Asking Non-Math Questions II

In my last post I said that non-math questions were important and interesting to ask because of the discussions that can arise from them. In this post (and the next) I want to share other examples of non-math questions I’m asking in my class, because I’m curious what other non-math questions there are out there being asked so that I can keep developing in this practice!

This activity was adapted from one that the really wonderful Jim Noble created and shared with me at a workshop, based on interesting research done by Ipsos-Mori on the Perils of Perception.

This activity fits within any unit on percentage change, for example, as percentage error uses the same concept. Students work in groups to first estimate the percentage of the population of their country who are Muslim, and then to identify the percentage error of countries’ estimations. Along the way, great discussions pop up. Here is the activity:

Again, the non-math part — the part that connects to geography, history, current events — is for me the key element. For example, when discussing how students decided which numbers should math with France versus Australia, students offered explanations like:

  • A lot of northern African countries (like Morocco) are former French colonies/protectorates, so there are a lot of Muslims who emigrate to France, because they can already speak the language. Woohoo, connections to History!
  • France is close to a lot of Muslim-majority nations so it is logical that more Muslims live in France than in Australia, which is far away. A counter argument to that is that Australia is close to Indonesia, the world’s most populous Muslim-majority nation. That fact alone was really interesting for the students, who don’t realize this (though they should know more about Indonesia as it is a former Dutch colony and famous Indo-Dutch people include Mark-Paul Gosselaar [Zack Morris!] and Geert Wilders [ironic?]). Woohoo, connections to Geography!
  • We hear a lot about Muslims in France, with things like the “burqa ban,” so since that’s in the news a lot, that must mean there are a lot of Muslims in France. Woohoo, connections to Current Events!

The great (sad?) thing is that you can connect this activity to current events no matter when you do it. I did this even earlier in the year, around the time of the Dutch elections, when Geert Wilders was decrying the Islamisation of the Netherlands and saying that he wants to ban the Quran.

I did it with a different class just a few weeks ago right around the time that controversy erupted because the head of the Amsterdam police force was considering reversing the ban on outwardly religious clothing for police agents in an effort to recruit Muslims (specifically hijab-wearing women) for a city that is 52% non-ethnically Dutch (and in which Islam is the second most popular religion; the most common being “none”) but boasts a police force that is only 18% non-ethnically Dutch.

There are so many rich conversations to be had here. You could avoid the Muslim issue altogether and pick a different issue highlighted in the Ispos-Mori survey (I always show the slideshow from Ipsos-Mori to wrap up the lesson — the kids LOVE making predictions and seeing how different countries react to all different things). You could discuss fake news. You could discuss the connection between percentage error and percentage change (so that rather than simply memorizing a formula, there is understanding of how these things work). The important thing is you can discuss.

So, what things are you discussing in math class?


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Asking non-math questions

I was recently asked to give a math lesson to a group of 7th graders, which I don’t normally do, on the topic of rates, which, again, I don’t normally do. But I was really excited by the prospect, because I’d noticed something weird at my local grocery store, namely the price of pine nuts (in Dutch “pijnboom pitten”).

I showed the students the three options for buying pine nuts at the main Dutch supermarket, Albert Heijn, and simultaneously highlighted and obscured important information for them:

Then I asked them the following three questions for them to work out in groups:

  1. Which of the three packages would you advise me to buy. Justify your answer (and buy justify I mean give a valid reason with some calculations to back up that reason).
  2. I believe your advice is good, but I choose to buy a different package anyway. Why might I do that?
  3. Why do you think that Albert Heijn has different pricing for these three packages?

I gave them ten minutes to work on it. Half of the groups had a good sense of where they should start and quickly came to the correct answer. Other groups didn’t have a good sense of where to start, so I asked scaffolding questions like “what makes something a better deal? How can you quickly tell if the medium is a better deal than the small?” and “what important information is given on the price tag? What do you think I covered up?”

