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When teaching students how to compute the difference quotient in a precalculus or calculus class, we need them to evaluate the expression

$$\frac{f(x+h) - f(x)}{h}$$

for various simple functions, like linear and quadratic functions. Let's say we are using the function $f(x) = x^2 + x + 1$ for example.

However, students find this to be more or less impossible. Specifically, when trying to find $f(x+h)$, students do all sorts of crazy things while computing the difference quotient. They replace $f(x+h)$ in the expression with things like:

  • $f(x^2 + x + 1 + h)$
  • $(x+h)^2 + x + 1$
  • $x^2 + x + 1 + h$

which are all clearly wrong.

However, if you ask these same students this question:

Find $f(\text{Tomato Soup})$.

they are happy to do so and do it successfully.

In fact, my experience is that they can actually also complete this question:

Find $f(xxx)$.

This is also easy for them:

Find $f(y)$.

However, this question is a whole different story for them:

Find $f(x+h)$.

Why do students see $f(x+h)$ as fundamentally different from the others? What series of questions or conversations can be used to help them?

Chris Cunningham
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    How are they with $f(2x)$? – TomKern Feb 15 '23 at 21:13
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    @TomKern I think a good answer to the question could suggest a better sequence of diagnostics like the one you suggest. I usually go from $f(3)$ to $f(17)$ to $f(☺)$ to $f(\text{Tomato Soup})$ to $f(y)$ to $f(x+h)$. But it's possible I could gain critical insight from some better sequence of diagnostic questions. – Chris Cunningham Feb 16 '23 at 00:05
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    Perhaps the problem is that not everyone sees that what is inside the brackets of the function is a single expression, that must be evaluated before the function is applied? That is, students might not see that $f(x+h)$ actually means $f\bigl(\thinspace(x+h)\thinspace\bigr)$, i.e. given $x$ and $h$ you must first add them to obtain "$x+h$" and then apply $f$ to the result of that? – printf Feb 16 '23 at 05:40
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    One question that comes to mind is whether your oral presentation is helping or hurting here. In particular, when talking to your students (casually, without trying to emphasize the difference), do you pronounce $f(x+h)$ and $f(x)+h$ the same way or differently? – Ilmari Karonen Feb 16 '23 at 14:21
  • @printf This is a very good point and gets to the idea of chunking. I think a good answer could be written focusing on the concept of chunking as it relates to this question. – Chris Cunningham Feb 16 '23 at 16:08
  • "Let's say we are using the function $f(x)=x^2+x+1$ for example." <<< I'm typically careful never to write a sentence like this when teaching. Instead I would write "Let's say we are using the function $f : \mathbb{R} \rightarrow \mathbb{R}$ (linebreak) $x \mapsto x^2+x+1$." (with the second arrow right under the first). There's something a bit wrong when you write "the function $f(x) =$ ...", and that leads the students to the conclusion that $x$ somehow has a special meaning, which will quite possibly lead to confusion when trying to evaluate $f(x+h)$. – Stef Feb 16 '23 at 17:00
  • Thanks @Stef ; the "x has a special meaning" does seem to be a potential problem. I think a good answer could exist that really goes into the dangerous special meanings we assign to the specific variable "$x$" when we do not mean to assign any such special meanings. – Chris Cunningham Feb 16 '23 at 18:25
  • Ask them to do $f(x@h)$ and $f(x&h)$ and $f(xSCRIBBLESOMETHINGVERYSQUIGGLYANDTIGHTh)$. Then ask them again to do $f(x+h)$. – Lee Mosher Feb 16 '23 at 21:27
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    I'm trying to be polite about this but can you all just post answers instead of answers in the comments? – Chris Cunningham Feb 16 '23 at 21:57
  • @ChrisCunningham and others: I wonder if asking them for f() or f(Tomato soup) and similar examples could not potentially generate new problems, since we are talking about functions from numbers to numbers, so what should f() denote? At least I'm confused by these examples (unless you are trying to use Smileys scribbles etc to denote a number). – Michael Bächtold Oct 12 '23 at 05:46

