When Plans Go Awry

This past weekend, on Saturday afternoon, I got to be part of a working group at Desmos headquarters in downtown San Francisco. The drive up there wasn’t too long – it stayed pretty dry, although I was sure the downpours would start any minute the entire trip. I wasn’t sure exactly what to expect, just that the plan was to work with other math education professionals, and have a Desmos activity for my Algebra I class by the end, ready to pilot.

There were 8 of us there, and after a while discussing some possible ideas, with a focus on quadratic equations and graphs, we split into two groups. The other group put together a really nice activity, exploring the symmetry of quadratics, and they included a paper and pencil worksheet to accompany the online portion of the activity. I’m pretty excited to use this activity when we get into the graphing of parabolas, as I’m sure it will make it easier for students to anticipate and simplify their graphs.

My group took on something that I was really excited about, mostly because I couldn’t quite see how it was going to work. We were going to use Desmos to explore the concept of completing the square graphically. I’ve covered the concept with algebra tiles, which students seem to get, and I’ve covered it algebraically, building a perfect square trinomial, but had no idea how to approach it using parabolas. Working with three other bright, creative, and experienced math teachers, we went to town, coming up with some interesting ideas and approaches. There were connections that we were making between vertically shifting a parabola to “sit” on the x-axis and adding a number to the equation to complete the square, and hopefully noticing the pattern between the b and c values.


It seemed so clever, so creative, until we started wondering – what are students going to get out of this? Will this help them make connections and understand the process of completing the square? Will it make things more complicated for them? In short, were we delving into this because it’s a good educational idea or because it’s interesting to us as people who enjoy mathematics?

For my upper level math courses, I’ve been more willing to experiment with them, to show them things that I wasn’t sure about, to see where things would go, just because the math itself was fun, and it was fun for the sake of itself. I’ve been less inclined to do that with my Algebra 1 students, by far the most differentiated of my classes in terms of previous knowledge, comfort with math and with their own abilities. Is this the same idea? Maybe showing something that is interesting, not because it reinforces a standard or because it introduces a new standard, but simply because it’s what I, as a mathematician, found interesting and curious. Isn’t there value in that, in and of itself?

Mind you, our working group WILL be getting a complete activity/lesson out of this, and it will be taught in a few weeks in my Algebra 1 class. Will it, in the end, be all about completing the square? Where will it fit? We will see.



No, I Won’t Just Give the Formula (Usually)

This week, I brought a nice activity to my geometry classes. At least, I thought it was a nice activity, and almost every student was fully engaged. The task itself doesn’t seem too exciting, maybe. Students used a compass to draw a circle, and then measured the diameters with a ruler. Then, they carefully used twine. Yes, twine, left over from my graph theory unit last year. I couldn’t find my string or wire, so twine it was. Anyway, they used twine to carefully trace around the edge of the circle, and then straightened the twine to measure the circumference. Then, they got together in groups of 3-4 and tried to find a relationship between their numbers. It wasn’t long before they discovered a similarity between the ratios of the circumference and diameter, which they then realized was π.

Then, we went a step further, and looked at the class averages, and also looked at the % error. Students then started to make a conjecture that there was a correlation between size of the diameter and the % error. I indulged them in this direction that was much more interesting than the standard C=πd lesson that this was supposed to lead to. We used Google Sheets to analyze all of this data that we collected. (See the scatterplots below).

Diam-PiEstimate%Error vs. Diameter

After this class, a student came to me. This student said, “What was the point of that lesson? I mean, wasn’t it just the formula for the circumference of a circle? The assignment was just about using that formula. Couldn’t you have just told us the formula, done some examples, and be done with it? That’s what teachers are supposed to do. The rest of it might have been interesting, but wasn’t it kind of a waste of time?”

So, where did I go wrong here? Or did I? I haven’t surveyed every student, so maybe the one student was an exception. I thought about this for a while today.

Then, tonight, I realized something. I’m a bit of a hypocrite. Not a complete hypocrite, but just enough to have an interesting contrast of lessons. Later today, during a trig class, after going through a few derivations of various identities (or telling the students that there’s an identity for a double angle formula, go ahead and find it), we got to a few identities where I told them, “I’m just going to give you these few identity formulas. We won’t go over the derivations, but we’ll practice using them a few times so you know how to apply them in different situations.”

Wait, what? That’s not who I am as a teacher, right? I thought I didn’t just give students the formula and tell them to use it. They have to work for it.

It all comes down to context, though. In some cases (Pythagorean formula, quadratic formula, exponential growth functions, angle sum identities), there are very, very rich and meaningful lessons, extensions, and mathematical conversations that can be generated. In other cases (half angle identities, product-to-sum formulas), maybe there are meaningful lessons that I just haven’t developed, but I haven’t developed them, and I do have limited time.

Common core has sometimes been referred to as a way to refocus mathematical education to focus on deeper understanding, and pull back from the mile wide, inch deep approach to math education in years past. I mostly agree with that, but every once in a while, it sometimes makes sense to just give a student a formula. Hopefully I can always do it only when there isn’t a rich concept or application that can come out of trying to derive that formula, though. Speaking of which, anyone have a really interesting lesson on half angle formulas that doesn’t sound completely contrived?


