A math problem

"The principal at Venn Elementary took a survey of 110 fifth and sixth graders to see what they did over the summer. She found 20 who went to sports camp, 30 who went to summer school and 45 who went to science camp. Twelve students went only to sports camp, and 4 went to sports camp and science camp. How many of the students surveyed didn't do any of the three activities?"

to see the answer.

This is a real 6th grade math problem. Here's the real answer:

Given:
A + B + C + D + E + F = 110
A + B = 30
B + C + D = 20
D + E = 45
C = 12
D = 4

We can figure out the rest as follows:
E = 45 - 4 = 41
B = 20 - 12 - 4 = 4
A = 30 - 4 = 26
F = 110 - 26 - 4 - 12 - 4 - 41 = 23

Conclusion: 23 kids did not attend any camp.

Just one problem with this: while this is the real answer provided in the textbook, it's wrong.

for discussion.

Here's the correct Venn diagram:

The problem is that the "answer" makes an unwarranted assumption in "solving" the problem. As can be seen by the correct diagram, there are eight possible combinations of the three camps, and therefore eight variables. The assumption that A and E don't intersect is implicit in the wrong diagram and not supported by the problem statement. In fact, there are a total of 125 different solutions with 27 different possible values for F. Students shouldn't be asked to guess additional assumptions to solve a problem. And this problem could have easily been fixed with one sentence: "No students attended both summer school and science camp." (I'll come back to the problem with guessing in elementary school math in a future post.)

Here's the sad part: when I raised this issue with the teachers responsible for assigning that problem, they said that it was just a question of making different assumptions in solving the problem and that both answers were correct. Uh-huh.

In math, there are right and wrong answers. And when we make assumptions in math it's a big deal: mathematical axioms or postulates are the foundation of mathematics.

Mathematics should not be taught like history and literature, where the interpretation is a matter of opinion. And it shouldn't be taught like science where theories are confirmed by evidence, but are always subject to revision as new evidence is found. Mathematics is about theorems, not theories.

[Originally published in three parts.]