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.

Do we Confuse Good Pedagogy for Good Teaching?

Our professional organizations (both MAA and AMATYC) have published references related to good pedagogy within the last two years.  MAA had the Instructional Practices guide (https://www.maa.org/programs-and-communities/curriculum%20resources/instructional-practices-guide), and AMATYC has IMPACT (http://www.myamatyc.org/).  Lots of good ideas.  References to decent research.  What could be wrong?

Let me use an illustration from the other side of our ‘desk’.  When a student uses procedures without understanding them, we uniformly provide feedback that this is not sufficient.  When a student has some understanding of procedures, but can not understand connections between topics … we tell them that the connections are important.  If a student gets those connections to a reasonable level but can not transfer the knowledge to a slightly different context, we tell them that their learning is not good enough.

However, we tolerate — or even encourage — corresponding misuses of teaching pedagogy.  We see a pedagogy at a conference and ask questions about what to to in the process but rarely a question about WHY does it seem to work.  Very seldom do we even reach the low standard of minimal understanding of the procedures.  Rarely do any of us reach the expert level of knowing how to transfer our understanding to a new situation.

Now, it’s true that ‘good pedagogy’ (like good procedures) create some correct answers … ‘learning’, even if performed without much understanding.  However, the same can be said for some ‘bad pedagogy’; certainly bad stuff has worked reasonably well for me (though I try to not do that stuff anymore).  How do we even identify a method as “good” for teaching?

Sadly, we seem to have only two standards we apply to the process of identifying good pedagogy:

1. It’s good if the method feels right to us.
2. It’s good if somebody has seen good results with it (either better grades or some ‘research’)

Of course, our students do a lot of bad learning by the first standard (such as ideas about fractions or percents).  Students don’t generally use the second standard, and the second standard is actually not a totally bad thing when the results are solid research comparing two or more treatments with somewhat equivalent students.  I don’t expect us to use a gold-standard for research prior to using a pedagogy; I do expect us to do a better job of judging elements of a pedagogy based on understanding the process and the validation of those elements in research over time.

We also fall in to the trap of saying that diverse teaching methods are good.  Now, it is reasonable to assume that a given pedagogy might be well matched to a certain situation; we might even believe that a pedagogy is especially suited to a given mathematical topic (though this is difficult to justify by research).

What should we do differently?   My advice is to keep the classroom pedagogy simple from a student point of view.  I’ve seen teachers use multiple complicated methods over a semester, which requires students to learn our methods which will necessarily have a reduction in their learning of mathematics.  [Students have a finite supply of ‘learning energy’.]

My teaching methods are very simple.  Every day (besides tests) are team based with two activities for learning (start and end of class).  We don’t have assigned roles, and we don’t create artifacts to share with the entire class.  There is only one criteria for measuring the value of our teaching methods:

• Every student learns the most possible mathematics with the highest level of rigor possible every day.

Making this simple method work depends upon my understanding of learning processes as they relate to each topic and concept we explore.  I have studied the learning process for my entire professional career (it’s what my graduate work was in), and what has been shown by research supported by theory is:

The amount and quality of learning are functions of the intellectual interaction of the learner with the material to be learned.

In other words, maximize a quality interaction for each student in order to impact their learning every day.  The learning needs vary with the individual, so the pedagogy must provide a structure for my intervention (based on instant interviews) during class.  My assessment of my methods involves global and individual progress in learning mathematics (including how much rigor is achieved).

One of my students commented last semester: “We could not help but learn.”  I have had more dramatic comments (usually good 🙂 ).  However, this ‘we … learn’ comment is the most valued comment I have received.

My concern involves the frequent copying of teaching methods (often based on the ‘seems right’ criteria).  If you don’t understand how it works … you don’t understand who you will harm with the method.  Although we don’t take a professional oath about this, seeing ourselves as a profession suggests a ‘do no harm’ standard of practice.  Any specific pedagogy has the capacity to harm students; some pedagogies have a decent chance of helping students.

We should not settle for “it works for most students”.  We certainly should not settle for “this generally works, but I do not understand how it works”.  Our lack of understanding will cause harm to students.  Being an expert means that we see simpler solutions that produce broad benefits; using complicated ‘solutions’ means that we don’t understand the problem.

Resist the temptation to copy ‘methods that work’.  Copying methods is not productive for our students; it’s harmful to students if we copy methods without understanding the processes involved.  Your best bet is to keep it simple and interact with students constantly.

