Grant Wiggins (who is not, as far as I know, a math teacher, but someone who likes to use math as an example of things) challenged us to make a distinction between what kids understand and what they know. I think that’s a useful distinction, and a hard one, and so I was intrigued by his set of questions that he hypothesized would show whether students had procedural knowledge or conceptual understanding (question: are there such things as conceptual knowledge and procedural understanding?). Reading the questions I realized that they are certainly evocative (if they don’t cause one to shut down and say “dunno, geez, leave me alone!”) and also that if I were to use them to assess students’ conceptual understanding, I would need to be primed to listen to a lot of ideas and make sense of a lot of different ways of thinking and understanding to truly get what students get. Coming up with answers, and even with a short list of what I’d want to attend to in kids’ responses was challenging. So: crowdsource! How would you answer these questions? What would you be attending to in listening to and assessing others’ understanding based on their answers? Go!

1) “You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)
First of all, what would it mean? Is anything divided by zero equal to infinity? That’s what I thought as a kid, but I’ve since encountered experiences that suggest that 3/0 = ∞ is thinking of infinity as a number and thus not really mathematically sensible. But I got to that idea by thinking of how I understood division: How many times does 0 go into 3? Or, what times 0 gives 3? But even infinity isn’t a sensible answer to those questions. The limit of the sum of a bunch of zeros as the size of the bunch approaches infinity is not 3. Infinity zeroes is not 3 or any whole number. So those are intuitive reasons why the answer isn’t zero, it’s not even defined.

Mathematically, though, the best reason I’ve heard for why 3/0 is undefined is that if it were defined, you could make any number equal any other number! For example, if 3x = 5x, does that imply 3 = 5. Nope: you have 2 possible moves you can make solving 3x = 5x. Either use the additive property of equality to write 0 = 2x, and then the division property of equality to write x = 0. 3x = 5x -> 0 = x. Or you can try to apply the division property of equality immediately: 3x = 5x -> 3x/x = 5x/x -> 3 = 5. You have two choices for not breaking math. Either accept that 3 = 5, or say that the division property of equality only holds when you’re not dividing by 0, and since 3x = 5x is only true when x = 0, 3x = 5x only implies 3 = 5 if the division property of equality allows dividing by 0.

That’s kind of related to the fact that if 3/0 = ∞ (or even if 3/0 = some special made-up number §) then that means 3 = 0 * ∞ or 3 = 0 * §… but then doesn’t 5 = 0 * ∞ also since 5/0  = ∞? And if 3 = 0 * ∞ = 5, then 3 = 5 again. Unless you want to invent this whole shadow number system so 3/0 = §₃ and π/0 = §_π… but then what the heck kind of arithmetic would you do with those numbers? How would you visualize them? Why would you do this? On the other hand, please see question 10.

2) “Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.

Well I know an easy example of this. Say I want to find the value of x that makes this statement true: 2x + 4π + 12 = 7(x + 2π) – 5x + 10π + x

I could guess values of x and see when I got a true statement. I could plot a graph of each side and look for the point of intersection. But easiest would be if I could simplify the above statement without changing which value of x made it true.

So… if 2x + 4π + 12 = 7(x + 2π) – 5x – 10π + x
That’s equivalent to 2(x + 2π) + 12 = 7(x + 2π) – 5(x + 2π) + x, by the distributive property of equality
Which is equivalent to 2(x + 2π) + 12 = 2(x + 2π) + x, by association and the distributive property and arithmetic
Which makes it easy to quickly guess and confirm that x had better equal 12.

Of course, I made up the example so I knew just which simplifications would get me quickly to a form of the equation that was easy to solve for x. There were many points at which I could have made other decisions and written the expression in other, equivalent ways, such as by expanding any terms with parentheses, combining like terms, adding and subtracting the same thing from both sides, etc.

As for what the = sign means, it means that the expressions on either side of it are equivalent aka have the same value. Because of some properties of equivalence relationships in our math, it’s “legal” to do certain kinds of moves to equal statements and you know you haven’t changed the truth of that equals sign.

I wonder what examples people might have come up with that didn’t have to do with the = sign? Or that had to do with the = sign but not solving for x?

3) You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.

Here is one reason it works. a/b ÷ c/d asks, “how many c/d’s are in a/b?” Because of what fractions mean, we can think of c/d as c 1/d’s. In other words, 3/5 is three one-fifths, 10/7 is ten copies of 1/7, etc. So… if we think of a/b ÷ c/d as asking “How many c/d’s are in a/b?” an easy way to calculate that is to ask first about how many 1/d’s are in a/b.”

Example: How many 3/5 are in 10/7? First lets find out how many fifths are in 10/7. There are 5 fifths in every unit, so there are 10/7 of 5 fifths in 10/7 of a unit. How many 11/20 are in 3/2? There are twenty 1/20s in 1, so there are 20 + 10 1/20s in 1 1/2. There are 20 * 3/2 or 30 twentieths in 3/2.

Now how do we use the fact that we know how many fifths are in 10/7 to find how many 3/5 are in 10/7? How do we use the fact we know how many 20ths are in 3/2 to know how many 11/20s are in 3/2? How do we use knowing how many 1/ds are in a/b to know how many c/ds are in a/b?