6 Answers6

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You have already applied some good diagnostic tests. I recommend the following additional diagnostics

What happens if you ask them to evaluate each of the following:

  1. $f(3y)$: Passing this test indicates that they can think of a monomial as an input. I am guessing your students will pass this test given the other diagnostics you have run.
  2. $f(3x)$: A student could pass the first test and fail this one if they don't think it is "kosher" to give $f$ an input of $3x$ when $f$ was defined by $f(x) = x^2 + x + 1$. They are experiencing variable scope anxiety. They are getting "the $x$ in $3x$" confused with "the $x$ in $f(x)$". To be more technical about it we should really say "The function $f: \mathbb{R} \to \mathbb{R}$ defined by $\forall x \in \mathbb{R}, f(x) = x^2+x+1$". The $x$ is bound by the universal quantifier.
  3. $f(t+3)$: Passing this test indicates that they can think of a binomial as a "single input". A student who fails this test is not thinking of $(t+3)$ as a "single thing". They might also have difficulty distributing $(t+3)(a+b) = (t+3)a + (t+3)b$ since they cannot conceptualize the $(t+3)$ as a single number to be distributed.
  4. $f(x+3)$: Again, a student could fail this test while passing the third test because of variable scope anxiety.
  5. $f(j+h)$: A student could fail this test while passing all of the other tests because they think that a function should apply to only "one variable at a time". They view $x+3$ as a legit input since it is a single expression involving one variable. They freak out at $f(j+h)$ because they are not sure which variable they should be applying $f$ to. In essence they are seeing a function being applied to an expression with two variables and think that this will make $f$ a "two variable function" even though it is only a one variable function. They don't know what to do with the "second input".
  6. $f(x+h)$ Again, a student could fail this test while passing test #5 because of variable scope confusion.

My success rate at diagnosing these issues is pretty good, but I have a poor cure rate...

Steven Gubkin
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    Ooh -- I like these a lot. I was definitely lacking diagnostics between "xxx" and "x+h." I am definitely fine with improving my diagnostics even if the cure rate is low, too. :) – Chris Cunningham Feb 16 '23 at 00:00
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    A warning: if you apply these diagnostics in order you will have a lot more students who pass than if you apply one at random. The "scaffolding effect" can get them there without any real understanding. – Steven Gubkin Feb 16 '23 at 01:33
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    @StevenGubkin I suppose if the goal is to teach them, that's fine; if the goal is to understand the teaching and learning process, it's not – user253751 Feb 16 '23 at 14:31
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    I think this kind of scaffolding can be an effective teaching technique, but you need to gradually remove the "supports". It can be tempting, as an instructor, to see your students' success after scaffolding and trick yourself into thinking that they now understand. – Steven Gubkin Feb 16 '23 at 14:37
  • @StevenGubkin how do you remove the "supports" in this case? – justhalf Feb 18 '23 at 09:35
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    Great answer and great question. Is there any place that actually conceptually thinks in this way about math education? I mean the diagnostic approach? It seems really smart. – DRF Feb 18 '23 at 09:53
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I know a teacher who (at least in the past) would require students to write underlined blanks in place of the input whenever they were evaluating a function from its formula:

[Examples with $f(x)=5x^2-3x+1$]

Evaluate $f(6)$.

Answer:

$$\begin{align} f(6) &= 5(\text{_______})^2 - 3(\text{_______}) +1 \\ &= 5(6)^2 - 3(6) + 1\\ &= \dots\end{align}$$

Evaluate $f(3+h)$.

Answer:

$$\begin{align} f(3+h) &= 5(\text{_______})^2 - 3(\text{_______}) +1 \\ &= 5(3+h)^2 - 3(3+h) + 1\\ &= \dots\end{align}$$

This wouldn't stop a student from mishandling the $(3+h)^2$, but it might get the $h$ in the right place.

