Modules or Clumps in Developmental Math?

A lot of people are talking about modules in developmental mathematics as part of an effort to ‘fix’ our programs.  Of course, the word ‘modules’ has implications that sometimes are not meant … so I was inspired when I saw somebody refer to them as ‘clumps or modules’. 

The word ‘module’ carries connotations, and also has a denotation.   You might be surprised to learn that ‘module’ does not have a denotation (definition) relative to the practice of clustering learning outcomes into small pieces (‘clumps’), nor with process of assigning a subset of ‘clumps’ to a given student.  Most dictionaries will not give an educational meaning to ‘module’.  What we are doing here is describing by metaphor — “this is like modules in electronics where sub-systems are replaced as a unit”. 

Which leads in to the connotations.  When we think about ‘modules’, we usually have positive images — easy, efficient, better.  “Modules” has a scientific sound, as if describing by metaphor automatically assigns a scientific basis.  I suspect many people think that ‘modules’ means that we are meeting students’ needs, and that the program is individualized.  Some people believe that ‘modules’ mean that students spend more time actively doing ‘mathematics’.

Using modules does not mean anything more than using ‘clumps’ would mean.  Of course, a particular implementation of clumps (or modules) might mean a great deal of good stuff.  Too often, using ‘modules’ means that we focus on the delivery system to the exclusion of critical analysis of the content (beyond creating clumps).

In practice, there usually is one difference between using ‘clumps’ and using ‘modules’.  With ‘modules’, there is not (normally) any summative assessment at the end of a ‘course’.  This means that there is no need, from the student’s point of view, to integrate knowledge and understand how parts fit together.  “Connections”, in a modular math environment, is limited to those that can be developed within a single module.  “Clumps” might share this property, but ‘modules’ almost always do.

As mathematicians, we have shared values — reasoning, application, relationships, representations, and even creativity.  Whether you call them ‘modules’ or ‘clumps’, do our values come through?  This really is important; imagine a freshman writing class where students learned about components of writing in isolation, and never had the opportunity to develop a position or argument.  Like writing, the purpose of mathematics is centered on communication.  Let’s build courses where our values are accessible to students.

 
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Math – Applications for Living XII

In our ‘math – applications for living’ class, we are reviewing what we have learned this semester.  Some parts (like probability) are still tough for students, partially because there is some memorizing to do with new material.  Truth is … I like to cover probability mostly because the process encourages reasoning about quantities.  [For example, we had a problem to solve about the probability of having 5 children — 2 girls followed by 3 boys; some of us wanted to look at this as dependent probability: 2/5 for the first girl, 1/4 for the second, and then confusion about what to do with the boys.  Clearly, knowing that events are independent is critical.]

The best problem we worked on today was one with almost no practical value: 

We had to really work on this problem.  The intent is to have students focus on the units (we need ‘square feet’ for area; we have cubic feet and feet … how can we do this?).  When students asked how to do this problem, I would ask them “How do you measure area?” (to get them thinking about units).  Every student (individually) said “length times width”; clearly, we are still too focused on one formula, and not thinking about what we are measuring. 

Of course, we could follow up on the “length times width” idea with something more reasonable. 

S: Area is length times width.
I: Okay, for a rectangle we calculate area that way.  How do we calculate the volume of a box?
S: Multiply (writes V = LWH)
I: So, the volume is L*W times H; right?
S: Yes
I: We know that L*W is the area of a rectangle.  Think of that volume formula as “V = area * Height”.  How would we solve this for the height, which is like the depth of the lake?
S: Hmmm (thinking) … we would divide
I: Yep — divide both sides by area.  Does that give you an idea how to solve the lake problem?

Most students originally decided that they had better multiply the numbers in the problem; of course, they only dealt with the value not the units.  They did not think about getting “feet to the fourth power”, and what this might mean.  A couple of students thought that the ‘cubic’ in ‘cubic feet’ meant that that value needed to be cubed.  [More evidence of a ‘messy landscape’ of math knowledge.]

The good news from today’s class was that students actually did a reasonably good job figuring out a complicated ‘unit conversion problem’ (given dimensions of a box, the flow in gallons per minute, and rate of gallons per cubic feet … how long would it take to fill the box).  Prolonged effort on related problems with diverse settings has paid off.  We are having more difficulties with geometry (concepts) than we are with proportional reasoning.

 
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Neat Knowledge? Messy Landscape?

We all spend quite a bit of time talking with students, and we also look at massive amounts of student work.  Sometimes, we get in to “homework system mode” where we only provide feedback on the answer.  The answer, by itself, is very weak as a communicator of the knowledge a student possesses. 

