## Tuesday, March 10, 2020

### A Lack of Algebra Skills

My pre-calculus (trig/analysis) students finished a test Monday, and the results were positive only to manufacturers of red ink.  While over half of my students scored A's and B's, the most frequent grade was an F.  And some of those scores were such low F's that a better grade would be a G.

I guess I could take heart that many students, even when making a mistake in a problem, got the trigonometry correct.  They understood the use of sum and difference identities, half-angle formulas, and double-angle formulas.  No, too many of the errors are algebra errors, e.g.:
• (cos x + sin x)^2 is not cos^2 x + sin^2 x, and similar errors
• not knowing how to manipulate numbers under a radical sign
• not correctly manipulating numbers, especially fractions within a fraction
• not "seeing" simple substitutions that would lead to a solution
Four or five years ago my district switched from the so-called traditional Algebra 1-Geometry-Algebra 2 sequence to Integrated Math.  While I'm not fond of integrated math--I prefer focusing on a set of topics for a longer time for deeper understanding--I don't think it's the integrated math per se that's causing such a precipitous drop in Algebra skills.  I think it's the integrated math curriculum that our district has adopted.  While it's big on visualization of topics, it's weak on actual algebra.  There aren't large problem sets divided into easy, moderate, and difficult problems that a teacher can choose from and help differentiate for students of different abilities; no, there are only a small number of practice problems on each topic, and there's no differentiation for the teacher (or student) as they're pretty much all bland and of the same difficulty.

I guess our integrated math teachers could try to augment the curriculum, but teachers will recognize that that brings with it its own set of problems.  And honestly, our teachers aren't paid to create curriculum, they're paid to deliver the district-approved curriculum.  Our district recognizes that there's a problem but takes only half-measures in response, for example, adopting an "augmentation curriculum" for our Integrated Math 1 classes--after how many years?  and with only how many years left in this current adoption before we rush to buy a new round of crappy textbooks?

I try not to criticize my school or district on this blog, but when I see the obvious damage that's being done to students by the district, it bears pointing out.  Our textbook adoption process is broken because our district is broken.  There is no leadership at all; in fact, the response from the district is often "we're letting the schools experiment to see what works best"--and then, if something does work well at one school, the district won't mandate it, but rather will allow other schools to "try" it if they want to.

In the meantime, too many students are learning math at a superficial level and are not prepared for STEM majors after high school.  We've closed doors on them before they ever even knew the doors were there.

Update:  I've checked with other teachers who administered these chapter tests (in their many versions) and all are horrified by the algebra mistakes.

Ellen K said...

My observation is that the educrats only read data.
As a result, when they see that certain key demographic groups are failing advanced math & science, instead of asking why they aren't succeeding, the answer they give is to water down complex subjects in order to give the appearance of success without really contributing much to the student's individual abilities.

Mr. W said...

everyone of those mistakes I have seen time and time again in all my math classes, from pre-Algebra to pre-calc.

My favorite story is that when I taught Algebra 1 I knew how important factoring was for the course, CSTs, and future courses. So in my daily we would have a factoring problem. Literally 4 days a week, for more than a semester, we factored a problem. So naturally by the end of the year they were pretty good at factoring.

Next year I have a student who took Geometry over the summer and was now in my Algebra 2 class, good kid. We get to factoring and he literally says I have no clue how to do this. I about lost it. Well after a few problems, it re-clicked and he was one of the better students with factoring. But for that moment I really thought I wasted all that time.

Ex-PFC Wintergreen said...

I continue to persist in the belief that the only way most people (basically, everybody who’s not, say, Gauss) master math is through repetition. You learn how to do new problems more-or-less by rote; you repeat your rote learning with variations; sooner or later (sooner, with a good teacher) you start picking up on the patterns, internalize them in your brain (to some extent, hijacking our inborn capabilities, such as how we hijack our language capabilities to form a counting system), and develop sufficient mastery to understand what you’ve just learned in some depth and be capable of extending it to new situations.

But there is no substitute for the repetition phase if the goal is mastery. Some students will need more, some less, but no amount of “teaching the concepts” will eliminate (or even much reduce) the need for repetition. It seems every generation of math pedagogy forgets this brutal lesson and has to re-learn it, to the detriment of the students. But we come to subjects like math with the brains that evolution has forged, not with what education “reformers” wish we had, and the more we take that into account, the better off we’ll be. Learning math above Algebra 1 will always be non-trivial though; there are no shortcuts.

Darren said...

"There is no royal road to geometry."