Why Prove Things? A Geometry Debate

Earlier this week, my geometry students took part in a new debate format. As is often the case, the topic was student generated. Every year, students inevitably complain about proofs in geometry. It generally happens based on what they have heard from parents  or older friends who have horror stories about their experiences with impossible wastes if time, and it comes up before we do proofs. Once we start doing proofs, students question why we are proving things that were already proven. “Shouldn’t geometry be about measuring angles and lengths and areas and volumes?”, they ask. “Why can’t we just apply what others have proven before us? What’s the point?”

Of course, I have my reasons, but  

Pythagorean Triples For The Whole Class!

After introducing my geometry students to the Pythagorean Theorem, I introduced them to the idea of Pythagorean Triples. I showed them that the numbers 3, 4, and 5 will work in the equation: 32+42=52. Then I had students pick a couple of numbers, and we generated a brand new Pythagorean Triple with some fun algebra. At least, I thought it was fun, and students were curious where I was going with my algebraic manipulations. There are several methods, but the one I used went like this.

Pick two interesting numbers. Make them integers, not too big, but not too small. Did I hear 7 and 19? Perfect.

First, multiply those two numbers, and double your answer:
2 ∙ 7 ∙ 19 = 266

Next, find the (absolute value of the) difference of the squares of those two numbers:

192 – 72 = 312

Now, find the sum of the squares of those two numbers:

192 + 72 = 410

And just like that, we have a brand new Pythagorean triple:

2662 + 3122 = 4102

Go ahead – check it on your calculator if you’d like. Pretty nice, huh? I’ll leave it to you to do the algebra to see that it will always work. The most amazing thing about this little task is that students were immediately convinced (and rightly so) that even though it may be hard to find Pythagorean triples, there are an infinite number of them.

And then the game started. I gave each group of 3-4 students a stack of 24 numbers that can be turned into 8 sets of Pythagorean triples. (Of course, there’s only one way to get all 8 sets, but since there are some doubles, it’s possible to match up some that will throw you off).

For both of my geometry classes, all students (no exaggeration!) were engaged bell to bell, furiously testing and playing with the numbers, discussing strategies, and trying to find the eight correct sets. Not only that, although students were in separate groups, they soon realized that they’d be more successful working together. I heard so many great ideas about how to attack the problem that I wasn’t surprised when, even though it came down to the wire, one of the classes found all 8 sets!

This, to me, was a great low floor, high ceiling activity. Every student was able to work on calculations – calculating squares, calculating sums, etc. And some students really worked through some great strategies – what units digits can sum to the biggest number’s units digit? Can we work backwards from that? And of course, the little bit of algebra review in the beginning was helpful as I found later, when I had students repeat the process to develop their own new Pythagorean triples. It was easy for them to check their answers at the end and then look for any mistakes they may have made, and there’s something about big numbers that makes students truly proud (and rightfully so, even when they are using a calculator to do the calculating work).

Student Generated Problems Are Best!

Around this time of my geometry course each year, students have learned a whole lot of geometry – parallel lines, congruent triangles, basic proof techniques. It is the perfect time for them to come up with challenge problems for me. Believe it or not, classes never disappoint in this fashion. I’ve had some great questions in the past. (One of my favorites – can you fold an 8.5 by 11 piece of paper into an isosceles triangle where the entire triangle has the same number of layers of paper. I’m still not sure about that one, but I really enjoy watching students try every year.)

I tell my students that there are two things that make a challenge problem great:

  1. It is relatively easy to explain to anyone.
  2. It is challenging to find a solution (or even figure out if a solution is possible).

This year, a student came up with a problem that fit those requirements perfectly: Is it possible to take a square, and cut it into isosceles triangles that aren’t congruent? We then restated it to require a finite number of non-isosceles triangles (though the fractal version is a pretty cool visual and a nice approach).