Nick C
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    This is nice, at least for functions of one variable! – Steven Gubkin Feb 15 '23 at 21:56
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    This is interesting. Of course to me, $f(x) = 5x^2 - 3x + 1$ looks very similar to $f(---) = 5(---)^2 - 3(---) + 1$, because I see the $x$ as a "blank" already. Perhaps many students have specialized "$x$" and do not see it as a blank. It seems like this concept should have a name... – Chris Cunningham Feb 16 '23 at 00:04
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    @ChrisCunningham In my personal experience, the concept of x being a "blank" is conveyed very poorly in pre-calculus classes. It is explained like once when the concept of functions is introduced, and then they use x in every single expression ever as the default sign of function variable. With this background, for many students it is quite nontrivial to realize that the x in f(x) can actually be anything, a variable can be denoted with something other than x, and that a variable denoted with x is not special at all unless told so. – Neinstein Feb 16 '23 at 14:53
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    @StevenGubkin It can work with functions of more than one variable by underlining the blanks in different colours – Stef Feb 16 '23 at 16:54
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    I had an instructor that made us use ( ) this way. You wouldn't see underlines. You had to watch it being written out to see it like this because when you're done it just looks normal. But to this day I think of parenthesis as things you drop "inputs" into. – candied_orange Feb 16 '23 at 18:16
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    @ChrisCunningham I think you have a key gap in your model of where students are coming from which this answer(+1) indirectly addresses, but which I'd like to call out explicitly: Students who can evaluate f() understand the x as a blank they can replace with anything, but all of the tests in your question are with expressions that basically don't require parentheses. And similarly most evaluation of functions in their prior experience is with numbers or single letters or maybe a power, none of which require them to invent or practice introducing parentheses in their evaluation of functions. – Mark S. Feb 17 '23 at 08:59
  • One thing I've seen trip people up is the idea that an x here and x there are different x's. I use phrases like: "the x from this" and "the x from that" to make their distinction clear. – candied_orange Feb 18 '23 at 11:35
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The issue here seems to be with substitution, a fundamental operation in mathematics.

Specifically, substituting $x$ with an expression that itself contains $x$.

The issue of how to perform substitution correctly actually arises in the lambda calculus and requires variable renaming to handle conflicts.

I propose using this idea of renaming to avoid confusion. Specifically, we "break down" the substitution $x \mapsto x + h$ into two steps: $x \mapsto y$ and $y \mapsto x + h$. Here, $y$ is what's called a fresh variable, that is, a variable that has not appeared before (and thus avoids any kind of conflict with the expressions we've working with so far). For example:

\begin{align} && f(x) &= x^2 + x + 1 \\ x &\mapsto y & f(y) &= y^2 + y + 1 \\ y &\mapsto x + h & f(x + h) &= (x + h)^2 + (x + h) + 1 \end{align}

Note: Make sure to emphasize that, when substituting a compound expression like $x + h$, it should be wrapped in parentheses, like $(x + h)$. This ensures that it's treated as a "single unit" in the underlying expression (e.g., avoids changing the order of operations).

In contrast, performing $x \mapsto x + h$ all at once induces an additional cognitive load. This cognitive load comes from having to keep track, as the expression is being written out, of which $x$s are "the original $x$s" versus the ones that come from the new expression. In particular, you have to make sure you don't miss an "original $x$" and make sure you don't accidentally replace a "new $x$". For example, consider the following thinking process:

\begin{align} & x^2 + x + 1 \\ & \text{We have to replace $x$ with $x + h$.} \\ & \text{Ok, let's start with the first occurrence of $x$.} \\ & (x + h)^2 + x + 1 \\ & \text{Hmm, which of the $x$'s am I supposed to replace now?} \end{align}

