## Tuesday, December 1, 2015

### Traditional algebra as "mere" symbol manipulation

Some people who critique the traditional approach to algebra instruction cite papers like this one, a “research brief” by Toni M. Smith, published last year by the American Institutes for Research and entitled "Instructional Practices to Support Student Success in Algebra I." Smith writes:

Typically, instruction in algebra focuses on symbolic manipulation and algebraic procedures, with little attention to the connections between these procedures and the underlying mathematical concepts (Chazan and Yerushalmy, 2003). When students are expected to memorize and operate with a set of rules that are seemingly meaningless, they may become frustrated and eventually fail Algebra I.
Smith is thus suggesting that your typical algebra class neglects to teach the meaning of the mathematical symbols. But times, she suggests, have changed:
For decades, the heart of algebra education has been learning to manipulate algebraic symbols to solve equations. Now, students are expected to know and be able to do more than solve equations. They are expected to demonstrate fluency in working with algebraic procedures, understand the associated mathematical concepts behind these procedures, and be able to articulate the connections between and among them.
If "articulating connections" means doing so verbally, that's clearly new. But I hadn't realized that it's only now that, for the first time, math classes are (a) expecting students to work through algebraic procedures fluently and (b) expecting them to understand the underlying math concepts.

More specifically, as far as "articulating connections" goes:
Not only should students be able to solve an equation such as 3x + 5 = 20 for x by performing one or more steps, but also they should be able to articulate the mathematical principles (e.g., order of operations, properties of equality) that support the procedure they used and critique a different approach for solving that same equation.
True, it's possible to stumble on the correct solution to this equation without understanding order of operations and properties of equality. But how about assigning a bunch more problems of similar and greater difficulty? Is it really possible to mindlessly follow the order of operations and perform the same operations on both sides of the "=" sign without understanding order of operations and the meaning of "="?

As far as understanding the underlying math concepts goes, now, for the first time, students are expected, when solving 3x + 5 = 20, to
understand that the solution they find represents the value for x in the function f(x) = 3x + 5 when f(x) is 21 [sic]. Not only should they be able to graph the function f(x) = 3x + 5 but also they should (a) understand why it would look like a line, (b) be able to identify the solution to the equation they solved on that line, and (c) know what kind of real-world relationship such a function would model.
All of this, apparently, is completely absent from traditional algebra texts and traditional algebra classrooms, which, apparently, are "focusing solely on skill in manipulating algebraic symbols" without attaching "meaning to the associated algebraic symbols." This is too bad, because, as Smith explains to us:
Research indicates that when instruction attaches meaning to the associated algebraic symbols, students develop procedural fluency as well as conceptual understanding. This can be done in a number of ways, including through the use of technology.
Luckily, though, technology is not essential:
One approach to assigning meaning to algebraic symbols involves algebraic expressions (e.g., 3x + 4). Students are given a list of algebraic expressions and asked to predict whether or not any of the algebraic expressions are equivalent. Some expressions are equivalent but look different (e.g., 2(x + y) and 2x + 2y) and others look similar but are not equivalent (e.g., 2(x – 3) and 2x – 3). After they make their predictions, students are asked to test them by substituting several numerical values for the variables in those expressions. This step emphasizes the meaning of variable. Once they have their results, students are then asked to provide a justification for what they found. Students who experienced this form of instruction on algebraic expressions produced higher pretest-posttest gains on measures of symbolic manipulation and understanding of variable than did students who received conventional, skills-based instruction (Graham and Thomas, 2000).
Presumably the control group was never properly taught the meaning of parentheticals, or they wouldn't have seen 2(x + y) as more similar to 2x + y than to 2x + 2y.

Of course, the idea that any textbook or half-way competent teacher would teach symbols without teaching their associated meanings—whether in math class, music class, or Chinese class—is ludicrous. Equally ludicrous is that the only way to for students to learn the underlying mathematical concepts is to make predictions, test them out by plugging in numbers, and then provide a verbal justification for what they found.

How about instead doing what traditional math textbooks used to do, and have students represent real-world situations as algebraic expressions, translating words into mathematical symbols? For some nice examples of problems of this variety, rare in today's Reform Math texts, but presented to students at the very beginning of a traditional algebra text, see here.

Brett Gilland said...

"Of course, the idea that any textbook or half-way competent teacher would teach symbols without teaching their associated meanings—whether in math class, music class, or Chinese class—is ludicrous."

No True Scotsman. 5 Yards and repeat of down.

