Developmental Mathematics Revival!

Math is Non-Linear

Within our current curricular structure, there is premise of linearity — that the topics need to be studied in a particular order (in general), AND that students will not be able to understand a later topic that depends on some earlier topic.

This is not true.

At the global level, we have “pre-algebra” → “beginning algebra” → “intermediate algebra” → “college algebra” → “calculus”. We have an existence proof that this is not true … students (in larger numbers than we’d like to admit) can perform as well in the college math class without taking the prior course ‘required’ according to placement tests.

At the micro level, each course is constructed (normally) by a list of topics and then outcomes within each topic. If the topics were basically linear, however, we would always teach these topics as connected to other topics (prior and future); we do not do so, making the non-linear nature evident.

You might consider both of these points ‘logic chopping’, and classify this as a worthless post. So … let’s move on to a different analysis.

One of the imbedded linear conditions is ‘fractions’. Let us assume, for the moment, that we have sufficient rationale to justify the inclusion of rational expressions in the algebra curriculum. Back in pre-algebra, we cover various operations; we suggest to our students that they need to master simplifying arithmetic fractions prior to simplifying rational expressions (‘algebraic’ fractions). In between these two topics, we covering factoring … and feel good about the parallels between prime factoring in arithmetic and polynomial factoring in algebra.

I see two reasons why the sequence of these topics is not linear. First, arithmetic fractions deal with place-value numbers; students need to transform these additive forms into multiplicative forms to simplify by factoring … and this is a more advanced topic than algebraic fractions (which are often multiplicative in the first place). [Just show a group of students the fraction “54/24” and see what fun they have with these hidden binomials… compared to ‘8x²/4x’ (obvious monomials) or ‘(6x+24)/(8x+32)’ (obvious binomials).]

Second, the fraction topics are not linear because of the extra rules often imposed for arithmetic — improper fractions and mixed numbers. These rules do not exist for algebra. Because arithmetic is a relatively advanced topic, we often cover these topics with a series of guidelines or procedures for each type of problem; few of these items transfer to algebraic fractions.

Personally, I would rather help my students understand a mathematical topic like ‘algebraic fractions’ without having to cope with layers of bad learning relative to arithmetic fractions. In this area, I can not expect this to occur.

I suspect that you have noticed some of what I am talking about. We cover linear forms before quadratic before exponential — and yet some students ‘get’ the more advanced topic while they still struggle with the earlier one. You might have noticed, in particular, that some students just don’t understand these properties of real numbers (associative, commutative, etc) … and then they start simplifying expressions with terms and parentheses, and they get an insight into what the properties were saying.

Math is not linear. We are not building machines that have a clear dependency in design; we are dealing with human beings working with ‘mathematics’ — a collection of scientific domains dealing with different types of objects. Our job is to identify the most important mathematical concepts appropriate for each course, and allow the course to be non-linear; we can revisit concepts, bring in a new perspective, look at a different context.

Math is not linear. Our curriculum tends to consist of a series of two to five courses, presumed to be linear … and we create many opportunities for students to leave math. We tend to put a lot of “preparing for the future” procedures in each course, which tells students that they will not study the good stuff of mathematics until much later (if at all). Our job is to show good mathematics to students in all courses; rather than seeing mathematics as a difficult set of hundreds of procedures, they might just see mathematics as interesting.

We can not expect to inspire students by using a linear curricular model which ensures that the early courses cover very little of interest … and even less of intellectual value.
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