Curmudgeon has taught math and science for thirty years and runs the Math Arguments 180 blog, an indispensable source of interesting prompts and questions.

Here are three images he’s posted in the last month:

In nine classes out of ten, you’ll find a teacher ask her students to calculate the area of those shapes. Maybe Curmudgeon would ask his/her students to calculate their area also. That’s a fine question. But Curmudgeon does an excellent job *developing* the question of calculating area by first asking:

- What is an easy question we could ask about the shape? A medium difficulty question? A hard question?
- What is the best way to find the area of the shape?
- What combinations of addition or subtraction of figures could you use to find the area?

Each question *develops* the next question. Earlier questions are informal and amorphous. Later questions are formal and well-defined. They all *develop* the main question of calculating area. They all make it easier for students to *answer* the main question of calculating area and they make that main question more interesting also.

This technique runs back to my workshop participant’s advice that “you can always add but you can’t subtract.” Once you tell your students your question, you can’t ask “What questions do you have?” Once you tell students what information matters, you can’t ask them “What information matters here?” Once you tell them to calculate area, it becomes very difficult to ask them, “What shapes combined to make this shape?”

*Tomorrow*: Why Graphing Stories does a pretty lousy job of developing the question.

*Preparation*: If the main question is “sketch this real world relationship,” what are ways we could develop that question?

One way to view schoolwork is as a series of answers. We want students to know Boyle’s law, or three causes of the U.S. Civil War, or why Poe’s raven kept saying, “Nevermore.”

Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question. But as the information in this chapter indicates, it’s the question that piques people’s interest. Being told an answer doesn’t do anything for you.

*Developing* a question is distinct from *posing* a question. Lately, I try to assume that every question I pose is more precise, more abstract, more instrumental, and less relational than it had to be initially, that I could have done a better job *developing* that question. If I do a good job developing a question, my students and I take a little longer to reach it but we reach it with a greater ability to answer it and more interest in that answer.

Over the next few days, I’d like to offer an example of someone doing a good job developing the question and somebody else missing the mark. I’ll be the one who misses the mark with my Graphing Stories lesson. Math Curmudgeon will be the one who gets it right. After those entries, I’ll encourage us all to make a couple of resolutions for the future.

**2014 Aug 13**. Daniel Willingham weighs in:

@ddmeyer Devoting sig time to explicating why something matters to the field, and doings so in a way that gets student buy-in.

— Daniel Willingham (@DTWillingham) August 13, 2014

@ddmeyer Prob may be made interesting via practical app, everyday life examples, but developing=deeper interest by situating 2days problem..

— Daniel Willingham (@DTWillingham) August 13, 2014

@ddmeyer ….. relative to what else we’ve learned.

— Daniel Willingham (@DTWillingham) August 13, 2014

]]>@ddmeyer Ideally, "developing" means getting s's to understand why it's important to know the answer & not just for an practical reason.

— Daniel Willingham (@DTWillingham) August 13, 2014

For instance, Dandy Candies tells students they’re going to package up 24 cubical candy boxes. It asks them, “Which of four packages uses the least amount of packaging? Which uses the least amount of ribbon?”

This is the usual house style. Concrete imagery. No abstraction. Contrasting cases. Predictions. Students make their guesses. Then they get the dimensions from the video. They calculate surface area and ribbon length. (Ribbon length is a little bit more interesting than perimeter but not by a lot.) They validate their predictions with their calculations.

But then we ask them to find out if *another* package dimension will use even *less* material.

So now the students have to think systematically, tabling out their work so they don’t waste effort finding the surface area of a lot of different prisms.

Contrast that against a worksheet like this, which is practice also, though rather less purposeful:

Where else have you seen purposeful practice?

I’d look to:

- Robert Kaplinsky and Nanette Johnson’s Open Middle project.
- Malcolm Swan’s tasks: impossible points & break 25.
- Bryan Meyer’s scientific notation task.

**BTW**. Given any number of cubical candies, what is the best way to minimize packaging? Can you prove it? I can handle real number side lengths but when you restrict the sides to integers, my mind explodes a little.

A workshop participant gave this algorithm. I have no reason to believe it works. I also have no reason to believe it doesn’t work.