user76284
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    This answer seems to me like a very good hypothesis for why $f(☺)$ is easy but $f(x+h)$ is difficult. To test this more precisely, I should try asking students to find $f(y+h)$, right? – Chris Cunningham Feb 16 '23 at 18:22
  • @ChrisCunningham Yes, that might be a good way to test it. – user76284 Feb 16 '23 at 18:27
  • Excellent answer! – Steven Gubkin Feb 16 '23 at 21:21
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    I think @Steven's answer is awesome, but yours is even more satisfying because it appears to be stating the obvious... so obvious that it ironically resides in our blind spots. – ryang Feb 17 '23 at 06:51
  • The other place where this comes up a lot is expressing the fundamental theorem of calculus. They don't understand why $\frac{\textrm{d}}{\textrm{d}x}\int_0^x f(x) \textrm{ d}x$ is problematic, and why we need to introduce the "dummy variable" $t$ and write $\frac{\textrm{d}}{\textrm{d}x}\int_0^x f(t) \textrm{ d}t$. I wonder whether using @user76284's suggestion regularly when we work with functions in algebra/precalc would resolve this confusion. – Steven Gubkin Feb 17 '23 at 11:11
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    Come to think of it, how heretical actually is it to write sentences like $g(x)=\displaystyle\int_0^x f(x) \textrm{ d}x$ and $h(k)=k\displaystyle\sum_{k=3}^7 a_k$ ? They're certainly shoddy, but is $g(5)=\int_0^5 f(x) \textrm{ d}x$ or $h(5)=5\displaystyle\sum_{k=3}^7 a_k$ more wrong than $\forall xPx\land\exists xQx\equiv \forall xPx\land\exists yQy$ ? @StevenGubkin – ryang Feb 17 '23 at 12:39
  • Incidentally, I used to screw up evaluations similar to this one often for another reason: flat-out forgetting which instances I had already substituted for as I referred to and copied down the perturbed version of an expression — for instance, for a function given by $f(x)=2x$, I might have forgotten $k$ times and wind up saying $f(2x)=2^{1+k}x$, typically just $f(2x)=4x$ though. Then again, most seem to consider me as still having a particularly severe case of reaching the end of a line in a book only to keep scanning over the same — essentially, shoddy carriage return. I do miss scrolls... – Vandermonde Feb 17 '23 at 17:28
  • @ryang, the qualm is less about whether some interpretation could in principle make all instances of such notation tractable and more practical considerations (incl. unpleasantness of reading and handling); example. – Vandermonde Feb 17 '23 at 17:28
  • @ryang To me it makes little sense to present a single answer then claim it authoritatively applies, in the same vein as how depending on the setting, division by zero, the limit at $x=0$ of $1/x$ (not just of $1/x^2$ or of $1/|x|$), and $\sup \mathbb{R}$ could all well be notions one wishes to discuss. Anyone might agree to a system of notation where use of the same symbol for the co-ordinate of integration as for the region is well-formed. Which circles back to matters of actual application and opinions on style. – Vandermonde Feb 17 '23 at 20:25
  • I will however confess that while I dislike the integral, the physicists write that quite regularly and apparently get along just fine and that while it had given me a double-take, the summation was far easier to swallow (having felt like $k\left(\sum_k a_k\right)$ so by the time that $k$ tries to be re-used outside of the bracket, it's just been freed up). I can see nothing objectionable in the last formula because the would-be offending scopes end and become disjoint rather than enclose one another and clash. – Vandermonde Feb 17 '23 at 20:26
  • @Vandermonde Your last comment makes sense (even though it sounds like it shouldn't, lol)! – ryang Feb 18 '23 at 04:45
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My solution is something like Nick's. I think it has helped my students, but I don't have hard data.

I scribble out the x's. So $f(x)=5x^2-3x+1$ becomes $f(scribble)=5scribble^2-3scribble+1$. And I talk about f of any mess gives us 5 times that mess squared minus 3 times the mess plus 1.

