Probem Solving in a Digital Age

Whether we ‘flip’ a classroom, use an online homework system, or refer students to Khan’s Academy … our students are using task-oriented videos.  In addition, students have a tendency to see ‘look it up online’ as a substitute for learning something.  As we become immersed in (and dependent upon) the digital age, can we still work on problem solving or critical thinking?

One of the sessions at this year’s AMATYC conference dealt with the topic ‘stop the assault on critical thinking’.  In the session, they played the roll of a short video on subject (related rates in calculus, maximizing a function in pre-calculus, or unit conversions in a liberal arts math class).  The audience experienced something like a typical 3 minute video on that topic, and then we talked about how this supported critical thinking (or not).

The next session was one by Jim Stigler on ‘using teaching as a lever for change’, though he talked more about the futility of identifying specific teaching activities as being ‘effective’.  Dr. Stigler did include 3 aspects of teaching that are connected to improved learning — productive struggle, connections, and deliberate practice.  Learning is a complex process, and the presence of these 3 factors in the learning environment are connected to improved learning.

So, there is a connection between this research-based observation and the concern about critical thinking, I think:

Discrete learning experiences like short videos focused on successful completion of a task, based on clarity and being easy to follow, are guaranteed to limit both overall learning and critical thinking.

Mathematicians hold critical thinking as a goal to be valued; we want students to be able to flexibly apply knowledge to novel situations and interpret results.  This seems to be a basic problem.  Our students expect math to not make sense, that they could not figure something out; task-oriented videos support this self-defeating belief.

We can not hide from the digital age, even if we wanted to.  However, we can improve our understanding of the factors that contribute to the learning opportunities for our students.  A balanced approach appropriate to each course can help students through the learning process — including the struggle, the connections, and the deliberate practice.  We might even see these digital resources as just-in-time remediation to be used occasionally, rather than seeing the digital material as the basic course.

We need a more subtle understanding of how our teaching can contribute to student learning.  A single belief or methodology will not succeed for our students, no matter how good of an idea we might have.

To see a presentation by Dr. Stigler similar to what he did at the conference, see http://www.salesmanshipclub.org/downloadables/scyfc-Stigler.pdf

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Percents as Evil … Percents as Good (Applications of Math)

Given a percent and another number, do we multiply … divide … or something else?

A few years ago, I was at a presentation about a pre-algebra course where the presenter classified percent problems as either growth or decay.  My initial response was that these are concepts too advanced for that course; after a few minutes, I liked the idea, and my experiences since have strengthened that opinion.

Within a few days, I had a chance to work on percents in both a beginning algebra course and in our applications course.  In both courses, the percent problems are varied; one thing that was constant — students ‘wanted’ to multiply a percent and the other number in a problem, regardless of the context.  Sales tax rate and marked price … multiply and add.  Sales tax rate and final price … multiply and subtract (wrong).  Percent decrease and old amount … multiply and maybe add.  Percent decrease and new amount … multiply and add (wrong).

We seem to have reinforced overly simplistic rules about percents to the point where students are impervious to a need to change; 40% wrong answers is not enough (even if I asked ‘8 is 40% of what?’).  It’s really that 100% value that is the problem.  The connection between a growth rate of 3% and a multiplier of 1.03 is a challenge.

In the applications course, I had students work in small groups on a sequence of problems to make a transition from a simple percent value to a multiplier.  They worked hard, explained to each other, and seemed to do well.  The next day … a quiz on percents where they could use the multiplier; result — not so good.  In the applications course, we use this multiplier again — in our finance work (1 + APR) and in our exponential models [y = a(1+r)^x].  I suspect that a deliberate focus on the multiplier in 3 chapters might result in some improvement.

I actually fault our presentations on percents as the root of this ‘evil’.  We do “2 places to the left”, “is over of”, and mechanical use of “a is n% of b”; sure, we include problems where students need to find the base (divide), but the work is too superficial.  Students do not generally understand the contexts where percents are used.  An initial approach on growth or decay, which means seeing the multiplier, might just help.

The most common uses of percents in developmental courses is usually in that pre-algebra course.  Based on long-term goals of understanding, if we are not going to cover the whole story of percents (with the multiplier), we should omit percents entirely.  However, percents are one of the richest zones of overlap between math and the world students experience that we need to see percents as a good thing — and do it right.  Fewer tricks, a lot more understanding!

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Do we Have 80-Year-Old Students?

I was at a meeting earlier this month (on my campus) about developmental education.  We had a broad conversation about ‘what works’ and what we would like to do.  The person leading the meeting has a background in writing — including developmental writing courses; I’ll call him George for convenience.

George told the story of one particular student he was having trouble with.  The student was polite and all that, but could not write a coherent paper.  After grading some papers with agony, including one responding to Angela Davis, the instructor (George) had a conversation with the student.  Based on that conversation, they decided to have the student write about a different topic — the student’s own experiences in a war (World War II, in this case).  The result, according to George, was a well-written essay (far longer than required).  The lesson George took from this was … let them write about something meaningful to them.

