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. (
See here for a discussion of problem complexity from a "higher-order thinking" perspective.)
Let's look at this whole thing cognitively and take as our first couple of examples what most would consider to be relatively simple statistics problems:
- 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.
- Jennie's Rugs has been aggressively marketing their products on the web with Google ads and popups over the last twelve months. Below are the ad costs for both types of ads and the sales revenue for the last twelve months, as well as the sales revenue (before Jennie's Rugs started advertising on the web) for the previous twelve months. First, use the appropriate statistical tests to determine if the web ads have had any statistically significant effect on the sales renenues. If so, use the appropriate statistical test to determine if the sales revenues for the second twelve months can be predicted from either of the web ad types. Report all relevant statistics, and if relevant, include the formula to predict the sales revenue from the web advertising.
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.
Let's return to formalism. 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.
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 these two:
- The Superbowl Company produces footballs. Superbowl must decide how many footballs to produce each month. The company has decided to use a 6-month planning horizon. The forecasted demands for the next 6 months are 10,000, 15,000, 30,000, 35,000, 25,000, and 10,000. Superbowl wants to meet these demands on time, knowing that it currently has 5,000 footballs in inventory and that it can use a given month’s production to help meet the demand for that month. (For simplicity, we assume that production occurs during the month, and demand occurs at the end of the month.) During each month there is enough production capacity to produce up to 30,000 footballs, and there is enough storage capacity to store up to 10,000 footballs at the end of the month, after demand has occurred. The forecasted production costs per football for the next 6 months are $12.50, $12.55, $12.70, $12.80, $12.85, and $12.95, respectively. The holding cost per football held in inventory at the end of any month is figured at 5% of the production cost for that month. (This cost includes the cost of storage and also the cost of money tied up in inventory.) The selling price for footballs is not considered relevant to the production decision because Superbowl will satisfy all customer demand exactly when it occurs—at whatever the selling price is. Determine the production schedule that minimizes the total production and holding costs.
- 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 two (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 these problems are 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 problems are more mathematically complex (the first requires linear programming and the second, a monte carlo simulation) and therefore the process to arrive at the solution is more complex, and unlike the first two, there are terms and concepts (sometimes with their own hidden calculations) students must know and understand: Planning horizon, inventory, demand, production and storage capacity, holding and production cost, production schedule, fixed cost (of developing a car, as opposed to fixed cost in general), 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 (eek! another one of those hijacked phrases!) and many scan for numbers and ignore everything else.
See? I didn't even get 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.
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.
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