick them on the graphic that is projected at the front of the classroom.

Think-Notice-Wonder

Writing about math helps students organize their thoughts, use important vocabulary terms, and express their ideas in depth—which leads to deeper understanding.

Think-Notice-Wonder (TNW) activities are open-ended writing prompts where students are required to complete I think… , I notice… , I wonder… , based on a given graphic related to a math topic.

For example, students observe the soda and popcorn price graphic above and are prompted: What do you think? What do you notice? What do you wonder?

Encourage your students to think deeply for a minute or two before putting their thoughts into writing. They can share their ideas about the relationship between the price of a bag of popcorn and a soda based on size. They can verbalize how they perceive the proportional relationship to behave, wonder about which option provides the most value, and question how the prices were determined in the first place.

Since TNW writing activities are open-ended and do not have a correct answer, they encourage full group participation. Teachers often have students share their responses in a math journal notebook, but you can also use this free TNW student response template .

If you are looking for free images and graphics to use as TNW writing prompts, here are a few helpful resources:

• Find math-related graphics and images using Google Image Search and display them at the front of your classroom.
• Access and share teacher-created TNW activities on Twitter by searching the math education hashtags, including #ITeachMath, #MTBoS, and #NoticeWonder.
• Free stock photo websites such as Unsplash have an excellent collection of photos that relate to math topics, including estimation, three-dimensional figures, and mathematical patterns in nature.

When you add more reflection questions into your math lessons, students will have more opportunities to participate and engage in mathematical thinking without fear, which leads to a most-desired outcome—accessibility and growth.

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What Is Reflection in Math? Definition, Examples & How-to

Whether you are just starting to explore reflections in math or need to brush up on your geometry, this simple, middle-school-friendly guide is for you.

Read on to find easy-to-follow definitions and explanations, solved examples, and resources to help you learn and master reflections.

What Is Reflection?

Reflection in mathematics is a geometric transformation where a shape or object is flipped across a line, known as the line of reflection . This results in a mirror image that is the same size and shape as the original but appears flipped or mirrored.

You encounter the concept of reflection every time you look in a mirror.

Try it now:

Step in front of a mirror and raise your right hand.

What hand is your mirror image raising?

If you raise your right hand in front of a mirror, your reflection, i.e. your mirror image, raises its left hand.

Transformations in Geometry

Reflection is one of the 4 types of transformations in geometry .

Other types of geometrical transformations are:

• Translation : Moving a shape without rotating or flipping it. It's like sliding the shape in a particular direction.
• Rotation : Turning a shape around a fixed point. It’s like turning a key in a lock.
• Dilation : Resizing a shape by enlarging or shrinking it evenly. It's like stretching or squeezing the shape while keeping its proportions intact, like inflating or deflating a balloon.

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Properties of Reflection

 To truly understand what reflection is and to distinguish it from other geometrical transformations, we need to know its properties.

Reflection has 5 key properties:

• Shape:  The shape of the original figure and its mirror image are identical.
• Size: The size of the original figure and its mirror image are the same.
• Orientation:  The orientation of the original figure and its mirror image are opposite.
• Distance:  The distance between any point on the original figure and the line of reflection is the same as the distance between the corresponding point on the mirror image and the line of reflection.
• Angle: Angles between intersecting lines are the same in both the original figure and its mirror image.

If any of these five properties are missing, the geometrical transformation is either not a reflection at all or is a combination  of reflection and other transformations.

Important Terms in Reflection

Let’s go over some language we use to talk about reflections.

We use these terms to explain how reflection works:

• Line of Reflection:  The imaginary line across which a shape is reflected to create its mirror image. It is also known as the axis of reflection or mirror line.
• Mirror Image: The image formed when you reflect a shape across the line of reflection. It is identical to the original shape but appears reversed.
• Congruent Figures: Two figures that have the same size and shape. In reflection, the original shape and its mirror image are always congruent.
• Symmetry: The property of a shape or object that remains unchanged when reflected across a line, known as a line of symmetry.

How to Do Reflections on the Coordinate Plane

To reflect a point or figure on the coordinate plane, we sometimes use the X-axis or Y-axis as the line of reflection.

