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Oct 6, 2017, 9:59:12 AM10/6/17

to mathf...@googlegroups.com

My fourth-grader brought home "algebra" homework.

Given a word problem, the student was asked to display:

(a) a SITUATION equation

(b) a SOLUTION equation

(c) evauate the solution equation

(c) check the arithmetic.

It turns out, the "situation" is a "change" situation, with two possible "situation" equations:

"change plus" (which I would call gain or increase): start + change = end

"change minus" (loss or decrease): start - change = end

There are also six possible "solution" equations, one for each variable

"chagne plus"

start + change = end

end - start = change

end - change = start

"change minus"

start - change = end

start - end = change

end + change = start

I had attempted to show the student how to derive the solution equations from the situation equations,

by adding or subtracting variables from both sides.

But it occurs to me that perhaps the teacher expected the students to memorize the various forms instead.

(That's the way we did it when I was in 4th grade, but then we didn't call it "algebra".)

My overall impression is that this was too much for one lesson.

I'd probably start with creating situation equations from word problems,

matching start, end, change, and deciding whether the change was gain (plus) or loss (minus)

that would probably be enough for one lesson.

But even the presupposes two other concepts:

(a) the notion of equations: the idea that "change" is a regrouping operation,

not creating or destroying, so what you have "before" change is the same (total) as what you have after,

just rearranges, e.g. between a store and the customer.

(b) the idea of "variable", that one can abstract a quantity into a names "role" without regard to the actual number,

e.g. "unsharpened pencils" vs "12", or even more generally, "starting value" vs "unsharpened pencils".

So my question:

Is it appropriate to teach "algebra" in 4th grade (or younger)?

And if so, how does one go about it?

Personally, i would answer "yes" to the first question, as I believe much of the difficulty of "word problems" is that students are expected to tackle them

without a foundation of algebra which makes them routinely solvable.

I would stress creating mathematical relations ("situation equations") from word problems as the first lesson,

then later teach the abstract methods of rearranging equations--adding or subtracting equals from both sides, for example.

I would definitely not expect students to remember all the "solved" forms of equations.

But there's another concept lurking in the background: the concept of an "equation" itself.

In science, an equation is typically a statement of a conservation law, a statement about the order in the universe.

In the case of the "change" equations above, it is a statement that, in a transaction, the total "stuff" remains constant,

just changes hands. And typically such an exchange is a zero-sum game: if I "lose" one thing, I "gain" another.

For example, if a student buys pencils from a store, the store loses pencils, but gains cash; the student gains pencils, but loses cash.

I'd say there is a philosophical, even ethical, argument that the two exchanges should not be separated.

And the relationship between pencils and cash presents the opportunity to discuss proportion, leading to equations involving multiplication and "fractions" (which are often more about "proportion" than "parts of a whole").

Joe Austin

Given a word problem, the student was asked to display:

(a) a SITUATION equation

(b) a SOLUTION equation

(c) evauate the solution equation

(c) check the arithmetic.

It turns out, the "situation" is a "change" situation, with two possible "situation" equations:

"change plus" (which I would call gain or increase): start + change = end

"change minus" (loss or decrease): start - change = end

There are also six possible "solution" equations, one for each variable

"chagne plus"

start + change = end

end - start = change

end - change = start

"change minus"

start - change = end

start - end = change

end + change = start

I had attempted to show the student how to derive the solution equations from the situation equations,

by adding or subtracting variables from both sides.

But it occurs to me that perhaps the teacher expected the students to memorize the various forms instead.

(That's the way we did it when I was in 4th grade, but then we didn't call it "algebra".)

My overall impression is that this was too much for one lesson.

I'd probably start with creating situation equations from word problems,

matching start, end, change, and deciding whether the change was gain (plus) or loss (minus)

that would probably be enough for one lesson.

But even the presupposes two other concepts:

(a) the notion of equations: the idea that "change" is a regrouping operation,

not creating or destroying, so what you have "before" change is the same (total) as what you have after,

just rearranges, e.g. between a store and the customer.

(b) the idea of "variable", that one can abstract a quantity into a names "role" without regard to the actual number,

e.g. "unsharpened pencils" vs "12", or even more generally, "starting value" vs "unsharpened pencils".

So my question:

Is it appropriate to teach "algebra" in 4th grade (or younger)?

And if so, how does one go about it?

Personally, i would answer "yes" to the first question, as I believe much of the difficulty of "word problems" is that students are expected to tackle them

without a foundation of algebra which makes them routinely solvable.

I would stress creating mathematical relations ("situation equations") from word problems as the first lesson,

then later teach the abstract methods of rearranging equations--adding or subtracting equals from both sides, for example.

I would definitely not expect students to remember all the "solved" forms of equations.

But there's another concept lurking in the background: the concept of an "equation" itself.

In science, an equation is typically a statement of a conservation law, a statement about the order in the universe.

In the case of the "change" equations above, it is a statement that, in a transaction, the total "stuff" remains constant,

just changes hands. And typically such an exchange is a zero-sum game: if I "lose" one thing, I "gain" another.

For example, if a student buys pencils from a store, the store loses pencils, but gains cash; the student gains pencils, but loses cash.

I'd say there is a philosophical, even ethical, argument that the two exchanges should not be separated.

And the relationship between pencils and cash presents the opportunity to discuss proportion, leading to equations involving multiplication and "fractions" (which are often more about "proportion" than "parts of a whole").

Joe Austin

Oct 9, 2017, 9:36:58 PM10/9/17

to MathFuture

Joe

Your post reminded me of the issues I raised in this post where I suggest we should maybe think of using different equal signs for the different **purposes** of the equality (or assignment of a value) we have in mind:

I'm now wondering whether we should think about these things at the 4th grade level and it's probably a case of why not? Perhaps students will have less trouble with algebra later if they know "=" doesn't always mean the same thing.

Regards

Murray

Oct 11, 2017, 7:58:20 PM10/11/17

to mathf...@googlegroups.com

Murray,

From my training in Computer Science, I am well aware of the "assignment" equal sign such as FORTRAN's

X = X + 1. ALGOL "solved" the problem by introducing another symbol, " := ". C kept " = " as assignment,

but substituted " == " for comparison.

But the "assignment = " is also the "equal" sign on the pocket calculator, so

7 + 3 = 10 + 3 = 13 − 8 = 5

is exactly the way you would do the problem on a calculator.

Another related but sometimes overlooked convention is that subtraction is left-associative.

10 - 5 - 2 = ( (10 - 5) - 2) = 3 not (10 - (5 - 2) ) = 7

But to my original point, these are pitfalls we learn to handle in "real" algebra.

But I'm wondering whether the 4th grader is being taught algebra,

or is just being confused by being shown an algebraic-style notation that the student doesn't really understand.

I believe true algebraic thinking is the key to solving word problems.

But I'm not convinced that my 4th grade student is actually learning "algebraic thinking".

I suppose I could go further, and ask whether it is appropriate to introduce "word problems" without teaching the tools and procedures to solve them properly.

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