When you teach a course and you are responsible for creating the materials, assignments, and exams, you look at problems differently — more analytically, and from various perspectives one wouldn’t normally. One of the ways in which you analyze problems is in terms of difficulty or complexity. And there’s more to it than most realize.
Let’s look at this whole thing cognitively and take as our first example what most would consider to be a relatively simple statistics problem:
Lessen Waist, Inc. produces low-fat cereals, which they sell in 12-ounce (weight) boxes. Because of settling and production scheduling, Lessen Waist cannot weigh every box of cereal, and 0.35 ounces (weight) is considered to be an acceptable variance from the advertized weight. Lessen Waist weighs a subset of boxes because the filling machines must be adjusted periodically. Use the sample weights below and the appropriate statistical tests to determine if the boxes of cereal are within the acceptable weight. If they are not, use the appropriate statistical tests to determine how much the filling machines need to be adjusted. Report all relevant statistics.
The first problem students have — because a problem is more complex than most realize — is parsing the text of the problem. Far too many students experience some kind of frustration just reading the problem, and find it even more frustrating to try to get past the first reading (sorry to be cliché, but if I had a dollar for every time a student has come to office hours and expressed exasperation at being required to figure out how to figure out the “story problem,” I’d have my own island in the Caribbean). And this problem is getting worse.
Students — again, judging from what those who come see me say and (don’t) do — shut down when math is involved (yes, even in a statistics class — as if they expected it to be, well, I’m not sure what, though I’ve often wondered), and from what I’ve observed, much of the reason is because they see math as some kind of abstruse knowledge expressed in some kind of foreign language. Students ten years ago were much more likely to understand something when you wrote equations on the board than now, when more and more students give you the deer in the headlights (we used to introduce linear programming problems mathematically, by drawing the x and y axes on the board, extracting the information, calculating the lines and slopes, drawing them, and looking at the corners — but we no longer do that, because it mystifies students).
I think that’s partly because it is a foreign language due to lack of exposure, and a de-emphasis of formalism.
But once they get past the first reading of the problem, they have to do a number of things: Decide how best to solve the problem, extract any essential information, determine what additional calculations they may need to do, then set up the problem and solve it. So yes, if you’re statistically literate, either of the problems seems almost childishly simple, but to an undergrad, both are actually pretty complex.
But the problems get even more complex. How about this one:
General Ford (GF) Auto Corporation is developing a new model of compact car. This car is assumed to generate sales for the next 5 years. GF has gathered information about the following quantities through focus groups with the marketing and engineering departments.
- Fixed cost of developing a car: This cost is assumed to $1.4 billion ($1,400,000,000). The fixed cost is incurred at the beginning of the year, before any sales are recorded.
- Unit Gross Profit: GF assumes that in year 1, the gross profit will be $5000 per car. Every other year, GF assumes the unit gross profit will decrease by 4%.
- Sales: The demand for the car is the uncertain quantity. In its first year, GF assumes sales – number of cars sold – will be triangularly distributed with parameters 100,000, 150,000, and 170,000. Every year after that, the company assumes that sales will decrease by some percentage, where this percentage is triangularly distributed with parameters 5%, 8%, and 10%. GF also assumes that the percentage decreases in successive years are independent of one another.
- Depreciation: The company will depreciate its development cost on a straight-line basis over the lifetime of the car.
- Taxes: The corporate tax is 40%.
- Discount rate: GF figures its cost of capital at 15%
The general process is the same, of course, as the first (decide how best to solve the problem, extract any essential information, determine what additional calculations they may need to do, then set up the problem and solve it), but this problem is even more complex because there is more information to extract, there are more calculations required that the students must perform (after they’ve figured out they have to perform them), the problem is more mathematically complex (it requires setting up and statistically interpreting a monte carlo simulation) and therefore the process to arrive at the solution is more complex, and unlike the first problem, there are terms and concepts (sometimes with their own hidden calculations) students must know and understand: Fixed cost (of developing a car, as opposed to fixed cost in general, and as opposed to variable cost), unit gross profit, triangularly distributed (and parameters), depreciation (and straight-line basis), cost of capital, and NPV (net present value).
Then there is the covert information in the problem, such as “The fixed cost is incurred at the beginning of the year, before any sales are recorded,” which actually is a hint on how to set up the problem, “The demand for the car is the uncertain quantity,” which is another hint to tell students what the input variables for the simulation will be, “GF also assumes that the percentage decreases in successive years are independent of one another,” another hint to tell students how to set up the problem. But students are poorly prepared in the problem-solving process and many scan for numbers and ignore everything else.
See? I haven’t even yet gotten to the mathematical knowledge required to know what additional calculations to set up, do them, or figure out how the information given fits together. But sure, they have to do that too. Solving a problem is much, much more than just coming up with the correct solution — though the correct solution is essential, and no, and estimate is not an acceptable substitute.
