02 Oct 2005 - 23:17

why do we need to drill basic math skills until they are completely automatic?

The short answer: because completely automatic tasks are cost-free.

Every math problem a kid does - any bit of work that anybody does -- takes something out of him. As a kid gets older, the problems get harder and involve more steps. It's automation of the earlier, simpler steps that keep the complexity of a task from getting too big to handle.

Think about it in the context of reading. When you, as an adult, with your long-term expertise in reading, contemplate the work involved in learning something new, such as Roman history, do you consider the cost of the mechanical act of reading the words on the page? Of course you don't. In fact, given a page of text to read, you almost certainly can't help but read it. The act of reading costs you nothing.

Any component of a problem that a kid has mastered to automaticity will be discounted; in other words, completely automatic (rote!) tasks are not a drain on a kid's intellectual energy budget. The kid can focus on what he's intended to learn in that lesson, and can go further before he has to quit.

an example

In order to get specific, let's analyze what needs to be done in order to do an algebra word problem of a type that most of us don't remember fondly; mixture problems.

You have two lemonade mixtures. Mixture A is 5 parts water to 1 part lemonade powder, and mixture B is 2 parts water to 1 part lemonade powder. How much of mixture A and mixture B should you mix tin order to get a quart of a mixture that is 3 parts water to 1 part lemonade powder?

Nightmares are made of this stuff, but let's look at the steps you must take to do this problem. There are different ways to do this problem, obviously, but I would guess that they boil down to the same set of steps, more or less.

Step 1. First, you must figure out what fraction of each mixture is powder vs. water. This involves converting ratios -- such as 1 part lemonade powder to 5 parts water -- into equivalent fractional parts: i.e., 1/6 of this mixture consists of lemonade powder. Mixture A is 1/6 lemonade powder, mixture B is 1/3 powder, and the mixture to be created is 1/4 powder.

Step 2. You have to identify what you want to find; in this case, the unknown is the number of quarts of mixture A (once you know how much mixture A you need, the remainder needed to make a full quart is mixture B). You have to give this quantity a symbol, say x, with an associated unit, say quarts. This step seems trivial, but it's far from it (see the endnote).

Step 3. You must derive an algebraic relationship between the amount of mixtures A and B that you can solve for the unknown. Most reasoning methods will lead you to a conclusion similar to this one: if you have x quarts of mixture A, then you have 1-x quarts of mixture B, and the resulting mixture will have a proportion of lemonade powder that is expressed as:

1/6 x + 1/3 (1-x).

The correct value of x will have to satisfy:

1/6 x + 1/3 (1-x) = 1/4.

Step 4. Manipulate the above equation until you have isolated the variable x and obtained x =1/2.

This involves first multiplying out the terms in the above equation, then isolating x on one side of the equation, then solving. Isolating x correctly will give you the equation

1/3 - 1/4 = x (1/3 - 1/6).

You must then perform the fraction computations, and finally solve for x.

Step 5. Interpret this solution correctly to yield: 1/2 quart of each type of mixture is needed.

So, solving such a problem involves at least 5 separately identifiable steps. The "deepest" one -- the one involving the most insight and the least plug-and-crunching -- is step 3, in which the student derives the relationships among the given elements of the problem, and figures out what must be done to finish out the problem. That's the part of the problem that one would hope would take most of a student's effort and energy.

However, I've taught a lot of kids (in 'college algebra' classes) how to do this sort of problem, and step 3 is not the step that really flattens the kids. It's mainly step 4 that does that; the manipulation of the symbols in the equation, and the addition and subtraction of the simple fractions involved. Not far behind step 4 in difficulty is step 1, conversion of the ratios of the mixture's components to the fractional part that's lemonade powder. They haven't learned this stuff to the point of automaticity.

What does a student see when he looks at the steps involved in doing this problem? If he knows he can perform step 1 -- the conversion from ratios to fractional parts -- then that task shrinks to a point. The student knows he can do it without any effort, and discounts it from his 'energy budget' -- the effort he knows he'll have to expend to solve the problem. If he knows he has mastery of the algebraic symbol manipulations and fraction calculations involved in step 4, then that step also becomes one that has no cost for him. He realizes that most of the cost of doing the problem will come in step 3.

Step 3 really cannot be completely automated, as every problem is unique, so that step will always impact a kid's cognitive energy budget. The other steps should have no cost for a student; they are completely automatable. They should be the easy stuff.

Now imagine that you are a kid facing ten such problems for homework. You know that the equivalent of 'step 3' is going to be a challenge for every single problem. For the kid who has practiced the other skills to automaticity, that's the only challenge he'll have to face; and that's as it ought to be, since it's presumably that higher-level 'step 3' functioning he's trying to learn in this lesson.

