After the jump, some examples, both fictional and real world.
Kids today are still taught their times tables, but the curriculum makes room for other skills as well, skills that help students apply those math facts to real world problems. One skill taught in new math that is particularly upsetting to traditionalists is estimating. For example, besides memorizing that 7x9=63, students may also be taught to estimate the answer. 7x9 is pretty close to 7x10. The answer to that is easy, 70. So, the answer to 7x9 is going to be less than 70. If you think the answer is, say, 93, you'll know you are wrong. This drives some parents batty: what do you mean, my child is being taught that 70 is a close enough answer to 7x9?!?
Another skill taught is to perform a sanity check on your answer. In science class, a student may be asked to calculate the distance from the Earth to the Sun, given, say, the circumference of the Earth's orbit. A student applying math skills learned by rote memorization might busily calculate away and come up with an answer of 93 miles. A sanity check should suggest to him that the correct answer is a lot more than that: the Sun is farther away from Earth than, say, Houston is from Dallas.
Ironically, estimating and doing a sanity check used to be critical even in the pre-calculator days, when a slide rule hung off every engineer's belt. If you ever used a slide rule, you know they are very good at giving you the first two or three digits of a calculation, but they don't do squat in telling you where to place the decimal point. Good estimation skills can help you decide whether the answer is 93 or 93 million.
Estimating and sanity checks can help with another source of error, one that was popularized with the introduction of computers to do calculations, called "Garbage In, Garbage Out." Even if you can perform lightning quick, 100% accurate calculations, your answer might still be wrong if you start with inaccurate facts.
That brings us to a recent real life example. I blogged about Richardson's state representative Stefani Carter and her disregard for the sad state of Texas public school finance. That, in turn, led one commenter to lob an accusation that public schools are ridiculously way overstaffed:
Dallas schools employ 157,000 to process 20,000 students through the current education system. That's over a 7:1 ratio. Seven school employees for each student. How can the taxpayer not question the wisdom there?Source: The Wheel.
The reader claimed a DISD employee to student ratio of 7 to 1. A quick sanity check would have suggested the claim was preposterous. Go to a school. A ratio of 7 employees to 1 student would require the classrooms and halls and front office to be overflowing with employees. You'd have trouble picking out a student in the mass of adults.
The reader's error was not in dividing 157,000 by 20,000 (although that answer is much closer to 8:1 than 7:1, but no matter). It was in his starting assumptions. Is it reasonable to believe that there are only 20,000 students in the DISD when the population of the City of Dallas is over a million? In fact, the reader got his facts swapped. There are over 157,000 students in the DISD and 20,000 employees, not the other way around. He got the mechanical long division part right, but he failed spectacularly in applying those mechanical skills to the real world.
Why did this happen? Perhaps he is old enough not to have been taught math skills other than rote memorization and mechanical operations. More likely, the mistake (believing a 7:1 ratio of employees to students) aligned perfectly with his preconceived notion (schools are way overstaffed) and so it escaped normal sanity checks that would have easily revealed the answer to be preposterous. Ideology trumped reason.
Outrageously inaccurate errors like this influence public opinion all the time. Politicians exploit this to advance partisan ideology over wise public policy. State legislators like Stefani Carter act as if voter ID, sanctuary cities and carrying guns on college campuses are more important issues than solving the sad state of public school finance in Texas. Nationally, Congress acts as if the federal budget could be balanced if only we defunded NPR and foreign aid. Rather than letting the facts guide us to wise public policy, our ideology leads us to give uncritical acceptance to preposterous "facts."
I don't know how to solve this problem. Oh wait, I do too. It's called education. Someone please convince Stefani Carter.