You end up with the sort of surprising answer that the medium package is the best deal, even though one assumes (or maybe 7th graders don’t yet have that assumption) that buying in bulk results in a better deal.

So the 3rd, non-math question became the most interesting. The students suggested that Albert Heijn had done market research on the most common amount of pine nuts people want to buy and priced accordingly. They also suggested that the plastic of the medium package, which looked flimsy, cost less than the nice, sturdy looking plastic box that the large package came in. They also suggested that maybe people like the resealable box, so that’s why it’s more expensive.

If I’d had longer, we could’ve delved into this idea more, that the price of something is not just the raw price of the goods itself but the materials used, the convenience of it, etc. I think these sort of discussions are important to have in today’s society and that in math class we should be grappling with the sort of questions of the real cost of goods and services. I could imagine a much larger unit about, among other things, environmental justice. Do you have any other good non-math questions that you’ve asked in math class?


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What is the purpose of math education?

I was reading Dan Meyer’s latest post Teaching for Tricks or Sense Making and felt compelled to post for a few reasons. One, I’ve tackled the topic of how to teach zero and negative exponents before and I’ve just taught this very topic in three classes. But two, it relates to why I haven’t posted in a year.

Last school year I started teaching a new-to-me course called Theory of Knowledge, which is basically an epistemology course that asks how we know the things we claim to know. This course has been very rewarding to teach but has also taken up a lot of time and thought-space for me, which is why I haven’t blogged at all.

The current area of knowledge we’re covering is mathematics and one of the questions that arose was what is the point of learning things beyond basic arithmetic. If the majority of the students will never again solve a quadratic equation, why do we spend so much time on factorization, completing the square, the quadratic formula, forms of a quadratic function, etc. In essence, what’s the point of schoolmath?

A student (bless her) said that we study it because it teaches us to think and to problem solve. And my response to her was, “But does it?” I think that is the idealized claim (and certainly that is the impetus behind the MTBoS’ Nix the Tricks), but I’m not 100% convinced that we’re achieving that. I think that despite our best efforts many students simply follow rules or procedures that they only half understand. I saw that play out recently.

This year in my advanced 8th grade classes, we covered the rules of exponents (in an algebra-only context) and scientific notation. In a discovery-style lesson on scientific notation, in one class the students came up with the idea that if 10 raised to the nth power was a 1 with n zeros behind it, then 10 to the 0th power was just a 1. They also managed to get to the idea that 10 to the -1 should be one-tenth. I felt like in that class that I had taught for sense-making for small numbers in scientific notation, whereas in the other class I had merely taught a trick for it (in the interest of time; they had also had the discovery-style lesson on scientific notation for large numbers).

On the test I put two sense-making questions. The first asked them to find (-1)97  (no calculator) and explain their answer. A majority were able to give a completely correct justification. But I also had a fair amount tell me that the answer was -97, two tell me it was -1 x 10 97, and one tell me it was (-1)100 – (-1)3. What mistakes of thinking are here?

The last question asked the students to do a mini-investigation. The first part asked them to calculate 25, 24, 23, 22, 21, then give the pattern as you go down in exponents, then use the pattern to find 20 and 2-1. The number of students who could actually use their pattern was limited, across both the trick-learning and the sense-making class. From the sense-making class, despite having seen that 100=1, many students still thought that 20=0. Among students who did get that part, lots of them then thought that 2-1=0.2! In one particularly egregious case, a student told me that 2-1=0.2, 20=2, 21=20, 22=200, 23=2000, etc. In other words, there was no sense-making at all! In fact, our class-wide explanation of how the powers of 10 worked had primed some students to not think about how powers of 2 might work, or how any power actually works.

I’m still excited about this question and can’t wait to go over the test with them to talk about how we should deal with pattern finding and its application, how to critically think and get over our biases of what we think the answer should be, and above all to prove to them that yes, shockingly, any number raised to the power of 0 is 1, by asking them to recall the rules for multiplication and division of exponents.