Let’s see how each type of reflection works.

How to Do   Reflection Over the X-axis

Reflection over the x-axis is a transformation where each point in a shape or a graph is flipped across the x-axis .

If you have a point (x, y), reflecting it over the x-axis will give you the point (x, -y).

In other words, the x-coordinate (how far left or right the point is) stays the same, but the y-coordinate (how far up or down the point is) becomes its opposite.

For example, if you have the point (2, 3), reflecting it over the x-axis would give you (2, -3), because the x-coordinate remains 2, but the y-coordinate changes from 3 to -3, flipping it across the x-axis.

How to Do Reflection Over the Y-axis

In reflection over the Y-axis, each point in a shape or graph is mirrored horizontally along the Y-axis .

When you have a point (x, y), reflecting it over the Y-axis gives you the point (-x, y). To put it simply, the y-coordinate remains unchanged, but the x-coordinate changes to its negative value.

For example, let's take the point (4, -5). Reflecting it over the Y-axis results in (-4, -5).

Notice that while the y-coordinate remains -5, the x-coordinate changes from 4 to -4 as it mirrors along the Y-axis.

Here’s a visual representation of reflection over the Y-axis.

How to Do Reflection Over the Line y = x

The line y = x represents all the points where the y-coordinate is equal to the x-coordinate.

It's a diagonal line that passes from the bottom left corner to the top right corner through the origin at a 45-degree angle.

Reflection over the y = x line means flipping a point or shape across this diagonal line .

During this reflection, the x-coordinate of each point becomes its y-coordinate, and the y-coordinate becomes its x-coordinate.

For example, if you have a point (x, y), its reflection over the y = x line would be (y, x).

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Revisiting Students' Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and Future Directions

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Consortium to Promote Reflection in Engineering Education

Campus: green river college, 9. math test reflection essay.

Educator:  Mike Kenyon, Faculty, Mathematics Context:  Out of class; MATH 141 Pre-Calculus Keywords:  mathematics, essays Student Activity Time:  30 minutes

After completing an exam, students wrote a reflection essay about how the test went.

Introducing the Reflection Activity

One way to prompt students to think about their test performance is to simply ask for their first response to the test and their score. An educator used post-exam essays to prompt students to reflect on and articulate their reactions about the exam and their score. The purpose of this activity is to assist students in verbalizing their concerns, thoughts, and reactions to their graded exam.

The educator administered the regularly scheduled math exam, graded it, and returned the test to students with a solution key. The educator offered students a chance to earn a small portion of their homework grade by completing an essay about how the exam went and their reactions to it. The educator required students to write at least one double-spaced, typed page for their essay. Students often turned in reworked problems with the essay. The educator then graded the student’s submission and included a short 2-3 sentence response to their essay. The response generally included validation, answers to open questions, or recommendations that would assist the student in the future.

The educator used the information shared in the students’ essays in individual interactions with them, either in class or during office hours. When multiple students expressed similar concerns, the educator chose to respond to those concerns in class. As a result, students experienced an opportunity to articulate their successes, misconceptions and errors on the exam. Students often take action based on the educator’s response and suggestions, such as visiting office hours or the tutoring center to improve future exam performance.

 Recreating the Reflection Activity

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Mathematical Reflections

Our free online journal.

Mathematical Reflections intends to fill the editor’s perceived need for a publication aimed primarily at high school students, undergraduates, and everyone interested in mathematics. Through articles and problems, we seek to expose readers to a variety of interesting topics that are fully accessible to the target audience.

Through the articles, we hope to expand the horizons of the student readers, introducing them to material outside the scope of most classes. For instructors, the articles provide an intriguing opportunity to move away from a structured curriculum, motivate the addressed problem, and guide students through to the invaluable moments of discovery.

Through the problem column, we challenge students to develop their creative problem solving and reasoning skills by devising solutions to the proposed questions. Exceptional solutions will be published, with the intent of encouraging students to formally write out and submit their work to be showcased in print.

The journal invites submissions from students and instructors alike for all sections.

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