We spent a few minutes on this at the end of class, but didn’t get anywhere right away. I sent this out via twitter, and almost immediately got a tweet back from Henri Picciotto:

What? That brought out some resolve in me to find a solution, and later that day I did. The next morning, before school, I had a student rush in to show me her solution, which was pretty similar to mine. I suggested to both my geometry classes that there may be more solutions – how many unique solutions or approaches may there be? Can we find an even number? Fewer than seven? I ended up waking up in the middle of the night thinking about this, and came up with even more solutions. So far, I’ve got solutions for 6, 7, 8, 11, 13, and 17 triangles. But the best part of this whole story is that a week later, I have students talking about this before and after school, at lunch, outside my classroom, totally invested in a problem that they made on their own.


When you are a teacher, these are the moments that you live for. Students of varying backgrounds and supposedly different ability levels collaborating and arguing with passion about squares and triangles. Sometimes, that question of “When am I going to use this?” or “How does this apply in the real world?” never come up because the math, by itself, is just fun.

So…can anyone find a solution with fewer than 6 triangles?

Reflections on a Busy Year

Yes, I’ve been busy before, and yes, all teachers are perennially busy, but this year has been different. The summer before the school year, I was asked to teach a second section of Algebra 1. The enrollment at our school increased (yay!) but not enough to hire another math teacher (boo!). I agreed, and am receiving a nice stipend to do it (which we’re pretending we don’t have and putting straight into savings). What I wasn’t prepared for was just how little time I would have. Teaching six periods, with no prep period at school, and spending most of my time before school, during lunch, and after school working with students has left me no time during the school day to do those other parts of my job that are important (if sometimes menial) – reviewing lessons from years past, preparing and updating lessons, reflecting on daily lessons, communications with parents, and oh so much grading and student feedback. I had a schedule like this in the past, but there were a few things that were very different.

  1. I was much younger, and had much more energy.
  2. My commute was much shorter (15 minutes vs. 45-75 minutes).
  3. I didn’t have any kids at the time. Man, having kids really eats up your time. Making breakfast, giving baths, story time, preparing lunches, putting the girls to bed. (Make no mistake about it – my wife does as much as I do, sometimes more, but we do try to split the duties evenly).
  4. I decided to implement standards based grading (which I really need to blog about), an endeavor that is mostly working really well, but needs some tweaks to push forward.
  5. For my Algebra 1 class, I finally decided to move forward with CPM, which has meant a lot of extra work and a very different approach to my teaching style.

I fully chose to do this, but it’s led to working from 8 to 11 or later most nights, while waking up at 5:30 every morning, working most of the time on weekends, mostly not blogging this semester, and committing to myself to not do this to myself again. I ‘m well past the point of being in danger of burning out (I think) in this career, but next year I want some more time for myself and my family, and maybe really think about doing that Desmos fellowship that I decided not to apply for this year (but instead followed longingly on Twitter).

So, for all you teachers in a similar position, I’d love to know what strategies you have. What do you cut back on? What do you just cut out? How do you survive? Any thoughts on how to be productive during commutes besides listening to podcasts, etc.?

Also, if you are interested, blog posts should be coming, in no particular order, on:

  • Some Thoughts on Implementing Standards Based Grading (So Far)
  • Debate #2: Discrete vs. Continuous Graphs
  • Debate #3: Do We Need to Prove Things in Class That People Already Proved?
  • A Great Geometry Question from a Student
  • When a Pythagorean Triples Warm-up Becomes a Class Long Activity

I’ve got a nice long train ride from LA to San Jose after Thanksgiving, and maybe after finishing my Asilomar Talk (11:00, Oak Shelter – Get Your Students Talking: Introducing Debate to Math Class, with Noirin Foy), sub plans, and grading, I’ll put some thoughts down. Thankfully, the in-laws will be coming with us on the train, and should occupy the girls while my wife and I get some valuable time to work.

Show, Don’t Tell!

