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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

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

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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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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.

Engaging Maths

Dr catherine attard, promoting student reflection to improve mathematics learning.

• by cattard2017
• Posted on July 16, 2017

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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My Reflection in Mathematics in the Modern World

Related Papers

Cheryl Praeger

as the wonderings about the status of school mathematics are becoming louder and louder, the need for a revision of our reasons can no longer be ignored. In what follows, I respond to this need by taking a critical look at some of the most popular arguments for the currently popular slogan, “Mathematics for all.” This analysis is preceded by a proposal of how to think about mathematics so as to loosen the grip of clichés and to shed off hidden prejudice. It is followed by my own take on the question of what mathematics to teach, to whom, and how.

Ten pages paper, will be presented at '5th International …

Mette Andresen

As the time enters the 21st century, sciences such as those of theoretical physics, complex system and network, cytology, biology and economy developments change rapidly, and meanwhile, a few global questions constantly emerge, such as those of local war, food safety, epidemic spreading network, environmental protection, multilateral trade dispute, more and more questions accompanied with the overdevelopment and applying the internet, · · · , etc. In this case, how to keep up mathematics with the developments of other sciences? Clearly, today's mathematics is no longer adequate for the needs of other sciences. New mathematical theory or techniques should be established by mathematicians. Certainly, solving problem is the main objective of mathematics, proof or calculation is the basic skill of a mathematician. When it develops in problem-oriented, a mathematician should makes more attentions on the reality of things in mathematics because it is the main topic of human beings.

Amarnath Murthy

There is nothing in our lives, in our world, in our universe, that cannot be expressed with mathematical theories, numbers, and formulae. Mathematics is the queen of science and the king of arts; to me it is the backbone of all systems of knowledge. Mathematics is a tool that has been used by man for ages. It is a key that can unlock many doors and show the way to different logical answers to seemingly impossible problems. Not only can it solve equations and problems in everyday life, but it can also express quantities and values precisely with no question or room for other interpretation. There is no room for subjectivity. Though there is a lot of mathematics in politics, there is no room for politics in mathematics. Coming from a powerful leader two + two can not become five it will remain four. Mathematics is not fundamentally empirical —it does not rely on sensory observation or instrumental measurement to determine what is true. Indeed, mathematical objects themselves cannot be observed at all! Mathematics is a logical science, cleanly structured, and well-founded. Mathematics is obviously the most interesting, entertaining, fascinating, exciting, challenging, amazing, enthralling, thrilling, absorbing, involving, fascinating, mesmerizing, satisfying, fulfilling, inspiring, mindboggling, refreshing, systematic, energizing, satisfying, enriching, engaging, absorbing, soothing, impressive, pleasing, stimulating, engrossing, magical, musical, rhythmic, artistic, beautiful, enjoyable, scintillating, gripping, charming, recreational, elegant, unambiguous, analytical, hierarchical, powerful, rewarding, pure, impeccable, useful, optimizing, precise, objective, consistent, logical, perfect, trustworthy, eternal, universal subject in existence full of eye catching patterns.

Journal of Humanistic Mathematics

Gizem Karaali

Katja Lengnink

Mathematics plays a dominant role in today's world. Although not everyone will become a mathematical expert, from an educational point of view, it is key for everyone to acquire a certain level of mathematical literacy, which allows reflecting and assessing mathematical processes important in every day live. Therefore the goal has to be to open perspectives and experiences beyond a mechanical and tight appearance of the subject. In this article a framework for the integration of reflection and assessment in the teaching practice is developed. An illustration through concrete examples is given.

Swapna Mukhopadhyay

Michele Emmer

It is no great surprise that mathematical structures and ideas, conceived by human beings, can be applied extremely effectively to what we call the "real" world. We need only to think of physics, astronomy, meteorology, telecommunications, biology, cryptography, and medicine. But that's not all mathematics has always had strong links with music, literature, architecture, arts, philosophy, and more recently with theatre and cinema

Liliya Samigullina

The article considers mathematics as a way of teaching reasoning in symbolic non-verbal communication. Particular attention is paid to mathematical ways of thinking when studying the nature and its worldview. The nature is studied through the theory of experimental approval of scientific concepts of algorithmic and nonalgorithmic "computing". Various discoveries are analyzed and the role of mathematics in the worldview is substantiated. The greatest value of mathematics is development of knowledge in order to express it in abstract language of mathematics and natural science, i.e., to move to the meta-pedagogical level of understanding of problems

Ecampus Course Development and Training

Providing inspiration for your online class.

