Proving the Pythagorean Theorem

As I type these first sentences, I am watching my students in full math mode. I do not know whether to cry, to jump for joy, to laugh out loud, or high five all the students as quickly as I can.

Cliff is explaining his group’s proof to his group members. Bianca and Ellen are struggling through understanding the algebra behind their proof. Stephen and Eloise are creating a detailed slide show presentation to explain their proof. Jennifer, Clay, and Thomas are moving around triangles they cut out from paper to get a grasp of their proof. Only Veronica seems completely distracted and uninvolved. (Not their real names, but I assure you they are real people.)

Next week, each of these groups are tasked with presenting a proof of the Pythagorean Theorem. This a class of tenth graders all of whom have run into a-squared plus b-squared equals c-squared at some point in their math lives. Last year, I did a project based on a MARS task on proofs of the Pythagorean Theorem. I liked it pretty well, but this year I decided to make it a lot more open ended and give my students a chance to practice being mathematicians.

Each group has to find at least one proof of the Pythagorean Theorem and then present it and teach it to their classmates. (See my last post on presentations.) How they teach the proof is completely up to them. They can do a computer slide show, they can project up sheets of graph paper. They can demonstrate how manipulating triangles and squares can justify the Theorem. They can show a clip from a Khan Academy video or make their own.

This project is another attempt at a “low floor, high ceiling” assignment. Jo Boaler discusses such assignments in her book Mathematical Mindsets. The idea is that an assignment is accessible to every student. Every student should have some way in to the problem and have the ability to produce something of quality that adequately addresses the prompt. At the same time, the assignment should have enough freedom so that students who are inspired or sufficiently motivated can extend the problem in different ways to make it more challenging and to grow more. I went so far as to suggest possible extensions of this assignment such as proving the converse of the Pythagorean Theorem, presenting real-life applications of the Theorem, discussing Pythagorean Triples, and researching the history of their proof. (President James Garfield anyone?)

One group has already started making a video in which they ask a people at our school what they know about the Theorem. They clearly are taking great pride in making a humorous and well-produced video.

I am very curious to see what Alex and his group do with the project. At times this year, Alex has expressed frustration with the difficulty of the work. He appears to want more challenging material. However, up to now, he has not shown the willingness to take risks and extend assignments. I have been gently nudging him, explaining the opportunity before him.

I think I am most pleased by the work I saw Cliff doing today. Cliff has been one of my struggles this year. He is often disinterested and shows little effort. He rushes through work without much thought and sees the course as a punch-the-clock type of class. But today, he was as engaged as anybody and working hard to explain a challenging proof to his group. His struggle is clearly increasing his understanding and I am thrilled to finally see Cliff leave his comfort zone and lose himself in the mathematics.

Presentations are next week, but those seem perfunctory almost at this point. Just watching my students work this week is all the evidence I really need to know how hard they have worked and how much they have learned.

Now here's something I hope you'll really like

High achievers are often high achievers in the U.S. system because they are procedurally fast. Often these students have not learned to think deeply about ideas, explain their work, or see mathematics from different perspectives because they have never been asked to do so. When they work in groups with different thinkers they are helped, both by going deeper and by having the opportunity to explain work, which deepens their understanding. [Jo Boaler, Mathematical Mindsets, pg 138]

In my quest to find meaningful assessments, presentations are presently high on my list. In my second year integrated mathematics course, presentations are a prominent component of the course.

Students learn in all sorts of ways. Once upon a time, I learned that there are three modes of learning: visual, auditory, and kinesthetic. That is, one can learn by watching, by listening, and by doing. Some students, I was told, will be strong in one mode and weaker in others. It was also explained that the best learning happens when all three modes are utilized, but that kinesthetic learning was generally considered the best, so should be very prominent.

This model of learning seems very simplistic and I believe is outdated. Notably, it does not mention learning by explaining. Perhaps one might argue explaining to be a form of kinesthetic learning, but in my mind these are two separate activities.

In my math class, kinesthetic learning happens when a group of students tries to make sense of a possible pattern by drawing a model graph. It continues as the group constructs a table of values or a model equation. And it happens while the group attempts to use its models to make predictions. Significant and meaningful learning will certainly take place during this process. The learning will undoubtedly be richer than had the students merely watched a teacher working at the board.

But this sort of learning is starkly different from learning through explanation. As Dr. Boaler’s quote notes, explaining work deepens one’s understanding. How can it not? When a student attempts to explain an idea to another, she must struggle to find the words to convey her understanding. She must think through the idea in a way that is much different than when she just wants to understand it for herself.