Well, if we know there are 50/7 fifths in 10/7, and we want to find how many sets of 3 of those fifths are in 10/7, we just break the 50/7 into groups of 3. Aka divide 50/7 by 3. The answer is 50/21. We got that by doing 10/7 multiplied by 5 and divided by 3, which is the same as 10/7 * 5/3.

Same with the 3/2 ÷ 11/20 example.

3/2 = One whole and a half.

There are 20 twentieths in the whole, and 10 twentieths in the half, for 30 twentieths in all.

How many sets of 11 twentieths are in 30 twentieths?

30/11 of course!

And finally: There are a/b * d 1/ds in a/b. There are c groups of 1/d in c/d. So there are (a/b * d)/c c/ds in a/b, aka a/b ÷ c/d = (a/b * d)/c = a/b * d/c.

4) Place these numbers in order of largest to smallest: .00156, 1/60, .0015, .001, .002

At least I don’t have to write an essay for this one… First of all, 1/60 = 100/6000 = (100/6)/1000 = (16 2/3)/1000 = 16.666666…/1000 = 1.66666…/100 = .1666…/10 = .01666…./1 = 0.016666…

So, in order, we have .002 > .00166666… = 1/60 > .00156 > .00150 > .0010

I think you were trying to trick me (and it did make bells go off in my head) with what to do when there are no digits after the first ones you compare. Like is .001 bigger than or smaller than .0015, since .0015 is fifteen ten-thousandths and .001 is one thousandth. And since Alum comes before Alumna when you are alphabetizing, since a blank after a letter is prior to any other letter after a letter.

But I wasn’t fooled! .0015 is five ten-thousandths bigger than .001 which is ten ten-thousandths!

5) “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

Gracious! It’s a good thing you included the “just” because it’s certainly false that multiplication is just-as-in-only repeated addition. Though repeated addition can be used to calculate and conceptualize many kinds of multiplication problems. Here are two examples where repeated addition doesn’t make sense.

-1 * -1 = 1.

Does adding -1 to itself -1 times make any sense? And if it does, would it really make sense that the result would be 1? Nah…

sqrt(2) * sqrt(2) = 2

What does it mean to add something to itself a irrational number of times? If you really get deep with it, you could define this using the distributive property as a limit: sqrt(2) * 1 + sqrt(2) * 4/10 + sqrt(2) * 1/100 + sqrt(2) * 4/1000 + … which (assuming you’re comfortable with adding something to itself part of a time) approaches 2. But then you have to be comfortable with defining sqrt(2) * sqrt(2) as a limit that approaches 2, and not just plain old 2, which seems kind of sad. Even though these days the limit and the thing itself are thought to be the same, such as 0.9999999999… = 1, I am still more comfortable with having a concept of multiplication such that sqrt(2) * sqrt(2) just is 2, no limits, based on, say, an area model.

6) A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?

Piece of cake. You think I’m going to be dumb about remainders or rounding or give an exact answer, but I know this trick!

243/8 = 30 tables with 3 people having no seats.

The school could rent 31 tables to be on the safe side, or they could take a risk and just rent 30 tables figuring that if 243 people say they’re coming it would be crazy to expect exactly all 243 to show up. All it would take is one family feeling under the weather and you’d have a whole unused table.

7) Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.

Umm, is this a math question or an assessment question? One thing that the mean hides is whether you know some things not some different things, or whether you kind of know a lot of things. For example, what if your grade was based on 2 assessments, one on each of 2 major topics? What if you were taking a class on how to be nice and clean and you had a test on cleanliness and a test on niceness. Say that you are a really sweet, stinky person and you learned a lot about niceness and nothing about cleanliness, and so you got a 100% on your niceness test and a 0% on your cleanliness test. Your classmate is an okay person and kinda messy about personal hygiene, and got a 50% on both tests. You both have the same final grade (50%) in the class even though you have really different profiles as nice, clean people, and really different needs for remediation.

Another thing the mean hides is change over time. Let’s say you’re taking a class on learning to ride a bike, and when you start the class you can’t stay on the bike for even 1 meter. By the time you’re done, you can stay on the bike without putting a foot down for balance for 1000 meters. And say you developed that skill all of a sudden late in the class, so your distance assessments looked something like:
1, 1, 1, 1, 2, 3, 5, 5, 3, 4, 2, 3, 1, 3, 5, 2, 5, 5, 200, 500, 800, 800, 1000, 800, 1000. Your mean would be about 206 meters. But what about your friend who started out the class as a fine biker but developed vertigo towards the end of the course: 1000, 1000, 1000, 1000, 1000, 0, 0, 0, 0, 0, 0, 0. Her mean distance is about 416 meters. Does that mean she’s a better biker than you are at the end of the course?

8) Construct a mathematical equation that describes the mathematical relationship between feet and yards. HINT: all you need as parts of the equation are F, Y, =, and 3.

The length of 3 feet is as long as the length of 1 yard. That does not mean 3F = Y though because F usually refers to the number of feet and Y usually refers to the number of yards, not the length of feet and yards. Since the length of 3 feet is as long as the length of 1 yard, the number of feet in a given distance is three times the number of yards in a given distance and therefore F = 3Y. If we defined F as the length of a foot and Y as the length of a yard, it might make sense to say 3F = Y but that would not help us know how many yards long a distance we had measured as a number of feet was.

