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 an example from an undergraduate operations class, taken predominantly by freshmen. Here is the problem:
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 the problem up and solve it 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, 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. 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. It’s a relatively complex problem that requires a linear, logical approach. 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 (rather, 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. 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.
Even by the standards of fuzzy math proponents, their teaching methods are a disgrace.