Reflection Assignments in Math Courses

Reflection assignments as an active learning strategy are commonly seen in humanities courses. The purpose of this writing is to share an example of how simple reflection activities can make a huge impact in two math courses. MTH 251 Differential Calculus covers five units, with one exam for each unit, counting 14% of the final grade. Before students attempt to take the unit exam, they are assigned to read textbook readings, watch instructor-created lecture videos, work on Canvas-based homework assignment and Adaptive Learning based practice assignments in Knewton Lab online platform. After assignment due date expires, students are assigned to complete a weekly written homework reflection. The weekly homework and the weekly homework reflection together count for 14% of final grade in this course, weighing the same as each of the unit exams.

MTH 341 Linear Algebra I has ten weekly modules. Each week, students  read textbook assigned readings, watch lecture videos created by the instructor (Dr.   ), complete post-reading questions in quiz format, work on graded group discussion questions to solve math problems in small groups, complete written homework individually, and in the following week, complete a written homework response activity individually in discussion format.   

The written homework reflection in MATH 251 and the written homework response in MATH 341 are both reflection activities designed to optimize student learning success, through comparing their own homework solutions with answer keys and evaluate whether they did it correctly or incorrectly and analyze where they did it wrong and how to get it right. The purpose of such weekly reflection is to help students develop meta-cognitive skills related to their learning. By looking back at students’ own work and learning from their mistakes, they develop an understanding of what is the proper way to solve a problem and what is not the proper way for solving a particular math problem. It also prompts students to plan for proper action in the future and exercises students’ executive functioning skills (CAST, 2018).  Here is what the instructions for the weekly reflection look like: 1. First answer the weekly prompt: Reflecting on the Unit 1 module, which topics did you struggle with the most? 2. Download the written homework solutions PDF: (Solution for each written homework in pdf format is attached here.) 3. Look over the solutions and compare to your submitted homework. Look for any problems where your solution differs from the posted solution.

• why you struggled with certain problems
• why each solution makes sense now
• what your misunderstanding was
• what will you do in the future when solving problems similar to these?
• what strategies will help you?
• what did you learn by making a mistake?
• what did you learn from looking at the solutions?
• If you are still confused about a problem, ask a question. DO NOT simply list which problems you got wrong.
• If your solutions are all correct then in the discussion board please discuss the problem that you found the most challenging. Describe what specific tasks helped you to complete that problem. Be as detailed as you can about your solution process.

Students not only posted their own reflections, but they also comment on or answer other students’ reflections as well. Additionally, the instructor and the four TAs in the course responded actively to students’ reflections, which makes the reflection more valuable since students get encouragement, praises, or corrections from the instructor and teaching assistants. Again, feedback from experts is critical in the success of a reflection activity (Vandenbussche, 2018)

Image 1: How reflection usually looks like and How reflection should look like ( Image Source )

Many students were reflecting on what they did wrong and asked for help. Some were reflecting on their time management in completing the homework assignments. And we were glad to see students completing homework, evaluating their own work, analyzing where they did wrong, and planning for future improvement. Overall, the purpose of this assignment is accomplished!

(Image by Dave_Here )

A great benefit that comes from these weekly reflection activities is increased or sustained homework completion rate. For MTH 251 winter 2021 week 1 to week 7, over 85% of students completed the weekly homework and the reflection activity on average. For MTH 341 Fall 20 week 1 to week 7, over 90% of students on average completed the weekly homework and the reflection assignments. All math teachers love to see their students practice with homework assignments before they attempt to take the quizzes or exams! And evidence-based research tells us that deliberate practice with targeted feedback promotes mastery learning (Ambrose et al., 2010).

So, if it works in math courses, it will work in Chemistry, Biology, Physics, Engineering and other STEM courses too! If you’re interested in implementing this technique in your teaching and have questions about setting it up, feel free to contact us. We’d love to help you figure out the easiest way to set it up in your course.

Ambrose, S.A., Bridges, M.W., DiPietro, M., Lovettt, M.C. , Norman, M.K., & The Eberly Center for Teaching Excellence at Carnegie Mellon University. (2010). How learning works: Seven research-based principles for smart teaching. San Francisco, CA: Jossey-Bass

CAST. (2018). UDL Guidelines. Retrieved from https://udlguidelines.cast.org/  

Vandenbussche, B. (2018). Reflecting for learning. Retrieved from https://educationaltoolsportal.eu/en/tools-for-learning/reflecting-learning  

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How to Write a Reflection Paper: An Easy-to-Follow Guide

Last Updated: May 1, 2024 Fact Checked

Sample Outline and Paper

Brainstorming, organizing a reflection paper, as you write, expert q&a.

