Category: math reasoning and applications

Problem Solving Skills

This is the story of what a student did when confronted with a procedural problem for which she did not ‘remember’ the standard procedure.

One of the in-class assessments I used is a ‘worksheet’; it’s like an open-note quiz over a set of material.  During our last class (intermediate algebra), the worksheet is longer than usual because it ‘covers’ the entire course.  Item 6 on this worksheet is:

Rationalize the denominator 

As I said, she could not remember ‘what to do’.  However, she did a great thing … she recognized that both numbers could be written as a power of 2:




I was very pleased that she did this, but the student was frustrated … she then could not see what to do.  This is pretty typical when novices dive in to the world of ‘non-standard problems’ — problems for which we lack a remembered process.

Of course, it was pretty easy to guide her through the remainder of the work:




Obviously, the expectation (this is our traditional intermediate algebra course) was that students would apply the standard procedure (multiplying top & bottom by the cube root of 4).  Students do not like that procedure, and I tell them that the procedure itself is seldom needed.

The alternate method worked only because there was a common base between numerator and denominator, and I doubt if the student will gain any long-term benefit from this experience.  This was more of a positive thing for me, as a teacher and problem-solver: Noticing a special pattern within a problem is a critical problem solving skill.

I’m sharing this story just because I had fun with it!

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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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Saving Mathematics, Part II … Diversions in our Curriculum

Among the threats to mathematics is the ‘diversion’ strategy, wherein colleges look for the least-mathematical option when choosing program requirements.  The original diversion course was “Liberal Arts Mathematics”, or early relatives.  Since then, “Statistics” has gained ground and “Quantitative Reasoning” is sometimes used as a title to make a Liberal Arts course sound more mathematical.  #MathProfess #MathVsStat

I am using the word ‘diversion’ in reference to courses that are used for a ‘math’ requirement, often offered as a ‘math’ course (academic department), while not being a ‘good math course’ (see below).  Quite a few of these Liberal Arts Math or Statistics courses are diversions from mathematics, just like basic math and pre-algebra are diversions (and dev math in general).  The developmental courses have, at least, the excuse that they are not claiming to meet a mathematics requirement … though that is not a always-true statement.

Think about this way of approaching the question of ‘what mathematics’ is required for a college degree … the student is taking course(s), which are samples from the population ‘mathematics’.  Like all good samples, this sample needs to be representative of the population in the important ways.  The question becomes: what are the important characteristics of ‘mathematics’?

Here is one possible list of characteristics:

  • use of standard mathematical language and symbolism
  • almost all content follows from use of properties in the mathematical system, applied in consistent manners
  • the content represents multiple (2 or more) domains of mathematics
  • the mathematical reasoning would transfer to other samples of mathematics
  • learning can be demonstrated in both contextual and generalized ways

The purpose of this approach is to assess whether a student’s general education math requirement provided them with a valid ‘mathematical’ experience.  If that sample was not representative, then the student experienced a biased sample and is not likely to know what mathematics is (making the reasonable assumption that most students do not have an accurate view of mathematics, prior to the course in question).

In the classic Liberal Arts Math (LAM) tradition, the content is either ‘appreciation’ or specialized with little generalized knowledge; in some cases, the majority of the course derives from proportional reasoning with applications across non-mathematical disciplines.  The tradition of LAM is based in both liberal arts colleges or in ‘math for non-math-able students’.  In the former case (liberal arts colleges), the LAM course would make sense as one of the capstone courses, with an earlier math course that is more of a representative sample.  The latter (‘non-math-able students’) speaks more to our problems in teaching than it does to student problems learning mathematics.

The Quantitative Reasoning (QR) tradition is fairly new, and the QR name is sometimes used as a re-branding of a LAM course.  A strong QR course meets the requirements for a representative sample.  The QR course at my college is our best math course, combining both contextual and generalized results.  However, some QR courses are arithmetic-based applications courses; learning can not be generalized because the symbolic language (algebra in this case) is not required nor utilized.

The comments about statistics being a diversion from mathematics might be the least-well received due to the current popularity of ‘introductory statistics’ as a math course for general education.  The intro statistics course has a lot to offer … in particular, the fact that it is a fresh start in mathematics for most students.  However, the content is mostly from one domain (stat) with just enough probability to support that work.  The primary ‘non-representative sample’ issue, however, is the one about properties — where the vast majority of the intro stat content deals with concepts (good), and reasoning (good) but without a unifying structure (properties).  There is, of course, the irony in suggesting that a statistics course is a non-representative sample.

When a math course is a non-representative sample, students are being diverted from mathematics for that course and the students reach invalid conclusions about mathematics.  Such diversions tend to reinforce negative attitudes about mathematics OR suggest that the student is now good at ‘mathematics’.

All of this is written from a general education perspective.  Some programs clearly need knowledge of statistics, and I suppose a few of these needs can actually be met by an introductory statistics course.  The most common use of statistics in general education is the same as the original “LAM” (liberal arts math): a course that looks like mathematics for students who we do not believe can handle a representative sample of mathematics.

A good QR course is a representative sample of mathematics; although most students in a QR course do not take another math course, the QR course itself is not a diversion from mathematics.

The primary drawback to “QR” is that we lack consensus about that is ‘covered’ in a QR course.  In general, I am likely to be happy with any QR course that meets the standards above for being representative. Sometimes we worry far too much about the ‘topics’ in a course, and attend way too little to the important criteria related to what makes a ‘good math course’.

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