# Embodied learning, and the teaching of quadratic functions and equations

Through this week’s reading, I was able to conjure a new way to teach quadratic functions, and equations going forward that incorporates technology. In the following sequence of planned activities, I plan to incorporate a greater amount of collaboration between students, more opportunities to physically manipulate the information they are working with, and focus more on an inquiry based approach.

One of the major concepts students taking Pre-Calculus 11 need to learn is the quadratic function, and how the equation of the function determines various attributes of the graph such as its direction of opening, the location of the vertex, and its width. In teaching these different concepts, I plan to utilize Desmos Calculator to have students play around with different forms of the quadratic formula using the calculator’s slider function: http://support.desmos.com/hc/en-us/articles/202528919-Sliders

This Desmos activity I plan to create will leverage the “interdependence” of embodiment, embededness, and adaptation” (Winn, 2002) as students will physically manipulate the artificial environment I have created to learn, and adapt their knowledge about parabolas.  An activity of this sort would also promote collaboration between students on a “mixed reality” type software where students have the opportunity for “physical interplay” of the graphs, which, according to Lingren (2013).

As for the choice of using Desmos Calculator over a standard graphing calculator, I am making the choice partly due to my own experiences with Desmos, but also to avoid many of the misconceptions related to quadratic functions pointed out by Powell (2015), many of which involves incorrect equation entry, and may be due to the standard graphing calculator’s small screen.

Students frequently have difficulties connecting the concept of solving quadratic equations and working with the graphs due to their failure of seeing the relationship between the two concepts. To solve this problem, I would utilize another activity using the Desmos calculator. This time, I wish to use the calculator to showcase the relationship between the solution to the equation of the form 0=(x-a)(x-b), and the graph of an equation in factored form: y=(x-a)(x-b), vertex form: y=a(x-p)^2+q, and standard form: y=ax^2+bx+c. I would again utilize an activity involving sliders. I believe an activity utilizing sliders again in this instance, would allow me to reap the same benefits as the previous activity.

My questions for colleagues:

1. What are your experiences learning Mathematics with technology? How has graphing technology (whether it is on a graphing calculator or not) helped you with learning?

2. What do you see to the biggest problem in Mathematics education today?

3. What misconceptions in Mathematics have you struggled with in the past? How did you manage to get past them?

• Lindgren, R., & Johnson-Glenberg, M. (2013). Emboldened by embodiment: Six precepts for research on embodied learning and mixed reality. Educational Researcher, 42(8), 445-452.http://www.move2learn.education.ed.ac.uk/wp-content/uploads/2015/04/Lindgren-2013-Embodied-Learning-and-Mixed-Reality.pdf
• Powell, J. (2015). Solve the Following Equation: The Role of the Graphing Calculator in the Three Worlds of Mathematics. Interpreting Tall’s Three Worlds of Mathematics, 52(2), 11. Read one of the chapters from: https://scholar.google.ca/scholar?as_ylo=2013&q=embodied+learning+and+graphing+calculators&hl=en&as_sdt=0,5
• Winn, W. (2003). Learning in artificial environments: Embodiment, embeddedness, and dynamic adaptation. Technology, Instruction, Cognition and Learning, 1(1), 87-114. Full-text document retrieved on January 17, 2013, from: http://www.hitl.washington.edu/people/tfurness/courses/inde543/READINGS-03/WINN/winnpaper2.pdf

1. MichaelHengeveld says:

Hello Gary,

Thanks for your post. It was an interesting read. I like how you are using Desmos in class. It is yet another app that a student turned me on to and I find it so democratic and available!

I am interested in your question “What do you see to the biggest problem in Mathematics education today?”. In a word, I think the answer is relevance. So much of the educational literature I have encountered points to the importance of motivation, engagement, and student-centered learning. In this most recent batch of reading refers to “umwelt” as a personalized universe shaped by genetics and and naturally varied experiences. Is it even possible to reach a student in a meaningful cognitive way when discussing how to factor a trinomial by finding the GFC?