For much of the early part of my teaching career, there was a strong focus on literacy from “the outside”. In other words, as soon as I stepped out of my own math classroom, I was bombarded with messages of how important it was to build literacy in students, and how the primary goals in schools needed to be about literacy above and beyond all other topics. To be fair, I started my teaching career at a school for students with various learning differences, but which at that time had a strong focus on dyslexia and other literacy-based disabilities. I am sure that had an impact on the messages I heard, from parents, from administrators, and from the speakers we had and conferences we attended during professional development. Still, there seemed to be a focus on literacy at the expense of math. As a beginning teacher and still in my mid 20’s, I took much of that message personally, and found myself both explicitly and implicitly arguing in favor of the need for more focus on math, which I saw as “the great equalizer” (as described by Edward James Olmos’ portrayal of Jaime Escalante in Stand and Deliver). 

Now, 16 years later, and in my very, very late 20’s, I have come to see math take center stage in prominence, thanks, for better or for worse, to Common Core. It has been literature refreshing to become more aware of the education world outside of math education, and see that we math teachers have a pretty prominent position in shaping (or trying to shape) the conversation about what education should mean in the 21st century. Thanks in part to Robert Kaplinsky’s #ObserveMe movement, I was reminded that I can learn a lot from every teacher, and at a small school like mine with a math department of 3, learning from teachers in other departments is almost an expectation.

Enter Laurie Miller, a veteran literature tLaurie Millereacher who led a small workshop during our inservice at the beginning of the year on the idea of “Show, Don’t Tell”. I’ve definitely heard the terminology before, but never spent any significant time on it that I remember in any of my English classes. We went through a nice activity where we read a passage and attempted to identify when the author was showing and when the author was telling. In the rich discussion afterwards, I found myself wishing we had more time to explore this idea, but also wondered when I would really use it (except in the written projects that I have my students do throughout the year).

A couple of weeks later, our other math teacher noted that she used the terminology to encourage a student to show their work, and things started to click. Asking students to show me their work in the past often went nowhere, because they had given the answer, and if the answer was right, why does the work matter? That can be a difficult argument to win as a teacher without resorting to authoritarian tones. But this new, simple phrase, one that students were buying into because they hear it in every class, really says it all. “Tell” is “give me the answer”, and that lends itself to a closed conversation of right or wrong. “Show”is along the lines of “convince me” or “prove it”, and leads to an open conversation about methods, efficiency, effectiveness, clarity, cohesion, organization, persistence, and all the stuff that we think of as important, and as transferable outside of our math class bubble.

I realize that this is a hastily written blog post, poorly edited, and probably rife with “tell, don’t show” examples, but in my defense, implementing SBG in Geometry and Pre-Calculus, switching to CPM for Algebra 1, and having 6 classes and no prep periods, plus a 2 year old and 5 year old at home, lead to lots of late night grading and very little time for blogging so far this year. And that’s unfortunate, because between SBG, CPM, and math debates, I have so many things to write about…over Thanksgiving break? Winter break? Hopefully sooner? Time will show.



Day 1 Debate: What Number is Best?

I know there’s a tendency to spend the first day of class talking about rules and expectations and grading policies and the syllabus and lots of other stuff that bores kids silly because, well, it’s boring. I mean, I know it’s all necessary for them to be familiar with, but do we really need to spend the first day doing that? It’s taken some time, but I’ve (mostly) changed to jumping right into something math related. Maybe it’s not a deep subject that we explore, but something to get students talking, and having fun, in class.

For my first day of Pre-Calculus, we didn’t hand out textbooks, we didn’t review the content standards, we didn’t sign out graphing calculators. Instead, the students fought. And laughed. And fought. And laughed. And laughed some more. Students entered the class and were told to think about their favorite number. (If they didn’t have a favorite, they were to make one up, and if they couldn’t, I’d give them one. Luckily, every student came up with one on their own.) I then told them to do some thinking, and come up with as many reasons why their chosen number was the best. Reasons could be from math of course, but also from pop culture, sports, numerology, other cultures, anthropology, mythology, religion, art, design, or whatever they chose. After five minutes of brainstorming, they discussed within their group of three or four which of their numbers was actually the best one. They then did further research for about fifteen minutes, using the Internet as a resource, to prepare arguments for why their own group’s number was awesome and the other groups’ numbers were boring. In the middle of their research time, I told students the story of the Hardy-Ramanujan number as well as the Interesting Number Paradox to give some incentive and inspiration.