There are plenty of traditionalist teachers out there who teach procedures without much emphasis (if any at all) on meaning. You don't just get to ignore them by claiming their existence is ludicrous.

Katharine Beals said...

Please note the qualifier "half-way decent." I'm not ruling out the existence of lousy teachers.

Katharine Beals said...

Sorry--"half-way competent." Same point.

Anonymous said...

I must be pre-traditional. I remember something called "word problems". But they used non-Dolch words, so I supposed that wouldn't fly today.

Anonymous said...

I understood everything, conceptually, about math up until Trigonometry. Then, although the teacher tried to ground the operations in conceptual understanding, I hit the wall. (Interestingly, beginning Calculus was easier to understand). So, yes, good teachers have always grounded the operations in understanding. But the more you did the operations, the more firmly the understanding was implanted in your brain. This was the mid-'60's.

owen thomas said...

congratulations on placing your piece at
such a widely-read site. it's an important
issue and the furor around it proves you
& barry g. did your jobs well.

the existence of lousy teachers is, alas,
the *heart of the matter*.

*most* elementary math teachers are
probably incompetent by my standards.
i don't propose any program to *change*
this, but i'd be lying if i pretended not
to believe it.

the community college i worked at for many
years had quite a few statistics teachers that
couldn't do high-school algebra; the algebra
teachers were generally quite a bit better
informed but there were more than a small
handful that didn't understand, for example,
set-builder notation. a lot of these had
high-school experience... and knew that
the elementary teachers that had passed
along their students to them had failed
badly in instilling such basic concepts as
"equality".

the thing about symbolism: there are right
and wrong answers. real life seldom offers
such clarity.

this is the great strength of maths.
and the reason that differing ideological
systems--nazis, anarchists, democrats,
republicans, communists, and for-god-
all agree that 1+1=2 and so on.

so you don't just *hamstring* a math teacher
by changing the subject to "how well does
this student color in the lines"; you cut 'em
off at the knees.

i had an elementary school teacher tell me
once that a frog was the opposite of a tadpole.
i must've been about 8 years old. but i knew
then, and still know now, that that teacher
*didn't know what she was talking about*.

the math teachers... i was very lucky... would
actually listen to you if you thought they'd
said something wrong. and if you were *right*...
and they were wrong... they'd *admit* it.
joyfully.

this was true in no other subject.

the *reason* the math teachers had this tremendous
advantage was that *the symbolism itself* was the
highest authority. the cold equations.

many of the the other teachers had learned that going-along-
-to-get-along is the *whole art* and had *nothing* to teach *me.

meanwhile.

all these arguments about how things *should* be.
are useless.

owen thomas said...

PS
the almost-unbelievably competent
mathematician-and-teacher david
hilbert very famously emphasized
that in a well-wrought axiom system,
the "meanings" of the undefined terms
are *not to be discussed*.

i'm lazy to look it up. but something like:
we should be able to replace the terms
"point", "line", or "plane", with "table",
"chair", or "beer-mug"... and have theorems
that are *just as true* as the those using

the point, again, is to *get rid of*
such vague things as "meanings"---
and get on with the geometry.

a mighty fortress is our mathematics.

owen thomas said...

PPS please allow me to reintroduce myself.
the entity once widely known in certain
small circles as "vlorbik". always a pleasure,
ms.~beals.
http://vlorbik.wordpress.com

Brett Gilland said...

Adding "half-way decent" doesn't evade the No True Scotsman charge. It reinforces it. When you add the disclaimer that no 'half-way decent' teacher would do such a thing, you are simply attempting to rule out the counter-example by fiat, essentially begging the question. The wikipedia page on No True Scotsman gives a pretty good example that should feel eerily familiar.

https://en.wikipedia.org/wiki/No_true_Scotsman

Katharine Beals said...

From Wikipedia:

When faced with a counterexample to a universal claim ("no Scotsman would do such a thing"), rather than denying the counterexample or rejecting the original claim, this fallacy modifies the subject of the assertion to exclude the specific case or others like it by rhetoric, without reference to any specific objective rule ("no true Scotsman would do such a thing").

Person A: "No Scotsman puts sugar on his porridge."
Person B: "But my uncle Angus likes sugar with his porridge."
Person A: "Ah yes, but no true Scotsman puts sugar on his porridge."

The difference here is that the modifier "half-way competent" was not added later to the subject of the assertion; it was there all along.

In any case, Brett, to the extent that you disagree with my statement that no half-way competent teacher would "teach symbols without teaching their associated meanings," we have different ideas about what constitutes half-way competent teaching.