# How Much Should Students Retain from High School Math?

So, after a wild foray into differentiation and applications of differentiation, we just started to learn how to deal with exponential and logarithmic functions, before starting our equally wild foray into integration. I gave a pre-test to see what they remembered about exponential and logarithmic functions from the past.

It was less than encouraging when I think about the quality of math education we give to high school students.

I gave them a little more than 10 minutes to fill in what they knew on a two page pretest. Most were just about as blank as the one below:

Most students could not sketch anything that even looked like an exponential function (these are four different student’s answers):

Very few knew how to evaluate a logarithm, or even knew what ln(x) meant:

Which might have also been because not even had any idea of what e was:

Okay, so I have fully embraced the whole idea of “teach the where they are.” I enjoy teaching this class because I have the liberty to slow down and fill in gaps in their math backgrounds and then can teach the Calculus material with the depth in conceptual understanding but without the depth in mechanical skills (if that makes any sense). Sure it’s harder to explain the idea of a limit as x goes to negative infinity for e^x when they don’t fully even understand what a negative exponent is, but I wouldn’t be in teaching if it was an easy job. So, I’m not complaining that I have to change my sacrosanct yearly plan and “lose time” – I’m just wondering if this is all they can remember about a major family of functions that I’m sure they learned about in both Algebra II and Precalculus, what did we accomplish in teaching them this in the first place? What do we expect our students to know coming out of high school? How big is that gap between what we teach and what they actually learn?

I’m not blaming their past teachers, and I’m especially not blaming my students (though my non-AP Calculus students are labeled “weak,” many of them are really quite talented but have yet to find that spark of loving mathematics or need a bit more time). Worse, I have no ideas for solutions, just those questions. I guess I can find comfort in knowing that other people are asking the same thing, and even better, that those are the questions that are driving (some of the) reform in mathematics education right now.

Posted on February 2, 2012, in Math Ed. Bookmark the permalink. 6 Comments.

1. I will be interested to hear how quickly students remember these concepts. While its concerning they didn’t make a table to graph the function I never expect good recall. What I do expect is the “oh yeah!” moment after a few well phrased questions and some quick recap. I don’t remember Cramer’s rule since I’ve never taught it but if I looked over an explanation I could easily comprehend the steps and reasoning. I hope your students have that deep understanding hidden just behind the curtain.

• yeah, good point – i wrote that right after i had looked at those pre-tests and was really frustrated. when we did review, they were okay, but a lot seemed to remember some of the procedural stuff, but have none of the understanding. like, for some reason, the log laws were in some of the kids’ heads who really didn’t know what a logarithm was. i would have kinda hoped it would be the opposite (like they know exactly what a log is, but are fuzzy on some of the details). i also just think the general shape of an exponential graph should be something that’s part of the deep understanding of exponential functions. i mean, anytime we looked at an increasing concave up graph, whey would call it “exponential” without the understanding that the word exponential really only applies in a very particular context. as further evidence that they didn’t understand the most important idea of exponential functions, we did a couple of problem solving activities that showed me they had no idea how to recognize or apply them in a situation. like basic stuff – coming up with the equation for the number of people helped in the movie “Pay it Forward” (where one person helps three people, and each one of them helps three ad infinitum) and it was disastrous! even after reviewing exponential functions for a few days, kids were still writing “3n” or “n^3” for their equation.
you are definitely right though that certainly we shouldn’t expect kids to remember everything right away.

• yeah, good point – i wrote that right after i had looked at those pre-tests and was really frustrated. when we did review, they were okay, but a lot seemed to remember some of the procedural stuff, but have none of the understanding. like, for some reason, the log laws were in some of the kids’ heads who really didn’t know what a logarithm was. i would have kinda hoped it would be the opposite (like they know exactly what a log is, but are fuzzy on some of the details). i also just think the general shape of an exponential graph should be something that’s part of the deep understanding of exponential functions. i mean, anytime we looked at an increasing concave up graph, whey would call it “exponential” without the understanding that the word exponential really only applies in a very particular context. as further evidence that they didn’t understand the most important idea of exponential functions, we did a couple of problem solving activities that showed me they had no idea how to recognize or apply them in a situation. like basic stuff – coming up with the equation for the number of people helped in the movie “Pay it Forward” (where one person helps three people, and each one of them helps three ad infinitum) and it was disastrous! even after reviewing exponential functions for a few days, kids were still writing “3n” or “n^3” for their equation.
you are definitely right though that certainly we shouldn’t expect kids to remember everything right away.

2. I have been asking myself this question quite a bit, lately.

3. One thing your post made me think about is the embarrassingly miniscule number of things I can remember (as a trained adult) after having been exposed to them only one or two times. The number of things I can remember from high school that I was not completely immersed in is even smaller.

I’ve been thinking lately that in many cases, we are not so much “teaching” kids concepts and skills in high school as “exposing them to these ideas” as they whiz by. While the kids are whizzing by, they are going through massive, system-wide developmental changes that are rewiring their entire beings (and brains) every night while they sleep. I frequently look out at one of my classes and wonder, who are you people and where have you been for the last week while I was trying to guide the class that looks like you through this material?

This is one of the reasons I have become so interested this year in the developmental psychology of learning mathematics. The things that have really stayed with me from my high school learning are the things that really arrested my attention — stopped it dead in its tracks. There was so much going on for me in high school (whole-life-wise) it’s a wonder so much stayed with me at all.

– Elizabeth (aka @cheesemonkeysf on Twitter)

4. You’re right that it is surprising that they don’t remember the basics. I’d be happier if my students could remember and explain the general shape, but not remember the log laws. Hell, I never memorized the log laws until I started teaching. But I think it just goes to show that many of our students are so used to focusing on the procedural aspects of any concept that they spend too little time and energy on understanding the main ideas. I am convinced that while some students rehearsed log laws until they knew them like their own phone numbers, probably none of them thought to write down and discuss their own explanations of what e really is.
As teachers, we can do something about this in terms of the activities we choose to do in class – we can focus on understanding and discussion rather than procedures – but then we must be consistent and assess the same things in exams. If our tests focus on procedures so will the students, naturally and unfortunately. I wish I could make my kiddos understand that deeper understanding will help them on procedure-based tests as well.

Just one more thing: from a cognitive psychological point of view, it seems much more likely that the students still have the knowledge about logs and exponents in their long-term memory, so it’s not gone, it’s just that they couldn’t recall it into short-term memory and from there onto paper. That’s why they’ll recognize the information if presented to it again, because they’re seeing something that they already have in their long-term memory. The way to improve recall is by practicing recall (if anything is certain in psychology it’s this little known fact) – and if tests are procedure based then students will have practiced recall of procedures more than recall of understanding.