Math topics that reward going beyond cookbook methods

Question

Students fresh out of high school are often under the impression that mathematics is a discipline based entirely in recognizing the type of problem and applying an algorithm or cookbook method. These students think there are finitely many types of problems, and each type of problem corresponds to a method. From this perspective, "doing math" is about, in order,

1. Identifying the type of the problem
2. Remembering the method that corresponds to the problem
3. Executing the method

However, one of the main goals of quality math courses is to carry students up to the next level of Bloom's Taxonomy, where they might comprehend instead of just remember what is going on. In fact, we hope they will comprehend strategies or facts that transcend specific "types" of remembered problems. I always hope to chip away at the idea of every problem being classifiable into memorized and cookbook categories.

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I am interested in which topics in College Algebra or Precalculus best reward a student who ventures up a level in Bloom's Taxonomy in this way. These should be topics where a cookbook/remembering student will be able to solve the problem, but a comprehending student will be able to solve the problem more easily.

Answer 1

One answer would be: any real-world word/application problems; insofar as one avoids making them in a repetitive template by "just changing the numbers". I find that most any textbook has an "Extensions" and/or "Real-World Applications" block of exercises in each section, after the repetitive symbol-manipulation exercises.

But I would caution about pushing this too hard. To the extent that Bloom's Taxonomy is a legitimate analytical tool, I've got a thesis, to wit:

Math is hard because by its nature it's taught at TOO HIGH a level compared to other classes.

When most (non-math) teachers gripe about "Level 1: Remembering" their typical examples are "Recite a policy. Quote prices from memory to a customer. Recite the safety rules." (link) As soon as we in the math discipline even "change the numbers", we're already operating a higher level, e.g., "Level 3: Applying" (examples from link: "Use a manual to calculate an employee's vacation time. Apply laws of statistics to evaluate the reliability of a written test."). Do we usually quiz students on whether they've studied our foundational definitions, or do we just take that for granted? I might posit that we should do more at a lower level, because our normal practice is blind to the difficulty at even that level. Arguably a problem where even the idea of conceptual pattern-matching is designed to fail/be insufficient, and a brand-new structure or pattern must be created for the one problem, is at the very highest level of the hierarchy.

It's not like "pattern recognition" is bad: many thinkers consider that to be the very definition of mathematics! For example, Ivars Peterson wrote in Islands of Truth (quoting from Harold Jacobs):

Mathematics is really the science of patterns. Mathematical discovery begins with a search for patterns in data -- perhaps numbers or scientific measurements...

For many of us, having our students be able to recognize patterns regularly would count as an epic victory.

Answer 2

One really good one is solving absolute value inequalities.

Students can memorize the following pattern:

When you have $|X| \geq Y$, the setup is $X \geq Y$ or $X \leq -Y$.

Students with more comprehension will have the following more universal strategy:

When you have an inequality, start by asking whether it is impossible, or always true, or sometimes true.

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So, if you ask the students to complete:

Solve for $x$: $|4x + 17| \geq 0$

You will get the following (correct!) solution from the pattern-recognizer:

$4x + 17 \geq 0$ or $4x + 17 \leq 0$

$4x \geq -17$ or $4x \leq -17$

$x \geq -\frac{17}{4}$ or $x \leq -\frac{17}{4}$

[number line with everything shaded in]

All real numbers are solutions.

But the comprehending student will give the following, better solution:

The absolute value of a number is always greater than or equal to 0.

So all real numbers are solutions.

Answer 3

I fear you will do your students a disservice if you focus too much on avoiding “pattern recognition”.

Firstly, most mathematical “discoveries” at the level taught in school (as opposed to later on, and even then to some degree) were found through people observing patterns and investigating them. The archetypal example being Gauss and summing series :o)

Secondly, it is a great benefit to anyone wishing to apply mathematical methods to as quickly as possible recognise if any of the toolkit they have at hand is useful “out of the box” rather than having to spend a lot of time working out what to do. This particularly applies to areas which use mathematical techniques in a routine manner — branches of engineering, physics, chemistry, and so on.

What does help, a lot, is simple practice at spotting patterns in progressively “harder” situations, i.e. moving from spoon-feeding algebraic methods, differentiation, integration etc. towards the more “real-life” examples, although I think that it can’t hurt to slip in (say) a few puzzles stolen from Gardner et al or the many, many web sites devoted to such things by enthusiasts.

I’d venture to say in fact that the main benefit of “understanding” the foundations of any method is that it becomes easier to recognise novel situations which might benefit from it.

Answer 4

Instead of giving precise instructions and objectives, open the question up to whatever kind of inference and creativity the students can generate.

Traditional Closed Area Exercise

A carpet is 4m wide and 6m long. What is its area?

Open Area Exercise

Three rectangular carpets have the same area but have different dimensions. What might the dimensions of those rectangles be? Find a simple triplet of rectangles that works, then a medium hard triplet, then a triplet that nobody else in the class will think of.

Traditional Closed % Word Problem

An investment of $5000 earns 6% interest per year for 10 years. How much will the investment be worth in 10 years?

