Reflecting on Mathematics

I spent the last two days in class reflecting on mathematics with my students. Here’ the context:

This week is state testing week. I only taught class two days, between testing days and field day Friday. I didn’t want to stress kids out while they were taking tests, but also wanted to be purposeful with my time. It’s also relevant that I teach Algebra I to 8th graders. We have carts of Chromebooks that I used for these lessons — made data collection easy, and I used Google Apps for Education to collect student writing.

My goals were:
Learn more about what my students think of math
Introduce the idea of a growth mindset
Articulate honestly the reasons that math is worthwhile
Provide some opportunities to feel successful with math

Day One:

Math Survey
I started with a survey. Two questions, looked like this:
Screenshot 2015-05-17 at 9.23.51 PM

Here were the results:
Screenshot 2015-05-19 at 7.51.59 PM
Screenshot 2015-05-19 at 7.51.25 PM

I used a Google Form for the survey so it would generate a chart automatically, and cut and pasted the words into Tagul to show students right away what they thought. There was pretty significant variability between classes — I think there was some hive mind going on, although I’m not sure the exact mechanism. Either way, the general trend is clear. My work was cut out for me if I wanted convince them that math is worthwhile. I showed the students the results. I think they really appreciated this honesty, but was also a significant risk to allow the class to see the collective feelings toward math. A few kids made a bit of a joke out of it. We can make excuses about plenty of things here — it was in the middle of testing week, the negative ideas tended to focus on a few words while the positive words were more diffuse — but they are only distracting from the basic idea, that I have not done a great job selling math to my students.

Growth Mindset
I started with an excerpt (first few paragraphs) of this article by Jo Boaler, and did a quick discussion and recap to make sure kids got the general idea. Then, I showed this video from Jo Boaler’s project at Youcubed. It’s a bit hokey in my opinion, but gets the big ideas across. Then I asked what students thought — had them write first to get ideas out, and then discuss with a partner, then shared out. The response was really interesting. The most vocal people were those who didn’t believe in the idea of a growth mindset. Those were the kids snickering during the video, and the loudest in the partner conversations. But when we started to share with the whole class, I was surprised and gratified with how many people said they absolutely believed it. “Some people might not like it but if you work hard you can get good at anything.” And plenty more. It felt really great to hear students say those things, especially coming from students who I least expected it from. There was definitely still plenty of skepticism, but it felt like a really positive moment.

I had students finish class with a few reflections. First I asked them to write about things they had used math for outside of math class and math homework in their lives, and to explain what math it was. This was interesting — some drew a blank, and one student who was kindof fooling around and ignoring the question answered me in a tone that I can’t capture in text — “I don’t do math” — as if it was the dumbest question he had ever heard. That one hurt a bit. Then, I asked students to write about their experiences with school math — their first memories, their positive experiences, their negative experiences, and everything in between. I didn’t do a great job revisiting this second part, but their writing was really interesting. Some of it was positive, but much was negative, in particular in the last few years of their math classes.

We then spent some time sharing and discussing what they used math for. The vast majority had to do with money — getting change, seeing if they could afford something, or deciding which product was a better deal. A bunch of kids mentioned statistics in video games or sports. Some mentioned planning out travel on public transportation, and there were a bunch more that came up once or twice.

As students shared out, I made a T-chart on the board, with N on one side and A on the other. Every tally in every class went under N. At the end of the share-out, as students were getting pretty curious, I told them that I was keeping track of whether you would be able to do that math without algebra. I noted that it seemed like much of the math they had learned from 5th-7th grade they used a great deal, in particular decimals, averages, percents, and proportions. I also acknowledged that it didn’t seem like folks were using the math we learned in Algebra in their regular lives. I then asked what they anticipated using math for in the future. This brought up more advanced math — interest, possible jobs, taxes, and more. At the same time, it was much tougher to generate these ideas, they came with much less enthusiasm, and seemed a bit hollow. At this point, class was wrapping up, and I left it with a cliffhanger — next class, my goal is to convince students that algebra, and higher math, is worth learning. Simple as that.