To quote Edelson (2000) “Thus, although a teacher can create a demand for knowledge by creating an exam that requires students to recite a certain body of knowldge, that would not constitute a natural use of the knowledge for the purposes of creating an intrinsic motivation to learn.”

To be sure, math is not the problem. Building a curriculum based on the assumption of requiring a steady progression to calculating surface integrals is. Any high school graduate has learned polynomial long division, but still show signs of being unable to contextualize the difference between millions and billions of tax dollars when it comes to political news. Perhaps if we rooted more mathematics in a more meaningful context, we would all be better off? Engineering types and physicists can chose the algebra elective.

Michael

1. Allen Wideman says:

Hi Gary,

Your planning involving Desmos looks very interesting, I’m curious to know how it all works out with your students. In your post you mention that you were aiming to incorporate a greater amount of collaboration between students, and I’m wondering how that might look in your mathematics classroom?

In terms of my own misconceptions in mathematics, one of the biggest that I had to overcome was my own personal belief that I wasn’t “good” at math. I see a similar attitude in many of my students every year, and they often approach mathematics with negativity at the beginning of the school year. For me, I believe that the way that I was taught mathematics as a student led me to believe that the difference between getting questions right or wrong is what ultimately made you good at math (or not).

I also feel that relevance is key when approaching mathematics. Another approach that we take is emphasizing the importance of the process, or exploration, of Mathematics while moving away from a need to constantly be “correct” in the work that we do. It seems that many of our students have been “wrong” in their mathematics thinking so often over the years that they just eventually start to lose confidence and tune out.

2. Gary Ma says:

Hi Michael,

I am in strong agreement with you in that in some respect, the Math curriculum steers students more towards abstraction than with dealing with practical reality. It was an understated problem that is only recently being fixed. With the new BC math curriculum, we are seeing a turn towards more “applicable” math being introduced back into the the standard stream as students are required to learn about budgeting, taxes, and interest.

I think the value brought about by a high school math course has a potential to be much more about the content. Just like reading different novels in English gives teachers a chance to teach students to be critical of language, and printed text, and to be culturally aware. I think Mathematics can teach students a lot of “soft skills” as well. Math courses have the potential to teach students the value, and the skills needed to present logical arguments, a way of presenting facts with numbers and statistics, and to understand numerical trends and patterns. It is difficult to directly teach a lot of these things to a class, because as you suggested, teachers artificially create a demand of knowledge by setting their exams. How teachers teach these alternative skills will depend on how the teacher goes about teaching the different concepts and what the emphasis is. Could more time be spent on a project, where students get the opportunity to present an idea, as opposed to simply cranking through worksheets? I believe so, but many teachers, and students believe that there is no such thing as a math project. Sadly, a lot of Math teachers deem presentation skills less imperative than skills like learning how to factor a trinomial, for example.

2. Gary Ma says:

Hi Allen,

Defeatist attitude is one of the more common problems that I see a lot of my students encounter, and you wonder why they can feel so strongly about being bad at Math, but not towards other subjects they may struggle in. Rarely do students say “they suck at English or Social Studies”. I believe a lot of it again goes back to the point I made above, and also like you said, where competency is judged by whether you get the answer right or wrong.

If you are good at building a mathematical argument through speech, or if you are proficient at presenting sound, logical arguments, and presenting it to a group of people in a clear, and precise manner, doesn’t that have value in a subject like Mathematics? In the field of Mathematical research, nobody has the right answers, in fact, most of the work lie in developing a way to get as close to the answer as you possibly can, and preparing to defend your work when you need to. Yet in most situations, we teach students, whether intentionally or not, that the answer is all that matters.

I spend a lot of time telling students that if I wanted the answer, I can go look it up in the answer key in the book, or I can punch it into the calculator myself.

In terms of collaboration, I think the key is to create situations where students would have to explore and discover the facts themselves, and for teacher to create a path to the facts that isn’t necessarily completely linear and direct. Leveraging the slider in the activities is a good way of doing that. The slider can be moved in different ways to create different results. When students move the slider on the screen, the graph will change, and students will have to observe the changes and make conclusions themselves. The task itself is straight forward, but I anticipate the collaboration will occur when students fact check among themselves to ensure they have reached the correct conclusions.