9) As you know, PEMDAS is shorthand for the order of operations for evaluating complex expressions (Parentheses, then Exponents, etc.). The order of operations is a convention. X(A + B) = XA + XB is the distributive property. It is a law. What is the difference between a convention and a law, then? Give another example of each.

Shoot… I’m surprised that X(A + B) = XA + XB is a law. I thought it was a property of our arithmetic system. Like maybe an axiom? Or a property of a ring or a field or some such. Aren’t there arithmetics without distributive properties? I think, though, that I could talk some about what is different between PEMDAS and the distributive property (although it would have been helpful to have said “the distributive property of multiplication over addition in the real numbers” or something more official like that).

So… what’s the difference between PEMDAS and the distributive property (aka the d.p.o.m.o.a.i.t.r.n-or-something?)

Well, I think it’s that changing PEMDAS wouldn’t fundamentally change the way numbers relate to each other, how we solve problems, how we do math… just how we write it. Changing the distributive property would make it really hard to solve equation, it would change what was equivalent to what, it would make it really hard to know how to think about arithmetic with integers or fractions, etc.

Back in the day, there wasn’t really PEMDAS because we didn’t use symbols much. If someone wanted to write 3x + 7 = 10, he (usually, though sometimes she) would write something like “A quantity is tripled and seven is added to the result. The sum is ten.” I think there are good reasons that mathematicians chose to invent a symbol system where multiplication and division are done before addition and subtraction, exponentiation is done before multiplication and division, precisely because of the distributive property, and I have a hard time believing that there are clearer, more compact ways to express “A quantity is tripled and seven is added to the result. The sum is ten.” than using our PEMDAS, but I imagine that just like there are many languages and ways to express the same thought, there can be many mathematical ways to express the same fundamental relationship. And there are different ways that we express calculations. There’s a computer notation called “Reverse Polish Notation” in which you press 3,5,+ to get the computer to add three and five. If you want to do 3 – 4 + 5, you press 3 4 – 5 +. It makes it so you don’t really need parentheses. The math is exactly the same though, it’s just how you tell the computer what to do that’s different.

Now what about the distributive property? Does it have to do with PEMDAS? Or is it independent of that? Can you use the distributive property in Reverse Polish Notation or does it require a system with parentheses? Answer: Yes, the distributive property is independent of how you write it, and has to do with two things being equivalent. The distributive property of multiplication over addition tells you that X(A + B) is equivalent to XA + XB. It doesn’t matter if you write it X(A + B) = XA + XB or A,B+X* = X,A*X,B*+. (A way better explanation of this is here: http://mathforum.org/kb/message.jspa?messageID=4538619)

One reason arithmetic would be different without the distributive property is that the distributive property lets you know that -1 * -1 has to be equal to 1. Because we know -1 * 0 has to be equal to -1 * (anything that adds up to zero if you do the adding first). Whatever notation you use to tell you to do the adding first, just make it clear to people. So without using PEMDAS I could write -1 * 0 is equivalent to adding 1 and -1 and then multiplying that sum by -1, since -1 + 1 = 0. Using PEMDAS I could write -1 * 0 = -1 * (1 + -1). Now, without the distributive property, I’d be done. I wouldn’t know if -1 * 1 + -1 * -1 was equivalent to -1 (1 + -1) and so I couldn’t simplify or further calculate anything about -1 (1 + -1) other than to go back and forth between -1 (1 + -1) and -1 * 0. But with the distributive property I can keep going. If I use words and not PEMDAS I can write -1 * 0 is equivalent to adding 1 and -1 and then multiplying that sum by -1, which is equivalent, by the distributive property, to writing doing -1 * 1 and -1 * -1 and adding the results. So I know now that -1 * -1 + -1 * 1 = 1 * 0 = 0. And I know that -1 * 1 = -1 because anything times 1 is itself. So -1 * -1 + -1 = 0. The only thing I can add to -1 to get 0 is 1, so -1 * -1 must equal 1. But without the distributive property, how would I have know that!?! Without the distributive property, we can’t prove a lot of the arithmetic we take for granted (like combining like terms, using place value-based algorithms, etc.)

10) Why were imaginary numbers invented? [EXTRA CREDIT for 12th graders: Why was the calculus invented?]

I actually know the history on this one. I’m skipping the extra credit. My understanding is that a LOOOONG time ago Italian guys wanted to come up with rules for finding intersections of cubic functions and lines. Every line intersects every cubic function, but sometimes their formula that usually worked spit out values for intersections that involved adding up expressions involving roots of negative numbers. “Yuck, our formula is broken,” many of them thought, but others thought, “Well, but what if there *were* a reasonable way to do arithmetic on these puppies?” They knew what their answers should come out to by carefully graphing and finding the intersections, and they saw ways to manipulate their answers using the usual rules of multiplication with just a light suspension of disbelief, and get to the known intersection. But… this line of reasoning was pretty well abandoned because even though it mostly didn’t break math (the distributive property, commutative and associative properties, and all the usual results of multiplication on the reals still held), having two positive numbers which, when multiplied, made a negative result, did seem to break math. And breaking math is generally frowned upon by mathematicians and math students alike.

