Act One. In my last post, I made the case for a math curriculum that demands higher order problem solving skills from the kids, starting in middle school. I was glad to come across a recent post by Scott McLeod that also addresses the what, rather than the how of teaching.
Scott’s main points are that “we’re herding many, many students through math classes that are largely irrelevant to their future life success” and that “high school math problems generally [are] either decontextualized pure math problems (students: who cares?) or pseudocontextual word problems (students: who cares?)…. There wasn’t much on the math test [ the ACT] that I think would be of interest to typical high school students outside of the artificial environments of classes and testing.” Commenting on the ACT (which he took again at age 44!), Scott wrote: “[The] exam does … focus heavily on procedural knowledge (and provides a variety of contexts in which students can show that knowledge). Occasionally it assesses – in a fairly limited bubble test way – some application, synthesis, analytical, or inferential skills. But for the most part, the exam does not get at higher-order thinking skills in any substantive, applied, hands-on, performance-based way.”
Scott is an Associate Professor at the University of Kentucky – his opinion carries weight as an education professional. There are other educators too who agree that there is a need to change the math curriculum to emphasize higher-order thinking skills – but how do we get change it?
Act Two. The current California Algebra II curriculum has 27 standards. In the textbook used at our school (Glencoe) – a fairly typical Algebra II textbook – these 27 standards are broken down into 97 lessons. California typically offers 180 instructional school days, even though many districts – under budgetary pressure – are shortening the school year. With 180 days, that works out to about 1.86 days per lesson. Taking account assessments, exam feedback and other non-teaching activities, this may well bring down the average to about 1.8 days per lesson. How can one introduce, for example ‘Exponential Growth and Decay’ – lesson 9.6, and then reach higher order thinking skills on this topic in 1.8 days? As the Brits would say, “Not bloody likely”. Besides that, we don’t let the concepts “marinate” in the minds of the kids either – the next lesson in the sequence is ‘Midpoint and Distance Formula’. Connections anyone?
So what happens in real life? The curriculum gets cut down – we teach only the “key standards”. We blow through some topics so that we can spend more time on others. At the end of the year, all teachers agree that it is virtually impossible to teach all the topics in the book well, or even teach well all the topics tested on the state exams and we fall back on phrases such as “mile wide and inch deep curriculum”.
Act Three. Since we do cut down the curriculum already, does an Algebra II course truly need to cover all these topics? What math should we teach? Specifically, do we make math relevant or do we say that math is intrinsically important, that it creates habits of logical thinking and that relevancy is not a concept that has merit here?
The truth, as I see it, is that this is not an either-or proposition. A student who has in mind a career as a dental technician probably does not need Cramer’s rule, multiplying and dividing rational expressions, or infinite geometric series. A student planning to go into engineering probably will need to get a good handle on all of these (except possibly Cramer’s rule, to be replaced by computer algorithms).
The point is that we should not have one curriculum for all students.
For reasons of ability, desire or any other factors our kids do not arrive at Algebra II with the same degree of math knowledge or math reasoning skills. In my experience, and that of my fellow teachers, this course serves to further differentiate students by performance.
The situation is analogous to that of a jeweler that receives raw materials from one source: chains, stones, rings, etc. Not all these raw materials are of equal quality and workmanship. The jewelry maker has two choices: either invest in a lot of pre-processing to raise the quality of the below-par raw materials to some acceptable level or differentiate the merchandise that s/he sells. If s/he chooses to differentiate, the jeweler can select the best raw materials, process them into beautiful jewelry and sell them to high-end stores. The below-par raw materials can be made into cheaper jewelry that can be shipped to stores that do not cater to the high-end clientele.
Differentiating your products and aiming these products to specific target markets is the technique that most businesses will use under these conditions – pre-processing is too costly and often does not result in high quality products anyway. In education, the equivalent of product differentiation is called tracking.
And…tracks can cross.