Day Two:

I started with the simple goal that I wanted to talk about what mathematics is for. I shared the pie chart and word cloud from above (I had only shown each class their own results the first day; this was the data from all of my classes). I also shared the range of ideas for what students had used math for. I acknowledged that a number of students didn’t like math, that while math seemed to be useful for many students, the math they used now wasn’t related to what we were doing in Algebra class. I then told them we were going to start with a game. I told them that the game didn’t have any directions, and it was their job to figure it out. Then I sent them to Game About Squares.

For the next 20 minutes or so. I did very little. Every student was engaged, even those who rarely do what I ask them the first time. If you haven’t played the game, go give it a shot real quick. It doesn’t have directions, and requires you to try something, fail, and try again — but is well scaffolded to a point where most students can work their way through the game for some time before reaching a point of frustration.

This was honestly the most fun I’ve had in my class all year. There was a constant chorus of “oohhhh!” and “I got it!” and “yesss!!” I spent plenty of time helping kids who asked for it. The vast majority of the time, the only things I said were, “what have you tried so far? What else could you try?” This was usually a great hint, and led to some great thinking. In cases where students got more stuck, I did my best to follow my guidelines for hints from my previous post. Kids spent plenty of time helping each other, and while I kept a close eye on it, I really enjoyed how many students gave hints that promoted thinking — and even if students were told how to beat one level, it was because they were tired of struggling on it, and wanted to try the next one, and do more thinking. Overall, fine with me.

After about 20-25 minutes of playing, I pulled the class together. These were my questions — they wrote their answers first, then shared in partners, then full class.

  • Is this math?
  • What parts are similar to math? What parts are different?
  • What did you like best about the game?

The discussion was great. There was plenty of disagreement about whether it was math, including one student who didn’t think it was, except that it was in math class and I try to trick them sometimes, so it has to be math. Lots of kids pointed out that it was logic and reasoning, which are part of math. Almost everyone loved the game, and it was hard to tear many of them away from it to keep them engaged in the discussion.

What is math for?
This is where I went into teacher mode. I took pieces from my elevator speech, and expanded on it, trying to be really honest with my students. I borrowed liberally from the ideas in Underwood’s What Is Mathematics For as well. I started by acknowledging the different reasons teachers use to justify math. We say it’s for future math, or for a test next week, which is true, but shouldn’t be enough. We say it’s all around us — but you don’t have to understand math to use the computer it powers. We say it’s elegant or beautiful — but many students don’t see it. We say you’ll need it for a job — but if we’re honest, if we only taught to prepare students for jobs, we wouldn’t teach most of what we do, and would instead be teaching shop, computer science, and accounting.

I then went in on the “real world”. I took these three problems from What Is Mathematics For to make my point:

An investment club decided to buy $9000 worth of stock with each member paying an equal share. But two members left the club, and the remaining members had to pay $50 more apiece. How many members are in the club?

Yea right. I’d just ask someone in the club how many members there are. That’s a “real world” answer.

You are a facilities manager for a small town. The town contains approximately 400 miles of road that must be plowed following a significant snowfall. How many plows must be used in order to complete the job in one day if the plows can travel at approximately 7 miles per hour when engaged?

Realistically, there are countless other factors involved. And there are already plows in town. Last winter, were there enough? If not, buy a few more. Problem solved.

Howmuch ice creammix and vanilla flavor will it take to make 1000 gallons of vanilla ice cream at 90% overrun with the vanilla flavor usage rate at 1 oz. per 10 gallon mix? (90% overrun means that enough air is put into the frozen mix to increase its volume by 90%.)

This is my favorite. I told my students I hope they get the chance to work at an ice cream shop over the summer sometime because the perks are awesome. And while this may be a problem you need to solve to make ice cream, my bet is that one person solved it, one time, then told everyone else the answer to save the complication of new hires doing it wrong.

So at this point I had been pretty hard on math. A few students defended math, and I validated them — I’m not saying math isn’t useful, just that we oversell that aspect a bit, and I think there’s more to it than that. I had plenty of students totally on board, but I needed to turn that energy into a belief in what we do in my class every day. Here’s where I launched into my spiel.