This article was co-authored by Alicia Cook . Alicia Cook is a Professional Writer based in Newark, New Jersey. With over 12 years of experience, Alicia specializes in poetry and uses her platform to advocate for families affected by addiction and to fight for breaking the stigma against addiction and mental illness. She holds a BA in English and Journalism from Georgian Court University and an MBA from Saint Peter’s University. Alicia is a bestselling poet with Andrews McMeel Publishing and her work has been featured in numerous media outlets including the NY Post, CNN, USA Today, the HuffPost, the LA Times, American Songwriter Magazine, and Bustle. She was named by Teen Vogue as one of the 10 social media poets to know and her poetry mixtape, “Stuff I’ve Been Feeling Lately” was a finalist in the 2016 Goodreads Choice Awards. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 3,814,896 times.

Reflection papers allow you to communicate with your instructor about how a specific article, lesson, lecture, or experience shapes your understanding of class-related material. Reflection papers are personal and subjective [1] X Research source , but they must still maintain a somewhat academic tone and must still be thoroughly and cohesively organized. Here's what you need to know about writing an effective reflection.

How to Start a Reflection Paper

To write a reflection paper, first write an introduction that outlines your expectations and thesis. Then, state your conclusions in the body paragraphs, explaining your findings with concrete details. Finally, conclude with a summary of your experience.

• These sentences should be both descriptive yet straight to the point.

• For lectures or readings, you can write down specific quotations or summarize passages.
• For experiences, make a note of specific portions of your experience. You could even write a small summary or story of an event that happened during the experience that stands out. Images, sounds, or other sensory portions of your experience work, as well.

• In the first column, list the main points or key experiences. These points can include anything that the author or speaker treated with importance as well as any specific details you found to be important. Divide each point into its own separate row.
• In the second column, list your personal response to the points you brought up in the first column. Mention how your subjective values, experiences, and beliefs influence your response.
• In the third and last column, describe how much of your personal response to share in your reflection paper.

• Does the reading, lecture, or experience challenge you socially, culturally, emotionally, or theologically? If so, where and how? Why does it bother you or catch your attention?
• Has the reading, lecture, or experience changed your way of thinking? Did it conflict with beliefs you held previously, and what evidence did it provide you with in order to change your thought process on the topic?
• Does the reading, lecture, or experience leave you with any questions? Were these questions ones you had previously or ones you developed only after finishing?
• Did the author, speaker, or those involved in the experience fail to address any important issues? Could a certain fact or idea have dramatically changed the impact or conclusion of the reading, lecture, or experience?
• How do the issues or ideas brought up in this reading, lecture, or experience mesh with past experiences or readings? Do the ideas contradict or support each other?

• Verify whether or not your instructor specified a word count for the paper instead of merely following this average.
• If your instructor demands a word count outside of this range, meet your instructor's requirements.

• For a reading or lecture, indicate what you expected based on the title, abstract, or introduction.
• For an experience, indicate what you expected based on prior knowledge provided by similar experiences or information from others.

• This is essentially a brief explanation of whether or not your expectations were met.
• A thesis provides focus and cohesion for your reflection paper.
• You could structure a reflection thesis along the following lines: “From this reading/experience, I learned...”

• Your conclusions must be explained. You should provide details on how you arrived at those conclusions using logic and concrete details.
• The focus of the paper is not a summary of the text, but you still need to draw concrete, specific details from the text or experience in order to provide context for your conclusions.
• Write a separate paragraph for each conclusion or idea you developed.
• Each paragraph should have its own topic sentence. This topic sentence should clearly identify your major points, conclusions, or understandings.

• The conclusions or understandings explained in your body paragraphs should support your overall conclusion. One or two may conflict, but the majority should support your final conclusion.

• If you feel uncomfortable about a personal issue that affects the conclusions you reached, it is wisest not to include personal details about it.
• If a certain issue is unavoidable but you feel uncomfortable revealing your personal experiences or feelings regarding it, write about the issue in more general terms. Identify the issue itself and indicate concerns you have professionally or academically.