So… why do we still use them? For some reason, someone wanted to think more about these roots, and had a reason to think of them on a Cartesian plane. They were able to show that these numbers could be thought of as vectors on a plane, and the algebraic operations on these numbers not only made sense visually on the plane but also made a hard problem (finding the results of rotating and stretching a vector) easier. So then people gradually came to accept that it was okay to break the idea of “a positive times a positive is a positive” because of thinking of multiplication in a new way, since this new way didn’t break any other ideas about multiplication, had a great visual analogy, and made life easier in an important new field of vectors.

11) What’s the difference between an “accurate” answer and “an appropriately precise” answer? (HINT: when is the answer on your calculator inappropriate?)

If my teacher gives a daily subjective participation grade out of 3 and a daily subjective homework completion grade out of 5 and I had some quizzes out of 10 and 20 points and some tests out of 100 points and my final point total is 538/635 then it is inappropriately precise to say that my grade is 89.449% and to quibble with me about whether that rounds to an A- or a B+. If we only went to the 1s place in our point calculations, then any decimal after the 1s place is meaningless, and maybe even after the 10s place. Best to round to the nearest 10 and say I got an A-.

12) “In geometry, we begin with undefined terms.” Here’s what’s odd, though: every Geometry textbook always draw points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually. So: define “undefined term” and explain why it doesn’t mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos will ensue if there are no definitions or familiar images for the basic elements.

I have no idea! I have hung out in Geometries where “planes” were not flat, they were hyperbolic or lived on spheres. I’ve met Geometries where circles and lines were equivalent, and called clines, and Geometries that included a “point at infinity” among other points. Geometry was still meaningful, that’s for sure, and the points, lines, and planes weren’t like good old Euclidean ones.

What I don’t know is why these different kinds of points, lines, and planes are “undefined”? How do we differentiate between this kind of plane and that kind of plane if they are undefined? Is the undefined part the part that makes 2 very different planes still planar in some sense?

Maybe points, lines, and planes (and other undefined terms) are like pornography, and we don’t have to define them because we know them when we see them? http://en.wikipedia.org/wiki/I_know_it_when_I_see_it

13) “In geometry we assume many axioms.” What’s the difference between valid and goofy axioms – in other words, what gives us the right to assume the axioms we do in Euclidean geometry?

I don’t know. I think that’s an open question. For a long time, didn’t we think that it would be goofy to imagine a geometry without Euclid’s 5th postulate? And then it turns out it wasn’t? Didn’t most smart people agree it was goofy to violate the “positive times a positive equals a positive” until they finally agreed that complex numbers were useful and meaningful? Isn’t negotiating whether an axiom will turn out to be goofy and valid part of what makes mathematics a living, breathing, fuzzy, human, creative domain? And what makes math so hard? You kind of are walking a line between madness and sanity when you make rigorous truly new mathematics. How do we know there will never be a mathematics in which dividing by zero is useful and meaningful? A geometry which angles can be trisected using only valid construction tools? I’m not sure we do!

Update:
David Radcliffe’s answers: https://docs.google.com/document/d/1FK8m_oVaS_UWS4grETMAfAhByN_vrQC0TR85ld5FXj8/edit
Erik Johnson’s answers: http://step1trysomething.wordpress.com/2014/04/25/answering-the-conceptual-questions/

Links and agenda for a workshop at Colonial School District, March 18 2014

Welcome:

• What do you know about the Math Forum?
• What’s a goal you have around problem-solving or communication?
• Anything else you’re looking/hoping for?

Three of my favorite open Math Forum resources:

• Ask Dr. Math: http://mathforum.org/dr.math
• Math Tools: http://mathforum.org/mathtools
• Teacher Exchange: http://mathforum.org/te

The Math Forum Problems of the Week (membership provided by Colonial School District)

• Finding good problems: http://mathforum.org/pows/
  • Find the “Problems of the Week Library” tab
  • Find the “Write Math: PoWs by Standard” tab
• Using the online submissions to collect student work and support student communication:
  • http://mathforum.org/pows/
  • Find “My PoW Work”
  • Find “Manage My Purchase”
  • Create Class
  • Create Students
  • Now what’s different about My PoW Work?
  • Now what’s different about the PoWs you view?
• Tracking and responding to student work
  • Each PoW has a “My Students’ Work” link
  • My PoW Work as a Teacher also has ways to view students’ work
  • I Notice, I Wonder responses
  • Rubrics

If you hang out with educators on a regular basis, then surely you have heard someone say a variation on, “My kids come to me not knowing stuff, and that makes it hard for me to teach them enough new stuff.” If you hang out with helpers-of-educators regularly, then you hear these helpful helpers saying things like, “Tough, you have to teach the kids you have.” And “Don’t remediate, accelerate.” and “You can’t go back and teach everything; if they’re behind it’s their responsibility to get extra help,” and “You have to differentiate and make your lesson work for everyone,” and a whole other mess of conflicting things. Because (duh) this is a hard problem. Personally, when I have a hard problem I like to look at a specific case, and guess what! One happened to me the other day. So let’s play “What would you do?”

I was guest-teaching a class that had been studying exponential growth (doubling and tripling, mostly), writing recursive and explicit rules, making tables, and graphing. Today, we were introducing the concept of compound interest as an exponential scenario, and the teacher’s goal was to have the kids recognize just that — compound interest can be modeled with exponential functions (and is cool and important, because free money!).