Brain-Based Math Learning

I have been amazed (and appalled) by the phrase ‘brain-based learning’.  The suggestion is that there is learning NOT based on how the human brain functions; like mathematics, the brain uses ‘existence proofs’ — if learning happened, the brain must have worked.

The point of this post is to talk about what we commonly report as facts about the human brain.  For example:

Learning occurs through modification of the brains’ neural connections.

This is just about the most basic statement we can make, and it is actually correct.  Of course, it does not lead to an easy-to-implement teaching method.

Take a look at the following statements with an eye towards truthfulness:

• Individuals learn better when they receive information in their preferred learning style (e.g.,
  auditory,visual, kinesthetic).
• We only use 10% of our brain.
• Differences in hemispheric dominance (left brain, right brain) can help explain individual
  differences amongst learners.

Each of these statements is false; these statements are examples of ‘neuromyths’, a phrase used by the Organization for Economic Co-operation and Development (in “Understanding the Brain:
Towards a New Learning Science”, 2002).  In other words, experts in neuroscience have determined that these statements are false.

The first myth listed is dangerous, because it leads to easy-to-implement teaching methods which will not help learning (and can reduce learning).  Even if “learning style” was a valid construct with a solid research basis, matching a trait to a treatment has shown to be a very difficult design strategy for learning (based on decades of research on attempts).  However, the recent summaries I’ve seen on “learning styles”  are still showing concerns about the construct itself.  The phrase ‘learning styles’ is most often used by educators trying to influence others; learning theorists and cognitive psychologists will seldom use the phrase (and often react very negatively to the phrase).

So, what would “brain based math learning” look like?  This is equivalent to asking what math learning would look like.  To me, the key is to keep focused on the basic statements about the human brain — like the one above about modifying neural connections.  Each learning task in a college math classroom is an interaction between new information and existing connections in the brain.

1. The default response by the brain is “what I have now is correct” and is reinforced by the new information
2. The need for modifying existing neural connections is based on some level of conflict
3. “Learning” occurs during the resolution of the conflict
4. The strength of this learning is based on multiple factors, including the use of verbal conclusions and practice (amount and variety)
5. The learning may create a new set of neural connections that store information in conflict with pre-existing information; which set is accessed in the future depends upon the processing of inputs
6. Resolving conflicting neural information takes the most effort but results in the most stable set of knowledge

As an example, we used about 6 class days last semester in my intermediate algebra course on a better understanding of rational expressions.  Most students responded based on their existing (incorrect) ideas about fractions. The classes created enough conflict (mentally) that most students developed some new information about fractions.  Later (on the chapter test, or the final exam) some students retrieved the new (correct) information while other students retrieved the old (incorrect information).  In a perfect world, students would have further learning experiences based on these assessments.

We seldom have sufficient time for students to learn math in college when they have existing incorrect information.  At the developmental level, the New Life project courses (Mathematical Literacy; Algebraic Literacy) focus on reasoning and communication with a more defined content — allowing some additional learning time.

As a profession, we need to move beyond pseudo-science so that our pedagogy is based on a body of knowledge accepted by scientists specializing on the human brain.  For a single-source, you might try Applications and Misapplications of Cognitive Psychology to Mathematics Education  at http://act-r.psy.cmu.edu/papers/misapplied.html

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Rational Expressions … Easy Reducing, or “They’re Just Symbols”

For our intermediate algebra course, I am grading the test on rational expressions; in general, this is my least favorite test to grade.  One of my students provided a bit of unintentional humor, however; at least, it was funny for a minute.

Here is the problem and the student’s work:

So, what you are seeing here is that the student combined the numerators (fine) and added a half space in one term in the denominator, which caused “4k – 5” to be seen as “4 k-5”.  Reducing fractions the easy way!!

Fortunately, most students are actually doing okay with this test.  This problem has been on my tests before, and this is the first time I’ve seen that ‘method’.  I think this illustrates a generality about our students and fractions of any kind:

Students will deal with fractions at the symbolic level only, whenever possible.  Meanings are not attached, in general.

If you are curious, I start our work with rational expressions with an activity where we explore the meaning of the ‘fraction bar’ and the role of factors in simplifying.  That activity does shift students thinking a bit towards the meaning, though clearly not always :).

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