• Avoid dragging someone else down in your writing. If a particular person made the experience you are reflecting on difficult, unpleasant, or uncomfortable, you must still maintain a level of detachment as you describe that person's influence. Instead of stating something like, “Bob was such a rude jerk,” say something more along the lines of, “One man was abrupt and spoke harshly, making me feel as though I was not welcome there.” Describe the actions, not the person, and frame those actions within the context of how they influenced your conclusions.
• A reflection paper is one of the few pieces of academic writing in which you can get away with using the first person pronoun “I.” That said, you should still relate your subjective feelings and opinions using specific evidence to explain them. [8] X Research source
• Avoid slang and always use correct spelling and grammar. Internet abbreviations like “LOL” or “OMG” are fine to use personally among friends and family, but this is still an academic paper, so you need to treat it with the grammatical respect it deserves. Do not treat it as a personal journal entry.
• Check and double-check your spelling and grammar after you finish your paper.

• Keep your sentences focused. Avoid squeezing multiple ideas into one sentence.
• Avoid sentence fragments. Make sure that each sentence has a subject and a verb.
• Vary your sentence length. Include both simple sentences with a single subject and verb and complex sentences with multiple clauses. Doing so makes your paper sound more conversational and natural, and prevents the writing from becoming too wooden. [9] X Research source

• Common transitional phrases include "for example," "for instance," "as a result," "an opposite view is," and "a different perspective is."

• For instance, if reflecting on a piece of literary criticism, you could mention how your beliefs and ideas about the literary theory addressed in the article relate to what your instructor taught you about it or how it applies to prose and poetry read in class.
• As another example, if reflecting on a new social experience for a sociology class, you could relate that experience to specific ideas or social patterns discussed in class.

You Might Also Like

• ↑ https://www.csuohio.edu/writing-center/reflection-papers
• ↑ https://libguides.usc.edu/writingguide/assignments/reflectionpaper
• ↑ Alicia Cook. Professional Writer. Expert Interview. 11 December 2020.
• ↑ https://www.trentu.ca/academicskills/how-guides/how-write-university/how-approach-any-assignment/how-write-reflection-paper
• ↑ https://writingcenter.unc.edu/tips-and-tools/thesis-statements/
• ↑ https://writingcenter.unc.edu/tips-and-tools/conclusions/
• ↑ https://www.anu.edu.au/students/academic-skills/writing-assessment/reflective-writing/reflective-essays
• ↑ https://academicguides.waldenu.edu/writingcenter/scholarlyvoice/sentencestructure

About This Article

To write a reflection paper, start with an introduction where you state any expectations you had for the reading, lesson, or experience you're reflecting on. At the end of your intro, include a thesis statement that explains how your views have changed. In the body of your essay, explain the conclusions you reached after the reading, lesson, or experience and discuss how you arrived at them. Finally, finish your paper with a succinct conclusion that explains what you've learned. To learn how to brainstorm for your paper, keep reading! Did this summary help you? Yes No

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Guide on How to Write a Reflection Paper with Free Tips and Example

A reflection paper is a very common type of paper among college students. Almost any subject you enroll in requires you to express your opinion on certain matters. In this article, we will explain how to write a reflection paper and provide examples and useful tips to make the essay writing process easier.

Reflection papers should have an academic tone yet be personal and subjective. In this paper, you should analyze and reflect upon how an experience, academic task, article, or lecture shaped your perception and thoughts on a subject.

Here is what you need to know about writing an effective critical reflection paper. Stick around until the end of our guide to get some useful writing tips from the writing team at EssayPro — a research paper writing service

What Is a Reflection Paper

A reflection paper is a type of paper that requires you to write your opinion on a topic, supporting it with your observations and personal experiences. As opposed to presenting your reader with the views of other academics and writers, in this essay, you get an opportunity to write your point of view—and the best part is that there is no wrong answer. It is YOUR opinion, and it is your job to express your thoughts in a manner that will be understandable and clear for all readers that will read your paper. The topic range is endless. Here are some examples: whether or not you think aliens exist, your favorite TV show, or your opinion on the outcome of WWII. You can write about pretty much anything.

There are three types of reflection paper; depending on which one you end up with, the tone you write with can be slightly different. The first type is the educational reflective paper. Here your job is to write feedback about a book, movie, or seminar you attended—in a manner that teaches the reader about it. The second is the professional paper. Usually, it is written by people who study or work in education or psychology. For example, it can be a reflection of someone’s behavior. And the last is the personal type, which explores your thoughts and feelings about an individual subject.