I then tell them to circle the x's. For f(x+h) we'll need an (x+h) inside each of those circles, and yes, we'll want those parentheses around it each time.

Then I give them a quiz on finding the derivative (and tangent line) from the definition. I let them take this quiz multiple times, hopefully until they get it right. Then they have to do it again on the first test.

Sue VanHattum
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    I like the scribble idea. The strange thing is that the $x$ is supposed to be that scribble already! So it seems like a symptom of something very strange that we somehow wore out $x$, giving it too much meaning, when it was supposed to be a scribble all along... :) – Chris Cunningham Feb 16 '23 at 16:07
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I think the problem originates from conflicting mathematical notation. In elementary school multiplication is explicitly written out $a \times b$. By middle school it becomes just $ab$, so $a(b+c)$ means $a$ times $(b+c)$.

Then functions come in during the last few years of high school. And the notation there is similar to multiplication, and perhaps student thinks $f(x)$ should mean $f$ times $x$. So $f(x+h)$ should be $f$ times $(x+h)$.

One solution is to insist on explicit writing of all operations. So $a\times b$ for multiplication and perhaps $f \circ (x)$ or $f \& (x)$ for $f$ of/at $x$ or something similar. Of course the downside is lengthy writing. Probably $a\times b$ will be a hard sell, but I think explicit math symbol for "of" or "at" (as in $f$ of $x$) is very much needed in high school. Using blank space to mean an operation when it is the first time you are introducing that operation and it conflicts with another usage is just questionable.

The symbol $f(x)$ emerged rather late (in 1734 by Euler) and exclusively for the practicing mathematicians. The attempt to sell it to high school students appears as a bit of premature activity or oversight.

I have used something like Nick C. recalls.

$$ f\circ (\square) = 5\square^2+ 3\square +2 $$ In my precalculus and calculus I classes (instead of the standard $f(x)=5x^2+3x+2$). This is to emphasize two points, (a) $f$ is the name of a function, as flagged by $\circ$ and it takes an input, (b) the input is whatever is in $\square$. The box helps instruct the student to be careful with substitution. It also avoids the overuse of $x$.

We use blank space to indicate an operation in other places as well, for example when it comes to exponents $b^n$. This is not as much of a problem because it does not conflict with an earlier usage. However when it comes to $f^{-1}$ to mean functional inverse or $f^{(2)}$ to mean the second derivative we run into similar issues in calculus.

Maesumi
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    The question says they handled Tomato Soup, $xxx$, and $y$ correctly, so the problem doesn't seem to be misinterpreting function application. The problem seems to be substitution (see my answer). – user76284 Feb 16 '23 at 18:14
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    This is an insightful answer and I'm glad to see it, but it does not line up well with the evidence I have collected. Specifically, it is very rare for a student to see $f(x) = x^2 + x + 1$ and then compute $f(x+h) = (x^2+x+1)(x+h)$, mistaking the function parentheses for multiplication. It's very possible this causes problems elsewhere in the curriculum though. – Chris Cunningham Feb 16 '23 at 18:24
  • @user76284 In$ f(x+h)$ student has to process two functions, the first is the instruction to add $x$ and $h$, the second is to apply $f$ to the output. This complication is not seen when the input is Pizza or xxx. There is nothing to process there. So there is less change of confusion. (If a recipe for an actual pizza is given the student, and I for that matter, are likely to be confused with the order of operation there as well!) – Maesumi Feb 17 '23 at 18:05
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You start with $f(x)=x^2+x+1$. How about having students determine $f(y)$ where $y=z +1$?

First, substitute $y$ for $x$ to obtain $f(y)=y^2+ y +1$.

Then, substitute $z+1$ for $y$ to obtain $f(z+1)=(z+1)^2 +(z+1) +1$.

Then expand and collect like terms to obtain $f(z)=z^2 +3z +3$.

After a bit of practice, they learn that they can do two steps at once as a "shortcut."

Dan Christensen
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