My response to this story was:

We should look for 80-year-old students in our classes, who happen to be stuck inside a 20-year-old body.

You see, my lesson from the story is a different.  Students are complex human beings (there is no other kind).  For ‘good students’, they can focus on academics and see what we expect of them.  For ‘struggling students’, they have difficulty keeping their history and current life challenges separate from what we ask of them in a classroom.  The student in the story was 80 years old at the time, and had much to deal with; of course, writing about something personally important is meaningful.  However, society in general … and occupations in particular … demand that we communicate about ideas that we do not necessarily care about.

The lesson, for me, is this: Students need to learn how to separate and focus.  Many of our students have had challenges in their lives; sometimes, this is just a math challenge.  Other times, they have faced significant life issues and trauma.  Just being able to talk about these will help a student focus in class.  Sometimes, they do not realize that the challenges they have faced will be a benefit in a math class.  To some extent, the affective factors can prevent cognitive work; just articulating the issues behind the affective can let the brain focus on the cognitive.

It’s tempting to say that “the lesson is to show students that we ‘care’ about them as people”.  Many of us do care about our students.  However, my observations do not support this conclusion in general.  I think the lesson is more subtle than ‘we care’ or ‘make it relevant’.   Maybe the lesson is more like “give them credit for making it this far.”

Our students have faced a lot of life, whether they are 20 or 80.  For some, they have overcome more challenges in 20 years than I will have faced in 80.  We seem to be more gracious to the 80-year-old than the 20-year-old.  I think we should look for all of the 80-year-olds in our classes, especially those who are stuck in a 20-year-old body.

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Towards a Balanced Approach

When I hear somebody suggest that we take a ‘balanced approach’, my first thought is that the speaker either does not have confidence in their judgments about what is important … or does have confidence but does not want to offend the audience.  The phrase ‘balanced approach’ is often used in reference to a reform model balanced with traditional ideas.

I suggest that we think about the phrase in a new way.  Let’s begin with the assumption that the traditional curriculum has limited value and that the reform curricula have limited value.  What would we build from a blank slate?  How would we use scientific evidence in the process?

A balanced approach looks at implementing two basic properties of human learning:

1. Understanding (connected information) results in more transfer of learning and facilitates long-term retention.
2. Repetition (deliberate practice) results in efficient recall and abilities to apply information.

Some reform curricula emphasize (1) almost to the exclusion of (2).  I have taught courses like this, and talked with my students; few of them have a good report about the experience.  We all have students who approach a math course in that fashion — the students who usually are in class, and do very little ‘homework’ because they understand what they are doing (occasionally they are correct).

As mathematicians, we are drawn to ideas with power — ideas that can represent relationships among quantities, communicate the information, and help reach conclusions about some future state of those quantities.  [We are also drawn to special cases, as well as mathematics that is aesthetically pleasing.]  Our students need the ‘basic ideas with power’ so they can handle the quantitative demands of academic and social situations.  I think we can have fairly strong consensus on the mathematics that most students need.

The balance we need is about pedagogy.  Having a better ‘table of contents’ will not help if students do not learn any mathematics.

I see this issue of balance as our basic problem over the next 5 years.  We know that our courses are going to change in basic ways.  We understand what mathematics is important for all students.  Our issue is to address both the understanding and repetition in the learning process.

Currently, an ‘understanding’ method is based on students dealing with a situation and using guided questions so that they discover the basic idea.  In some cases, this works surprisingly well.  However, discovering an idea has little connection with understanding mathematics.  Here’s an example:  By looking at a set of ordered pairs (bivariate data for a situation), students are led to the idea of slope so that they can predict another value.  This forms the beginning of understanding, not the end: understanding takes extended work with diverse views of the same idea.  Students often over- or under-generalize.  In the case of slope, students think this applies to any set of values … or that it does not matter which ones ‘go on top’ … or that slope is like an ordered pair.  Understanding is a natural human process, but does not happen spontaneously with correctness.

As for ‘repetition’, we seldom get this right.  Textbooks often confuse ‘any sequence of problems’ with ‘repetition for learning’.  Much is known about properties of practice that result in different degrees of learning — a sequence can highlight the most important idea (or hide it), a sequence can reinforce good understanding (or prevent it), and a sequence can reinforce accurate recall (or prevent it).  We somehow make the mistake that good authors can design good assignments; these are vastly different sets of expertise. We also make the mistake that computer systems provide appropriate repetition.

We can (and need to) focus primarily on the big ideas in mathematics; our courses need to match the amount of material with reasonable expectations for students learning with understanding and repetition.  With a balanced approach, we can help students succeed.  With a balanced approach, we can show policy makers that we have the professional skills to solve the problems that they are concerned with.

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