- Take the cube root of the volume.
- Floor that to the nearest integer factor.
- Square root the remainder factor.
- Floor that to the nearest integer factor.
- With the remainder factor, you have three factors now.
- The smallest of all three factors is your height.
- The other two are your length and width. Doesn’t matter which.

Note to self: test this against a bunch of cases. Find a counterexample where it falls apart.

]]>A person lives inside a room that has baskets of tokens of Chinese characters. The person does not know Chinese. However, the person does have a book of rules for transforming strings of Chinese characters into other strings of Chinese characters. People on the outside write sentences in Chinese on paper and pass them into the room. The person inside the room consults the book of rules and sends back strings of characters that are different from the ones that were passed in. The people on the outside know Chinese. When they write a string to pass into the room, they understand it as a question. When the person inside sends back another string, the people on the outside understand it as an answer, and because the rules are cleverly written, the answers are usually correct. By following the rules, the person in the room produces expressions that other people can interpret as the answers to questions that they wrote and passed into the room. But the person in the room does not understand the meanings of either the questions or the answer.

This is a pretty perfect parable.

**Featured Comment**

Joel Patterson, talking to commenters who’d send this parable along to their students’ parents:

]]>I like this parable. Before you condense it, and give it to all your parents, consider whether those parents have science/engineering backgrounds. It’s a pretty complicated picture to envision. I think S/E people would grasp it (if they haven’t heard of it already) and would get your point. But if the parents have less of an S/E background, the complicated parable is likely to bore them and not convey your point.

Have an explanation at hand that is more like the guitar players who can improvise, not just repeat the 5 songs they’ve memorized. Or cooks who can put together a soup without the recipe because they know which spices and foods have good flavors together.

**tl;dr**. I made another digital math lesson in collaboration with Christopher Danielson and our friends at Desmos. It’s called Central Park and you should check out the Walkthrough.

Here are two large problems with the transition from arithmetic to algebra:

**Variables don’t make sense to students.**

We give students variable expressions like the exponential one above, which they had no hand in developing, and ask them to evaluate the expression with a number. The student says, “Ohhh-kay,” and might do it but she doesn’t know what pianos have to do with exponential equations nor does she know where any of those parameters came from. She may regard the whole experience as one of those nonsensical rites of school math which she’ll forget about as soon as she’s legally allowed.

**Variables don’t seem powerful to students.**

In school, using variables is harder than using arithmetic. But what does that difficulty buy us, except a grade and our teacher’s approval? Meanwhile, in the world, variables are responsible for anything powerful you have ever done with a computer.

Students should experience some of that power.

**One solution.**

Our attempt at solving both of those problems is Central Park. It proceeds in three phases.

*Guesses*

We ask the students to drag parking lines into a lot to make four even spaces. Students have no trouble stepping over this bar. We are making sure the main task makes sense.

*Numbers*

We transition to calculation by asking the students “What measurements would you need to figure out the exact space between the dividers?” This question prepares them to use the numbers we give them next.

Now they use arithmetic to *calculate* the space width for a given lot. They do that three times, which means they get a sense of the parts of their arithmetic that *change* (the width of the lot, the width of the parking lines) and those that *don’t* (dividing by the four lots).

This will be very helpful as we take the next big leap.

*Variables*

We give students numbers *and* variables. They can calculate the space width arithmetically again but it’ll only work for *one* lot. When they make the leap to variable equations, it works for all of them.

It works for sixteen lots at once.

Variables should make sense and make students powerful. That’s our motto for Central Park.

**2014 Jul 28**. Here is Christopher Danielson’s post about Central Park on the Desmos blog.

**Featured Comment**

In thinking further about your complaint about “Write an expression” I think what is also going on in this app is a NEEDED slowing down of the learning process. The text (and too many teachers) are quick to jump to algorithms before the students understands their nature and value. Look how long it takes to get to the concept of an appropriate expression in the app: you build to it slowly and carefully. I think this is at the heart of the kind of induction needed for genuine understanding, where the learner is helped, by scaffolding, to draw thoughtful and evidence-based conclusions; test them in a transfer setting; and learn from the feedback – i.e. the essence of what we argue understanding is in UbD.

