Using Mathematics: It’s Not Always About ME !!

In the traditional college mathematics curriculum, mathematics is used to solve problems which students do not care about.  Some reform curricula involve mathematics only for problems which most students care about.  Is one of these extremes naturally superior to the other?

Perhaps some researchers are already working on experiments to test that hypotheses.  My own conjecture on this might surprise a few people:

The net gain for students is higher in a curriculum which solves problems which students do not care about, compared to a curriculum focusing on problems students do care about.

The traditional curriculum normally focuses on individual students creating a symbolic statement (equation or function) for the problem, and then using this symbolic statement to determine all answers.  The reform curricula often engage students with informal group work around a context, looking for alternative strategies to find the answer; symbolic work comes later (often on a different class day).

Most reformers will assert that the group work in a context provides definite advantages in student learning.  The etymology of this assertion often has its roots in a constructivist point of view; the original researchers in this area were more interested in the social context and juvenile development.  We often conflate the issue by speaking of a ‘constructivist theory’ — there is no constructivist theory (since a theory provides predictions that can be tested with either positive or negative results); I’ve never seen research supporting constructivism in learning mathematics with adults.

However, there is a non-trivial advantage to the reform work with work on problems which students care about:

Students having the novel experience of working on problems they care about is exciting and motivating.

Seeing that process in class is exciting for instructors; sometimes, we become addicted to this experience to the point that we think students have to be dealt with in this manner all of the time.

Is a math class all about ME?? (a student)

Of course it isn’t.  Students are in college to either get an education or training (or both).  Getting an education is all about “not me” — understanding other points of view, analyzing problems, and solving … often with the person deliberately left out (objective point of view).  We might think that ‘training’ should deal with just problems which students care about … this view has two fatal flaws.  First, let’s assume that training exists to get a job (employment); how much of any job is something that the student personally cares about?  Sure, the student picks a program that they care about in general — but their job is going to involve a large portion of specifics which they don’t care about.

The second fatal flaw in the training point of view is ‘stability’ (or lack there of).  How many workers deal with the same types of problems for years at a time?  We are hearing from business and industry that they need a flexible work force — not one constrained by ‘it’s important to me’.

When I teach our traditional algebra courses (beginning & intermediate) I almost always make a statement such as the following:

Passing this math course means that you can apply mathematics to problems which you don’t care about, but you did so because somebody else said they were important.

The main downfall of the traditional curriculum is that it does not modify the pre-existing negative attitudes about mathematics [though I try 🙂 ],  Students have a negative attitude about mathematics and especially about ‘word problems.  Using problems which students care about can provide some scaffolding to get students out of their negative attitudes.

We can’t stop there.  For each problem students care about, we should have them deal with 2 or 3 which they don’t care about.  We need to make the connections between the processing done on the ‘care about’ problems and the symbolic tools of the trade (expressions; functions; known relationships [such as D=rt]).

At the developmental level, students will be proceeding to college courses.  College courses have a general expectation of dealing with symbolic statements.  Being able to determine solution to a specific problem is often a trivial exercise in itself.  Students need to see quantitative relationships and use appropriate symbolism to state that relationship.  We have no confidence that the majority of these situations will be innately important to the student; we do them a diservice to imply that the only mathematics they need is to find solutions to problems they care about.

We need to get rid of the traditional curriculum, recognizing that we achieved some good results within that.  We also need to moderate our use of ‘problems students care about’, and we need to make sure that we always keep the focus on the tools of the trade (relationships, symbolic statements, representations).

 
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Discovery Learning versus Good Learning

As people look at improving mathematics courses in college, we tend to look at some methodologies as naturally superior to others; we often fall in to the trap of criticizing faculty who use “ineffective” methods (traditional ones).  Some of my discomfort with the current reform efforts in developmental mathematics is the focus on one category of teaching methods … discovery learning.  #CollegeMath

At the heart of the attraction for discovery learning (and it’s cousins) is a very good thing — an active classroom with students engaged with the material.  It’s no surprise to find that research on learning generally concludes that this type of active involvement is one of the necessary conditions for students learning the material (in any discipline).  We can find numerous studies that show that a passive learning environment results in low learning results for the majority of students.  One such study is “The Effects of Discovery Learning on Students’ Success and Inquiry Learning Skills” by Balim (http://wiki.astrowish.net/images/e/e1/QCY520_Desmond_J1.pdf). In this study, the control group was (perhaps intentionally) very passive; of course, discovery learning produces better results.

It feels good to have our students engaged with mathematics.  By itself, however, that engagement does not produce good learning.  Take a look at a nice article “Correcting a Metacognitive Error: Feedback Increases Retention of Low-Confidence Correct Responses” by Butler et al (http://psych.wustl.edu/memory/Roddy%20article%20PDF’s/Butler%20et%20al%20%282008%29_JEPLMC.pdf) The role of feedback is critical to learning, but most implementations of discovery learning suggest that the teacher not intervene (or even correct errors).

Good learning does not happen from constantly applying one teaching method; teaching needs to be intentional, and modern teaching tends to be diverse to the extent that our work is research based.  I can see the benefits of incorporating some discovery learning activities within a class, along with other teaching modes.  See a study of this for college biology “The Effects Of Discovery Learning In A Lower-Division Biology Course” by Wilke & Straits (http://advan.physiology.org/content/25/2/62)

I use some discovery learning activities in my classes, and have found that I need to be very careful with them.  Here is my observation:

When students are asked to figure something out, they tend to apply similar information they have (correct or erroneous) and the process tends to reinforce that prior learning.