I was expecting that the hard bits would be the vocabulary of compound interest, the concept of investing, and the concept of the bank giving you more money each month because you have more money each month. And all of those things were hard for some subset of kids, but what was really hard for the majority of kids was doing mathematical operations with the quantity 8%.

Here’s the lesson plan:

• Pair kids and ask them to discuss, “You won the lottery and there are two prize options: would you rather have $10,000 now or $20,000 in ten years? And what would you do with the money?”
• Individuals answer the above question by making a graph of the money they’d have over the next 10 years based on their choice.
• Share a few different graphs on the document cam and discuss the story they tell.
• Pretend to get a call from a banker offering to hold the $10,000 in a CD earning 8% interest, compounded annually.
• Ask kids “What would you do now? Take $10,000 now, $20,000 in ten years, or invest the $10,000 in the 8% interest rate account for 10 years?”
• Define the question: will earning 8% interest on $10,000 over 10 years earn you more money than just waiting and letting the lottery folks give you $20,000 in ten years.
• Turn students loose to use a table (strongly encouraged), graph, or rule to help them answer that question, using their knowledge of percents.
• Come together as a class to compare & discuss tables, and convert tables to graphs and tables and graphs to rules to help us understand what’s going on in compound interest situations.
• Practice on another compound interest situation.

So it turned out that the students right away thought of investing. On their warm-up graphs, some saw investment as a linear scenario (I’ll get $1000 a year from investing in the stock market) and others saw their growth rate increasing as their money grew (exponential graphs).There was engaged discussion about spending vs. saving vs. investing, and more than one student knew what compounded annually meant when the banker called.

Here was the part I wasn’t quite prepared for. When 8% was mentioned, I right away heard fear about the use of percents, and students asking tentatively, “doesn’t that mean we have to divide something by 8 or zero-point-eight?” Maybe half the students knew that 8% was the decimal 0.08 (most went right away to 0.8, which sounds a lot like point-zero-eight but is a lot different!), and only one could state clearly that 8 percent was 8/100 which was 0.08, since 0.8 was 8/10 or 80/100. And worse, even once we’d established that 8% could be typed on a calculator as 0.08, more students wanted to divide than multiply (I guess because they wanted a smaller number)?

So the conversation about making the table wasn’t around what I was hoping — identifying those students who’d found 8% of the principal but forgot to add it to the principal and so who got 10,000, 800, 64, for the balance, and identifying those students who’d found that she made $800 in interest the first year and so reasoned she’d get $10,000; $10,800; $11,600; $12,400; etc. Then we could have talked about linear vs. exponential growth and what that has to do with compounding, and we could have talked about 8% vs. 108%, and gotten into some key ideas that would let us make sense of an exponential rule.

Instead we got bogged down in 8% vs. 80%, and number that didn’t make any sense, like $10,000 / 8 or .8 or .08. Even though we talked through those issues, I lost a lot of kids in talking through those issues, and we didn’t have very good conversations about the other, more connected-to-the-concept-of-exponential-growth conversations.

So, what would you have done, with 20/20 hindsight? Here are my questions:

1. Can a class of students, the majority of whom don’t understand how to calculate with percents, learn compound interest? Should they be asked to? If so, how? If not, what should we do instead?
2. Would there have been a way to differentiate this lesson so that more students got through the right amount of it at their own pace? What could I have expected everyone to have gotten? What could I have done to support the most struggling? The most advanced?
3. Could technology have helped students focus on the concepts, rather than calculations? How? What technology?
4. Is there a way to present this concept more conceptually, so we avoided getting bogged down in the calculations? Were the students conceptually ready but lacked fluency, or were there underlying conceptual issues I needed to address?
5. How long does it take a good understanding of percents to sink in? Did I need to stop and teach a conceptual foundation for percents and then come back to compound interest? What would that do to my pacing guide (already pretty far behind with snow and 4 rounds of state testing instead of the 1-2 we used to do)?
6. And finally, does thinking about this particular story give us any more insight into the general question of teaching kids who come in with gaps in their understanding, fluency, or both?

Talking about race, class, gender, sexual orientation – stuff like that – in relation to math class feels like tricky territory. Race, class, gender, sexual orientation, language, physical ability, religion — they shouldn’t matter for student achievement. Does focusing on social identities in thinking about math class help or hurt students who are outside of the mainstream?

I think it’s really important to be aware of and responsive to students’ outside-of-math-class cultures and experiences. Here’s why. I want all my students to work as mathematicians in my class. I believe that all students have a playground in their brain for shapes and quantities, that all students can wonder, observe, conjecture, explore, refine, reason, explain… And I also believe that all my students are pretty novice about these things. I have to be able to listen to them and hear them as mathematical thinkers, even when their expressions are very, very novice — and therefore different from what I’m used to hearing.

Students’ outside-of-math-class experiences change the “accent” in which they speak math. Not necessarily their actual accent (though some of our students have accents or speak math in a different language than we do!) but more the way they think and express their thinking.

We don’t realize we all have accents. We think of people with accents that are strongly different than our own as being “weird” “different” “cool” “special” — and rarely think of ourselves as being people with accents (unless we have an accent that’s different from the people around us so it gets pointed out to us a lot).