3. sarah winkler says:

Hello Gary,
Interesting post. I was struck by your question “What is wrong with mathematics today? as what I believe is behind the question is something that a number of teachers and I have been talking about. Why it is okay for a parent to say “I was bad a math, that is why my son/daughter is.” It seems to be the one subject that parents give their child permission to be bad at. As we start teaching math in a more interactive way, parents understand it even less and continue to support their children being bad at it. I have certainly had a few parents talk about how they don’t understand how I’m teaching and why does it have to be so different from when they went to school, they learned why can’t their kids learn that way too? We know more about how kids learn now and can connect in better ways, we just have to catch the rest of society up to understand why it is important. This blog by Nikki Lineham, a super fabulous math teacher, talks about optimizing student learning in Math. She is great and has lots of ideas to support math instruction in its newer form.

https://educatingnow.com/blog/306206/seeing-is-believing

1. Gary Ma says:

Thanks for sharing Sarah! I also have similar experiences with parents as well and believe that some times the defeatist attitude is passed down by parents. At the very least, parents who are unwilling to help students with Mathematics put their own kids at a disadvantage. One could argue that at the elementary school level, the easiest topic for parents to get involved in is Mathematics since most adults should remember how to add subtract multiply or divide. I wonder how involved these parents get despite their own perceived weaknesses in Mathematics.

4. momoe says:

Hi Gary,

Great post. It would be really interesting to study student learning based on the Desmos calculator vs. a standard graphing calculator in your context.

As for my own math learning, I have to admit that it makes me laugh remembering those days. Let me paint you a picture: immigrant asian parents, Kumon math, Saturday Japanese school. . . . this equals repetition, rote memorization and competition for speed and accuracy. As for technology, my parents were appalled when I had to go purchase a TI 83 for math class. In Japanese school, we used an abacus! So my math learning with “technology” was very different from what students in current day classes are exposed to. . . at least this is what I gather based on what you describe. I think the greatest challenge in math education is getting over this generational divide. We are in this interesting time in our history where you have parents, such as myself, that were taught the importance of “knowing” these abstract math concepts, which for the most part meant memorization and not understanding. This competes with current day thoughts of the importance of application of knowledge. It scares some parents when they are told students no longer need to “know” the multiplication table. I have recently talked to some moms who feel strongly about this and they are choosing schools outside their designated community based on the math curriculum. Now this may just be a small set of parents that I am exposed to (Type A, academic parents in my medical community), producing this skewed view that I have. . . has this been a challenge for any of the math teachers out there?

1. Gary Ma says:

Hi Momoe,

I can absolutely relate to your experience with your parents, and also what you have observed in these “type A” parents. Out of all the subject areas, Mathematics is the one subject that so many of these parents want to accelerate their kids. Many of them want their child pushed up a grade level, or sometimes 2, in Math because the kids are “bored” or find the material too easy. Most students are in this situation because they have taken some part of their mathematics education in Asia, or have taken extra-curricular Math sessions, where the focus is likely on rote memorization, a focus that does allow students to learn more material quicker than their counterparts in North America.

However, there are currently efforts to reform Math education in places like China to a include more of an inquiry, and student focused approach (Xie, 2009), in efforts to motivate students to better utilize their mathematical knowledge outside of the test situations they they are often pushed into. What does this tell us about educational research, and western methods if there seems to be this simultaneous effort for Western, and Eastern schools to convert to student centred approaches?

I believe there is value in memorization, as speed is a very good motivator for young students, and a good confidence booster that pushes many students to like Math. But I don’t think that should be the sole focus. There needs to be a focus on how to apply the knowledge in different ways such as the ability to communicate, and present Mathematical ideas.

Xie, M. (2009). Mathematics education reform in mainland China in the past sixty years: Reviews and forecasts. Journal of Mathematics Education, 2(1), 121-130.