Math as a Historically Collaborative Effort

As Isaac Newton famously wrote in 1676, “If I have seen further, it is by standing on the shoulders of giants.” Many great, great bits of math are just making progress on problems, discovering new things about problems, even without coming to the actual solution. Perhaps most famous theorem in math, Fermat’s Last Theorem, was suggested in a margin of a book in 1637, and had bits and pieces proven by many great mathematicians (including the French mathematician Sophie Germain, my dog’s namesake). In the 1950’s, two Japanese mathematicians (Yutaka Taniyama and Gomo Shimura) introduced the Taniyama-Shimura conjecture (later the Taniyama-Shimura-Weil conjecture, and then the Modularity Theorem when it was finally proven a half century later). Sadly, Taniyama committed suicide, and in his suicide note said that he had “lost confidence in his future”.

It was discovered in the 1980’s that a proof of this conjecture would lead to the proof of Fermat’s Last Theorem, even though on the surface and to the layperson, they seem to be parts of completely unrelated fields of mathematics. However, when Andrew Wiles introduced the proof of Fermat’s Last Theorem, it was by working on, advancing, connecting, and completing, the mountains and mountains of work produced by giants (great and small) over a period of almost 350 years. I am sure that nobody studying this proof will see it as a straightforward proof of start at A and end at B, but rather as a meandering and complex journey from the margins of an amateur French mathematician’s text to a series of 3 lectures given by a British Princeton professor in June of 1993 at the Isaac Newton Institute at University of Cambridge.

So why this little history lesson? It’s a reminder that the process really is just as important as the result. Isn’t that part of why we ask students to show their work? What else can we do to encourage the process over the solution? In my Pre-Calculus class, we frequently encounter trig equations that don’t follow a cookie-cutter formula. One activity that I’ve been working on with students is to limit the amount of time they have on a problem to something very short. I explain that they will probably (hopefully) not have enough time to get the right answer, but that we would discuss different approaches taken, why they may or may not work, and what we can learn from them. One equation may lead a class of students in many different directions. Now, knowing that nobody will have the right answer at the end of 1 or 2 minutes, all students feel more comfortable writing something/anything down. It took time for most students to buy into this process, and some still want me to just give them an algorithm, a single method that will always work, convinced that I’m hiding away a magic formula or something. Still, the risk taking, the increased and productive conversations, have been well worthwhile.

A second activity that I’m going to try out next week is going to be even more collaborative. Each student will be given a different problem, and will be given a short period of time to write down one thought, question, or step in a solution. They then pass the paper on to a peer, who will also have a short time to write one thought, ask (or answer) a question, write a second step in a solution, or write a first step in a different solution. They then pass it on to a third student, or back to the first student. We will continue this process until significant progress has been made on several problems, before discussing. This is a silent activity, where students collaborate solely by writing, and giving each student in a group a different colored pen/pencil/marker can help to see who builds on whom, and how that happens.

In both of these activities, students get a chance to see several different approaches to a problem, and can discuss the merits of each one – some positives, some negatives, some suggestions, some ideas where an approach that doesn’t work in one case may work somewhere completely different. There is still a lot more work to do to adjust these activities, by starting them at the beginning of the year next year, and collecting some data to see just how well it seems to be working. Initially, though, many students who used to leave problems blank are starting to put some writing and math on paper, and taking more risks to develop their ideas, even if they don’t know for sure where their work will take them.

When We Make Math Look Too Easy

Many math classes go something like this: Teacher reviews a homework problem or two, introduces a new idea, gives a couple of examples, and then assigns a problem set to students. Students try those problems after class, at home, and what seemed to make sense in class is suddenly a complete mystery. As teachers, we often forget that the first time we try a problem, it isn’t always easy. In my case, of course Algebra I appears easy for me when I do it. I’ve been playing around with basic algebra skills for more than 25 years, so it would be pretty inexcusable for me to falter. And yet, when I first started teaching, I used to go out of my way to be sure I would appear infallible to my students. I selected problems that worked out nicely and quickly, perpetuating the idea that for some of us, math is just easy. As I progressed, I realized that students needed to see understand the true work of a mathematician, and I had several examples from my pre-teaching career to draw upon. But all the stories in the world were no substitute for seeing me make mistakes. I’d then fake mistakes, but no matter how good my acting, I knew it was fake, and students surely figured it out as well.

Now, I seek out truly challenging problems, the kind of low floor/high ceiling problems that are easy to explain, but difficult to solve, and most importantly, new to me. On a few occasions, I have introduced some of these problems to my classes, either as challenge problems or problems that arise in the context of a lesson. A recent example from my geometry class involved trying to find the area of a trapezoid in a different way than normal. I suspected that we would be able to derive the common formula, but I wasn’t positive the solution would be there, nor was I sure how we could arrive at the solution. Still, this gave my class the opportunity to see me in a vulnerable state, wondering out loud to them what we could do, soliciting ideas from students, and exploring every possible pathway. In the end, this turned into a really nice derivation. (That’ll be another blog post, don’t worry).


So, my challenge to myself is to continue to not just show math to students, not just to ask them to do math, but to actually collaborate with them on math problems for which I don’t already know the answer. Because I want them to see how we actually do math as mathematicians, and to see why the focus needs to be on the process, and not just the answer.

Moving to a New Blog

I’ve had a blog that I’ve used from time to time on the teacher page of the school where I teach (Mid-Peninsula High School), but I’ve decided that I want to reach out to a larger audience. Plus, although I’m very happy teaching where I am teaching, and would love to retire here many years from now, life sometimes happens, and I want to be sure that I keep my blog separate from my job. So, here it is – my inconsistent musings and thoughts about mathematics, and how it finds its way into every part of my life, not just teaching.

I’ll copy in old blog posts (in sequential order from my earlier ones) – not sure if I can backdate them or not, but hopefully it will be clear enough, especially once I start adding new posts. I’ve got a few in my head, and just need to get them on virtual paper when I get the chance.