Contextualized: Does Everything Need an Application?

In some corners, ‘algebra’ is getting a bad reputation.  Algebra weeds students out of programs, prevents completion, and is not identified as needed for most jobs.   Some of us have responded by taking a very  contextualized approach to algebra, so students can see how useful it is.

This is the first week of our summer classes, so I have been working with my introductory algebra class on basic concepts.  We actually do very little with operations on signed numbers (traditionally the start of an algebra course); instead, we spend 3 class days mostly working on the language of algebra.

My interest is in having each student understand the objects we are using.  When we see ‘3x’, I want them to have multiple and correct ways of expression this verbally.  When we see ‘the square of a number’, I want them to have at least one correct symbolic expression they can write.  I deliberately do all of this work without any context for each problem; in other words, the problems are not framed in terms of a situation with physical objects or meaning.

In our Math Lit course, we also do some of this same work.  The difference there is that we introduce algebraic reasoning by talking about some contexts where algebra might be helpful, and then deal with understanding the objects when there is no context.  Does it help to have the context first?  Not really.  It’s fun to have a context, and it motivates some students (though not most).

What seems to happen with context is that ‘understanding the context’ takes quite a bit of energy; I think the brain tends to then organize related information as being connected to that context.  Making the ‘math visible and general’ is not easy, when students begin in a context.  In some ways, beginning in a context comes across as just being a more complicated puzzle word problem (“two trains left at the same time …”).  Students seem to feel like the context was just there to give them another word problem.

One of the myths seems to be that “we need to make it relevant”.  In some cases, we have gone so extreme that we refuse to cover a topic if we can not show students a context that they can see the math within.  I think we have confused math education with something else — having a context for everything is a basic property of occupational training.  Unless we are teaching an occupational math class, context is a tool to use when it helps; context should not be a cage that prevents good mathematics from being learned.

Whatever we might call a course (introductory algebra, mathematical literacy, whatever), a core understanding of basic ideas is critical.  Think about this problem:

2x+4x=??

Without further learning, something like 30% of students will give either 6x² or 8x² as an answer.  [Even among those who generally give the correct answer, their confidence may not withstand a little questioning about ‘why’.]  I’m not talking here about understanding operations on rational expressions, or factoring trinomials with a leading coefficient greater than 1, nor about simplifying radicals with an index of 3 and a radicand containing constants and variables.  The issues here deal with the initial constructs of an algebraic language system.

A related issue is ‘transfer of learning’ — context generally creates barriers to transfer.  Context is a concrete approach, and serves an instructional purpose when used appropriately.  However, an initial learning (in context or not) does not enable transfer to situations where the knowledge is needed.

In reforming the math curriculum, we need to keep aspects of the prior design that have benefits for students.  Think about (1) Transfer of learning and (2) Student confidence.  Known factors support transfer of learning — ease of recall, connections, and flexibility.  Student confidence seems to be impacted by feedback and repetition.  The presence of repetition can support both transfer and confidence — it’s not the presence of any repetition; rather, it’s purposeful repetition (including the use of mixed repetition) that provides the benefits.

When people say that algebra is not needed in occupations, this is often based on people in those occupations looking at a list of typical topics in an algebra course.  I think different results would be obtained if we asked about a different list — variables, algebraic reasoning, functions and models, graphical interpretation, etc.

I’d encourage us, as we re-build our curriculum, to incorporate more context — but not be limited by context.  I’d encourage us to help students learn deeply by providing sufficient repetition (with mixed practice especially).

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