For example, I use an activity in my intermediate algebra class to help students understand rational expressions at a basic level — focusing on the fraction bar as a grouping symbol and on “what reduces”.  The activity provides a structured sequence of questions for a small group to answer.  Each group tends to use incorrect prior learning, even when the group is diverse in terms of course performance.  Even the better students have enough doubts about their math that they will listen to the bad ideas shared by their team; the only way for me to avoid that damage is to be with each group at the right time.

So, I have taken the discovery out of this activity; I now do the activity as a class, with students engaged as much as possible.  Even when done in small groups, students tend to not really be engaged with the activity.

I notice that same self-reinforcing bad knowledge in our quantitative reasoning course.  I use an activity there focused on the basics of percent relationships — percents need a base, and percent change is relative to 100%.  Many students do not understand percents, and the groups tend to reinforce incorrect ideas.  I continue to use that particular activity, as the class tends to be a little smaller; I am able to work with each group, during the activity.

Some of the curriculum used in the reformed courses are intensely discovery learning (often with high-context).  We need to avoid the use of one methodology as our primary pedagogy.  Do not confuse the basic message of replacing the traditional math courses with the pedagogical focus used in some materials.  To achieve “scale” and stability, our teaching methods need to be more diverse.

 
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Developing Grit and Recognizing Grit

One of the recent emphases in education, especially college mathematics, is ‘student grit’.  Grit is what allows students to succeed when there are barriers, and we can recognize success (usually).  However, the concept of grit is not productive unless we can recognize and develop grit prior to that point.  #gritmath

The context for this post is a recent test in our quantitative reasoning course (Math119).  Our first set of topics dealt with dimensional analysis; every conversion in class was completed by that method.  Overall, students did about as well as I’ve seen.

However, some students did their work in an indirect fashion.  Take a look at this first example:

And, this example:

In both cases, many of our math classes would say “Just move the decimal point”.  I did have a few students complete the problem that way.

More importantly, many of us would tell these students that their method is wrong.  However, the first example is conceptually perfect; the error in the answer is strictly due to the rounding of the conversion facts.  The second example is also pretty good … except for the inversion of a basic conversion.

I think both students showed significant ‘grit’ in working these problems.  Although I don’t generally want students to do a problem in a complicated way when a simpler way exists, it is impressive that both students were able to salvage a problem begun in a non-standard way.

I’m not suggesting that any grit shown in these two cases is equivalent to the level needed to complete a math course.  However, I do think that developing grit is the same as developing other traits:  We start small, make it explicit, and practice.

One of the wonderful things about a good quantitative reasoning course is that there is a focus on non-standard problems.  Methods are emphasized, but we don’t focus on procedure as much as we do reasoning.  This environment lets students explore and develop in ways that traditional math courses don’t.

I suspect that our traditional math courses either discourage grit or prevent much development.  With such a strong focus on procedures and correct answers, students are often doing the ‘instructor dance’ — following steps because it will please the instructor.  Student traits can not develop in a overly structured environment.

It is important that we recognize the difference between “incorrect thinking” and “different thinking”.  Different thinking is part of trait development, like grit.  Students can not show, nor develop, grit unless I provide them opportunities to work differently.

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Building Understanding in Algebra

Like most of us, I have a tendency to assess student learning with an emphasis on “doing” problems … simplify this, solve that, etc.  We risk missing critical information by this practice — information that would help us build a stronger understanding of algebra.

On a small-group activity this week in our beginning algebra class, I used this question:

Paraphrase the expression  5x² – 4x + 5

This was the first problem of a set of 3, with a heading that included “properties and order of terms & factors”.  Because students have a hard time accepting a math expression as an object (and not always a directive to ‘do something’) many students struggled with the problem.

However, there was one particular error that was quite common, leading to this answer;

21x + 5

Since no work was shown, I was puzzled; I asked each student how they got this result.  Their answer?  The square meant 5 squared, so 25x … then 25x – 4x = 21x.

This is exactly the same issue we deal with when we present “-8²” and “(-8)²”; many of us see those problems as unnecessary.  I don’t agree, as many of my students have struggled mightily with “what does that exponent apply to”. These students can get a majority of correct answers when we say “simplify” because they have memorized the rule about like terms; it’s not that they believe it is wrong to get 21x for the problem — they just know that they are not supposed to do that when the directions are ‘simplify’.

If our students are not clear on “what the exponent applies to”, their understanding is limited to  first degree objects.  Now, we waste a lot of time on polynomial arithmetic that would be better spent on exponential models & numeric methods (to complement symbolic methods).  I have to say, though, that a beginning understanding of our symbolic language is based on the answer to that question “what does the exponent apply to”.

If you teach any algebra (beginning, intermediate, college, or pre-calculus), consider giving your students some open-ended questions about the meaning of our expressions.  Don’t assume that  correct answers is an indication of correct knowledge; the human mind is capable of much memorization and disconnected information.

Helping students build a strong understanding is a labor-intensive process.  Individual and small group dialogues are the most powerful tools to correct bad ideas; just getting feedback like “not correct, it means this” will not be effective.   [This is the reason why about 33% of my class time is spent using those tools.]

Remember that assessments don’t have to involve points or grades.  The best learning in my classes occurs when individuals and small groups struggle through stuff they did not understand correctly.  Every human comes with a drive to understand, and that can be harnessed in our math classes — if we use assessments that create those opportunities for deeper learning.

 
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