When research time was up, we did a round-robin debate. I started with one group, and went around the room in a circle. Each group had up to 30 seconds to present an argument either for their number or against another number. I assigned each argument a subjective score of 1 to 3, and added it to that group (or subtracted from another group if they were arguing against another group’s number). I told them I’d keep going in a circle until I got bored with their arguments, but in both classes, the students were really impressive in their research and thought processes. During their research phases, I was also able to wander and listen to what they were thinking, how they discussed the math with their peers, and how they worked in groups.

I had some interesting arguments presented, and though I don’t remember even close to all of them, these were some highlights:

  • 4 is the only number that is spelled with its own number of letters.
  • 2 is the only even prime number.
  • 13 is the sum of the squares of the first two prime numbers.
  • 11 is both the number of points on the maple leaf of the Canadian flag and the number on the Loonie (Canadian one dollar coin).
  • There are 8 “quadrants” in 3d space (split by the xyxz, and yz planes). By the way – anyone know what they are called? Not quadrants, surely, since that’s how we refer to the 4 regions of the xy plane, but I don’t know right off.

In the end, students on winning teams got 10 points each, and runner up students received 5 points each. They aren’t extra credit, mind you – I’m doing away with extra credit, but that’s another blog post. These will go towards…something to be determined.

This was a great low floor, high ceiling activity where the richness of the mathematics was unbounded, but there wasn’t a single student who felt too intimidated to take part. One day doesn’t make an entire year, but this was a really fun way to spend our first day.

What Makes Mid-Pen Unique?

What Makes Mid-Pen Unique?

In anticipation of the transition to a new head of school, we were asked what we don’t want to change about Mid-Peninsula High School. To me, that means defining what it is that makes our school stand out from any other school. Having taught in a few schools before this one, there are a number of qualities we have that differentiate us. After I put together my list (the first five bullet points), I joined three other staff members to put our lists together. Interestingly enough, we all came up with different things, and we all agreed about each of them. 

  • This is a truly student-centered school. Decisions are made based on what’s best for students, individually and collectively. Other schools often create programs that students should fit into, but I feel that we try to develop programs (curriculum, support staff, opportunities, late start time, etc.) around our students.
  • It is largely assumed that teachers know what they are doing professionally. Once a teacher has been hired, their methods aren’t constantly scrutinized. They are supported in their curriculum development, textbook choices, professional development opportunities, grading policies, and classroom practices.
  • Students are included in all parts of the school. Whether this means skills level to allow for differentiation in classes, no-cut sports teams, or the encouragement of all students to take part in the arts offered here (visual, music, drama), every student has a chance to participate in every aspect of the school.
  • Focus on development is very strong, especially for such a small school. At other schools where I have taught, generally of a similar size, we did not have a single person dedicated to raising money for the school, and as a result, were unable to fund some of the important things that we do fund here.
  • Our dedication to true diversity is one of the highlights of the school, and very important to me personally, as well as many of the students that I talk to. We have diversity in race, ethnicity, economic status, disability, and geographic background that is practically unheard of in small private schools. We offer financial aid to a substantial number of students, and that is something that we should be proud of.
  • Classroom and curriculum freedom and autonomy
  • Ending staff meetings with victories
  • Alumni graduation speakers
  • Video contest
  • Senior dinner, Quaker style
  • Major award selection/discussion – only saying nice things about them
  • Kindness and acceptance of teachers and students
  • Lack of bullying
  • Small classes
  • Speaking compassionately about individual students
  • Interaction w/students outside classrooms
  • Individual student support
  • Using support staff (Randy, Wendi, Heidi, etc.)
  • Safe environment for everyone (students and teachers)
  • Benefits for all by having deaf staff member, learning ASL language AND culture
  • Keeping open mind for opportunities for staff to learn as well
  • Foreign students can integrate via ASL into culture

I think this is a pretty impressive list, and makes me proud to be a part of this community.