]]>One reason I like this activity so much is that it hits the sweet spot where “What can you do with it?” and “What does it mean?” overlap.

In one case, a teacher was teaching a lesson about division with remainders and the example was packaging meatballs in pack of 4. When faced with the problem of having 13 meatballs and needing 4 per pack, one student’s solution was “I would eat the extra meatball and then they would all fit.” It was so funny and joyful to see that all thinking was welcomed and the teacher artfully led them to the general thinking that she wanted by the end of the lesson.

I can trace my development as a teacher through the different reactions I would have had to “I would eat the extra meatball,” from panic through irritation to some kind of bemusement.

**BTW**. The comments here have been on another level lately, team, including Simon’s, so thanks for that. I’ve lifted a bunch of them into the main posts of Rand Paul Fixes Calculus and These Tragic “Write An Expression” Problems.

Two quick meta-items about blogging from the last week:

- I attended Twitter Math Camp 2014 in Jenks, OK, in which 150 math teachers who generally only interact online get together in person. I gave a keynote that could probably best be described as “data-rich,” in which I downloaded and analyzed details on 12,000 blogging and tweeting math teachers. Here are links to my slides and speech as well as the CSVs if you want to analyze some data yourself. (Who doesn’t!)
- A doctoral student in Canada is interested in blogging as “unmediated professional growth” and sent me a survey about my blogging. Here is a link to my responses. How would you have answered?

If you have one person in the country who is, like, the best at explaining calculus, that person maybe should teach every calculus class in the country.

It’d be helpful if we could work through the idea that good teaching is *just* good explaining and vice versa. Someone here at Twitter Math Camp mentioned she conducts a math night for parents at the start of school. “I wish I had learned math like this as a kid,” they tell her. That realization, that there is and *should* be a difference between how math was taught then and now, is a giant first step.

**Featured Comments**

This shows the idea that children’s minds are empty vessels that need to be filled with knowledge and teachers are the keepers of that knowledge, whose sole job is to effectively pour said knowledge into the vessel. And if their minds didn’t get filled with our knowledge the fault must lie with our explanations.

This flies in the face of what we know about teaching and learning.

]]>None of these reforms about math education can happen in a vacuum. There’s always a political side to what happens to people’s children, and if the way you help children learn math is important then the way you communicate with parents is also important.

The original title of this post called them “horrible,” but they’re truly “tragic” – the math education equal to Julius Caesar, Othello, and Hamlet – full of potential but overwhelmed by their nature.

Here’s the thing about variable expressions: they’re used by programmers and students both, but those two groups hold variables in very different regard.

Ask programmers what their work life would be like without variables and they’ll likely respond that their work life would be *impossible*. Variables enable *every single* function of whatever device you’re reading this post on.

But ask students what their school life would be like without variables and they’ll likely respond that their school life would be *great*.

**What can we do?**

The moral of this story isn’t “teach Algebra 1 through programming” or “teach computational thinking.” At least I don’t think so. I’ve been down that road and it’s winding.

But in some way, however small, we should draw closer together the wildly diverging opinions students and programmers have about variables. Ideas? I’ll offer one on Monday.

**2014 Jul 25**. I appreciate how Evan Weinberg has thought through this makeover (now and earlier).

**Featured Comment**

Dylan Kane restates our task here in a useful way:

In terms of making these problems a little better, students should feel a need for the expression. I think this question stinks in part because the expression it’s asking for is so trivial — it’s extra work, compared to just multiplying by 3/4 or doing some simple proportional thinking.

Jennifer offers an example of that kind of need:

I like to introduce the idea of expressions by having the students playing the game of 31 with a deck of cards!their goal- play until they can predict how they can win every time! This will take less than 15 minutes, and a whole class summary of verbal descriptions on ‘how to win’ are shared. Verbal descriptions become cumbersome to write on the board, so ‘shorthand’ in the form of clearly named and defined symbols are used to make the summarising more efficient. the beauty of this is that the idea of equivalent expressions presents itself.

i think the ability to generalize and write a rule with variables is really important, but you can come to that through lots of nice activities and investigations as well.

for example, i did dan’s “taco cart” with my students with a few notable changes. instead of telling the students how fast dan and ben walked, i had each group decide on the speed of the two men themselves and list that along with other assumptions they made in the problem.