But let’s look at the problem-solving process in more detail from a “higher-order thinking” perspective.
The popular educrat myth is that education needs to focus on “higher-order thinking,” to the exclusion of developing actual knowledge. Looking at this purely from a “higher-order thinking” context demonstrates this to be a falsehood. Let’s choose another example from the same undergraduate class:
A customer requires during the next 4 months, respectively, 50, 65, 100, and 70 units of a commodity, and no backlogging is allowed (that is, the customer’s requirements must be met on time). Production costs are $5, $8, $4, and $7 per unit during these months. The storage cost from one month to the next is $2 per unit (assessed on ending inventory). It is estimated that each unit on hand at the end of month 4 can be sold for $6. Determine how to minimize the net cost incurred in meeting the demands for the next 4 months. Use named ranges, and list all named ranges and addresses, as well as all functions and formulas and their addresses.
The students are given a blank Excel file. They must analyze the problem, decide what kind of problem it is and therefore how best to solve it, extract the relevant information, construct a logical solution — all by reading the text of the problem — then use that information to set up and perform any necessary calculations, and solve the problem on the blank Excel worksheet.
So what kind of problem is it? Reading through the information in the problem, and the words chosen (”Determine how to minimize the net cost . . .”) reveals that this is an optimization or linear programming problem, that is, a problem where the goal is to find the best solution given the data and a set of constraints.
We now know what the goal of the problem is. Now we need to extract the information:
- Orders over the next 4 months: 50, 65, 100, 70
- Production costs per unit, for the 4 months: 5, 8, 4, and 7
- Storage cost per unit from month to month: 2
- Price per unit, units on hand: 6
Extracting the information means we also have to infer what the constraints are. Reading through the problem, we see nothing about budget (maximum total cost), or maximum units to produce, or a minimum total profit; in fact, we can infer only one constraint:
- Units produced must be greater than or equal to the number ordered, over each of the 4 months
In order to construct a logical solution, we need to know what kind of problem it is (we do), we need to have extracted all the information from the problem (we have), and we need to know what our goal is (we do). We also have to determine what calculations we need to perform, including what information we need as a result or solution. We have done neither of those things.
We know what our goal is: Minimize the net cost (note that that means we must calculate the net cost when we set up the problem, something many students miss). What we don’t know is what data our solution needs to include. Sometimes, the problem will overtly state it; other times (like in this problem), it will not. To infer what our solution needs to be (since our problem does not overtly state it), we need to read through the problem, asking ourselves a question: What information do we need in order to solve the probem that isn’t given in the problem?
The number of units produced for each of the 4 months. That is the information we need to find.
And what calculations will we need to perform? Well, since our goal is to minimize the total net cost, we will have to calculate the total net cost. In order to do that, we have to multiply the number of units produced by the production cost per unit. But that’s not all. Scanning through the problem, we see that there are storage costs from month to month. Storage costs are for units produced in excess of the orders, and there’s another calculation. You get the idea.
The student’s worksheet should look something like this (click on the image to open a larger, legible one):
Looking at this problem in the context of “higher-order thinking” reveals that it is relatively complex, despite its mathematical simplicity (for a linear programming problem). It’s a relatively complex problem that requires a linear, logical thought process to solve. But for the moment, let’s put that linear, logical approach on the back burner.
Fuzzy math proponents are forever claiming that what they term “rote memorization of algorithms” (or some similar sneering phrase) is not necessary because students can use their calculators. Calculators aren’t relevant here, but Excel — which we can think of as a super-calculator — is. And this problem demonstrates that the fuzzy math proponents are completely wrong.
Excel won’t do any of these things for you. Not one. Excel won’t analyze the problem, decide what kind of problem it is and therefore how best to solve it, extract the relevant information, construct a logical solution, or set the problem up and solve it on its own worksheet. The only tool that will do those things is the student’s brain, and if students don’t have the mathematical knowledge — all that “rote memorization of algorithms,” or some similar sneering phrase — they can’t solve the problem.
It’s as simple as that.
But there’s another, even more fundamental “higher-order thinking” reason the fuzzy math proponents are wrong. In order to solve the problem, students must approach it with a linear, logical process. Students who have gone through a “discovery-based” math program that eschews linearity and logic for Burger King math (“Have it your way!”) have been crippled. They do not have the cognitive tools other students have, and must either develop them quickly (unlikely) or fail (likely). I realize that letting Suzie cut up a piece of construction paper to determine estimate the area of a circle may make teachers feel good about themselves, but Suzie will be mathematically retarded when she gets to the university.
Math hasn’t been taught since before Aristotle just for its own sake. Math has also been taught from the time of the Golden Age of Greece because it teaches those linear, logical thought processes that are so crucial in so many aspects of daily life — not to mention the university classroom.