However, reflection paper writing will stop eventually with one very important final paper to write - your resume. This is where you will need to reflect on your entire life leading up to that moment. To learn how to list education on resume perfectly, follow the link on our dissertation writing services .

Unlock the potential of your thoughts with EssayPro . Order a reflection paper and explore a range of other academic services tailored to your needs. Dive deep into your experiences, analyze them with expert guidance, and turn your insights into an impactful reflection paper.

Free Reflection Paper Example

Now that we went over all of the essentials about a reflection paper and how to approach it, we would like to show you some examples that will definitely help you with getting started on your paper.

Reflection Paper Format

Reflection papers typically do not follow any specific format. Since it is your opinion, professors usually let you handle them in any comfortable way. It is best to write your thoughts freely, without guideline constraints. If a personal reflection paper was assigned to you, the format of your paper might depend on the criteria set by your professor. College reflection papers (also known as reflection essays) can typically range from about 400-800 words in length.

Here’s how we can suggest you format your reflection paper:

How to Start a Reflection Paper

The first thing to do when beginning to work on a reflection essay is to read your article thoroughly while taking notes. Whether you are reflecting on, for example, an activity, book/newspaper, or academic essay, you want to highlight key ideas and concepts.

You can start writing your reflection paper by summarizing the main concept of your notes to see if your essay includes all the information needed for your readers. It is helpful to add charts, diagrams, and lists to deliver your ideas to the audience in a better fashion.

After you have finished reading your article, it’s time to brainstorm. We’ve got a simple brainstorming technique for writing reflection papers. Just answer some of the basic questions below:

• How did the article affect you?
• How does this article catch the reader’s attention (or does it all)?
• Has the article changed your mind about something? If so, explain how.
• Has the article left you with any questions?
• Were there any unaddressed critical issues that didn’t appear in the article?
• Does the article relate to anything from your past reading experiences?
• Does the article agree with any of your past reading experiences?

Here are some reflection paper topic examples for you to keep in mind before preparing to write your own:

• How my views on rap music have changed over time
• My reflection and interpretation of Moby Dick by Herman Melville
• Why my theory about the size of the universe has changed over time
• How my observations for clinical psychological studies have developed in the last year

The result of your brainstorming should be a written outline of the contents of your future paper. Do not skip this step, as it will ensure that your essay will have a proper flow and appropriate organization.

Another good way to organize your ideas is to write them down in a 3-column chart or table.

Do you want your task look awesome?

If you would like your reflection paper to look professional, feel free to check out one of our articles on how to format MLA, APA or Chicago style

Writing a Reflection Paper Outline

Reflection paper should contain few key elements:

Introduction

Your introduction should specify what you’re reflecting upon. Make sure that your thesis informs your reader about your general position, or opinion, toward your subject.

• State what you are analyzing: a passage, a lecture, an academic article, an experience, etc...)
• Briefly summarize the work.
• Write a thesis statement stating how your subject has affected you.

One way you can start your thesis is to write:

Example: “After reading/experiencing (your chosen topic), I gained the knowledge of…”

Body Paragraphs

The body paragraphs should examine your ideas and experiences in context to your topic. Make sure each new body paragraph starts with a topic sentence.

Your reflection may include quotes and passages if you are writing about a book or an academic paper. They give your reader a point of reference to fully understand your feedback. Feel free to describe what you saw, what you heard, and how you felt.

Example: “I saw many people participating in our weight experiment. The atmosphere felt nervous yet inspiring. I was amazed by the excitement of the event.”

As with any conclusion, you should summarize what you’ve learned from the experience. Next, tell the reader how your newfound knowledge has affected your understanding of the subject in general. Finally, describe the feeling and overall lesson you had from the reading or experience.

There are a few good ways to conclude a reflection paper:

• Tie all the ideas from your body paragraphs together, and generalize the major insights you’ve experienced.
• Restate your thesis and summarize the content of your paper.

We have a separate blog post dedicated to writing a great conclusion. Be sure to check it out for an in-depth look at how to make a good final impression on your reader.

Need a hand? Get help from our writers. Edit, proofread or buy essay .

How to Write a Reflection Paper: Step-by-Step Guide

Step 1: create a main theme.

After you choose your topic, write a short summary about what you have learned about your experience with that topic. Then, let readers know how you feel about your case — and be honest. Chances are that your readers will likely be able to relate to your opinion or at least the way you form your perspective, which will help them better understand your reflection.