I have been thinking about how messy student knowledge is (about mathematics); perhaps this is true for all domains.  In my classes, I try to create an atmosphere where people feel safe asking questions and participating; of course, sometimes the questions asked indicate significant misunderstandings … which can temporarily cause some damage to the overall learning in the class.  In this post, I want to focus more on the implications of errors in student work.

Okay, in our intermediate algebra class we just had a test on ‘quadratics’.  The material is a mixture of procedural and conceptual, with a few ‘applications’ included.  One of the applications involves compound interest (annual) for 2 years — so it creates a quadratic equation like this:

 Most students managed to write this (based on the verbal description and the provided formula).  The most common error?  Subtracting 4000 from each side, a disturbing error.  Since this problem was on a test, I did not have an oppportunity to discuss the error with the students to identify the source.  My primary suspect:  An early rule (early in education) to the effect that “large and small numbers mean divide, similar size numbers mean subtract”.  Every  one of these students can solve linear equations requiring division (like -4x=30), and most solved a quadratic involving a common factor (like 2x²=100).  What triggers  the “we must subtract” response?

In another class (the quantitative reasoning course), we have been doing geometry this week.  As for other topics, the formulas are provided — we are much more interested in the reasoning involved.  One of the problems dealt with finding the perimeter of this shape:

Two consistent problems came up.  First, students thought that ALL perimeter problems dealt with “P = 2L + 2W” … causing them to count the widths of the rectangle (which are interior, therefore not part of ‘perimeter’).  Second, even when convinced that some perimeter problems dealt with a different relationship, they did not see  why we should omit the perimeter (they still wanted to include the interior dimension).  Since I was able to discuss these issues, I have some idea of what is behind them. 

My point here is not to complain about what students do or don’t do, nor to suggest that they have had bad teaching  [I mean, beyond having ME as a teacher :)].  Rather, perhaps we need to think more about the root cause for many student difficulties: 

The basic problem is an oversimplification of a connected set of ideas down to a single ‘if-then’ clause.

Students are sometimes desperate to learn math, and we want to help them.  Too often, this results in only dealing with one thing (or one thing at a time) — which simplifies the learning, but which avoids building connections (contrast, similar).    The geometry instance of this is easily described:  by only dealing with one shape for ‘perimeter’ we encourage students to connect just one formula with that work — and to disconnect this from the concept involved (‘distance around’).   Our procedural work in algebra is even more prone to the oversimplification, because the correct learning is abstract about a combination of ‘properties’, ‘undoing’, and ‘maintaining value OR balance’. 

I’ve got to anticipate what some readers are thinking (due to that last sentence) … are manipulatives the solution for this problem in algebra?  Not really.  The research I have read on ‘kinetic learning’ suggests that this might provide some of the scaffolding to abstract ideas — but is counter-productive if the manipulative remains a focus.  This is a very difficult task, to shift from manipulative to concept; the stages vary with each learner.  If you can do manipulatives on an individual basis with continual feedback (conversation) with a highly skilled faculty, then I believe this can support better learning.   The skills involved are complex, and most faculty will not have them — including me; this is combining the skills of a cognitive researcher, qualitative researcher, and math faculty.  The problem with most manipulatives in learning is that the entire process is over-simplified to be practical in a group setting; a group of 3 students is too large, and possible 2 students is too large.

What is the answer?  We need to clearly articulate the few critically important learning outcomes in a math class (perhaps a dozen for a course), and provide diverse learning experiences around those outcomes.  “Simple” is not the solution; simple is part of the problem.  The solution is a planned sequence of activities to deeply explore the ‘good stuff’ of mathematics.

 
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Slope … Fast?

Well, I’ve been somewhat discouraged by the latest test in our ‘math – applications for living’ course.  This test is all about understanding linear (additive) and exponential (multiplicative) change.  In spite of approaching these models in different ways, followed up by repeated work, students had considerable difficulty translating verbal descriptions to anything quantitative or symbolic.

One basic problem seems to be that students did not start with much understanding of slope for linear functions.  Every student in the class had completed beginning algebra (usually recently), where we do the usual work with slope — calculating it, using slope to graph, and finding the equation of a line given information about slope.  When faced with non-standard problems with information on slope, the transition to quantitative or symbolic statements was not easy.

One part of this difficulty is the connection between input  & output units and units in slope.  Like most traditional algebra books, ours does not make an issue out of units for slope in ‘applications’; as you know, slope must involve two units.  Because of this difficulty, students would see a percent change as a linear change.

Mostly, this post is a “note to self”:  Learning slope is not really a fast thing.  Repeated use, alternate wordings, and non-standard problems need to be used to build a more complete understanding.  We too often spend a week or two on all linear function topics in a beginning algebra course; to get that ‘sufficiently good’ understanding, we need to allow more time so we can really dig in to what slope and the linear form are doing.

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