when we did the whole class summary, i told them that i had written down a formula on my paper that would allow me to check if their answers were correct and that i needed that since everyone used their own speed. i should’ve asked them all to take a minute to try to come up with the formula i used, but instead i elicited it at that moment and one student gave me the correct formula. the need for two variables (speed on sand and speed on pavement) was obvious.

in part two of taco cart, when the students were trying to figure out where the taco cart should be so that the two men reached it at the same time, one group did seemingly endless guess and checks. i suggested to them that this wasn’t a good method and asked if maybe it wasn’t better to write an equation with a variable. again, they could see the need. once they started with a variable, the rest of the problem started to fall into place.

here’s the thing: these “write an expression” problem want to train students to learn to generalize and write a rule. they want them to be able to see a situation that would best be tackled with a formula of some sort, write said formula with various variables, and use their formula to solve complex problems. but the issue is that these problems don’t actually train students in that way because they’re so artificial and one-dimensional. what they do is teach students to “translate” from english to math (an important step along the way, i do believe), but not to recognize a situation in which a formula would be helpful or necessary or how powerful it can be.

]]>Two thoughts from a computer science teacher’s point of view:

1) When introducing programming to 15 yrs olds for the first time, we use Python interactive prompt as a calculator first. And the first point is to show the advantages. Variables and funstions (one-liner formulas) simply save work. That is it, that is one of the goals of the whole programming topic anyway. When they do quadratic equations in maths at that time, we are headed that way. And students realize pretty fast: aha, I have to understand how to solve them, but once I do and I describe it properly, I never have to do it by hand anymore. Their understanding of q.e. deepened, they interest in programming increased, and I could naturally introduce a load of important CS concepts on the way. In younger age we do simpler formulas, also like BMI calculation (not only the “area of the circle” kind of stuff, which they, um, do not prefer), but the point is the same: the kids need to see at least some hypothetical benefit for themselves. Having to introduce variables in maths, I would in principle search for a similar approach. (Note: for even younger kids, fun and creativity achieved with Scratch, turtle etc. overweigh the “practical benefits”; but when practicality leads to more fun – win-win!).2) A good and often forgotten tool between calculation on paper and programming is a spreadsheet. It can store lots of numbers in a structured way and perform basic calculations, what is well understood by kids. And when we want to do anything more complex without getting beards grown, we absolutely need formulas and “variables”. Their advantages are imminent. And the whole time, everything is in plain sight, the level of abstraction is way lower than with programming, making it very accessible for kids. I am of course interested in it “from the other side” – after some decent work in spreadsheets, many more advanced concepts are a step away (for-loop, data type, input-output, function, incremental work on more complicated calculations, debugging etc. etc.). But I believe that thoughtful use of spreadsheets can improve understanding in appropriate topics in maths.

Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them.

She also tours through some of the best ethnographic research you’ll read in math education but doesn’t cite two of them explicitly (that I counted) so I will.

- A Revolution in One Classroom: The Case of Mrs. Oublier (Cohen, 1990) describes how one teacher’s transformed
*beliefs*failed to transform her*practice*. - Candy Selling and Math Learning (Saxe, 1988) describes the informal math Brazilian children used to sell candy and how it failed to translate into formal classroom math.

**Featured Comment**:

It might be worth noting that the paragraph about ‘answer getting’ seems to be referring to Phil Daro and his whole take on answer-getting.

Simon Terrell writes about his trip to Japan with Akihiko Takahashi.

Dan Goldner on his resolutions:

]]>Of all the great things to focus on in this article, this is the one that spoke to me where I am now. Student-initiated in 40%, not 100%. 41% of time practicing, not 5%. Half the time on invent/think, not all the time on invent/think. I’ve been working so hard on making “invent/think” the dominant activity in my room, that practicing, which is

alsoa cognitive requirement for learning, has been de-emphasized. The next paragraph in the article acknowledges that Japan isn’t perfect, either, and these percentages certainly aren’t a perfect recipe. But as my personal pendulum finds its equilibrium it’s great to read this and take from it the encouragement that thatallthe modes of learning have to have a place during the week.