Which brings me back to Theory of Knowledge, because knowing in mathematics starts with a conjecture (I recognize a pattern that it seems that any number to the power of 0 is 1), followed by a rigorous proof using deductive logic to verify that it’s always the case. Using tricks robs students of what my student says the actual purpose of math is — to learn to think logically!


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What makes a good colleague

A new math teacher at my school is finishing up her study, and as part of it she sent the question to the math department “what makes a good colleague?”

I thought a lot about it and this is what I came up with:

I think a good colleague is someone who:

1.       Listens and sympathizes with you when you’re struggling so that you feel you’re not alone in your feelings or problems (since teaching can be a very lonely profession in some ways, as most of the day you’re the only adult in the room). While they critically listen, they don’t simply commiserate with you (because then you wallow in your troubles and things can start to fester). Rather, they offer understanding and friendly suggestions (without making you feel like you don’t measure up) that help improve your teaching or your mood.

2.       Brainstorms with you the best ways to present material, the best questions to ask (and what your goal is with each question you ask), the richest tasks you can set, the best ways to motivate students, etc. They think deeply about their subject material and how to connect it to the real world, other subjects, and students’ lives. They help make you enthusiastic for the subject you teach and excited by the possibilities of what can be done by a certain class.

3.       Is reliable, trustworthy, and always does what they say they will do. They pull their weight and expect you to pull yours, but will step in to lend a helping hand when you’re faltering. They expect the same from you, which means that they trust you in return.

Luckily I think I have many colleagues who meet at least one of my descriptors above.

What’s on your list of attributes of great colleagues?

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Discovery vs Investigation

Yesterday Michael Pershan tweeted the following:

He was pretty critical of this worksheet, which is definitely not as strong as Kate Nowak’s trigonometry investigation (or my own if saying that doesn’t sound too conceited). But what interested me was that he was also critical of the idea that anything that is worksheet-driven or teacher-led could be classified as “discovery learning.” He seems to suggest that it’s only discovery if the student comes up with it completely on their own.

I’ve already looked at discovery learning on this blog once before, and I think I can say that I’m really not for this narrow idea of discovery learning and think that it can actually be detrimental to a student’s mathematical development. I am, though, very much a fan of investigation.

Which brings me to the real reason for this post:

First, this is an investigation I did into Zero & Negative Exponents with MYP3 (US 8th grade age). The students did the first part (worksheet included on third page) under test conditions in class. Upon receiving their graded work back they wrote the essay (fourth page) where they not only used their understanding of a pattern to write a rule but verified that their rule was correct by using the rules of algebra.

A year later, in MYP4 (US 9th grade), I do the following investigation on Rational Exponents with the students, though it is more informal. They work in groups and are not graded on their work for this.

Scribd is doing some weird things with my document…there are no red question marks in the original. So here’s a picture of what it really looks like for me:

Screen Shot 2015-05-05 at 10.25.46 AM

These investigations are absolutely not “discovery learning,” because there are specific steps (suggested by me) that the students follow to ultimately “discover” the rule. But the steps ensure that the students have a focused approach, and it models for them what I think is an essential chain of mathematical inquiry:

  • 1. Use what you already know in simple examples to
  • 2. find a pattern so that you can
  • 3. generalize what’s happening/write a rule and finally
  • 4. verify your rule using algebra
  • Finally, not an investigation per se on exponents, but more of a hat-tip on the many wonderful ideas I steal from this community, here’s something we spent a class discussing and delving into, which I culled from various blogs and twitter feeds:

    Again with the red question marks, bah! (anyone know why it’s doing that?) Here:

    Screen Shot 2015-05-05 at 10.27.45 AM

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    Improving Tests & Eenzaamheid

    Last night I went to my first-ever in-Dutch lecture (spannend!), by top Dutch history teacher Jelmer Evers. One of things he talked about was how eenzaam (literally translated “onesome” or solitary) teaching can be. It felt great to make some connections here in the Netherlands, maybe find a bit of a community.

