Algebra best practices for students

Question

One thing I notice frequently is that students don't have 'best practices' for doing algebra. Let me given an example:

If students are trying to differentiate, say, $f(x) = (x^2 + x)^2$, they will often write: $2 (x^2 + x) [2x + 1]$, where the brackets are parenthesis that they don't write, so their final answer is $2 (x^2 + x) 2x + 1$. Observing them, I notice that this is because they tend to write $2 (x^2 + x)$ first, then they start to write $2x + 1$ next. But they forget to open a parenthesis between the two expressions!

I always write the $[$ first, but I didn't notice this until I started to think about why students make this mistake. I'm sure there are lots of other similar best practices - I think it took me a lot of making mistakes to learn those best practice techniques. They are now mostly subconscious, so it can be hard even to notice that I'm doing them and students are not.

Sometimes I try to tell students about specific mistakes they make, or give them tips for organizing algebra, but it's hard to catch all of them. So I'm wondering what other best practices people have noticed. I imagine that there is probably some research into this. (With a 'proof' that these best practices always work.)These are the kind of responses I'm looking for:

1. Suggestions of some similar best practices to point out to students (hopefully with helpful, battle tested exercises to help students learn these best practices)

2. References on this (something systematic)

Answer 1

Correct the mistake and then drill them with similar problems. Vary the methods: Lots of problems in class. Lots of problems at home. Kids to the blackboard, calling on students in class, maybe even chanting a problem together. But the key is probably just volume of problems at home.

[Use a problem source that has the answers so they can check their own work. This gives them INSTANT feedback to check their own errors ("Oh duh, I made that parentheses mistake again.") If you are worried about them doing enough than have them hand in worked problems sets (checked for answers by them) and then just grade based on completion. Even very small percentage participation grades can drive behavior. If you are worried about honesty, then have a verification signature block at the top. (If a few lie on that, then don't sweat it, they will get killed on the tests and it is not that much grade %. Plus it will help the non-liars.) Using a source that has the answers also takes off problem examination drudgery from you. But most important feature is to accelerate the feedback time to students and allow them to be more engaged by correcting own mistakes.]

This sort of mistake is typical of students that have not drilled enough. Teachers should not assume that poor explanation (finding the exact killer explanation to make a lightbulb go on) is the problem. It is often lack of care by the students. They get the concept. They just aren't careful enough. So you need to get them to do problems step by step (writing every step, not skipping some...you should do this as well until the whole class has proved efficacy...then you can skip some steps.) And then they need to practice that a lot. And on problems of easy to medium complexity (lots of one star and two star problems. (Math teachers want to emphasize 3 star problems since they are more interesting, but this is like working on complicated football plays for players who need to concentrate on basic skills...a misallocation of time.)

P.s. I would not go back and do a review. Kids will rightfully balk at that. So keep teaching them calculus. But even when they have LEARNED the calculus concepts and apply that part OK, if they are still making more algebra mistakes, you can work on that with lots of calculus drill. In other words, you're sneakily improving their algebra while making them do calculus problems. (Just like you get trig practice in physics or algebra practice in stoichiometry.)

P.s.s. If you have several mistakes happening, think to hand out a written list for their reference. At the beginning of each class. But keep it action oriented. NOT "distributive law" but instead "algebra mistakes that will bite you in the butt in calculus class". Do you capisce the difference in approach? Former is an abstraction requiring a conversion from the idea to the practice. The other is more of a "tips" type of coaching. More immediately understandable and more "in the trenches with the troops". [Could be a fun pedagogical project for you to develop this little handout.]

P.s.s.s. I also think the framing of the question is a little lacking. It's a GREAT job of explaining the concept of the mistake. But you don't give us enough info on the students or their training. How good are they? How much homework are they doing? How long is the class? What approach have you been using to teach them? Etc. I'm not sure if my answer would differ, but I think you have to see this as a multivariable problem and limitations of the student population (smarts, time) and class setting (year length, change of instructors by semester, etc.) are variables. That said, sometimes using something that works in one setting maybe should even be used in another (who cares if it "seems like high school" if it gets the training done best.)

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Good luck!

Answer 2

One thing I'll sometimes tell students is "if you write a minus sign, and then some 'large' expression is going to follow the minus sign, trap that large expression in parenthesis just to be safe." I tell them this (with more of less specific phrasing depending on the class) after having seen many typos in exercises like

Use the definition to calculate the derivative of $f(x) = 3x^2 +1$.

Students will often write something like this without the red parenthesis:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac{3(x+h)^2 +1-\color{red}{\boldsymbol{(}}3x^2 +1\color{red}{\boldsymbol{)}}}{h}$$

Now after seeing your example (and having seen that same mistake you're pointing out many times myself recently), I think it might be a good idea to generalize this line of advice and instead say something like

Anytime you're going to stick a new expression into an expression you're working with, like substituting for $f(x)$ or like using the chain rule or like changing variables (in integral calculus), be sure to trap that new expression in parenthesis just to be safe.

