Saturday, September 9, 2017

Shortcuts in Teaching Mathematics Lead to Quicker Dead Ends

There are lots of times in life where shortcuts lead to efficiency. Efficiency is great, provided it is actually effective at achieving your goals (or the goals you should have). On the other hand, you can sometimes efficiently achieve a short term goal only to find yourself at a dead end later on.

If you're in a maze, not every step closer to straight-line proximity with the cheese is necessarily actually getting you closer to eating the cheese. In a maze, you can be right beside the cheese with just a single wall between you and the cheese and you might be as far away as possible from being able to eat the cheese. Sometimes you need to head in a direction that seems to take you further from your goal, in order to be closer to achieving it. Do you want to have a really strong smell of cheese or do you want to actually eat cheese?

Take weight lifting as an example. Improving your technique often takes you back to lighter weight. Your goal is to lift lots, right? Well, lighter weight seems like the wrong direction if you're thinking naively. But improved technique will take you further along a path that can actually lead to "having cheese" rather than just "smelling cheese", both because you will be less prone to injuries which will set you back and because you will be training your muscles to work along a more efficient path. So, suck it up!—and reduce the weight if you have to.

What follows is a series of common math shortcuts and suggestions for avoiding the pitfalls of the shortcuts (like, avoiding the shortcuts 😜). Some of these are statements that arise from students who hear a teacher express a proper truth the first time. But, when asked to recall the statement, the student expresses an abbreviated version of the statement that differs in a way that makes it not true anymore. Sometimes the student really did understand, but was experiencing a "verbally non-fluent moment" or just didn't want to expend the energy to explain. The teacher, trying to be positive, accepts this as a token of the student paying attention and then gets lazy themselves and doesn't add a constructive clarification.

In any event, the quest for simplicity and clarity has pitfalls. Merely making something appear simple is not a sure path to understanding. Go deeper.

Two Negatives Make a Positive

Why it's bad: It is a false statement. If I stub my toe and later hit my thumb with a hammer it is not a good thing, it is cumulatively bad.

How it happens: Quick summary (not intended to be fully accurate) and humor. Unfortunately, the quick summary can supplant the real thing if students are not reminded consistently about the full version of the statement (below).

Better: A negative number multiplied by a negative number, produces a positive number.

Expansion: Negative numbers are a way of expressing a change in direction, as in, (sometimes figuratively) a 180° change in direction. $10 is something you want to see coming your way. -$10 can go to someone else.

With this in mind you can explain that subtraction is equivalent to adding a negative number. A negative number is equivalent to multiplying the corresponding positive number by -1. The negative on the front of the number means you move to the "left" on the number line (a metaphor I think is sound). To put it a little differently, if you pick a random number on the number line and multiply it by -1 it is like mirroring it about zero (0). Also, multiplying by -1 is equivalent to taking its difference with zero. That is,
$$0-z=-1 z = -z,$$ which are more or less a part of the definition and development of negative numbers in a technical sense.

Displacement is one of the easiest illustrations to use. Suppose I am standing on a track that has chalk lines marked from -50 m to 50 m. I am standing at the 0 m mark facing the end with positive numbers. Instructions show up on a board which are in a few different forms:
  1. Advance 10 m. 
  2. Go back 10 m.
  3. Move 10 m.
  4. Move -10 m.
It is easy enough to see that 1 is equivalent to 3 and 2 to 4. Following a list of such instructions will result in a predictable finishing position which can be worked out one step at a time in order or can be put into a single mathematical expression. Commutativity and associativity can be explored by comparing differences in order applied to the mathematical expression as well as the list of expressions. I can reorder the sequence of instructions and produce a new expression that gives the same result or I can tweak the expression an spit out revised instructions that parallel the revised expression and produce the same result. Arithmetic is intended to express very practical things and so if something you do with your math expression causes a disconnect with the real life example, you are guilty of bad math. The problem will not be with commutativity or associativity, but with the implementation of it. It is worth investing a good deal of time on this, but it will probably have to be brought up again and again when you move on to other things because, I think, students sometimes get slack on their efforts at understanding when something appears too easy.

