Category: math reasoning and applications

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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Word Problems and Reasoning … Rule 42

In my introductory algebra class, I gave a quiz today; this quiz included this question:

Some milk having 1% fat is being mixed with milk having 4% fat; the mixture will be 100 gallons, and have 2% fat. How much fat is in the mixture?

Now, I always review the quiz right after we complete … so sometimes I get to see some interesting reactions.  Please understand that all of the items on this quiz dealt with ‘puzzle’ word problems (not much real context), so students were not feeling mellow … many were feeling quite a bit of stress.  Besides the stated reactions about the question being ‘tricky’, I got to see students respond when they realized what they were supposed to do.

A typical wrong response to that question was to start working on solving the ‘problem’ they expected to see … how much of each type is supposed to be mixed?  Quite a few of these students wrote the correct equation including the “0.02(100)” for the milk fat in the mixture.  However, not many of these students realized that they had the answer to the question.  Our entire approach to these mixture problems in the class prior centered on “value = rate(quantity)” as a basic concept.

I was pleased that some students just wrote down the answer (2 gallons), perhaps with a note “0.02(100)”.  These students got that basic idea that value = rate(quantity).  I’d say that this 20% compares with the 40% who tried to ‘solve’ the problem but never realized that they had the answer … and the 40% who had no idea what to do.

One of the culprits for the difficulties is the inadequate way percents are done in math classes.  We focus so much on correct answers that we do not make it clear that percent is not ‘how much’ … and that every percent is a rate which is multiplied by a base.  For my question on the quiz, just knowing “percent times base” is sufficient to get the right answer (and show some understanding).

The other culprit is based on the high-anxiety suffered by students when faced with “word problems”.  I’d like to think that my class presents word problems as a reasonable use of language and algebra, even if the problems are either trivial or uninteresting.  Further, I’d like to think this positive approach helps students be more comfortable dealing with these problems.

Some readers might wonder “why do those puzzle problems at all” … perhaps we should “make the content relevant to the students”.  With all of the focus currently on ‘alignment’ and ‘context’, those are reasonable questions.  Based on my understanding of the learning process (along with some sociology), the question is not easily answered.  I am pretty sure that covering ONLY relevant applications is not a good idea for a mathematics course serving a general purpose; it might work in occupational math, or specialized math, but not so much when there is so much diversity among the students.  One student’s relevant problem is another student’s puzzle problem, and another student’s life survival issue; in addition, high context in problems can localize the learning and interfere with general reasoning and understanding.

So, I will continue to work with quite a few puzzle problems in our introductory algebra course — and keep a focus on the basic ideas that allow us to understand and solve them.  My goal is to help students develop a deeper understanding and develop connections.

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