I have a hard time recognizing my students doing math and being mathematicians when I am distracted by their “accent,” and I’m more distracted by it the more different it is from how I conjecture about, reason about, and explain quantities and shapes. In fact, I don’t always realize I have an accent — I believe I’m fluent in math and speak the universal language of math. Which makes what my students are saying and thinking sound “weird” “special” and even wrong.

Here are some concrete examples that I’ve read about or experienced:

• Dealing with uncertainty and the issue of right/wrong — different cultures, and in the US, different economic classes, have different ways of thinking about uncertainty, especially when it comes to uncertainty in school. In some cultures (for example, the latte-sipping liberal elite!) school is a place to learn to argue, to poke holes in accepted ideas, to get troubled and cause trouble. In other cultures school is a place to learn the right way to behave, talk, etc. School is your ticket into proper society… so encountering uncertainty, challenging authority, poking holes, etc. is bad news. This doesn’t mean that students whose cultures relate differently to authority and knowing should be told what to do and made to just memorize procedures… but it does mean that they’ll engage differently with the challenge of checking on their own whether a solution is correct, that they will need explicit coaching & parameters around times that they are debating the relative merits of different arguments when there is a teacher in the room who could be the voice of correctness but is choosing not to be.
• Written vs. oral communication — like everything, there is as much variation among individuals of the same “culture” as there is between cultures on this… but… that said: the role of writing vs. talking is different in different cultures. In some cultures the word on the page has a different kind of authority than the spoken word. In some cultures reading and writing is more of a communal, oral exercise where people talk about what they’re reading or writing as they read and write it. In some cultures girls are brought up to be more reserved, to not blurt — and so they may find it easier to use writing to marshall their thoughts before speaking. In some cultures, you talk things out before you put anything down on paper. So a “think-pair-share” task where students write silently and then share orally may go awesome in one classroom and bomb in another, because the students in the other classroom would have to “talk-think-write” to be successful on the same task.
• Consensus building & arguments — this one I’m more out on a limb about. I don’t know the research and have only read anecdotes but… it seems to me like one thing that happens in math class is there’s a tension between knowing things because you’ve deduced them through rigorous logic, knowing things because you have faith in the person telling them to you, and knowing things because they seem right inductively. Back in the Dolciani days, in theory everything was deduced through rigorous logic, and in practice a lot was taken on faith from the teacher. Student-led investigations these days in theory begin with intuition and inductive reasoning (wow, in all these examples, making a triangle just knowing the side lengths guaranteed that my triangle was exactly the same as yours) and end with rigorous deduction (that will always be true because I can prove there will always be a rigid transformation that maps the three sides onto their corresponding sides and since rigid transformations preserve shape and angle measure, all the corresponding parts will be the same). In practice, I think a fair number of investigations build up students intuition and inductive reasoning, but then the conclusion that this always works is taken on faith again. And it seems to me that there’s a cultural piece at work here. What is the way to win an argument in students’ out-of-math culture? Convincing others? How? By logic? By coercion? By force of personality? By showing lots of examples? By offering counter-arguments? Which is more important — being right or everyone agreeing? If students’ out-of-school experience is that agreeing is important, and that it’s not polite to bring a counter-argument if everyone else agrees, then where does that leave them in math class? How do we honor their experience of argument and also help them participate in doing math, as mathematicians?

It’s so, so important to me to meet mathematicians and math teachers from many cultures, and hear them talk about their experiences with school mathematics, their children’s experiences in school mathematics, and their students’ experiences with school mathematics, to help me hear my own accent.

As a white, American, mostly East Coast, mostly male, mostly middle-class person, I am so used to argumentation and picking things apart and valuing logic and trying to see “why does that work?” and “will that always work?”, and doing that kind of argument everywhere from the dinner table to friends’ houses that when I don’t see those same “accents” in my students I think they aren’t mathematical. I don’t see my students’ use of different kinds of mathematical knowing as they use their spatial intuition (not my strength) and can just “see” why something works, let alone when they decide to agree on a conclusion that doesn’t follow deductively because it preserves the feelings of everyone in the group (even if some of them “know” it doesn’t work). I don’t hear the mathematics in my students’ accents so I don’t know how to support them to be accountable to the math even as they’re being accountable to their group, or accountable to their lived experience of the world.

So… if you have a different perspective about your students’ math, about your own mathematical habits of mind, if you come from a culture where school math is a little different, where arguments happen differently, where authority and rightness are interpreted differently… I want to talk to you at TMC14, and I want to talk to you explicitly about your background and experience and how it’s made a difference in your math classroom!

One of our PoWs this week involved finding three consecutive numbers that added to 63. The four main approaches were:

1. Divide 63 by 3 and use that as the middle number
2. Guess and check, generally starting in the teens or low 20s
3. Write an equation: x + (x + 1) + (x + 2) = 63*
4. Think of 63 and 60 + 3 and find three consecutive digits that add to 3 (o, 1, 2) and then divide the 60 into three 20s, yielding 20+0, 20+1, 20+2. (This was the rarest strategy).

I’m always trying to define for myself what the heck Mathematical Practice #7, Look for and Make Use of Structure, means in actual kids’ thinking. I think strategy #4 is a particularly good example of a middle- or elementary-school version of making use of structure, taking advantage of place value.

