### The Homework Vs. The Test

When (and to what extent) should math tests start deviating from the practice problems?

For example, if I were observing an elementary school class learning how to multiply two-digit numbers by two-digit numbers, I would expect the test questions to deal almost exclusively with multiplying two-digit numbers by two-digit numbers. Maybe a few questions dealing with one-digit numbers. I would be very surprised to see a test question where both the multiplier and the multiplicand* had at least three digits. Sure, someone with a good understanding of place value and the distributive law could extend the algorithm, but how many elmentary school children would have a firm enough grasp of those concepts to do that under test conditions? Definitely if that were a test question, I would hope that its effect on the children's grades would be negligible.

But about a decade later (some of) these children will be ending up in calculus courses much like the one I teach. At that point, the test questions are not all clones of the homework problems. I structure my tests so that at least 70% of the points available do come from homework problems with the numbers changed, the worked out examples in the textbook, and other similar questions. The remaining points come from applying the concepts to somewhat novel situations; the degree of novelty varies depending on the difficulty of the topic. At most 5% of the points will come from questions that require a really deep understanding of the material.**

At some point there needs to be a shift in expectations: from "learn this procedure and use it" to "learn this procedure and determine when it can be used and how to apply it in a variety of situations."

I bring this up because Catherine at KithenTableMath is frustrated at how her son's math test differed from the math homework (she mentions this in a comment dated November 8, maybe half a dozen down from the top). (

According to Piagetian theory, it is during the middle school years that children can start making the transition from concrete operational thinking to formal operational thinking. This theory also postulates that this transition is not automatic, that students will not acquire these higher order thinking skills unless they are challenged with tasks that are just barely beyond what they can currently do. (My understanding is that children are not expected to be successful at these unfamilar tasks on their first tries -- they need a few attempts to figure it out.)

In other words: when do we start testing the flexibility of mathematical understanding?

*This is how you can tell that I've taught Math For Elementary Ed: I can use words like multiplier and multiplicand and minuend and subtrahend. And here you thought that arithmetic jargon topped out at dividend, divisor, quotient, and remainder.

**Recall: This is just a description of my tests. Different rules apply to quizzes and graded homework.

For example, if I were observing an elementary school class learning how to multiply two-digit numbers by two-digit numbers, I would expect the test questions to deal almost exclusively with multiplying two-digit numbers by two-digit numbers. Maybe a few questions dealing with one-digit numbers. I would be very surprised to see a test question where both the multiplier and the multiplicand* had at least three digits. Sure, someone with a good understanding of place value and the distributive law could extend the algorithm, but how many elmentary school children would have a firm enough grasp of those concepts to do that under test conditions? Definitely if that were a test question, I would hope that its effect on the children's grades would be negligible.

But about a decade later (some of) these children will be ending up in calculus courses much like the one I teach. At that point, the test questions are not all clones of the homework problems. I structure my tests so that at least 70% of the points available do come from homework problems with the numbers changed, the worked out examples in the textbook, and other similar questions. The remaining points come from applying the concepts to somewhat novel situations; the degree of novelty varies depending on the difficulty of the topic. At most 5% of the points will come from questions that require a really deep understanding of the material.**

At some point there needs to be a shift in expectations: from "learn this procedure and use it" to "learn this procedure and determine when it can be used and how to apply it in a variety of situations."

I bring this up because Catherine at KithenTableMath is frustrated at how her son's math test differed from the math homework (she mentions this in a comment dated November 8, maybe half a dozen down from the top). (

**Update:**She has expanded her comment into a post.) Her kid is in middle school, I think. The situation she describes (all or most of the test questions are unlike the homework) seems inappropriate for middle school students. But I do think that in middle school the students should be faced with situations where a few of the test questions are unlike the homework.According to Piagetian theory, it is during the middle school years that children can start making the transition from concrete operational thinking to formal operational thinking. This theory also postulates that this transition is not automatic, that students will not acquire these higher order thinking skills unless they are challenged with tasks that are just barely beyond what they can currently do. (My understanding is that children are not expected to be successful at these unfamilar tasks on their first tries -- they need a few attempts to figure it out.)

In other words: when do we start testing the flexibility of mathematical understanding?

*This is how you can tell that I've taught Math For Elementary Ed: I can use words like multiplier and multiplicand and minuend and subtrahend. And here you thought that arithmetic jargon topped out at dividend, divisor, quotient, and remainder.

**Recall: This is just a description of my tests. Different rules apply to quizzes and graded homework.