## “I have no idea why they would do this”

In the midst of a very good post, Julie writes:

They also distributed the exponent to both numbers inside the parenthesis.  I have no idea why they would do this, because I don’t even teach them the power to a power exponent rule.  I have them expand any set of parenthesis with an exponent.  They do love the distributive property, but we-e have never, ever, ever, distributed an exponent.  Sigh.  I failed.

There are a lot of wonderful reflective moments in that post, but I find the above fascinating. Kids are distributing exponents inside the parentheses.  You might be tempted to explain this mistake as confusion between the power rule for exponents and the distributive property, but seeing as the kids haven’t yet learned the power rule that explanation seems false, at least for Julie’s kids.

Distributing exponents is tempting for kids. Why?

Julie has a strategy for helping her kids move past these sorts of mistakes:

So, starting Monday, I am going to have one problem of the day for both classes posted on the board.  It will look like a variation of this.  Evaluate  -3x^2 – 2x + 5 when x = -2.  I’ll throw in fractions, decimals, and any other basic, easily forgettable concept.  This should help them quickly practice evaluating algebraic expressions, exponents, and the order of operations, EVERYDAY.  And we will go over it, together, everyday.

Will repetition work? Will she need something else? Here’s hoping that there’s a follow up post from Julie letting us know how her experiment goes. The question for me is, what are the returns on any given genre of problem? Do they diminish with repetition? How different does a problem need to be in order to promote real learning?

## Midterm Analysis: Inverse Trigonometry

Another area of struggle on my PreCalc midterm is inverse trig functions:

Students had a completed unit circle (radians, degrees and coordinates – they earned them on the exam by completing one on their own the previous week) but could not use a calculator for this section. What types of mistakes do you think students would make?

Feel free to check out the exact mistakes my students made at the Math Mistakes blog.

What I noticed:

While inverse trig was a struggle all year, I was concerned how many students over-thought $sin^{-1}(sin(\dfrac{5\pi}{6}))$. When my co-worker and I wrote the midterm we thought that question would be a gimme, but student after student evaluated $sin(\dfrac{5\pi}{6})$ (to varying degrees of success) and then found the arcsin. It didn’t occur to me to include many of the correct answers I saw (π/6 and even 17π/6!) on my answer key but luckily I was paying attention when I saw the first one and added solutions to my key as I found them. While 17π/6 isn’t a mistake, I am curious why students didn’t notice that the sine and arcsine cancel.

For the next problem I anticipated students would struggle since inverse tangent isn’t easy to read directly off of the unit circle. Not only do they have to recall that tangent is $\dfrac{sine}{cosine}$, but they also needed to be able to simplify a fraction within a fraction. The red flag was students who didn’t recognize that there was an intermediate step of simplification; instead they expected there to be an angle where sine=1 and cosine=$\sqrt{3}$ (apparently I didn’t get a picture of that mistake, sorry). The mistake I did photograph was an interesting one though. The student recognized that the -1 meant inverse, however they misattributed it to the inverse of the ratio rather than the inverse of tangent. That shouldn’t be a mistake I see on the midterm though. Especially after explaining countless times that for tangent the input is an angle and the output is a ratio, while for arctan the input is a ratio and the output is an angle.

On the final problem I saw good attempts and students who couldn’t solve it generally forgot that triangle trig was an option. I think those issues were more indicative of a lack of studying than a misconception. Or maybe an inflexibility in their approach? I’ve certainly seen students start a problem only to find that they can’t solve it. At this point they call for help assuming they made a mistake somewhere. Many times their work is accurate and what they actually need is to consider a different method. They could do with a good dose of productive struggle.

Requests for feedback:

I am continually surprised by how much difficulty students have doing anything backwards. In geometry we will study angles formed by parallel lines until they can fill in all 8 angles without a second thought. But then when I give them angles and ask if the lines are parallel, they look at me like I’m asking them if unicorns have purple stripes. The same thing happened here, students are great at finding values of sine, cosine and tangent if I give them the angle. But once I switch it up and ask for the angle instead, they start giving nonsensical answers. What is it about working backwards that students find so challenging? How do you help students recognize the structure?

p.s. Do unicorns have purple stripes?

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## Midterm Analysis: Inverse Functions

After attending the middle school data analysis meeting, I decided to do some analysis of my own.  I started with the inverse function problem on the PreCalculus midterm since it was the only topic students did well on during the semester but struggled with on the exam.

What types of mistakes do you think students would make?

Have a hypothesis?  Okay, now you can view the exact mistakes my students made at the Math Mistakes blog.

Back?  Great!  So here’s what I noticed:

Many students made small errors in basic algebra that may have been due to fatigue (this was question 23 of 25 and most students had to stay past the scheduled 90 minutes to finish the exam).

It was a “check if these are inverses” question but many students attempted to check by finding the inverse themselves, rather than composing the functions. When their solution didn’t look exactly like the given inverse they said “not inverses” and moved on. In fact, the inverse many students found was equivalent to the one given (they found $\dfrac{1}{x} + 3$).

Next time I teach/assess this:

In class/homework I need to give more examples of equivalent functions that don’t look equivalent so students are compelled to check more carefully.

I wonder if giving the functions in reverse order would change the results for the better (so students who found the inverse themselves would start with $\dfrac{1+3x}{x}$ and work from there).  I’m inclined to think I would see more algebra mistakes but fewer students who found the inverse correctly and didn’t see that their solution was equivalent to the given one.

My co-worker and I found it difficult to come up with a problem that requires students to solve the inverse since many of our students could immediately “see” the inverse without having to write anything down.  I will need to spend more time finding good problems with interesting inverses that aren’t too complex.

Requests for feedback:

Did the mistakes my students made match the ones you expected to see?  What would you do to remediate this year or prevent such mistakes next year?

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