On a website promoting Common Core tests being superior to other tests, ExcelinEd cites some examples of questions. In all fairness, some of the questions are indeed better than those which can result in correct answers by guessing or other means. But in other cases, they seem to mince words.

For a high school question they cite an example of the “Previous math question”, in which “previous” means pre-CC:

If 3(y-1) = 8, then what is y?

They feel that this question “is an example of solving equations as a series of mechanical steps.”

The CC type question however is in their minds much better: “What are two different equations with the same solution as 3(y-1) = 8?”

They state: “This question is an example of solving equations as a process of reasoning.”

Well, I guess, but they still both, at the end of the day, involve a series of mechanical steps. And both require that the initial equation be solved.

Seems like a distinction without a difference. And there are some tests that have been used for years that have done a nice job of distinguishing the abilities of students, despite the “mechanical’ nature of the questions.

It’s good to bear in mind what Zalman Usiskin, a noted anti-traditional math person had to say about testing:

*“Let us drop this overstated rhetoric about all the old tests being bad. Those tests were used because they were quite effective in fitting a particular mathematical model of performance – a single number that has some value to predict future performance. Until it can be shown that the alternate assessment techniques do a better job of prediction, let us not knock what is there. The mathematics education community has forgotten that it is poor performance on the old tests that rallied the public behind our desire to change. We cannot pick up the banner but then say the tests are no measure of performance. We cannot have it both ways.”*

Zalman Usiskin What Changes Should Be Made for the Second Edition of NCTM Standards. UCSMP Newsletter, n12 pp. 10 (Winter 1993)

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The CC version given by the cited piece does no require the equation to be solved. From 3(x-1)=8 you can get 6(x-1)=16 and 3(x-1)+1=9. A child could get either without actually solving, or for that matter knowing how to solve, the equation. If the learning or testing objective is to establish that a child can solve simple 3-step equations like this, the question is a failure.

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Thanks; hadn’t thought of multiplying both sides to get an equivalent equation, but yes, that would not require the equation to be solved at all.

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There is nothing wrong with learning to perform straightforward procedures mechanically. A primary objective of mathematics is systematic reasoning. I see that being undercut continually by the fuzzy math folks. Systematic procedures are not a weakening of mathematics — they are the *essence* of the subject. To quote Alfred North Whitehead:

“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”

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My complaint is that many feel the goal of state testing is to examine understanding or flexibility or something else that is best evaluated over time, tests, and homework by teachers who get to know the students. If you wait to find this out on the state test, you are a year late and many tutoring dollars short. The best use of state testing is to check to see if something is fundamentally wrong at the school, and that is done using a simple test for mastery of basic skills. That provides more concrete feedback than vague scores on problem solving. I was in a parent/teacher meeting once where our state score on problem solving was down. The answer was to work harder on problem solving. If a school does poorly on fractions, however, then one can create specific solutions and simple tests to fix and check the problem.

Also, many attempts I’ve seen on creating “reasoning” math problems fail completely. Perhaps it’s a wording problem or maybe the key idea isn’t clear enough. My son used to overthink these problems. He would get that problem wrong because it’s just plain dumb-ass stupid. Is y-1 = 8/3 a “different” equation? How about y = 8/3 + 1 or y = 11/3? If he just solved the problem, then he should get it correct! Perhaps my son would “reason” duh, that’s the whole point of the process – what else could they be looking for? You would have to understand and think like an education pedagogue to know what they are looking for. This is not about mathematical reasoning, but educator reasoning.

These rote pedagogues can’t see where the real rote learning problem exists.

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