Student: Will the test be cumulative?
Parker: Yes.
Around the second term test, or final exam, students often ask whether the test will be cumulative. The short answer is “yes.” The longer answer is that mathematics itself is inherently cumulative and so there is no way to design a reasonable term test for my courses which is not cumulative1.
In mathematics, we are constantly climbing the ladder of abstraction: we start with a simple concept like addition, generalize it to multiplication, and then keep going to exponentiation. Each step in this ladder assumes a certain fluency and mastery of the material that comes before it.
How does this work in practice? Let’s imagine a course on arithmetic with whole numbers. If the first test cover addition, and the second test covers multiplication, then should we say that the second test is cumulative or not? Certainly, addition and multiplication are distinct concepts. One can, in principle, understand the notion of multiplication without ever having learned addition2. In this sense, the second test is not cumulative. However, practically speaking, one is forced to add numbers in order to carry out any moderately complicated multiplication. A test which consists entirely of multiplication questions would implicitly assume fluency with addition. In this sense, the second test is cumulative.
And so, my tests are always cumulative in this inevitable sense. When I write a second test, I try to make the content “independent” of the previous test. However, I assume fluency with the previous material and the ability to employ it without hesitation. To continue with our hypothetical course in arithmetic: A fictional second test, entirely on multiplication, wouldn’t have any addition questions on it but would assume that one can do addition.
In a course where the topic changed radically every few weeks, one could design non-cumulative term tests. ↩︎
A friend of mine once taught an experimental course to Grade 1s that worked this way. She taught them all about multiplication by making grids of small objects. The kids got a deep intuitive sense for notions like prime numbers without ever having learned addition. For those kids, a number $p$ was prime if could only form the grid $1 \times p$. Compsite numbers could form other sorts of grids. ↩︎
Published: Nov 10, 2026 @ 10:36.
Last Modified: Nov 10, 2026 @ 17:42.
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