SOAP, PEMDAS … Is there some MATH here?

For some reason, I have always found mnemonics to be irritating.  Perhaps this is based on a worry that understanding was being condensed to a ‘word’ that referenced a ‘phrase’ that had no connection to the mathematics involved.  Because we cover order of operations prior to algebra (questionable), we introduce one mnemonic to almost all students — PEMDAS.  Something about Aunt Sally (we all have one?) being excused.  The mathematical statement might be Grouping, Powers, Products and Quotients L to R, then Sums and Differences L to R.  Somehow, we don’t see “GP(P/Q)(S/D)”, even though it is better mathematically.

Another idea is reduced to a mnemonic — SOAP, for ‘same, opposite, always positive’ in factoring binomials involving two cubes.  This one at least refers to a memory process; this factoring is essentially a formula application.  The mathematics that is lost is ‘binomials of cubes’.  Perhaps this one should be ‘cubic SOAP’.  Of course, cubic-SOAP still is incomplete … it fails to capture the binomial going with the ‘Same’, and the trinomial first term (another always positive).

However, I wonder about the MATH mnemonic.  Perhaps you’ve heard it:

Man, Anything That Helps!!  (“MATH”)

Are we so desperate that we offer incomplete or inaccurate memory aids?  Perhaps we confuse correct answers with understanding mathematics.

Instead, I would like us to consider what this means:

Students should learn good mathematics in every math course.

A list of nice topics does not create a set of good mathematics.  In conversations, I usually find a good amount of consensus on the phrase ‘good mathematics’; we might have trouble articulating a single definition, but we have a good idea what it looks like at various levels of student mastery in various domains of mathematics.

Not everybody in the world uses ‘math’ as a label.  The label ‘maths’ is better, since our field has a plural nature; there is not one mathematic … there are fields of mathematics.  Perhaps if we kept using the word ‘mathematics’ instead of the inaccurate ‘math’ it would help us maintain our focus on why we are here … what we are helping our students WITH.  We are not here to get students to produce a minimal number of correct answers; we are here to help them learn mathematics with value.

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Just For Fun …

We have a traditional intermediate algebra course, and my classes are currently working on factoring.  Of course, these topics are only appropriate if a student is headed towards a STEM-type field; most of my students are done with this class, so there is a basic mismatch.  [That problem relates to the current work on the Michigan Transfer Agreement, which may take intermediate algebra out of the general education mix.]

However, we try to always have fun in class, and my students know that I don’t mind looking at other ideas.  One of those ‘ideas’ happened today; this is not radical, nor important in our class — but it was just plain fun.

We were working on factoring by use of formulas.  This particular problem dealt with a perfect square trinomial, with fractional coefficients.  Like this:

¼(a²) – (2/3)a + (4/9)

I’ve already told students that we are doing this much factoring just because it is on our departmental final; we are looking at them as puzzles.  This problem got us into looking for squares of fractional terms.  We got through it, and showed the factored form.

So, one of the students says:

Can we clear fractions?

Of course, I said.  “What would you do?” The student replied “Multiply by 36”.  Now, we have been focusing on what I call the 3 big rules of factoring — write as an equivalent product, use integers unless the problem had fractions, and each factor must be prime.  Since multiplying by 36 clearly changes the value, we need to do something to ‘keep it balanced’.  The solution is to show a division by 36:

(1/36) * 36[¼(a²) – (2/3)a + (4/9)]

So, we distributed the 36 and factored the resulting non-fractional trinomial … and kept the (1/36) factor in front.    To me, this was just plain fun; I know most students don’t agree — but at least they got to see somebody have fun with algebra.

This particular issue has been a problem; it seems like a few students would ‘clear fractions’ but without keeping the balance on the assessments for this material.  These students tended to be those I expect to do better — willing to think and reason, trying to connect information, etc.  I’ve not felt okay about just bringing up the clearing fractions method, because most students do not think of it in this context.

I just hope that I have more students like this one, who will be willing to ask a good question … and we can have some fun!

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Applications for Living — Geometric Reasoning

We are taking a test in our Applications for Living class, and I am struck by two things.  First, students have made major improvements in how they deal with converting rates (like pounds per second into grams per hour). Second, how bad geometric reasoning is, before and after our work on it.

Just about the simplest idea in all of geometry is ‘perimeter’.  Students have very little trouble with a rectangle as a stand alone object.  This problem created a speed-bump:

As a class, we ‘passed’ on that item (in terms of proportion with correct work).