So to reinforce this, whenever I show students the chain rule now I'll be sure to write it with parenthesis:

$$ \frac{\mathrm{d}}{\mathrm{d}x}\left(f(\text{stuff})\right) = f'(\text{stuff}) \color{blue}{\bigg(}\frac{\mathrm{d}}{\mathrm{d}x}\text{stuff}\color{blue}{\bigg)} $$

Answer 3

Here is an idea that I just came up with and haven't tried. (So take it with the appropriate amount of salt.) Try a game of "telephone". That is, form one or more chains of students. Give the first person in the chain some multi-step problem and instruct them to do one step, write the result on a new piece of paper, and pass it to the next person in the chain.

The reason that I'm suggesting this in particular is the following: I think that it may be possible that when students write something like $2(x^2+x)2x+1$, they may be mentally grouping the $2x+1$. The failure to write the parens is simply (over?)confidence in their own ability to remember that those terms were grouped and a too-strong desire for brevity. In fact, they may be completely able to carry the grouping in their minds, without writing it correctly, in an environment where they are done with the problem quickly and their attention never breaks.

Alternatively, break that mentally carried grouping by activities which return them to their own written work after some time has passed.

I do not think that more drilling, as others have suggested, is worth anything; at least with old material. If they are at the level of calculus, they have had plenty of drilling in that stuff already and it didn't help.

Answer 4

I try to model the idea of creating blanks to fill in when I do my work on the board. The general idea is that there are certain procedures that we know are going to lead to terms that need to be grouped, so we should probably make sure that we establish those groups first, then fill in the blanks. Two examples:

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Determine $\cos(15^\circ)$ using the angle addition formula for cosine.

In my own board work, would start the the problem by writing something like \begin{align} \cos(15^\circ) = \cos(60^\circ - 45^\circ) &= \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ}) \\ &= (\hspace1.3cm)(\hspace1.3cm) - (\hspace1.3cm)(\hspace1.3cm). \end{align} The idea is to set up the form of the next step of the computation such a way that we can just "fill in the blanks". Once that is done we might, for example, determine that $\color{red}{\cos(60^{\circ}) = \frac{1}{2}}$, so the work written on the board becomes \begin{align} \cos(15^\circ) = \cos(60^\circ - 45^\circ) &= \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ}) \\ &= (\hspace0.45cm\color{red}{\frac{1}{2}}\hspace0.45cm)(\hspace1.3cm) - (\hspace1.3cm)(\hspace1.3cm). \end{align} Next, $\color{blue}{\sin(60^{\circ}) = \frac{\sqrt{3}}{2}}$, giving \begin{align} \cos(15^\circ) = \cos(60^\circ - 45^\circ) &= \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ}) \\ &= (\hspace0.45cm\color{red}{\frac{1}{2}}\hspace0.45cm)(\hspace1.3cm) - (\hspace0.45cm\color{blue}{\frac{\sqrt{3}}{2}}\hspace0.45cm)(\hspace1.3cm), \end{align} and so on. Again, the big, important idea is to write down the form of the answer or solution before doing anything else.

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Determine $f'(x)$ where $f(x) = (x^2+x)^2$.

This can be addressed in a similar manner: if $f(x) = (x^2+x)^2$, we want to use chain rule. If I am feeling like writing lots of letters, I might let $g(x) = x^2$ and $h(x) = x^2+x$ so that $$ f(x) = (g\circ f)(x), $$ or I might simply refer to the inner function and the outer function. In either case, I would write something like $$ f'(x) = g'(h(x)) \cdot h(x) = ( \underbrace{\qquad\qquad}_{\text{$g'$ at $h(x)$}} ) \cdot ( \underbrace{\qquad\qquad}_{h'(x)} ) $$ or $$ f'(x) = ( \underbrace{\qquad\qquad}_{\text{outer$'$ at inner}} ) \cdot ( \underbrace{\qquad\qquad}_{\text{inner$'$}} ). $$ The "outer" function is the $x^2$ function, which has derivative $2x$, be we want to evaluate this function at $x^2+x$ rather than $x$ (or, if I want to be even more clear, I'll use a dummy variable, and say $$ g(t) = t^2 \implies g'(t) = 2t, \qquad \text{evaluate at $t = x^2+x$}, $$ or something like that). The implies that the derivative of the outer function, evaluated at the inner function, is $\color{red}{2(x^2+x)}$, so on the board we now have $$ f'(x) = ( \underbrace{\color{red}{2(x^2+x)}}_{\text{outer$'$ at inner}} ) \cdot ( \underbrace{\qquad\qquad}_{\text{inner$'$}} ). $$ The inner function is $h(x) = x^2 + x$, which has derivative $\color{blue}{2x+1}$, so we finish the first step of the problem by plugging this into the second blank and getting $$ f'(x) = ( \underbrace{\color{red}{2(x^2+x)}}_{\text{outer$'$ at inner}} ) \cdot ( \underbrace{\color{blue}{2x+1}}_{\text{inner$'$}} ). $$