This is my primary motivation behind having students present to the class. It is one thing for a student to use mathematics to figure out how long it will take for a garden hose to fill a water tank or how many stars there are in the universe. It is altogether different for a student to attempt to explain his work to others, to verbally justify his results and describe the evidence and reasons. It is very common in my classroom for a student to struggle to explain his reasoning even when I know he understands very well how to do the problem.

It is this struggle that is so important. This is no time to invoke the lifeguard metaphor. A student struggling to find the words to explain herself is not drowning. She is no way about to die. There is no need for the teacher to dive in and save her. No, she is holding the map, she has a compass, she is making her way through the woods and she has all the tools she needs to find the lake. She does not need saving. She needs a chance to find her way without a teacher’s help.

Three to four times a semester, I ask my students to prepare a 5-10 minute lesson in which they must try to teach their classmates. Typically, I provide them with a problem that will be the focus of the lesson. The students are tasked with showing a full solution to the problem, but they must also find creative ways to improve the understanding of even the students who find the problem easy. Their teaching techniques might include extending the problem, or looking at common mistakes, or linking the problem to other ideas from the class.

A discussion is always a part of this presentation. It generally happens at the end, but sometimes it happens along the way. Sometimes, the discussions are started purposefully by the student teacher. Other times, a student in the audience brings up an interesting point. Otherwise, I will pose a question or make a conjecture for the teacher and students to consider. Because the student teacher has prepared in depth for the presentation, I know I have at least one student who has thought deeply about the topic to help grow the discussion. The discussions we have after these presentations are often some of the best of the year. As the student teacher wrestles with answering, I swear you can hear the synapses firing.

Certainly, one difficulty in presentations is the anxiety many feel towards public speaking. Add to that a dose of math anxiety and it can potentially be a very frightening experience. Thus, I work intentionally to help students become more comfortable. I provide time in class for preparation. I assign the lessons a week in advance so that they have ample time to get help should they want it. We discuss what effective presentations look like and what it means to be a helpful audience member. And generally, we work to create a classroom atmosphere where all voices are respected, where mistakes are celebrated and not shamed, where it is expected to express confusion so that all can help.

In my experience, these presentations have succeeded in deepening the learning of the presenter. Students gain confidence and comment favorably about them.

In the beginning

To begin with, let me state that this is neither the beginning of a course nor the beginning of my journey through creating meaningful learning. It is a couple weeks into the second semester of the school year and I have been intentionally working on creating meaningful learning for many years. But, I recently read Jo Boaler's book, Mathematical Mindsets, and I am currently blogging about my experiences in my first year teaching AP Statistics, and I kept thinking I should blog about what I am doing in my classroom to teach mathematical mindsets, and so here we are. It is the beginning of the blog.

What is "meaningful learning"? Students learn all the time whether teachers help them or not. Learning happens all the time in all sorts of ways and it is very presumptuous for a teacher to think that students cannot learn without them. No, my job is not to make my students learn. My job is to help them discover the subject matter and to develop their skills as learners in general. Math is the subject of my classroom, but I teach students. My goal is that they not only learn the skills and topics of mathematics, but that they also learn how to be better learners and are inspired to learn more. I aim to create opportunities in my classroom that allow students to grow.

I do not have a certification in teaching. Though I was a product of public schools, I have only ever taught in private schools and my first employer was happy that I had majored in math and so hired me to teach high school. In those first years, I simply tried to emulate what my teachers had done and what other teachers at my school were doing. And my students learned. But, in reflecting back on my work, I see now that there were many places where my teaching could have been better and that my students could have grown much more. My math homework was of the "drill and kill variety". I followed the textbook pretty closely, following the sequence that the textbook authors thought best. As a department, we did pick and choose a bit which topics from the textbook were to be taught and the pacing, but the curriculum was largely determined for us by the textbooks we used.

My courses nowadays are dramatically different from those many years ago. I am now teaching in my fourth independent school (the more savory way to say "private school"). I have worked with many different colleagues, gone to conferences and workshops, read books, and reflected on my teaching a lot. I have a better grasp of my role as teacher and I have developed and learned new means of teaching, ones that I believe create deeper and more meaningful learning. That is the point of this blog: to share with the readers what I have done and am doing and for me to reflect on my practices so that I might continue to improve and grow.

I have a lot to say on this matter. I plan to blog at least once per week. I would love to engage in discussions with other teachers out there, so please, comment away.

Finally, I will try to share materials and workshops and conferences that I have found instrumental in my growth. Dr. Boaler's book is one such item. Coming in February is the Learning and the Brain conference in San Francisco. Dr. Boaler and Carol Dweck (author of Mindset: the New Pyschology of Success) will both be presenting and I get to go this year. I believe the conference is full, so it is too late to register. But, there are three Learning and the Brain conferences each year and other one-day workshops.