A Trajectory in Math

A story about what students are capable of doing … and how resistant prior learning can be.

In our applications course, we took a test recently on numeracy and finance; of course, I did not call it ‘numeracy’ for the students.  They saw phrases like ‘percents’, ‘scientific notation’, and significant digits.  One of the non-standard problems on this test was:

My age is 5 · 10^9 seconds.  Is this reasonable?

We had done a couple of problems involving changing quantities in scientific notation to different units.  However, this combination had not been seen before — notice the unstated ‘change this to years’ part of the problem.  The majority of students did a good job with the problem.  Since this ‘age’ is about 159 years, the answer is ‘no’.

On the same test was a percent question:

The retail cost of a computer is 27% more than its wholesale cost.  Which of these statements is true?(A) The retail cost of the computer is 127% more than the wholesale price.
(B) The retail cost of the computer is 27% of the wholesale price.
(C) The retail cost of the computer is 127% of the wholesale price.
(D) The wholesale cost of the computer is 73% of the retail price.

Notice that the stem of the question is a direct conflict with choice (A).  Sadly, choice (A) was the most common incorrect choice; most students did not select the correct response (C).  Even though we had explored percent relationships in different ways, pre-existing knowledge seemed to trump recent learning.

So, here is the question:

Will students have significant long-term benefits from the college math experiences?

In other words, are we lining up trajectories in math … or are we just enjoying a shared experience with no impact of importance?  I would like to think that our courses are building reasoning, understanding, and structure; that we are aligning trajectories.  Of course, yes, I know — this is unlikely; perhaps I am hoping for too much.

I’m reminded of all I have read and studied about memory formation related to organized learning.  The human brain does like to organize information about the world; unfortunately, it seems like much of this ‘information’ is really an oral narrative related to experiences.  Perhaps this is due to the high emotional load many people in our culture experience in ‘mathematics’.

And, I think about all of the effort on ‘remediation’ of arithmetic and algebra.  The students who need the most tend to have strong connections to past stories with good endings, stories that contain bad mathematics.  [Cross multiply … PEMDAS … LCD … and other tag lines for stories.]

Perhaps we would be wise to focus on what mathematics students will need in courses they are likely to take.  Perhaps we can have some success in dislodging prior ‘learning’ if we create more intense environments for learning — with a focus on reasoning and connections.  Remediation might be possible given enough time and enough resources; a more reasonable goal would be building capacity and quantitative functioning.

If we focus on basic themes for the course or two with students, perhaps we can help our students get a positive trajectory in math.

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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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Active Learning: Rhetoric and Propaganda

I spent some time looking for research on ‘flipped classrooms’, which turned out to be non-productive time.  [I found one study showing negative attitudes from students about a flipped college class, and one study showing improved learning outcomes for a high school class.]  My search was for sound research on the methodologies; sadly, most of what I found was rhetoric and propaganda.  You might try a search yourself; let me know if you find more research with reasonably sound design.

The zeal these days is about two ideas (at the college level): Flipped classes, and “MOOC” (massive online open classes). Most of us will not make a choice to do a MOOC, and most of our community college students will not take one.  My concern is more with the flipped classroom ideas.

The narrative about flipping almost always centers on two phrases: active learning and collaborative processes.  I will not argue that active learning is a bad thing.  However, here is a truism:

Learning is always active.

Learning is in the brain, and the brain needs to be active for learning.  [I’m not being strictly correct here, as some researchers include memory alone as a learning activity:  people can remember a surprising amount without their brains being actively focused on that material; ‘large’ here is a comparison to none or to random amounts above none.  Like most faculty, I am mostly concerned about learning that exceeds memory of information.]

Using a concept of ‘active learning’ is to imply that learning can be something else.  My impression is that the use of the phrase is meant to convey “observable activity by students”.  Do students learn better when chairs are turned, when they move within  the room, when a product is created?  The problem here is that we often have students who are not truly attending within the class; if we design some method that creates more attention, learning is very likely to improve.  Flipping a class may be one method to get students to attend to the material; it’s not the only method, and may not be the best method, of doing so.

We treat collaborative learning as a certain “Good Thing”.  I’ve read about research and theory related to this for a while now, and I think we tend to over-simplify the issues involved with group processes: language, culture, and power all need to be managed to create the benefits of collaborative learning.  Some of these can be managed by using very structured processes; I suspect that most of us do not have the background to use those methods, and our easier methods can damage student learning.  [Most commonly: Students focus on the stated outcome for the group, rather than the learning we intend that they attend to.

All of this reminded me again of the erroneous use of “Dale’s Cone of Learning”.  See http://raypastore.com/wordpress/2012/04/bad-instructional-design/  for a brief review of that.

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Formulae As A Disguise

How do we know what a student knows?  More often than not, the use of formulae (such as perimeter or area) serve as a disguise for the lack of knowledge … a disguise which allows a person to achieve a preponderance of ‘correct answers’ in spite of having no relational or procedural knowledge. 

My motivation, sadly, is personal therapy.  Our beginning algebra classes took a test today, dealing with polynomials.  This is a traditional class, though our work together has focused on meaning and understanding.  One problem on this test is a contrived operation question:

Find the polynomial that represents the perimeter of the figure. [Figure shows a triangle with sides 3a+2, 2a+1, and 6]

A minority of students added the sides.  Two responses predominated the incorrect work — P = 2L + 2W, and A = LW.  Students retrieved these formulae in spite of the visual stimulus indicated that this was not a rectangle.  It is likely that most students had achieved ‘success’ by using these formulae in prior math courses, perhaps where the material was ‘blocked’ (all problems of a similar type, not mixed).

This thought led me to question something at the heart of our current work in this course:  ‘rules’ for operations with exponents.  The formulae for this work have been stated verbally, not symbolically; our class time has been focused on the reasonableness of our rules.  Based on the types of mistakes I see on other items, I suspect that students are storing some of their knowledge in those “formula files” just like the geometry ones.

I am suspecting that a formula in the hands of a novice math student is dangerous, just like some power tools in the hands of novice craftsmen (like myself).  Perhaps we would be better served by avoiding rules in most cases, and avoiding formulae as long as possible, so that all work is done based on some understanding.  Perhaps a student stops learning as soon as there is a rule or formula to remember.  This concern with formulae is related to concerns with PEMDAS:  The presence of a rule which provides sufficient correct answers stops the learning process, and may prevent deeper understanding.

If we are talking about finance formulae involving 6 input variables, I do not see a problem with the formula stopping the learning process.  However, when there is a key mathematical concept involved — whether perimeter or exponents — I think the formulae create enough problems to approach them with reservations.  If anybody knows of related scientific research on the impact of formulae on learning, I would love to hear about it.

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