I usually offer the following reasons for using variables:

(1) To avoid repetition. If we write the same number many times throughout our code, but later change our mind, we have to go and update every instance. Annoying and error-prone.

(2) To improve readability. When you give your variable a meaningful and descriptive name, you can more easily understand your calculations, etc. This was mentioned above along with the suggestion to use a better name than “x”.

(3) To deal with the fact you have no idea what the value will be. At first, this doesn’t come up, as we are only drawing static images, but once we introduce user input, we need a way to store that input somewhere so we can do something with it.

Maybe these points offer some clues to the question at hand.

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.

Dan, you should check out Evan’s comments above. Evan’s a spreadsheet fan also for similar purposes.

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.

@all:

why not represent the data? “Can we plot the data of number of remembered dreams….”

what about hidden equations: how many dream we dreamt if we remember just 370?

Here is what variables matter. The rest is pure mechanic illusion.

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.

For example: Think of a number, multiply by 2, add 10, divide by 2, subtract the number you thought of. What is your answer?
What if you start with a different number? I wonder why this happens?
Can you change the instructions to end with a different number?

Using a variable (and I prefer not to use x)
n –> 2n –> 2n + 10 –> n + 5 –> 5.

}I would love to have more such puzzles if any one can supply them.]

The other introduction to variables I use is through patterns in a calendar month: What patterns can you find and why do they work?

It seems to me that if variables are necessary for understanding a situation, then students are more likely to appreciate them as valuable in solving problems.

The issue with Dan’s textbook problem is that it is not a problem, but a series of instructions! I wonder how Dan will make it into a problem that is rich enough to need solving with a variable? I look forward to Monday’s post.

@Andrew, I’m not sure how your proposed makeover does anything different with the variable, though it certainly does involve students in more interesting aspects of modeling.

@Joshua Zucker, “reuse” seems like a potent theme to explore here. But if we do a ton of the same problem, writing down a variable expression to summarize that work just seems like The Next Problem here. For programmers, that expression carries with it some power that I’d like students to experience also.

@Mark James, exactly. You’ve restated the problem nicely. We’re looking for a scenario where students experience the ease and power of variables.

@Howard, go for it. Cheers.

@Dylan Kane, “extra work.” Right, that’s exactly what I was trying to convey to Joshua above. Thanks.

@Jennifer, nice introduction to the concept of a variable. Added to the post above.

FWIW, I’ll throw my chips towards Evan’s post on the matter. (Be sure to click through to his earlier one also.)

Here’s Evan:

Programmers use variables because they want to build a program that produces a correct output for every possible input that might be used to solve a given problem or design.

And the value of that variable expression grows the more times the programmer runs it. In the textbook original here, the student runs the expression once. That seems like a major shortcoming of this kind of problem to me.

@Pam, the biggest trouble was that the intersection of school Algebra and programming isn’t as large as we thought.