For example: After watching a TEDx episode on Wim Hof, I was able to reevaluate my preconceived notions about the negative effects of cold exposure.

Step 2: Brainstorm Ideas and Experiences You’ve Had Related to Your Topic

You can write down specific quotes, predispositions you have, things that influenced you, or anything memorable. Be personal and explain, in simple words, how you felt.

For example: • A lot of people think that even a small amount of carbohydrates will make people gain weight • A specific moment when I struggled with an excess weight where I avoided carbohydrates entirely • The consequences of my actions that gave rise to my research • The evidence and studies of nutritional science that claim carbohydrates alone are to blame for making people obese • My new experience with having a healthy diet with a well-balanced intake of nutrients • The influence of other people’s perceptions on the harm of carbohydrates, and the role their influence has had on me • New ideas I’ve created as a result of my shift in perspective

Step 3: Analyze How and Why These Ideas and Experiences Have Affected Your Interpretation of Your Theme

Pick an idea or experience you had from the last step, and analyze it further. Then, write your reasoning for agreeing or disagreeing with it.

For example, Idea: I was raised to think that carbohydrates make people gain weight.

Analysis: Most people think that if they eat any carbohydrates, such as bread, cereal, and sugar, they will gain weight. I believe in this misconception to such a great extent that I avoided carbohydrates entirely. As a result, my blood glucose levels were very low. I needed to do a lot of research to overcome my beliefs finally. Afterward, I adopted the philosophy of “everything in moderation” as a key to a healthy lifestyle.

For example: Idea: I was brought up to think that carbohydrates make people gain weight. Analysis: Most people think that if they eat any carbohydrates, such as bread, cereal, and sugar, they will gain weight. I believe in this misconception to such a great extent that I avoided carbohydrates entirely. As a result, my blood glucose levels were very low. I needed to do a lot of my own research to finally overcome my beliefs. After, I adopted the philosophy of “everything in moderation” as a key for having a healthy lifestyle.

Step 4: Make Connections Between Your Observations, Experiences, and Opinions

Try to connect your ideas and insights to form a cohesive picture for your theme. You can also try to recognize and break down your assumptions, which you may challenge in the future.

There are some subjects for reflection papers that are most commonly written about. They include:

• Book – Start by writing some information about the author’s biography and summarize the plot—without revealing the ending to keep your readers interested. Make sure to include the names of the characters, the main themes, and any issues mentioned in the book. Finally, express your thoughts and reflect on the book itself.
• Course – Including the course name and description is a good place to start. Then, you can write about the course flow, explain why you took this course, and tell readers what you learned from it. Since it is a reflection paper, express your opinion, supporting it with examples from the course.
• Project – The structure for a reflection paper about a project has identical guidelines to that of a course. One of the things you might want to add would be the pros and cons of the course. Also, mention some changes you might want to see, and evaluate how relevant the skills you acquired are to real life.
• Interview – First, introduce the person and briefly mention the discussion. Touch on the main points, controversies, and your opinion of that person.

Writing Tips

Everyone has their style of writing a reflective essay – and that's the beauty of it; you have plenty of leeway with this type of paper – but there are still a few tips everyone should incorporate.

Before you start your piece, read some examples of other papers; they will likely help you better understand what they are and how to approach yours. When picking your subject, try to write about something unusual and memorable — it is more likely to capture your readers' attention. Never write the whole essay at once. Space out the time slots when you work on your reflection paper to at least a day apart. This will allow your brain to generate new thoughts and reflections.

• Short and Sweet – Most reflection papers are between 250 and 750 words. Don't go off on tangents. Only include relevant information.
• Clear and Concise – Make your paper as clear and concise as possible. Use a strong thesis statement so your essay can follow it with the same strength.
• Maintain the Right Tone – Use a professional and academic tone—even though the writing is personal.
• Cite Your Sources – Try to cite authoritative sources and experts to back up your personal opinions.
• Proofreading – Not only should you proofread for spelling and grammatical errors, but you should proofread to focus on your organization as well. Answer the question presented in the introduction.

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How To Write A Reflection Paper?

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Reflect a Point

Across x axis, y axis and other lines.

A reflection is a kind of transformation . Conceptually, a reflection is basically a 'flip' of a shape over the line of reflection.

Reflections are opposite isometries , something we will look below.

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