I wonder what would have happened if the three consecutive numbers had added to 60 or 61.

I wonder if the fact that the structure they found doesn’t make every type of problem easier invalidates it as a strategy somehow?

Hmm….

Update: an upper-elementary student used a strategy that combined algebraic representation and looking for structure:

Y + (Y+1) + (Y+2) = 63.
Y + (Y+1-1) + (Y+2-2) = 63 – 1 – 2
Y + Y + Y = 60
3Y = 60
Y = 60 ÷ 3 = 20

Lots and lots of things matter a lot in classrooms. Curriculum matters, standards matter, assessments matter, the physical set-up of the school and classroom matter, the school climate and culture matter, the various cultures the students come from matter… But I’m coming more and more to agree with Steve Leinwand that good instruction a) really really matters, and b) is something that teachers can and should have control over.

Perhaps surprisingly, that’s a controversial statement. First of all, there are folks who want to make political statements about teaching and argue that the only way to improve learning in schools is to end poverty or extend the school day or to write harder assessments or to fire more teachers or to pay teachers based on test scores or… And then another kind of controversy is among teachers themselves, as they talk about what their jobs are. Should teachers be writing standards? Writing curriculum? Writing lessons? Writing assessments? And if so, what kind and how many and when?

I’ve been working in two pretty different school districts. Both have high-needs populations and would like to see improvement in their test scores. Both have a lot of initiatives going on, and a lot invested in those initiatives. PLCs, data-driven instruction, computerized formative assessment benchmarks, literacy across the curriculum, students in career-based academies, lesson-planning and curriculum mapping packages, and more.

Each district gives teachers an average of 70 minutes of teacher planning time per day (not counting lunch, because people should get to eat lunch without multi-tasking). There’s an expectation that some of that time will be in structured collaboration, and some of that time will be individual work-time. What the teachers I talk to want to be doing, first and foremost, and don’t ever have time to do, is:

• The basics (before the school year, perhaps): Looking closely at the existing curriculum (i.e. with the the book open in front of them, reading carefully and talking over the details) and figuring out:
  • What is being taught/emphasized
  • How kids are expected to figure it out based on the experiences
  • Why it works that way
  • How to tell if it’s going well
• Implementation details: Working with the curriculum they’ve got to figure out:
  • How to engage their kids in the experiences they’re supposed to be learning from
  • What scaffolding their kids will need to get from the initial experience to the expected outcome
  • Anticipate multiple methods students might come up with for making sense of the experience in the curriculum
  • Anticipate and structure activities that incorporate multiple approaches and multiple representations that kids can/will use to make sense of their experience
  • What will be hardest for the kids and what kids might do/say to show they’re struggling
  • How to meet kids’ struggle so it’s productive
  • Planning concise interventions for when struggle gets unproductive
  • Avoiding sloppy language that gets in the way of precise understanding
  • Good questions to engage students, focus them, redirect them, etc.
  • Key vocabulary that will help students organize their thinking and communicate better with others
• Classroom management and routines: Looking at the curriculum experience and content to decide on & find or invent:
  • Figuring out how to make complex experiences (Barbie Bungee! Centers! Algebra Tiles!) go smoothly
  • The right blend of whole-group, small-group, and individual work for their kids on this experience
  • Classroom routines for engaging kids in the work & the discourse to most efficiently experience and harvest the fruits of the curriculum tasks
• Looking at student work: Differentiating instruction, adjusting whole-group instruction, attending to the social-emotional needs, etc:
  • Reflecting back on student talk, written work, etc. to figure out if the kids learned from the experiences
  • Making decisions about revisiting topics, moving ahead, doing small-group interventions, asking kids to stay after school, etc. based on kids’ work
  • Noticing individual students and how they’re feeling, engaging, thinking, and learning
  • Noticing the group and group norms, behaviors, etc
  • Planning social/emotional interventions as well as mathematical interventions
  • Making notes about what to do differently next year

And sometimes, because teachers are creative and wise and fun:

• Identifying gaps in student understanding, numeracy, etc. and figuring out if and whether to address them, and if so, finding additional resources to do that
• Bringing in an extra activity that they found online, heard about at a conference, dreamed up, etc. because it seems like it would swap in or supplement nicely something in the curriculum
• Making time for students to pursue projects or topics that they are interested in — sparking mathematical passions

Here’s what the teachers do instead during the structured collaboration time, which eats up most of their “planning” time (leaving about 0 minutes for making copies, grading, using the restroom, jotting notes on what just happened in class that you want to remember for next period, tomorrow, or next year, posting the homework online, responding to parent calls and emails, attending IEP meetings, making handouts/visuals, organizing manipulatives, setting up the classroom for today’s experience, and all the other tasks teachers need to do as individuals):

