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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?

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  • $\begingroup$ (1) As is common with a bazillion questions here, you need to describe the SKILL level of your kids if you want advice (and also to optimize your teaching strategies). (2) In terms of practical advice, give them lots of practice with transforming into different forms. Do this before you ask too many questions requiring transforming (to answer some other question). $\endgroup$ – guest Nov 19 '18 at 0:25
  • $\begingroup$ Did you already came up with a definition of "form" for this purpose? $\endgroup$ – Jasper Nov 19 '18 at 16:40
  • $\begingroup$ "Then I ask "in the second expression with numbers, what number is in the same place as A? As B?" - can be either A or B because of commutative property. $\endgroup$ – Rusty Core Nov 19 '18 at 16:46
  • $\begingroup$ to both @Jasper and RustyCore, "form" is not easy to define because it's malleable. We would start with a very literal definition: some number or variable in the exact position of the "prototype form." Later flexibility can be introduced: understand implied 1, knowing that some "letters" can be replaced with zero without changing the form, while other zeros do change it (in parabola vertex form, $h$ and $k$ can be zero, but $a$ cannot). Accepting commutation when it's appropriate. Not sure how to do this. $\endgroup$ – composerMike Nov 19 '18 at 17:10
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    $\begingroup$ "form diagrams": these are just transformations using standard arithmetic rules. This is akin suggesting memorizing three formulas s=vt, v=s/t, t=s/v instead of one. $\endgroup$ – Rusty Core Nov 23 '18 at 6:33
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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.

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  • $\begingroup$ "y−y-coord of point=(slope)(x−x-coord of point)" -- in this equation, where slope comes from? I'd say, it is more straightforward to define slope first. Draw a line, then define slope as rise/run, calculate it, then plug it into the ratio to get the formula of linear function that describes the line drawn. Use any letters in the process. Proof of linear function graph being line and vice versa is extra point. $\endgroup$ – Rusty Core Nov 19 '18 at 23:01
  • $\begingroup$ @RustyCore It comes from the fact that, by definition, if $(x,y)$ and $(2,3)$ are both on a line whose slope is $6$, then $\frac{y-3}{x-2} = 6$. That's what slope is: the ratio of rise to run. Now just multiply both sides by $(x-2)$ and boom, point-slope form appears. Obviously you can do this just as well for a generic point designated $(h,k)$ and slope denoted $m$, but as I wrote it seems to me always better to begin with multiple specific examples to establish the pattern and then go to the general case as a summary/synthesis. $\endgroup$ – mweiss Nov 19 '18 at 23:09
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    $\begingroup$ "That's what slope is: the ratio of rise to run." - exactly. And this to me seems much simpler: draw whatever line you want, calculate its properties like slope and y-intercept, then plug them into proportion and get the equation. Quite in the vein of "discovery" so fashionable in modern school math, and is very straightforward and easy to understand. No need to establish patterns if you can go straight from definition of slope. $\endgroup$ – Rusty Core Nov 19 '18 at 23:27
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    $\begingroup$ In some cases the form does matter, though. As a Calculus instructor I am forever surprised by how many students insist, when finding the equation of a tangent line, on using the slope-intercept form to first write (for example) $y=6x+b$, then plugging in values for the $x$ and $y$ coordinates of the point and tangency, then solving for $b$. So much simpler to just use the data you have, rather than find an irrelevant parameter like the $y$-intercept. PLUS it extends naturally to quadratic approximations, and then Taylor polynomials and Taylor series in general. $\endgroup$ – mweiss Nov 20 '18 at 0:04
  • $\begingroup$ @mweiss, thanks for your considerate answer. Partly my fault, but you seem to be addressing the teaching of specific forms, when my question was really about teaching general form. See revisions. $\endgroup$ – composerMike Nov 22 '18 at 8:55
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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.

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  • $\begingroup$ The problem, @Jasper, is that in math education we often expect students to do things which require a fairly deep and abstract understanding of the domain. It appears to me that math educators shy away from teaching such things directly, but in my opinion that makes it harder. If we expect students to implicitly understand a concept, then we should make it explicit no matter how abstract it is. If it's not an easy concept, then we should introduce it progressively through examples, gradually making it more broad. $\endgroup$ – composerMike Nov 22 '18 at 8:54
  • $\begingroup$ The deep and abstract understanding of the domain is indeed required, I just argue that the domain is "variables and operations" and not necessarily "form". $\endgroup$ – Jasper Nov 22 '18 at 15:11
  • $\begingroup$ A lot of manipulating equations involves acting on only one side of the equation: applying the distributive property to expand expressions, combining like terms, etc. These change precisely the form of an equation without altering the quantities on either of the two sides. $\endgroup$ – mweiss Nov 25 '18 at 1:18

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