Should young math students be taught an abstract concept of form?

Question

Should a more general concept of the "form" of an equation or expression, be taught to math students as young as elementary school? I'm a fairly new tutor--do more experienced teachers think this would be useful (or perhaps have done it already)?

EDIT: Some people are answering as if I was asking about teaching specific form, but my question is really about teaching general form for multiple purposes. Perhaps it is unfortunate I gave all my examples as specific forms. I'm editing all of the following text to make my pedagogical suggestions clearer.

Specific forms appears in modern teaching in several ways. In Algebra I, students are taught the "point-slope form" of a line, the "slope-intercept" form, etc.

In Algebra 2 they are taught the standard and vertex forms of parabolas, the standard form of an exponential equation, and so on.

Yet, my typical tutoring student doesn't understand form as a general concept well.

If I give an Algebra 2 student a question like this: give them vertex form of a parabola,

$$ y = a(x-h)^2 +k $$

then a specific parabola such as

$$ y = -(x+1)^2 -1 $$

and ask what has been substituted for $a$, $h$, and $k$ (here, -1, -1, and -1) they usually are confused about what I'm asking. Then, if they have some idea, they get the sign wrong on $h$ or have no idea that $a=1$ in a parabola like $y=x^2$.

If they understand in a general way what it means to express the form of an equation as $y= a(x-h)^2 +k$, that is if they saw each letter and each mathematical structure as having a higher meaning to them, then working with a new specific form would be trivial.

Then what I do is take them to remedial discussion of form. This could actually be done at the pre-algebra level, such as giving a student an expression like $$\frac{AB}{C}$$ then asking which of the following expressions are in that form? $$ \frac{1}{2 \cdot 3}, \frac{1 \cdot 2} {3} , \frac{1 \cdot 2 \cdot 3}{1}$$ My 7th grade students usually are confused by what I mean for a second, but quickly get it. They answer "the second one."

Then I ask "in the second expression with numbers, what number is in the same place as A? As B? As C?"

Again they are confused for a moment, but if I use that language "in the same place as..." they see.

From there is takes off rapidly. I can give them laws of manipulating expressions like $$ \frac{AB}{C} = (A)\frac{B}{C} $$

and have them do three things: (1) write a similar expression by substituting specific numbers for A, B, and C, (2) identify matching patterns and name A, B, and C, (3) put the rule to use by manipulating expressions.

There are several goals here. Partly I'm working toward an understanding of specific forms. But more generally I'm training them to see the visual structure of a form and training them to conceive of "manipulating" the expression as "changing the form."

I'm also working up to giving them the rules of form manipulation in a way they can understand. A student who has been trained in general form from an early age will know right away what to make of a new factoring rules such as $$ a^2 - b^2 = (a-b)(a+b) $$. They'll know to ask if this rule is constrained by the independent variable (i.e. whether any $x$ is untouchable or not part of the form parameters) or whether this form identity is true for any substitution into $a$ and $b$.

They can learn to recognize ways of manipulate an equation and learn how to express them as "form diagrams": for instance $$ r(x-a) = b \implies x-a = \frac{b}{r} $$

They can practice writing their own "form diagrams" to communicate their thoughts back to the teacher.

Students will "get" much more deeply what it means to say that "the vertex form is" $ y = a(x-h)^2+k $, the main goal is far beyond learning specific forms.

So what I'm wondering is whether this approach (teaching general form) is used in any schools. My students, who are in both private and public schools in the Southern California area and range from 6th grade to college, pretty much universally have not been introduced to form as a general concept.

Would it help all math teaching to introduce form in this way?

Answer 1

I have not done this, but now that I consider it I think it might be helpful. I have certainly had students who, when applying the quadratic formula, will look at an equation like $3x^2+5x-10$ and write $A=3x^2, B=5x, C=-10$. Then they plug those monomials into the quadratic formula and stare helplessly as they try to figure out what to do with all of those $x$s.

Really, though, I think the best way to teach things like the point-slope form of a line, or the vertex form of a parabola, is to first do many specific examples, and then observe that they all have the same pattern to them. Take the point $(2,3)$ and a slope of $6$, and show them (don't just tell them the end result) why any other point $(x,y)$ on that line should satisfy the equation $y-3=6(x-2)$. Then repeat, using a different point and a different slope. Do this 3 or 4 times, and then ask: You notice how these all have the same basic pattern to them? What do they all have in common? Then write the general form, following the students' responses. You can point out that the letters you use ($h, k, m$) really don't matter -- it's just as fine to write

$$y - \textrm{y-coord of point} = \left(\textrm{slope}\right)(x - \textrm{x-coord of point})$$

as

$$y-y_0 = m(x-x_0)$$

or

$$y-k=a(x-h)$$

A new form should never be taught before examples; rather, the form should be a generalization or summary of a number of examples that have just been encountered.

They should, of course, be taught what the standard conventions are, and you should make a habit of using them yourself so that when they are in other contexts they speak the same language as is used in other classrooms.

Answer 2

I think an "abstract concept of form" is not helpful in mathematical education up to university level.

To quote YiFans comment,

I honestly think all this hype about "point slope form", "vertex form", etc. is really unnecessary

and I'd like to expand on this. One of the core concepts of algebra (or math in general) is the equation. Students have enough trouble grasping the concept of variables and if you add "form of a term" as another concept, this will lead to one extra thing to learn for the students.

But there is no need to add this layer of abstraction if the students really understand the meaning of variables and terms.

My guess is that students that are "fluent" in algebra can easily understand an abstract concept of form, but those who struggle to interpret (let alone manipulate) terms will only be confused by this new point of view.

Quoting OP:

They can learn to recognize ways of manipulate an equation [...]

Again, this boils down to "applying the same operation to both sides of the equation". This is what matters most and there is no need for the extra step of introducing the abstract concept of form.