Math Applications for Living XX: When a Foot is NOT 12 Inches

Our Math Applications Course is finishing the semester with reviewing and the final exam.   Students have made considerable progress in problem solving and reasoning, although they often can not see their own progress.  One concept from the course continues to create problems, though … so I’ve been thinking about what causes this difficulty.

Here is an example of a problem dealing with the concept:

A poster is 18 inches by 24 inches (rectangular).  Find the area, and covert it to square feet.

Finding the area’s numerical value was easy; knowing that the unit is ‘square inches’ was simple enough.  Converting to square feet?  Not nearly so easy to see.  Most students kept saying “a foot is 12 inches”, so they divided the area by 12.  One student suggested that we convert the original numbers to feet, and find the area in square feet.  This suggestion was seen as being reasonable, so  we did that … and then came back to the original problem.

As we struggled with this problem, we went back to the ‘a foot is 12 inches’ statement.  After a bit, we drew a square on the board — one side labeled ‘1 foot’ and another ’12 inches’.  Yes, we said, those are the same.  We labeled two sides ‘1 foot’ and two sides ’12 inches’.  The area?  1 square foot, or 144 square inches (a few students then understood what to do with our problem).  Some did not see the implication for converting, so I started drawing 1-inch strips in the square.  That might have helped a little; perhaps not.

A foot is not a foot when we are talking about area (or volume). In some ways, this is another example of prior learning being built in an overly simple space … we say ‘1 foot is 12 inches’, instead of saying ‘1 foot long is the same as 12 inches long’.  Conditional statements are critical for accurate learning, and enable problem solving skills to develop; unconditional statements impede future learning as the price for short term results.

Where am I presenting learning without conditional statements, when there should be some?  I fear that my classes routinely omit qualifiers for statements, sometimes due to the focus on the present problem … sometimes out of relative ignorance of where else the concept is used.

Sometimes, we create our own problems by deliberately omitting “if” and “when” statements.  Yes, these statements can impede current results; yes, we can become obsessed with technical accuracy to the point that only mathematicians can understand what we are saying.  However, I suspect that the price for simplicity in the ‘now’ is a set of problems in the future.

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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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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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