    I’m grateful to the MTBoS for giving me an online community, but I also realize that we each only share a small snippet of what we do. And what we rarely share (for probably good reason) are our tests. But this has got to be one of the most difficult (and considering the weight placed on assessment, most important) things we do, actually, and it would probably be better if we weren’t so alone in that endeavor.

    I really had no idea what the heck I was doing when I started making tests–I’m embarrassed now to think of how rag-tag and unprofessional they looked, all cut and pasted and mismatched. And as a new teacher (or a veteran) I don’t recall anyone ever sitting with me and saying, “this is how you make a good test,” or asking me why I choose a particular question (or the particular numbers in the question), or suggesting a different question (or way to ask something) instead.

    These days I’m trying to look at my own tests critically — looking for the things I can turn into an open-ended or an open-middle question, for example. And I’m constantly bookmarking problems on Twitter or in other people’s blogs as “GREAT QUESTION ON ___.” I’m also trying to find questions that expose misconceptions, but I don’t know how good I am at that yet.

    But the point is, although I’m trying it, I’m doing it mostly alone, with the occasional assist from my wonderful coworker (who never blogs, for shame). And sometimes another coworker will hand me their test, like FYI, this is what I’m giving to the class, but I rarely say, “Hey, I’m not sure about this question,” or “Have you thought about asking this instead?” or even “Do you want to work on developing test questions together?” It’s sort of difficult to initiate that conversation.

    But I think maybe I’ll propose it to the department, because teaching–and testing–should be less eenzaam, I think. Do any of you do test study and have any tips?

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    Introducing Trigonometry

    When I first started teaching trigonometry, I just introduced it the way the textbook did, sort of matter-of-factly and heavily reliant on SOHCAHTOA. But students were always asking things like, “Ok, but what is sin?” and thinking they could divide by sin to solve for x. Algebra is already pretty abstract, but the whys and hows of trigonometry made about as much sense to the students as the plot of Interstellar.

    For the last few years, I’ve been introducing it differently and I feel like it sort of addresses the too abstractness and trick-oriented way I used to do it. Now I always first begin with an investigation into special right triangles. I do this with MYP4 (US 9th grade):

    In it, students use their knowledge of Pythagoras’ Theorem and simplifying radicals to discover patterns in the relationships between the sides of a 45-45-90 triangle and the more complicated 30-60-90 triangle, write and check algebraic rules, and apply those rules in contexts of varying difficulty.

    I do this as a summative assessment under test conditions, though if a student has gone totally off track (for example, by not correctly simplifying radicals), I sit with them to discuss their error and then give them a second go at it.

    The idea is that students will see that if we know the angles of a special right triangle and we know one side, we automatically know all the sides and vice versa. My hope is that this primes them to see where trigonometry comes from. My follow-up lesson involves another investigation, but this one done informally in groups.

    I ask the students to draw a few lines of varying steepness and investigate the gradient (slope) of the line and the size of the angle of elevation and find a pattern or rule. I’m doing this on Tuesday with my current group, but last year they were really clearly able to say that a gradient of less than 1 gave an angle of less than 45 (and the smaller the gradient, the smaller the angle) and a gradient of more than 1 gave an angle of more than 45 (and the bigger the gradient, the bigger the angle). I then tie it back to their discovery of the sides of a 45-45-90 triangle.

    It is only then that I introduce tangent as the rule relating the angle of elevation to the gradient. I have them use the calculator to find the tangent of the angles of elevation that they measured and check them against the gradients that they had. Only after they feel comfortable with using tangent do I introduce the other ratios, again referring to the findings from the investigation on special right triangles.

    I hope in this way that trigonometry is less of a black box and a procedure that the students do, but don’t know why they’re doing it. How do you introduce trigonometry? How do you mitigate its abstractness?

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