The next step is to understand that reversing direction twice puts you back on your original direction. We can see how this works on the track by working with the displacement formula:
$$d = p_f - p_i,$$ where \(p_f\) is final position and \(p_i\) is initial position. It is easy to illustrate on a chalk/white board that if I start at the -20 m mark and travel to the 50 m mark I will have traveled 70 m, not 30 m. Using the formula to get there will require combining the negative signs in a multiplication sense.

I would love to have a simple example that involves two negative non-unity numbers, but while I've done this arithmetic step countless times in the solving of physical and geometrical problems, I have trouble isolating a step like this from a detailed discussion of some particular problem and still retaining something that will lead to clearer understanding than the displacement example.

Heat Rises

Why it's bad: It is a false statement.

How it happens: Unreasonable laziness.

Better: Hot air rises (due to buoyancy effects).

Expansion: The general way that heat transfer works is not related to elevation change. A more accurate but still cute, snappy saying to summarize how heat moves is "heat moves from hot to cold" or, my favorite, "heat moves from where it is to where it isn't" (relatively speaking).

So, why does hot air rise? The cause is density differences. Denser gasses will fall below less dense gasses and "buoy up" or displace the less dense gasses. Density is affected by both atomic weight and configuration of particles. Configuration is affected by temperature. The warmer the temperature the faster the particles move and bang into each other and tend to maintain a sparser configuration making them less dense (collectively).

It might be better to think of the more dense gas displacing the less dense gas. The gravitational effect on the more dense gas pulls it down more strongly than it does the less dense gas and the more dense gas pushes its weight around so that the less dense gas has to get out of the way. "Out of the way" is up. (A bit of a fanciful description, but a good deal better than "heat rises".)

Bottom line, if you want a shortcut, "hot air rises" is better than "heat rises". It's a small difference, but it is still the difference between true and false.

Cross Multiply and Divide

Why it's bad: It does not require the student to understand what they are doing. For some students, this is all they remember about how to solve equations. Any equation. And they don't do it correctly because they don't have a robust understanding of what the statement is intended to convey.

How it happens: Proportions are one the critical components of a mathematics education that are invaluable in any interesting career from baking, to agriculture, to trades, to finance, to engineering, to medicine (or how about grocery shopping). Teachers are rightly concerned to turn out students who can at least function at this. It breaks down when it is disconnected from a broader understanding of equations. Students may look at the easy case of proportions as a possible key to unlocking other equations, which has a degree of truth to it. However, it is important to emphasize the reason why cross multiply and divide works (below). Without understanding, there is little reason to expect the simple case to spillover to help in the hard cases.

Better: Always do the same thing to both sides of an equation. If both sides are equal, then they are equal to the same number (for a given set of input values for the variables). If I do the same thing to the same number twice, I should get the same result. The result might be expressed differently, but it should still be equivalent.

For simple equations, often the best operation to do (to both sides, as always) is the "opposite" of one of the operations shown on one or both sides of the equation. Cross multiply and divide is a specialization of this principle that is only applicable in certain equations which must be recognized. The ability to recognize them comes from having good training on the handling of mathematical expressions (see remarks on BDMAS).

Expansion: Provided "do" means "perform an operation", the above is pretty valid. The other type of thing you can "do" is rearrange or re-express one or both sides of an equation such that the sides are still equivalent expressions. Rearrangement does not have to be done to both sides of the equation because it does not change its "value". When operations are performed, they must be applied to each side in its entirety as if each entire side was a single number (or thing). Sometimes parentheses are used around each side of the equation so that you can convey that distributivity applies across the "=" sign, but from there, distributivity (or lack thereof) needs to be determined based on the contents of each side of the equation.

Most Acronyms (but Especially FOIL and BDMAS)

Why they're bad: My objection is qualified here (but not for FOIL). An acronym can sometimes summarize effectively, but it is not an explanation and does not lead to understanding. In rare cases, understanding may not be critical for long term proficiency, maybe. But an acronym is a shoddy foundation to build on. If you're trying to make good robots, use acronyms exclusively.

How it happens: Acronyms can make early work go easier and faster. This makes the initial teaching appear successful—like a fresh coat of paint on rotten wood. Teacher and student are happy until sometime later when the paint starts to peel. Sometimes after the student has sufficient understanding they may continue to use certain acronyms because of an efficiency gain they get from it, which may lead to perpetuating an emphasis on acronyms.