Variable Credit and Standards Based Grading

I’ve been soaking up a lot of information about standards-based grading (SBG), and today came across this in my twitter feed:

I am already very focused on encouraging growth mindsets, so the idea of assessing it was very much something I wanted to know more about. With some follow-up tweets, I got a link to the presentation, which included a SBG rubric template that I’m excited to look at more closely.

The school where I teach, Mid-Peninsula High School, has a lot of things that make it different from other schools where I’ve taught, observed, or attended. One thing stands out that may be unique to Mid-Pen, though. We have a variable credit system, rather than a standard all or nothing system. The VCS (because we need another acronym in education lingo) was the most confusing system that I encountered, although in retrospect it is because it was such a foreign concept.  Each semester class is typically worth 5.0 credits, but students receive 0.5 increment credits throughout the semester based on the amount of satisfactory work that they turn in. This allows for differentiation between two students that may look the same on paper in a traditional system.

Suppose you have two students, Alicia and Benjamin. Alicia finds Geometry class very easy, and halfway through the semester has an A in the class without too much effort on her part. She then decides to blow off the rest of the semester because she has x, y, and z priorities that are more important to her. She doesn’t turn in homework, doesn’t study, fails most of her quizzes and tests, and bombs the final. When all the grades are averaged out, in a traditional grading and credit system, she has a C and 5.0 credits.

Benjamin has always had a tough time in his math classes, but works persistently and diligently, and always completes his work on time, even if he usually gets mediocre grades. Throughout the semester, he maintains a consistent C average in the class, and at the end of the semester, in a traditional grading and credit system, he also has a C and 5.0 credits.

On their transcripts, these two students look identical, but there are very different stories behind those grades. In our VCS, Alicia may end the semester with an A, but only 2.5 credits. Benjamin would still have a C and 5.0 credits. This means that there are two meaningful measurements on the transcript. The credits reflect the amount of work that has been satisfactorily completed, and the grades reflect the quality of that work.

So, what does this have to do with standards-based grading (SBG)? For several years before I arrived at Mid-Pen, I was interested in implementing some sort of SBG but wasn’t sure how to do it. Halfway through my first semester teaching at Mid-Pen, I realized that basically what I was doing was SBG. I determined, in 0.5 credit increments, what amount and type of work qualified for those credits. Inevitably, each unit was either 0.5 credits or 1.0 credit, but it was mostly based on what had been covered by the end of each grading period. We have grading periods after 1.5, 2.5, 4.0, and 5.0 credits. Students are able to make up missing credit at the discretion of each teacher, but generally it’s encouraged, and even expected, for students to make up missing credit during the semester. Otherwise, it gets pushed to an independent study or summer school, both of which can be a huge pain for both students and teachers.

During that first semester, I took a few weeks off for paternity leave, so I was a bit preoccupied and didn’t put in the necessary time to formalize each half credit. I’ve had three more semesters at this point, but no other excuses, so my big summer project (besides deciding on an Algebra 1 text) is formalizing SBG for our VCS. More on my progress (and credit given to whomever I steal from) later in the summer.



Brainstorming a Penalty Kick 3 Act Task

I came across the following video in my twitter feed, and think it would make a great 3 act task for my Trig/Precalculus class, and maybe could work something out with my geometry class as well, although probably without the algebraic equations. Maybe something we could hash out nicely in Geogebra.

The biggest question is how much information to give. My initial answer is usually none, but especially at the beginning of the year, as students are just getting used to my style of making them work for the information they need, is it necessary to give them a bit more to go on? I think a lot will depend on the students that I have in the classes, their cultural and mathematical backgrounds.

