Andrew – I love your modeling idea. It would be a great way to set the stage for establishing the equal ratios idea that the original problem was about (aside from “writing an expression”). I can easily see where it could build from there.

Dan – I showed your “Pick a Point” Vimeo to a group of teachers today at a tech summit. They loved it (as they always do). That idea comes to mind here: Pick a Point makes it clear why mathematicians make things way easier for themselves by naming points. Can we find a parallel situation here that shows how programmers use variables to make things way easier? I think that’s what we’re after here. I only wish I knew enough about programming to offer up an idea.

Andrew, Joshua, I love your ideas as far as good pedagogy, but I can’t for the life of me see how they relate to programming-variables-vs-algebra-variables.

FWIW Andrew, I think you’re imagining variables as being much more high-level than they really are in practice. If someone is programming a gesture-based app, they might have one variable that stores a Gesture data object. But that Gesture object will likely also contain other, lower-abstraction variables, like x-y co-ordinates of where the finger swiped on the screen. And if you drill down the layers of abstraction enough, every programming variable is holding some set of numbers (because that’s all computers know at a hardware level).

I wonder whether this comparison is even fair to begin with. Why are we comparing programmer attitudes to math student attitudes? I’d be more interested to see how programming student attitudes compare to math student attitudes.

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.

That said, I think the best tool here is to ask students to answer the question, without requiring an expression, a few times for small or simple to calculate numbers, and then ask them to make predictions for much bigger numbers (dreams in a year? dreams in a lifetime?). Expressions have value because they are reusable for different values of the variable; that’s why a student should be using them.

I also think it’s worth comparing and contrasting expressions with a table of values/graphs/written explanations depending on the grade and having students compare and evaluate the usefulness of each.

I see “write an expression” problems more as a lesson in grammatically interpreting sentence structure than in understanding the purpose of the algebraic expression in the first place. That being said, however, it is this very exercise that enables me to identify in one session those students whose literacy will interfere with their understanding of algebra.

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.

And, even before algebra is introduced, we use symbols to support the simplification of fractions using cancelling down. Also works when explaining prime decomposition!

As for programming, yes, a wonderful but winding platform for teaching maths, but, isn’t algebra itself the original program language for code that explains our world, and everything in it?
I’d love to go back to encouraging more of our students to go back to studying pure maths to support more of the scientific research into unravelling the mysteries of our universe!

Programming variables are rooted in need, right?

There is some function that this program does, and we need that program to do that thing under a wide range of conditions and user inputs. The function is the decisions about what’s actually being done, the variable holds the thing/quantity/position/whatever it’s done in relation to.

The thing I find myself struggling with as I look at this problem is how in the world we create some kind of need to recognize that 3 out of every 4 dreams we have involve people we know, and how in the world you compel any kid to figure out how many dreams out of 300 would contain people they know.

Think, I will.

I had my students doing the hour of code in math class this year. The problem solving and algebraic thinking involved in “if/then” statements is immense and then they got to see if their work was successful or not successful.

Can’t we just take the program from Hour of Code and mathematize it in other ways, maybe so the situations are more “real life” than a zombie eating flowers or whatever. Just an idea.

@joshg

I wonder if teaching slope intercept form with more descriptive variables for some length of time initially would help lead to some better understanding:

y_coord = slope * x_coord + y_intercept

Still, without a conceptual understanding of slope, just writing out the whole word won’t magically bring about any more meaning than the letter “m.”

@everyone

Any thoughts on Bootstrap? A cursory look through the materials seems to promote mathematical understanding through a fun, computational framework.

http://www.bootstrapworld.org/

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.

I wonder if we get kids thinking about variables/placeholders soon enough. Seems to me that a prime point to do it would be operator precedence – it would work well visually and it’s conceptually similar. You need the student to view that line of numbers and operators as discrete items that match up in more than a left to right manner.

So why not write 4 + 9 * 3 / 2 on your board and rather than subbing in 21 you write A, with a note off to the side, rather than circling it or erasing it and replacing it with 21. It can be a stealth exposure – oh, let’s just make this easier to think about by putting this A in here rather than the parenthesis. Then we can do the same for the next one, and write it over on the next line saying that B is equal to A / 2. May as well say C = 4 + B then.

Now you’ve created representations and created a nice little shorthanded list to the right of one-operation things to do to get your answer. Building it worked on teaching precedence and you make it easier than visualizing the line and doing mental or on-board “stacks,” which is how my distant memory of it being taught was done.