But if you are a kid for whom each of these steps demands high-level mental energy, you are going to run out of steam sooner, get much less out of the current lesson, and remember (in all probability) not what you learned on this occasion, but how hard and painful each and every 5-step problem was. With memories of mathematics like that, no wonder you run from math at the first opportunity.

This is why drilling these procedural skills to the point of automaticity is so critical.

endnote

Step 2, identifying the unknown, is also a real hurdle for many students. I discovered when I was teaching algebra that it takes intellectual gumption to select an unknown and give it a variable name, and that sheer timidity made this necessary step difficult for practically all of my algebra students. Practice in identifying and naming unknowns seemed to help here -- simply having "permission" from the teacher to take this step seemed to help as well. If I were teaching now, I'd probably spend some time just drilling this one step for a number of types of equations.

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Comments

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I love it!

Cost free!

That's IT!

-- CatherineJohnson - 02 Oct 2005

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The analogy to reading is brilliant!

I love it!

BUT AUTOMATICITY IS NOT ROTE!!

IT'S PROCEDURAL!!

-- CatherineJohnson - 03 Oct 2005

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uh-oh

On half a beer, I can't read this post!

(No drinking while doing math....)

Up to the point of the analysis of the problem itself, BRILLIANT!!!!

-- CatherineJohnson - 03 Oct 2005

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Catherine, point on rote vs procedural is well taken; I've changed it.

-- CarolynJohnston - 03 Oct 2005

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Yes, rote is knowledge devoid of understanding, so this is not rote.

Now imagine complicating the problem by giving one powder in kgs, the other powder in ounces, one liquid in quarts and the other in ccs, while asking for the solution in completely different units like what you have to do in those messy "real world" problems our educators are so fond of. Lots more steps, calculations, and places for error, while filling up valuable slots of working memory. Less room for unclear conceptual knowledge, especially in the basic skills area.

-- KDeRosa - 03 Oct 2005

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i end up making an enormous fuss about "step 2":
"defining the variable(s)" (units included, please!).
the beginners that "have no idea how to start"
seem to need to be told many times to start here.
they just won't believe it. "intellectual gumption"
takes lots of handholding. meanwhile the faster
students are undermining my efforts by writing down
(essentially correct) equations without having
taken this step. which is cool (as long as they
then remember to put th' proper units in the "answer"s)--
but given the reality of "way too much to cover",
i've found that probably the single most important
intervention i can get across to the class at large
is the simple advice: "look, it's a textbook.
the authors are telling us how to define
our variables!" look at the question:
how many gizmos? -- x = # gizmos!
determine the price of a gadget? -- x = cost of a gadget!
(probably in $ even though \cents is usually more convienient.)
etcetera.
this bugs me somewhat since it's not really mathematics
-- it's "how to read the author's mind" --
but i want my classes to kill on the final ....

-- VlorbikDotCom - 03 Oct 2005

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It's always a joy to see what is on the Kitchen Table.

-- SmartestTractor - 03 Oct 2005

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V, good point on the variable units!... I'll throw that in too.

-- CarolynJohnston - 03 Oct 2005

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I agree and appreciate the steps as presetned here, but I believe an element is missing - a step where the student recognizes the procedural need for the problem. I beleive that when studetns get confused at problems of this sort, it is becasue they miss the simplicty of what is being asked (mathematics aside): - in order to get thing C, what do i need of thing B and Thing A? Once a student can ID the situation and can then apply the correct procedure, then the steps are logical. What I ask for in my students is to see problems like this, ones that can be initally scary, as hinting towards a process, then in order to find the process, lets look at it simply (thing c is made up of A and B). One of the best ways to find the simplicity is to work from the end of the problem to the front. Then you are usually seeing the answer first(as opposed to last when reading from start to finish) so you know the simplist side of the equation - then you think and read about the parts it is broken into. Seeing the simplicity of the problem allows students to Identify the realtionship between C, B and A, without having to worry about the math. Once the relationships are Identified, and the procedure determined, the steps represent substitutuions into the original, simple framework. Some nice examples of seeing the simple (or the core) occur in pre-calculus texts, where they focus on Core Graphs and how changing certain components causes shifting in the graph. Why math books don't do this more often escapes me.

-- RobWinky - 11 Oct 2005

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Seeing the simplicity of the problem allows students to Identify the realtionship between C, B and A, without having to worry about the math.

What you're saying sounds intriguing -- can you show how this idea would work in the case of the problem I used as an example? I.e., where would the step go and what would actually happen at that step.

-- CarolynJohnston - 11 Oct 2005