Better: Teach students to understand first. Give the student the acronym as a way for them test if they are on the right track when you're not around. Very sparingly use as a means of prompting them to work a problem out for themselves. (My ideal would be never, but realistically, they need to be reminded of their back up strategy when they get stuck.) Never, ever take the risk of appearing to "prove" the validity of operations you or others have performed by an appeal to an acronym (unless it is a postulate or theorem reference)—that's not just bad math, it is illogical.

Expansion: Certain acronyms, if you stoop to use them, can possibly be viewed as training wheels. Maybe BDMAS qualifies. But is there a strategy for losing the training wheels or are the students who use the acronym doomed to a life of having nothing else but training wheels to keep from falling over?

So, BDMAS is a basic grammar summary. But you need to become fluent in the use of the language. A good way to get beyond the acronym is to have clear, practical examples of things you might want to calculate that involve several operations. Calculating how much paint you need is a good way to help convey how orders of operations work. Before you calculate the amount of paint you need, you get the surface area, \(s\). The total surface area is a sum of the surface areas of all surfaces I want to paint. If I have the dimensions of a rectangular room, I can get the area of each wall and add them together. To make the example more interesting, we will omit to paint one of the long walls. Because of order of operations giving precedence to multiplication over addition, I have a simple expression for a simple thing:
$$s = lh + wh + wh = lh + 2wh = h(l + 2w).$$ If you explain how to arrive directly at each expression without using algebra (with reference to simple diagrams), the meaning of each expression can be understood at an intuitive level. Understanding the geometry of the situation gets tied to understanding of the sentences you are making in the language of math. To get the number of cans of paint \(N\), you need coverage, \(c\) in area per can. Then \(N = s/c\). And now, if you didn't already demonstrate how parentheses support the act of substitution in the surface area development, now is a good time, because now you can use substitution for one of the ungainlier expressions for the surface area and get:
$$N=(lh + 2wh)/c.$$ If you also walk through how to do the calculation in multiple simple steps you can draw the parallels with the steps you would take in calculating using the above formula. I realize substitution showed up much later in the curriculum I received than order of operations but I believe this is a mistake. Even if the student is not expected to use substitution in grade 5, why not let them have an early preview so it doesn't seem like it's from outer space when they need it?

Oh, yes, and FOIL. Don't use FOIL outside the kitchen. Better to teach how distributivity applies to multiterm factors which will again be something like a grammar lesson and can incorporate some substitution (e.g., replace \(x + y\) with \(a\) in one of the factors) or "sentence" diagramming, which is beyond the scope of this post.

Using Keywords to Solve Word Problems

Why it's bad: It does not require the student to understand what they are reading which masks long-term learning problems, and leads to long-term frustration for the student.

How it happens: Students normally want something to do to help them get unstuck. Telling them they have to understand what they are reading isn't the most helpful and giving a bunch of examples of similar expressions and finding ones they already understand seems like a lot of work to go through. Keywords are fast and easy to tell students and are often enough to get stronger students started.

Better: Find analogous expressions that are already understandable to the student. If you can find statements that the student already understands at an intuitive level, you may be able to point out the similarity between the statement they are having trouble with and the statements they already can relate to.

Expansion: I am not aware of a standard treatment of this issue that meets my full approbation. We use language everyday and we don't use keywords to figure out what people mean by what they are saying. We shouldn't use language any differently with a word problem. It's the same language!

The words used in grade 6 word problems are all everyday words. What is needed is the ability to understand and use the same words in some new contexts. Providing a lot of examples is probably the way forward with this. Being able to restate facts in other equivalent ways may be a good indication of understanding and accordingly a good exercise.

It's important to recognize that language is complex and takes time to learn. Not everyone will learn it at the same rate and having a breadth and variety of examples with varied complexity is probably necessary for students who struggle more with it. Unfortunately, school doesn't support this kind of custom treatment very well (Cf. "growth" as per Franklin, Real World of Technology).


Explanations, examples, and exercises that lead to genuine understanding are much needed by math students at all levels. I do not believe in the inherent value of making students suffer, figuring everything out for themselves by not giving them the best possible chance of understanding the material with good instruction. But undue opportunities to opt out of understanding are a disservice to them. Training wheels have their place, but we should make every effort to avoid seeming to point to training wheels as any student's long term plan for achieving competency in a subject area.

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