However, we struggled with this problem:

We did not pass on this item, as a class.  The most common error, of course, was counting the ’12 inches’ (which is completely internal to the figure).  Not as many included the ‘8 inches’, which is also internal.  We always say that perimeter is the distance around a figure, but that is not internalized as strongly as the “2L + 2W” rule.

A bonus question on the test looks like this:

This problem combines the reasoning about perimeter with some understanding of right triangles as components of shapes.  A few students got this one right.

We spent parts of 3 classes working on our reasoning and problem solving.  These compound geometric shapes are common objects in our environment (at least in the USA).  I’d like to think that our students would be able to find the amount of trim or edging to install.

We are a bit too eager to pull out a formula for perimeter (where it is never required for sided-figures); when we talk about circles, it’s not connected well enough to other ideas like perimeter.  One of the problems we did in class caused a lot of struggle:

We used this problem as a tool to work on reasoning about perimeter (and area).  Much scaffolding was needed; since we only spent 3 classes on geometry, we did not overcome prior mis-conceptions in most cases.  Our better results with dimensional analysis (rate conversions) is due mostly to the fact that students had few things to unlearn.

Let’s do a little less variety in geometry, with more focus on reasoning.  Formulas are fine for area and all-things-circular, but have no business in the perimeter of sided figures.

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Symbols as Window

Like humans in general, our students develop expectations based on experience.  Habits form, often without awareness or conscious effort.  Behaviors exhibit, which are used to measure knowledge.  In assessments, we often confuse correct behavior with correct knowledge.

Symbolic work can be difficult for novices.  We (experts) see large amounts of information in short symbolic statements.  For a novice, symbols are like a map to a city never visited — yes, we can remember how to get from point A to point B on the map … but without any understanding of what these points mean in the city.

On a recent test in my beginning algebra classes, two mistakes were made by at least 20% of the students (one or both):

• -3² + 5² = 2^4 = 16
• 8^6 divided by 8^2 = 1^6

The first error is a coincidental ‘right answer’ for a very wrong method.  The second one, not at all.  Both involve over-generalizations of ‘same number’ rules.  Obviously, there is a very high probability that the students making one or both of these errors have low study skills or habits (like not doing any practice outside of class).

My concern is not these particular students, nor these particular errors.  My concern is our overall approach to mathematics.  We tend to take one of these approaches to symbolism in mathematics:

1. Emphasize symbolic procedures, and measure understanding by correctly completing more complicated problems.
2. Emphasize context and reasoning, and measure understanding by correctly completing related problems with differences in details.

Some reform models take approach #2 to the extreme — very few symbolic procedures are introduced, and most of what is done is arithmetic; algebraic models are used but carried out with technology more than symbolic procedures.  We need to learn how to balance the ‘symbols’ and ‘reasoning’ aspects of mathematics — and be willing to embrace both as critical in all mathematics courses.

Clearly, there is much (perhaps a majority) of our traditional algebra curriculum that involves symbolic work without a purpose now or in the student’s future.  I seriously doubt that solving a radical equation by squaring each side twice will ever be a survival skill in a student’s future.

Just as clearly (to me, at least), many of our students will need good understanding of various symbolic structures in mathematics, in future science courses (hard science and soft science).  Terms, exponents, coefficients, subscripts, groupings, equations, inequalities … are involved in stating properties in sciences and in using predictive models.

When we assess the mastery of symbolism, we need to deal with much more than ‘correct answers’.  In the ideal situation, assessments would be done in a one-on-one verbal interview so the expert can probe into the novice’s understanding based on the individual learner.  Lacking that luxury, we will need to use diverse assessment tools that deal with process and connections, as well as answers.

Sadly, I had integrated some of this assessment into the beginning algebra class about two weeks  ago — dealing with the adding terms error (first error above).  On a worksheet, students were faced with adding like terms (10x^4 + 6x^4) before we had dealt with them formally in class.  Something like half the students added the exponents as well as adding the ‘terms’ (coefficients).   About 40% of these students apparently maintained this erroneous method up until the test.

Correct answers are only correlated with correct knowledge; students are always seeking the simplest rules for achieving correct answers — which can lead to totally wrong rules.  Mathematical symbolism can be a window into the houses where students keep their math knowledge.  Too often, however, symbolism is confused with the knowledge and correct answers stop the assessment process.  

We need to slow down our courses.  Learning mathematics is not a fast or spontaneous activity.  Learning mathematics is hard work for both us and our students.

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