Open % Word Problem

Here are some investment prospectuses. [Hand them fictitious and simplified investment prospectuses.] They are all trying you to persuade you that something will happen to your investment over the next 10 years. What is that something? What if they're wrong by a little? What if they're wrong by a lot? Generate examples with numbers you consider plausible to illustrate your thinking.

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Many math questions can be opened up like this. It's much harder to assess and grade, but I don't think there's any question as to which type of question requires more reasoning, mastery, and understanding.

Answer 5

I would think problems on proving trigonometric identities would be well up that alley. These are the problems I think the "memorize" students tend to find most difficult, as such problems often require the use of a healthy variety of identities and algebraic tools.

I guess these days there are apparently list of steps to solving them, but a student who can take a step back and try to map out a rough path based upon the unique terms in a given question will usually find things easier.

The common false starts and occasional dead ends are perhaps more reminiscent of complex physics proofs/derivations rather than other typical concepts.

I'm not sure precalculus courses spend much time doing such problems, particularly when accelerated to a college setting. I think it's often one of the first subjects to be cut back on, given its limited short-term utility. But having spent an extra semester doing them in a trigonometry course in high school, the practice at comprehending and cognitively applying tools proved mighty useful down the road to me.

Answer 6

This is not a direct answer, but too long for a comment. As some of the comments mentioned, the problem isn't patterns, it is blindly applying cookbook solutions. In fact I would argue that mathematics is mostly about patterns, but not about applying them but about finding them. Use this to your advantage, teach them how to do mathematics instead of how to use mathematics, let them make the patterns their own. I would suggest that you try something like the following:

Give them a few problems with a similar pattern to solve. I assume since you ask about topics that work against them, you know enough topics that have them. But then don't stop there. Instead ask them, if they see the pattern. Maybe give them another similar problem, but ask them not to solve it, but instead to tell you how they would solve it without performing the steps. Then help them to formalize those steps a bit. If this works well, you could even ask them to do the same steps, but with constants instead of numbers. Finally congratulate them on having effectively discovered and proven a mathematical theorem. It might not be a big one, but we all started small.

If done right, this will lead to a far better understanding of what they are doing then solving another dozen of the same problems. The downside for you of course is that its not knowledge you can check directly on a test and it might be a bit more labour-intensive, but I think it could be worth a try.

Answer 7

If they are used to smaller fraction calculations they may solve a more complcated version such as, $$\large{\frac{\frac{1}{2} - \frac{1}{3}}{\frac{1}{4} + \frac{1}{6}}}$$
by working out the details of the top and bottom separately as they have practised these before:

taking common denominators, $$\large{\frac{\frac{3-2}{6}}{\frac{3 + 2}{12}}}$$

now simplifying the top and bottom, $$\large{\frac{\frac{1}{6}}{\frac{5}{12}}}$$

then turning the bottom upside down and multiplying to give the answer, $$\frac{1}{6} \cdot \frac{12}{5} = \frac{2}{5}$$

It is faster to look at the whole fraction:

remove the fractions from the top and bottom line first using the lcm of all the denominators,
$$ \frac{12}{12} \cdot \large{\frac{\frac{1}{2} - \frac{1}{3}}{\frac{1}{4} + \frac{1}{6}}}$$ now simplifying the top and bottom lines, $$\frac{6 - 4}{3 + 2} = \frac{2}{5}$$

Answer 8

A. Defense of recipes:

Even with your modified question wording, I don't quite share your disdain for algorithms (methods, tools, tricks, etc.). Much of human effort was spent developing these tools, they have wide utility and students benefit from learning them.

Even though, to you, this stuff is simple and intellectually boring that does not mean it is so for the students. To whom the methods are new and to whom they open large arenas of problem solving that previously were closed.

Also, even though you list the three steps as mechanical (identify, remember, calculate), I don't think they really are. It still takes some mental ability and exercise to execute these three. Even for simple problems. (If the methods are new to the student.)

B. Possible solutions

I do think there can be space to develop some extra difficulty and sophistication. For example "identify" can be made more difficult in some problems by wording or choice of boundary cases. Or problems may require multiple methods (making all three steps harder.)

Also, don't make it digital: recipes versus hard problems. But include some tough problems at the end of tests or as extra credits or the like.

C. Caveat:

But even with B, you need to make sure the students have the basics first. If I've never done exponent manipulation before, let me learn it and practice it a while before you subject me to tricky sophisticated problems (boundary cases, etc.) This is basic pedagogy. Let me perfect the back flip for a while before you tell me to do a double back. Even the single flip is new to me!

Also, look at the class level of performance on the recipe-based cooking. Don't make things tougher if they are struggling with the basics. Get that first. But if your kids are kicking butt, fine, introduce some trickier problems (gently and watching not to do too much.)

Also, look at the time you have, the objectives of the students/course, and the skill level of the students. It doesn't make sense to go off the Campbridge Tripos deep end with nursing students at a community college. It just doesn't. (And I'm not saying Tripos is "better"--I like nurses!)

Plus they have different goals and needs. You should have some sympathy interest in their track, in their struggles with chem, etc. Not just wanting everyone to be a math major.