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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The Logic of Change: Do the same, expect different results

You have likely heard the quick definition of insanity:  Doing the same thing, and expecting different results.  Presenters often apply this statement to teaching, frequently stated as “Why should we teach it the same way as they had before, when that obviously did not work?”.  The interesting thought in this logic is the ‘same’ descriptor can be applied to many aspects of the current environment; in spite of this, most discussions focus on the pedagogy and on the teacher behavior in particular.

What about the content?  Perhaps we can improve our results if we first improve the content.  The appropriateness of the current content is questionable, and some have argued that the current content is damaging.  You might take a look at the New Life course outcomes (MLCS Course Goals & Outcomes Oct2012  and Algebraic Literacy Course Goals & Outcomes Oct2012).

However, perhaps we are tragically over simplifying the conversation.  What do we mean by content?  “Algebra” does not always refer to the same content, nor do we use it to refer to the same assessment standards.  We also ignore, I believe, the issue of student perceptions of content.  If you want to trigger a uniform reaction to content, put a simple problem that involves fractions and variables in front of students; in developmental courses, most students will perceive this type of problem as a threat and as something they can not understand.    We could improve our courses tremendously if we would invest time in improving the accuracy of student perceptions of content.  Yes, this takes time, and we would have to give up something … look at it this way:  Most students do not achieve deep understanding of most topics anyway; perhaps the net result would be better if we went a lot slower, with fewer ‘topics’.

You might try this experiment:  After you have covered a topic in a class like you have usually done, where the class went as well as you normally see, ask your students to write their answer to this question: “What are we learning about?”  [I ask individual students this question, and suspect that your students will struggle to provide a good answer just as mine do.]

I see still another over simplification in this conversation: is our content described by objects and procedures, or is our content better described by concepts and relationships?  We do not share perceptions of content, which makes it harder on students.  My hope is that we can, through many professional discussions over an extended period, involving all parts of the country, develop shared language to communicate our perceptions of content.  Of course, I would like us to emphasize reasoning and mathematical problem solving (beyond ‘real world’ problems).  In any case, our students would benefit from our accurate use of a shared language for content.

In many cases, speakers who use quotes like ‘do the same, expect different results’ are using this as a rhetorical device in their efforts to convince us to adopt their solution.  Our profession needs us to take a deeper look at the situations and problems.  If simple statements could solve the problems, they would have been solved long ago.  Making progress at scale (in location and in time) depends on broadly shared conceptualizations and collaboration on solutions.  We each are part of the solutions.

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Assessment in Mathematics Classrooms

I’ve been doing some thinking, and writing, lately on the roles of assessment in mathematics classrooms.  For many of us, assessment is a means to assign grades to students; we see assessment as following the learning.  Certainly, we need to use assessments for grading purposes.  However, learning can not be separated from assessment … learning without assessment is simply wandering in a city hoping to find that nice hotel or restaurant; little good is likely to result, and some damage is likely.

For general reading on assessments for learning mathematics, try one of these sources:

• CUPM Guidelines for Assessment of Student Learning (at http://www.maa.org/saum/maanotes49/279.html)
• Assessment and the Process of Learning Statistics (at http://www.amstat.org/publications/jse/v5n1/hubbard.html)
• The Assessment Principle (NCTM)  at https://www.usi.edu/science/math/sallyk/Standards/document/chapter2/assess.htm
• IMAGINE: Mathematics Assessment For Learning (at http://www.gatesfoundation.org/edextranet/Pages/ImagineConvening2009.aspx)

The ideal assessment in my view would be a series of interviews with each student, where their work and comments are prompts for questions and discussion; an expert talking to the learner can identify problems and reinforce partially correct understanding, which are difficult goals for mass assessments.  I have not managed to design a class to achieve this, although I have managed to create tools that enable some interviewing to support learning.

I’ll describe some of the assessment tools I am currently using to support student learning (and student motivation):

QUICK QUIZ
This is a traditional quiz, though very short (4 questions), with half of the items being concept or ‘no work’ items.  The quiz is given at the start of a class, covering the learning that ‘should’ have occured prior to class (prior to any homework questions).  Ten minutes is plenty of time, and then we review the quiz — by students explaining each item, and I reinforce & correct as needed.  Since the Quick Quiz uses ideas and simpler problems, the process encourages students to attempt the learning; in class, we often say the “quiz is part of the learning process” to support good learning attitudes.

NO-TALK QUIZ
As a variation, a No-Talk Quiz involves students working 2 or 3 problems focusing on key processes or ideas.  Their quiz is then reviewed by two other students, who can only write comments on the quiz.  The student then has an opportunity to re-do any problem where they think the feedback suggests that they were wrong.  As an assessment, this process involves every student doing two necessary steps:  Critically reviewing work for accuracy and completeness, and explaining.  In addition, each student has to judge their own understanding compared to the feedback they get; provides a little ‘meta-cognition’.

TEST DRIVES
During class, we develop ideas and reasoning as well as master procedures.  After the large group discussion (5 to 10 minutes), I have every student try a Test Drive … their chance to try out what we have been learning.  During a Test Drive, I talk to individual students about their work; this mini-interview does not involve every student for every Test Drive, but involves most students on any given class day.  We review the Test Drive as a class, based on students explaining what we should do.

FOCUSED WORKSHEETS
My classes know these simply as ‘worksheets’ — they include material from all sections since the last test, and normally involve 5 or 6 items.  My goal during worksheet time (20 to 35 minutes at the end of class), is to talk to every student at least once about their learning.  Over a semester, I will ‘mini-interview’ every student enough to understand some of their learning needs ; this helps to inform my teaching decisions.  As part of the worksheet process, students have an opportunity to work in groups; for my own purposes, I do not structure this nor require group work … though it is strongly encouraged.

Among the assessments I no longer use regularly, ‘writing’ is the primary category.  I’ve tried different methods, such as explaining steps or sentence completion.  I am sure that other people have developed a pattern with these so that they support learning; my own attempts seem to either frustrate students without benefit or actually reinforce learning wrong ideas.

For the curious, every class involves multiple “Test Drives”.  Each class day involves either a quiz or worksheet; I have tried doing both a quiz and a worksheet in the same two-hour class, but the price is a considerable amount of stress for students.  Overall, I try to do about 40 minutes of assessment activity in every 110-minute class period — not counting ‘test days’.  I don’t label this time as ‘assessment’ for students, because they, too, view assessment as ‘grades’.  Instead, I talk about improving our learning (which is exactly the goal of assessment).

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