• Teachers writing or aligning curriculum maps, scope and sequence, etc. Unless teachers are getting extra money and extra release time and extra professional development and working with experts because they like this sort of thing, it’s the wrong level of granularity. It’s like asking physicians to be medical researchers, too. People who write curriculum are smart enough that teachers shouldn’t have to rewrite it to make it usable (one hopes, and if not, buy a different curriculum) and it’s not that hard to make sure the stuff on the end-of-year test is covered before that test by re-ordering chapters with a sparing hand. Teachers should be focused on implementing curriculum to meet the needs of their specific group of students, which is NOT trivial!
• Teachers writing tests, called formative assessment tests, but which aren’t used to figure out why kids are struggling and what to do about it. If all the time is spent writing traditional tests and quizzes that won’t be recorded in the gradebook, and not talking about why students are making mistakes and what can be done to help them the next day then writing paper-and-pencil formative quizzes is not so useful.
• Teachers meeting to learn about new school climate, routines, rules, initiatives, etc. This stuff is important, but good instruction is probably the main contribution to school climate. If classrooms aren’t places where kids are invited to work hard and supported to succeed, then they won’t want to be there, which impacts climate. Of course, like eggs and chickens, it’s hard to get kids to work hard and succeed if their overall experience of school is chaotic, unsafe, etc. I’d caution administrators to make sure that teachers have LOTS of time to plan & reflect on instruction, and be frugal in the amount of time spent engaging teachers in face-to-face meetings about school climate.
• Teachers going over statistical data about which broad topics students don’t know. Data-driven instruction is only as good as the data. I think most teachers already know what their students don’t know. I think it’s harder for teachers to know what their students do know, and why their students can’t do the things they can’t do. Do students have “buggy algorithms” that they think work? Are they giving up before they begin? Do students know they don’t know but are trying something anyway? An exercise I love to do is ask, “What way of thinking about the math makes this student’s work coherent to them?” I can’t easily do that if all I know is that they chose C instead of B on this problem, and I definitely can’t do that if all I know is that 25% of them missed 80% of the “measurement items”
• Teachers filling out paperwork, like lesson planning templates. Teachers are often good at following instructions and crossing t’s and dotting i’s because they’re told to. It’s how you survive a job in a bureaucracy! Lesson-planning forms are only useful if they’re part of a focus on good instruction, and every nuance of the form is pointed towards encouraging dialogue about good instruction, and grounded in teachers’ concepts of good instruction. I’d lean towards introducing forms the way I’d introduce any procedure: start with the concept of the good instruction, solicit teachers’ current lesson-planning methods, talk about what works well and what we could improve, and generalize to a form that works for our community and our definition of good instruction.
• Teachers meeting in cross-discipline groups to discuss student behavior problems or school-wide, non-instructional or supplemental initiatives. Again, fine stuff, if there’s time for it. But right now, there’s not enough great math instruction happening in so many high-needs schools that I’d prioritize instructional planning and reflection over all this stuff. Squeeze in the conversations about tricky kids, how the 4th grade is doing overall, or cool cross-curricular projects after planning and reflecting on good instruction. Otherwise, planning and reflecting on good instruction (which is kinda the whole point of being a teacher) is what gets pushed to the side.
• Teachers doing work for extra initiatives such as finding readings for the reading across the curriculum initiative. Again, cool stuff, but not as important as good instruction.

(Almost) none of the above is a total waste of time, but none of it is the stuff that research on teacher collaboration actually suggests is useful. Considering how little time teachers get to plan, let alone collaborate, it’s frustrating to me how much of that time is wasted time. It’s also frustrating to me how much time teachers spend writing lessons or searching the web for lessons. I’m certainly very guilty of that — I tend to start with a vague sense of the topic, Google for it, and then try to do what I find on Google. My main focus is on what to do, not how to do it.

What I should be doing is starting with the curriculum, figuring out what kids are supposed to be doing & learning from that doing, and then applying my particular expertise to how to get my kids from A to B. It’s really concrete — should they be reading the questions and writing answers, or is this piece best done orally? Should students work in small groups on the main task and then share out, or should I lead the group through a demo and then have them try on their own? If small groups are right for this task, what will it take to keep them working productively, and how long can I expect each group to stay on task? Will it work if the groups are random? Will I need to make sure Keion doesn’t work with Marvin, or can they get it together today? If a group finishes early what should they do? If a group gets stuck, should I tell them what to do, ask a question, have another group share out, or something else? How many different “kinds of stuck” can I anticipate and how many different responses can I prepare? How long can I wait for a group to catch up before moving on? Will students’ struggles with integers get in the way? Will I let them use calculators? If students use Guess and Check to solve all the problems on this page, what are three things I could say/do to help them look for and find a more efficient approach? The list goes on and on… And I haven’t even scratched the surface of student-work questions like, based on what I asked them in their exit ticket yesterday, who needs more time grappling with yesterday’s concept? How can I help Deshaun get more fluent with division facts so he can make sense of one-step equations? What does this class believe counts as a right answer to an inequality — do they get that it’s a range of numbers that make the inequality true? If not, is there something in my language that could help them access that, or is there some challenge I could engage them in that would bring that point home? How does my curriculum make that idea “pop” and why did they miss it the first time through? Why does Emily consistently evaluate 2x when x = 5 as 25, but can solve 2x = 10?

If teachers don’t have enough time to ask, answer, reflect on, and revise their thoughts about the questions above, then we shouldn’t be filling their time with things other people get paid to do, like writing curriculum, writing fancy-schmancy benchmark tests, looking at data that’s not useful on the individual student level, or discussing which minutes of the day the bathrooms will be open to students.