In my beginning algebra classes, I made a number of changes for this year. Some are related to pre-algebra as a prerequisite for success in beginning algebra. For quite a while, I have concluded that a pre-algebra (or basic math) class is an inappropriate prerequisite to beginning algebra based on analyzing data; my experience this semester might provide an alternative. Students need not spend an entire semester ‘getting ready’ for algebra.
At the start of our course, we have a chapter which reviews operations on signed numbers with a minor emphasis on very basic algebraic expressions (like terms, distributing, etc). This chapter is essentially a typical pre-algebra course contracted down to one chapter; historically, I concluded that both experiences (a course, a chapter) did very little to enhance the readiness of students for algebra.
Instead of spending any class time on signed number operations, we spent every minute of class time on algebraic language, syntax, and concepts. We talked about adding changing coefficients but never exponents, and about multiplying changing both (depending) … and followed this up with a variety of problems for students to struggle with. Much time was spent on translations, but not just in to algebra: we did a bit of work on translating algebra into words; even when students remember the ‘right thing to do’ (procedure) there is often a misunderstanding about what the expression meant.
Given the equation concepts we will be studying, we covered zeros in adding and subtracting. Take some time to interview your students about a simple problem like this:
5y – 5y = ??
When this would come up later in the course, something like 20% of my students would report that the answer is ‘y’ — the fives cancel out, leaving the y. Curiously, textbooks do not have problems involving zero for combining like terms, even though this is critical for later work.
Class spent a lot of time (very frustrating for students) dealing with the different uses of the ‘-‘ symbol: opposite, negative, minus. Some of this was imbedded in the translation work, and others in procedural work. As instructors, we are incredibly careless about reading the ‘-‘ symbol, tending to say ‘negative’ when it is ‘opposite’ (like ‘-x’). The central issue here is often “do we have any options about how to treat this particular ‘-‘?”
To assess this change, I used the same test from prior years with some added ‘difficulty’ — 3 added questions on expressions (including the zero in adding). The initial assessment is that the new emphasis (pre-algebra just in time) helped students with algebraic proficiency without harming numeric skills. The average score on the somewhat harder test was almost identical to the average on the prior (easier test). Obviously, this is not enough to assess the merits of the new approach: if the change does not help students later in the course, then the new process is not good enough yet. I am especially eager to see if the ‘zero’ in adding has been improved.
My belief is that we could improve the outcomes of developmental mathematics by a fairly simple change: do not require any ‘math’ or ‘pre-algebra’ before beginning algebra, just some basic numeracy is good enough. Some students have a direct need to know arithmetic skills for an occupation, but this is a different need than ‘algebra’. By placing almost all students directly into beginning algebra, we eliminate a math course for a large group of students — without producing harm. [From general data I’ve seen, the chance of success in algebra AFTER a pre-algebra course is statistically equivalent to the chance of those below the placement cutoff … and the numbers are not good for either group.]
However, I am also sure that our current algebra courses need to do much more about basic literacy issues — translations, syntax, paraphrasing, etc — as well as a more complete treatment of procedures (zero in particular, but also adding vs multiplying). We tend to move too quickly into applications of literacy (procedures for solving equations, for example) without building the conceptual foundation required for understanding. We need “Pre-Algebra Just In Time”!
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Professional Development | Jack Rotman | 
September 13, 2012 12:20 pm | 
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Jack Rotman
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