Memory, Learning, and Teaching Math

One of the macro problems in our profession is the relative ignorance we have (as teachers) of sound scientific evidence and theory related to the target of all of our work: the human brain.  In particular, we want our students to show that they learned something by using their memory of information; however, we design our efforts around the surface features of ‘doing math’ with too little attention to how a person (like a student) will actually remember information.

I am talking here of ‘memory’ in the scientific sense: something is stored in the brain, and memory refers to both this storage and the retrieval.  We might get “memorization” confused with “memory”; one refers to a specific process for building memory … the other refers to all factors involved.

Through a connection (on LinkedIn, of all places) I encountered a surprisingly good summary of research on memory.  The readable source is http://www.spring.org.uk/2012/10/how-memory-works-10-things-most-people-get-wrong.php which is based on a more technical anthology of research on memory.

Three of my favorite summary statements are these:

Forgetting helps you learn.

Recalling memories alters them

When recall is easy, learning is low

Other items in the list deal with learning in context and productive organizations for learning new skills.  All items in the list have direct applications for our classrooms and learning mathematics.

We all have our preconceptions about how memory works.  As teachers, we develop ‘intuitions’ about our students and their learning.  Like most domains, intuitions are valuable but actually incorrect more often than not; partially, this is due to the fact that organic processes have a large number of variables.

I encourage you to at least read the 10-item summary; that article contains a link to an online copy of the anthology of original research … you might find that interesting as well.

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Points, Lines, and Mental Maps

Do we consistently use different directions for problems involving discrete points as opposed to equations or functions?  Does “plot” mean something different than “graph”?  Most importantly, what mental maps do beginning students create for graphical representations of points and lines?

I am observing some patterns in my students’ behavior:

1. “Plotting” two distinct points is treated as if they represented a straight line.
2. Graphing a linear equation is done inflexibly, often by the first or last method seen or practiced.
3. Comparing and contrasting basic methods of graphing linear equations is seen as ‘confusing’.

We are doing our test today on linear equations in two variables, in our beginning algebra class.  The very first item on the test is plotting two distinct ordered pairs.  Some students have difficulty with the concept of ordered pairs, but most plot the two points accurately … and the majority of those connect the two points with a line.  This issue came up in our discussions, based on student questions … still, the programming seems too strong to resist: got two points?  Draw a line!!

Our course is quite traditional in coverage and outcomes at this time.  This chapter includes graphing equations by using specified input values, then by intercepts along with a 3rd point; we use a table of values via a calculator, and then graph using slope.  We’ve done quizzes with directions about a method, and worksheets with both method directions and ‘choose the most efficient method’.  However, each student tends to graph every problem by the same method.  Many students are using slope to graph every problem, even when intercepts would be easier … and even when the method is misapplied to a special case (missing x or missing y); graphing y=4 often results in a graph of y=4x.

Although a comparison of methods was part of our sequence of activities in class (as a secondary point), we tried to have a discussion of the methods as part of our review process in class.  This was not popular … most students did not want to think about how methods compared, only about how they can work problems correctly.  This is the same challenge in metacognition that students face when asked about what their improvement plan is … “I will do better” is confused with thinking about how to do better.

I’ve been working at this type of teaching and learning for quite a while now.  More students seem to have a simplistic view of learning in which having one method that gets mostly correct answers is seen as better than understanding how to choose the tool for the problem at hand; consistently getting one type of problem wrong is not seen as feedback about the learning process … it’s seen as an acceptable price to pay for getting other problems correct.

One way to look at these issues is to view them through a literacy framework.  Instead of listing behavioral outcomes for linear equations in two variables, we would ask the question: “How would we determine that students understand linear equations in two variables?”  We certainly include too many topics in the existing courses to assess understanding of most topics, and are left with simpler performance measures.  The mental map developed by each student responds to the conditions of learning, including assessment.

My own teaching and assessment is part of the problem I am seeing.  In some zones, I have limited flexibility … the basic content of the course is fixed.  However, I need to see how I can change the conditions of learning to encourage the more complex mental map that I think is important.

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Math – Applications for Living XVI: The Evils of Line Charts

In our Math – Applications for Living class, we just finished our first chapter on statistics, which included some of the standard graphical displays (frequency charts, bar graphs,  and line charts) as well as scatter diagrams for bivariate data.  Based on observing students working with line charts, I commented to the class that line charts have a risk — human perception may suggest a much stronger pattern than is really present in the data.

Here is an arbitrary example of a line chart (pseudo-random data):

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The same data presented in a bar graph looks like this:

A quick search on research related to this yielded a study done by Jeff Zacks and Barbara Taversky at Stanford (1999); see      http://dcl.wustl.edu/pubs/ZacksMemCog99.pdf.  Briefly, their study found that bars tended to processed discretely while lines tended to be processed as a pattern.  My examples above are not ideal examples … the correlation is fairly high, perhaps higher than the bar graphs suggest. 

It seems that scatter diagrams have some of the same risk as line charts, in the sense that my students want to visually connect the dots as in a line chart.  We are working on identifying patterns correctly in bivariate data, though my expectations are not high.  Human perception might have a strong desire to establish a pattern, even when some data needs to be excluded.

Context in Math: Do Students See It?

I am frequently asked (yesterday, for example) whether students have changed during my time working with developmental students.  People often assume that I will say ‘yes, their math skills are even worse’ … I don’t say this, because I do not believe it’s true.  Certainly, there have been changes — areas of ignorance and zones of bad ideas have shifted, but not in a fundamental way and not in a significantly increasing way.  My answer is this:

Students have changed, especially in the past 10 years, in terms of their academic and social skills.  More effort is needed — in all of my classes, not just the lowest — on these soft skills.

Among these soft skills is an awareness of surroundings and knowledge of context.  Last week, I noticed a comment over at the Chronicle’s page (http://chronicle.com/article/So-Many-Hands-to-Hold-in-the/134454/?cid=cc&utm_source=cc&utm_medium=en):

I have a grandson who at 22 cannot literally read a real map!  He cannot navigate himself to and from a doctor’s appointment or an auto parts store without the directions capability of his smart phone.  Technology is a wonderful thing; however, it does have a down side.  In the grandson’s instance, he does not pick out reference points as to where to turn or to retrace his steps, ie, being aware of his surroundings.

If you look for this comment, be sure to load older comments (this one is from about Sept 19).  Context is critical in mathematics, like most other academic areas; one can not know context if one is unaware of surroundings. 

In math classes, this means that students are even more likely to apply a strategy or concept when it does not apply (context) … students are more resistant to analysis of special cases (surroundings).  The student magic and silver bullet used to be ‘know all the formulas’; now, the silver bullet is ‘know the one formula. 

For those of us who like to teach math ‘in context’, I wonder if the nature of our students makes this approach less desirable.  If students have trouble identifying context accurately, and we teach from context, they may not see the intended context. [In many cases, I think students see our ‘context’ as just a longer word problem that is more interesting.]

For all of us, I wonder if our students see their mathematical surroundings in the intended manner.  Learning is constrained by perception; incomplete perception can not lead to quality learning of mathematics.

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