Teaching manipulation of expressions and equations before introducing variables

In addition to the video I’ve been toying with a sample first chapter of this project idea.

It is quite usual in Algebra books to introduce variables early — they seem to be what makes algebra Algebra, that is. However, a good amount of early stages of manipulating expressions and equations is not dependent on having variables. I am wondering about possibly spending longer than usual on concrete number manipulation with problems like–

Isolate the 253:
253 + 23 = 276

Isolate the 3:
2(3 + 4) + 5 = 18 + 1

Show why the two sides of the equation are equivalent:
6 * 4 = 3 * 4 + 3 * 4

Theoretically, one could provide a strong enough background in manipulating expressions and equations to allow variables to be introduced while hitting the ground running, with students quickly able to isolate variables in multi-step equations.

This has the advantages of:

1. It’s clear that (using the first problem as an example) 253 = 253 is the goal. The point becomes the process, and the usual student tactic of writing the answer without explanation (I did it in my head!) would look silly.

2. Students who have “symbol overload” can use their concrete sense to help focus solely on operations, a la Bret Victor (sort of).

3. The meaning of the equality sign is reinforced; it’d appear much odder to a student to perform an operation on only one side of the equation when starting with 20 + 1 = 21. Also, the restrictions to operations on an expression would make more sense when starting with, say, 5 rather than 5 + 2x.

4. Answers would have a built-in sanity check.

Unfortunately, it means applications would be forestalled. While some of the manipulations represent tricks for mental arithmetic, I don’t know if that would be enough motivation to carry a struggling student through.

Has anyone tried something like this, even on a small scale?

11 Responses

  1. I love this! I teach intro algebra to 6th graders. It is the first time they have worked with variables. I usually jump right into variable and most get it. However, I feel that they could all get it by approaching it this way. I feel that this is getting them used to the processes before throwing letters at them.

    Getting them to show me the process is very difficult. I love #1 for just that!

    Another take on the “isolate the 3” could be to illustrate distributive property. The usual non-variable examples are so basic they really aren’t helpful when you try to transfer their knowledge and apply it to a distributive w/ variable problem. However, make it complicated and have to equal a problem on the other side of the equation and I think it would be more worthwhile.

  2. Actually, this is already done (at least when I was a little kid, it might have changed since then). Very early in elementary school, don’t kids have to solve questions like this: “5+___=8” (fill in the blank). It’s always struck me as odd that algebra teachers don’t use this as a launching board, explaining that “5+___=8” and “5+x=8” are the same exercise.

    • That’s still common, but it isn’t the same thing as forcing the demonstration of the steps of taking inverses to isolate a term.

      The “blank” technique really is using a variable, only with different formatting. My idea here is instead to do away with having a variable at all.

  3. I am so glad to see this, Jason! This is exactly the approach we take in our book, Prealgebra, by Bach & Leitner. We establish number patterns before defining and using letter variables. Students develop ‘number sense’ with estimation, distributive law, and order of operations, so later it makes more sense when combining like terms or factoring polynomials. You seem to go a step further and actually nail the number down like a variable. Great idea!

  4. I like this approach Jason. Have not seen it used by others, but have taken similar approach when walking my kids through a problem. That is, I would substitute the actual numbers for the variable, and then walk through the solution. It never occurred to me that it might be a better way to introduce the whole subject. Great idea.

  5. When my adult learners see 5 + x = 12, they often trip over that plus sign; it’s supposed to mean add. Sometimes “five plus what is twelve” at least gets the right answer, but many of them don’t have even that much number sense or arithmetic facility.

    WHen I draw it out or ask a simpler problem — “9 plus what is ten” they jump on the answer. It’s one, of course!

    Well, how did you get that?

    Almost always, the answer is, “I added!”

    So — teaching to manipulate equations is a magnificent idea🙂

    • There’s lots of analogues to “9 plus what is 10” being solvable but “5 + x = 12” being baffling; I think this sort of buildup often is assumed to be done by the tutors and not the responsibility of the books.

      • And that is because, of course, our math teachers use books to support their strong Math pedagogy backgrounds? I didn’t think so… so all the students need tutors…

  6. Absolutey dead on. Makes up for the idea of “visual algebra”.:)

  7. […] – I’ve been thinking about Jason Dyer’s ideas on Visual Algebra – in particular, the representation of an equation as a bunch of boxes that […]

  8. […] but a more sophisticated packaging of this code might conceivably be worked with directly.  Jason Dyer’s blog has examples of how this might work. Maybe twenty years from now people using math would expect to […]

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