I’m thinking about finding a couple students from the soccer team at the beginning of the year to practice penalty kicks while I record it. Not sure exactly where this really fits, though conic sections seems to be a good bet. I tend to do conics later in the year, and have generally had students in four groups lead the lessons on each of the conics. It’s a nice setup for the unit that I’ve found really successful, but sometimes it’s worth trying something new, just because. Plus, I’ve had a number of my students on the soccer team, so maybe this is just a good way to help them score some more points, win some more games, and answer for themselves, “When am I going to use this?”2015 Mid Pen Coed Soccer 0017.jpg

It would also pair nicely with Henri Picciotto’s Soccer Angles problem that I was already planning to add to my toolbox for next year, so putting together a combination of tasks related to soccer, even across different classes (Trig/Precalculus and Geometry), would be a lot of fun. Plus, hopefully it will help us maybe win the league championship that we’ve been close to the past couple of years. Go Dragons!



I Can’t, Shouldn’t Be the First to Believe

There is a comment I’ve gotten frequently from students, and it’s an absolute compliment, but one that tears me apart.

“You’re the first teacher who believed I could do math.”

In one way or another, that’s one of the most common responses I get in my end of year report card from students each year, or in cards or emails from students. I am thrilled on one hand that they felt I believed in them, because that is one of my biggest goals. I absolutely believe that everyone can learn math, that everyone can always learn new and more challenging math. I’ve held this belief for just about all of my teaching career, and it’s driven so much of my approach to my classes. So, I’m glad that that got through to students.

And yet…

Am I really the first teacher that believed in these students? Some of these students were in 9th grade, some were in 12th grade, some in between, but am I to really believe that each of them had teachers all the way through elementary/middle school that gave up on them? In fact, I know for a fact that many of the students did have other teachers that saw their potential and did believe in them, but somehow they did not get that message. Why is that?

Sometimes, the words of a single teacher can have such an impact that they color the words of every teacher that comes after. A 3rd grade teacher who tells a student that maybe she’ll never be any good at math because she can’t memorize her times tables. A 5th grade teacher who tells a student who struggles with fractions that at least he is good at other things like reading and art. An 8th grade teacher who tells a student that maybe Algebra just may be too hard for her, but at least she’s cute and can get help from boys on her homework. These are all stories I’ve heard from students, and their impact can be life changing. The words of a teacher, at the right time and in the right context, can dramatically change the perception that a student has about his or her abilities – for better or for worse.

It is easy to throw elementary school teachers (and to a lesser degree middle school teachers) under the bus. After all, they’re the ones who did all this damage, right? And they aren’t even really math teachers. Most of them don’t like math, and have a phobia about it, and shouldn’t be teaching. It is so tempting to complain about their inadequacies, which is what I often did early in my career. I even had anecdotal evidence – when I’d attend a math education conference, high school teachers were there because they wanted to go, and elementary school teachers were there because their administrators made them go. It was such a tempting mindset, until I realized that I was doing nothing to improve things, just complaining and making a case for why my job was so hard and why I couldn’t succeed with every student. This isn’t uncommon. As Kaneka Turner pointed out during ShadowCon16 at NCTM this year, how many of us as high school teachers at math education conferences ignore the elementary school teachers sitting nearby?

We are part of the problem! We need to be inviting elementary school and middle school teachers to our world. We need to be reaching out to all of them, and asking them to join us in showing all students that every single one of them is a mathematician.

On a related note, one of my Trig/PreCalculus students said a few weeks ago that she and her classmates weren’t mathematicians, that I was the only mathematician in the room. It was in the context of a debate, and I think she felt my glare and knew what she had said, but I didn’t address it until earlier this week.

“In English class, when you write, what are you?”

“A writer.”

“In art class, when you make art, what are you?”

“An artist.”

“In music class, when you make music, what are you?”

“A musician.”

“Then in math class, when you do math, what are you?”

“Math students.”

And then it sank in for them, and for me. There’s more that I need to be doing, because despite all the success I think I have sometimes, at the end of the day, my most advanced math students don’t see themselves as mathematicians. I need to send them better invitations.