We need to do something about these problems, which recur all throughout school Algebra.

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.

Kate Nerdypoo:

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.

Dan L:

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.

a/k/a [Makeover] Painted Cubes (See preview.)

That’s a very helpful comment from, Alyssa Boike, a recent workshop participant. Textbooks don’t have that same luxury.

Here’s an example. Watch how Connected Mathematics treats the classic Painted Cube problem:

Here are elements the textbook has already added:

1. A central question. (“How many faces of the little cubes have been painted?”)
2. A strategy. (“Look at smaller versions of the cube.” It also tells you by omission that it’s impossible to find more than three faces painted.)
3. A table. (For organizing your data.)
4. Table column headings. (Edge length, total cubes, total cubes of each kind.)

If you subtract those elements and add them in later, you get to ask interesting questions and host interesting conversations with your students. Like this:

1. A central question. (“What questions do you have about Leon and his cube?” And later: “Guess how many cubes don’t have any paint on them at all?”)
2. A strategy. (“What are all the possibilities for the number of faces that could have paint on them? Could five faces have paint on them? Can I tell you how mathematicians work on big problems? They look at smaller versions of the big problem. What would that look like here?”)
3. A table. (“All of the numbers from our smaller versions are getting out of control. How can we organize all these loose numbers?”)
4. Table column headings. (“What kind of data should we look at? What about these cubes seems important enough to keep track of?”)

You can always add those elements into the problem — the questions, the information, the mathematical structures, the strategy — as your students struggle and need help. But you can’t subtract them.

Once your students see the table, you can’t ask, “What tool could we use to organize ourselves?” The answer has been given. Once they see the table headings, you can’t ask, “What quantities seem important to keep track of in the table?” They know now. Once you add the strategy (“Look at smaller versions.”) their answers to the question “What strategies could we use?” won’t be as interesting.

In sum, much of the problem has been pre-formulated, which is a pity, seeing as how mathematicians and cognitive psychologists and education researchers agree that formulating the problem leads to success and interest in solving the problem.

So again I have to remind myself to be less helpful and be more thoughtful instead.

BTW. Of course I’m partial to Nicole Paris’ setup of the task:

Great Help From The Comments

I’m reprinting Bryan Meyer’s entire comment:

I don’t know that I have anything terribly insightful to add, but this seems like a fun conversation.

I don’t really see too much that is wrong with the problem/puzzle itself, which (to me) is something like:

I have this cube (show picture/tangible) made up of smaller cubes. If I dipped the whole thing in paint, how many of the smaller cubes would have paint on them? Is there a rule or shortcut we can create that would allow us to answer that question for ANY sized cube?

To me, the issue seems to be that the version we see in your blog post attempts to steer the direction of student thinking and leaves little room for play and divergent thinking/approaches. It “scaffolds” away all of the rich mathematical thinking and play in an attempt to cover standards. In particular, the unspoken assumption in the way it has been printed is that writing and graphing linear, quadratic, and exponential functions is the real “Math” in the task (things we can easily point to as belonging to the discipline/standards).

But, at it’s core and without all the mechanical scaffolding (as re-posed here), the question allows room for many mathematical strengths and habits of mind to be valued and sends different messages about what the real “math” might be: taking things apart and putting them back together, creating systems of organization, assigning variables, making generalizations, posing extension questions, etc. In addition, because it doesn’t dictate how to proceed, it encourages students to trust their own thinking and allows them to “see themselves” in the work that develops. The work of the teacher becomes to follow the student, looking for mathematically ripe opportunities in their work and thinking.

2014 Jun 2. Christopher Danielson brings his perspective to the task as writer of CMP.

2015 Nov 2. Great example from Jennifer Michaelis.

2014 Jun 2. Here’s the makeover.

Painted Cubes is a classic task, canonized right alongside the Pool Border task and Barbie Bungee, but that doesn’t mean it’s beyond help, or that everyone treats it exactly the same way.

Here’s a treatment from Connected Mathematics. What would you do with this and why would you do it? (Click for larger.)

Many thanks to Mr. Weiss for reminding me to compile all of this summer’s makeovers. Here’s every revision principle we applied this summer, ranked from most frequently occurring to least.

• (9) Add intellectual need.
• (6) Raise the ceiling on the task.
• (5) Add intuition.
• (5) Lower the floor on the task.
• (4) Reduce the literacy demand.
• (4) Show the answer.
• (2) Put students in the shoes of the person who might actually experience this problem.
• (2) Start the problem with a concise, concrete question.
• (2) Ask a better question.
• (2) Delay the abstraction.
• (1) Offer an incentive for more practice.
• (1) Enable pattern-matching.
• (1) Get a better image.
• (1) Add modeling.
• (1) Change the context.
• (1) Open the problem up to more than one possible generalization.
• (1) Justify the constraints.

If you’re looking for a dy/dan house style, for better or worse, that’s it right there.