Finally, fuzzy math proponents are wrong by teaching students that a problem can have many different solutions, teaching them to estimate instead of solve. There is only one solution — the correct solution — and all others are equally wrong. Students are often mystified when they turn in an often wildly incorrect solution and are first confronted with this grim reality. And they are mystified because their pre-university teachers have scratched them behind the ears and patted them on the heads like their little pet poodles and given them so-called partial credit for incorrect solutions.
You don’t get partial credit in the real world. You get fired if you can’t do the job, and do it quickly and efficiently. A student can use “what-if analysis” — plugging in numbers at random until coming up with what seems to be an ideal solution — but the student would get fired for it, only because of the gross inefficiency and wasted time and money involved. Sure, there may be several ways to solve — not estimate — a problem, but only a subset of those are efficient. Pedagogically, only a subset of those teach students the linear, logical thought processes they will need to succeed.
We aren’t here so much to teach students how to solve the problem as we are to teach students the quickest, most efficient route to solving the problem. If Suzie uses “what-if analysis,” she will get no credit (though if she uses the correct process but gets the wrong answer, she will get partial credit, because we are here to teach her the process). Allowing Suzie to cut up construction paper to estimate the area of a circle teaches Suzie that an estimation is as good as a correct solution, and that how she gets to the solution doesn’t matter, two falsehoods. Worse, “discovery-based learning” does not develop the linear, logical cognitive skills students must have if they are to succeed.
So even by the standards of fuzzy math proponents, their teaching methods are a failure. They teach none of the “higher-order thinking” processes that are required to actually tackle and solve even a simple problem, much less a complex one. They scorn the mathematical knowledge students must have to set up and solve the problem. Students are learning nothing useful, for the real world or the university, in these fuzzy math classes — and it shows in my classroom.
But forget “higher-order thinking.” Let’s turn to basic mathematical knowledge that every sixth-grader should know, but many of my students (more and more each semester) do not. And I know they don’t know these things because I have to explain them in class. Students do not know
- what a rate is: I have more than a few students who do not understand why they cannot just add the tax rate to the item price to get the total sale price.
- basic addition and subtraction: I have more than a few students who do not understand that you subtract the cost from the revenue to get the gross profit margin, or do not know that to get the total costs, you add the fixed and variable costs.
- basic multiplication and division: I have more than a few students who do not know that they must mutiply the number of units by the unit cost to get the total cost. I have more than a few students who do not know that because the interest rate is annual, they must divide it by 12 to calculate the monthly amoritization table.
- the relationship between multiplication and division: When we start doing optimization problems in Excel Solver, I have to tell students that because Solver does not like division, they must construct their problem with multiplication instead, and I have many students who do not know or understand how to do this (I also have more than a few students who do not know that you cannot divide by zero.)
- what an arithmetic mean is: I have more than a few students who not only do not understand what a mean is, but seem unable to grasp the concept. It goes without saying that they also do not grasp any statistical concept beyond the arithmetic mean.
All of these — with the exception of the mean — are concepts all students should know before they get out of elementary school. These are undergraduates, the best of high school graduates, yet they cannot do simple arithmetic. Even if they do get past the “higher-order thinking” and figure out what kind of a problem, etc., they cannot set the problem up and solve it, because they do not know basic arithmetic. Why am I explaining to undergraduates that they cannot divide by zero? Why am I explaining to undergraduates that they must multiply the tax rate by the item price then add it to get the sales price (actually, the question here is how do they not know this just from daily life?) Why do I have to explain to students that to get the total cost they have to add the fixed and variable costs? Why do so many of my students have less arithmetic knowledge than any sixth-grader should have?
It’s not just that students don’t know what 24 + 7 is without a calculator, because the solution isn’t the only point — something those who sneer at mathematical knowledge don’t seem to grasp. Knowing the relationships between arithmetic operations is just as important, if not more so. Knowing that you can’t divide by zero is just as important, if not more so. Being able to mathematically manipulate fractions is just as important, if not more so. And by sneering at “drill and kill,” educrats ensure that students will be mathematical morons.
At the end of the day, the only way to solve these problems without jumping off the roof of the nearest dormitory (actually yes, students do that — I’ve had two students die during the semester, but neither committed suicide, I’m glad to say) is to approach the problem with the process that traditional math pedagogy has been teaching for several thousand years now. What kind of problem is it? What is the goal of the problem? What information is in the problem and what information is not in the problem? And so forth.
You can always tell the students in class who have been rigorously trained in formalism: They’re the ones who immediately begin asking the questions and cutting it up into its components, and then solve it first, usually without much trouble at all. They read the problem and they know how to attack it. The students who have never mastered the thought processes behind solving problems are the ones that start then stop, start then stop, start then stop, and eventually give up, because they find it too frustrating just to try and get past reading the problem.
What happened to their “higher-order thinking” skills?
« Close it
Posted by rightwingprof on February 13, 2007 at 4:43 pm under Math, Education.
4 Comments.