Math helps us live our everyday lives. Much of that math is behind you, but practicing and furthering your knowledge can only enrich your life. Math will open doors for you, and keep options open that you may not now know you will need. Math will likely be an element of your job, and even if you could be taught that math at the time, mastering it now will only help, and improve your qualifications. All of that is well and good, but it’s not the reason I think math is worth learning. Watching everyone play the game about squares, and the joy in figuring things out, and the trying, and failing, and trying something else, and slowing down and thinking carefully — that is mathematics. Reasoning, considering all of the aspects of a problem, representing ideas in multiple ways, making connections and just the process of figuring things out are valuable no matter where you go in your life — and math is the best way we have of practicing that. I notice that the times when you seem to enjoy my class the most seem not to be when it is most connected to the real world, or most relevant to their lives. Instead, it seems to be when you are figuring something out that is at that just-right level of difficulty — not too hard, and not too easy. That, to me, is the heart of what it means to do and learn mathematics. That is what I believe is growing your brain and making you a smarter and more capable person, if you are willing to be open to it. That is why I teach math.


I stopped here, and asked students to reflect on two more questions:

  • Is math learning? Why or why not?
  • Are you looking forward to continuing learning math through your high school experience? Why or why not?

The answers were interesting. Many were positive on whether math is worth learning, and I was happy about having moved the needle a bit. I clearly hadn’t converted a significant bunch, and regretted not doing a pre-survey for comparison. Much more interesting, many who thought math was worth learning were still not interested in learning math in high school — a great reminder of the gulf between what I want my math classroom to be like, and how it actually operates on a day-to-day basis.

I really, really enjoyed this. It felt awesome to be so honest with my students, and to put aside a lot of the front I put on on a regular basis to cajole and coerce them through lesson after lesson. It felt awesome to be really honest about why I love math, and why I think it is worth loving. It felt awesome to really authentically hear what my students have to say about math. And watching my students get so totally and completely engaged in Game About Squares was pretty awesome too.

My Challenge:
If you’ve made it to the bottom of this post, I have a challenge for you. Put aside some time for something like this. Could be just a few minutes to give the survey. Could be to talk more about growth mindset. Could be to just play games. It felt like a huge risk for me, and was a bit terrifying, but also incredibly gratifying, and it felt therapeutic in some ways. Especially at this time of year, I can’t think of anything else our time would be better spent on, and I am optimistic that, while I doubt I changed the lives of all of my students, I made an impact on a few over these two class periods.

4 thoughts on “Reflecting on Mathematics

  1. Claire

    Thanks for sharing this conversation! I took Jo Boaler’s course two summers ago and as valuable as it was to changing my perspective, I’ve struggled with conveying growth mindset to my students. I’m curious, now that you’ve had this conversation with your students this late in the school year, if you will start the year next year with this conversation?

    Now that it’s the end of the year, I’m trying to gather all my thoughts about what I want to change next year and how I want to do that. Thanks, this post is helpful.

  2. dkane47 Post author

    Good question! I definitely want to start the year with conversations along these lines. I think it’s a really great touchstone to come back to over the course of the school year. That said, I’m a bit skeptical of the impact of a one-off lesson like this. I think the message needs to be embedded on a daily basis to really make an impact on students. I want to do some more thinking about how to make that happen.

    1. Jen spencer

      Exactly. One offs are incredibly challenging, unless you can use them as anchors. Have you thought about spacing it out? Carefully throwing a whammy in every once in a while to catch them off guard? Like asking 5times a year, what is math good for? And posting their answers over the year to have a (potential) view of their growth (or not). I have been thinking a lot about progressions, planting seeds, etc….

  3. dkane47 Post author

    I have. I’m not sure what the structures are — this was the motivation behind my recent post on my “elevator speech” for why math is worth learning. That is one piece — being able to articulate it clearly and succinctly when it comes up. I love your idea for comparing answers over the course of a year — I want to figure out a stronger set of questions that would be useful to show change over time. Not sure what that would look like.


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