Nested and Sequential: Not in Math, or “What’s Wrong with ALEKS”?

Much of our mathematics curriculum is based on a belief in the ‘nested and sequential’ nature of our content — Topic G requires knowledge of Topics A to F; mastery of topic G therefore implies mastery of topics A to F. A popular platform (ALEKS) takes this as a fundamental design factor; students take a linear series of n steps through the curriculum, and can only see items which the system judges they have shown readiness for.

Other disciplines do not maintain such a restricted vision of their content, whether we are talking about ‘natural’ sciences or social sciences … even foreign language curricula are not as “OCD” as math has been. As a matter of human learning, I can make the case that learning topic G will help students master topics A to F; limiting their access to topic G will tend to cause them to struggle or not complete a math class.

Whether we are talking about a remedial topic (such as polynomial operations) or a pre-calculus topic (function analysis), the best case we can make is that the topics are connected — understanding each one relates to understanding the other. Certainly, if a student totally lacks understanding of a more basic idea it makes sense to limit their access to the more advanced idea. However, this is rarely the situation we face in practice: It’s almost always a question of degree, not the total absence of knowledge.

At my institution, this actually relates to our general education approach (as it probably does at most institutions). In our case, we established our requirements about 25 years ago; the mathematics standard (at that time) was essentially intermediate algebra. The obvious question was “how about students who can place into pre-calculus or higher”. One of my colleagues responded with “these courses are nested and sequential; passing pre-calculus directly implies mastery of intermediate algebra”. My judgment is that this was incorrect, and still is incorrect. Certainly, there is a connection between the two — we might even call it a direct correlation. However, this correlation is far from perfect.

Learning is a process which involves forward movement as well as back-tracking. We are constantly discovering something about an earlier topic that we did not really understand, and this is discovered when we attempt a connected topic dependent on that understanding.

Some of my colleagues are very concerned about equity, especially as it relates to race, ethnicity, and social status. Using a controlled sequence model has the direct consequence of limiting access to more advanced topics and college-level courses for groups of concern … students in these groups have a pronounced tendency to arrive at college with ‘gaps’ in their knowledge. A mastery approach, although a laudable goal, is not a supportive method for many students.

In some ways, co-requisite courses are designed based on this mis-conception — we ‘backwards design’ the content in the co-req class so that the specific pre-requisite topics are covered and mastered. I don’t expect that these courses actually have much impact on student learning in the college-level courses.

Back in the ‘old days’ (the 1970s) a big thing was programmed learning, and even machine learning. The whole approach was based on a nested and sequential view of the content domain. My department used some of those programmed learning materials, though not for long — the learning was not very good, and the student frustration was high.

Our courses, and our software (such as ALEKS), are too often based on a nested and sequential vision of content — as opposed to a learning opportunity approach. By using a phrase “knowledge spaces”, ALEKS attempts to sell us a set of products based on a faulty design. Yes, I know … people “like ALEKS” and “it works”. My questions are “do we like ALEKS because we don’t need to worry about basic decisions for learning?” and “do we think it works because students improve their procedural knowledge, or do they make any progress at all in their mathematical reasoning?”

Obviously, there if a basic fault with a suggestion to remove the progressive nature of our curriculum … there are some basic dependencies which can not be ignored. However, that is not the same as saying that students need to have mastery of every piece or segment of the curriculum. No, the issue is:

Do students have SUFFICIENT understanding of prerequisite knowledge so that they can learn the ‘new’ stuff?

This ‘sufficient understanding’ is the core question in course placement, which I have addressed repeatedly in prior posts. I am suggesting that the ambiguity of that process (we can never be certain) is also valid at the level of topics within a course. It is easy to prove by counter-example that students do not need to have mastered all of the prior mathematics before succeeding; they don’t even need to necessary have the majority of that mathematics. Learning mathematics is way more messy — and much more exciting — than the simplistic ‘nested and sequential’ view.

There is a substantial literature based on ‘global learners’. I definitely prefer the concept of ‘global learning’, as I think our own ‘styles’ vary with the context. However, that literature might help you understand the ‘ambiguities’ I refer to; see https://www.vaniercollege.qc.ca/pdo/2013/11/teaching-tip-ways-of-knowing-sequential-vs-global-learners/ as a starting point. As a side comment, ‘global learning’ is also used to describe the goal of having students gain a better understanding of ‘global’ societies, cultures, and countries; in that context, they really mean ‘world’ not ‘global’ (global refers to a physical shape, while ‘world’ refers to inhabitants).

A nested and sequential structure, by design, limits opportunities to learn. This, in turn, ensures that we will fail to serve students who did not have good learning opportunities in their K-12 education. Just because we can lay out a logical structure for topics and courses from a nested & sequential point of view does NOT mean that this is a workable approach for our students.

Drop as much of the sequential limitations as you can, and start having more fun with the excitement of having more learning for our students.

Developmental Mathematics Revival!

Bringing a new vitality to college mathematics 2011 to 2019