Math by analogy is when teachers substitute ideas completely unrelated to math in order to make some concept "easier". Usually, this is because they themselves do not understand the meaning behind what they are teaching, so they cannot explain it accurately. Math by analogy substitutes presumed common context for reasoning. Yet most young students don't share enough common context to build the analo

You see math by analogy in both big and little examples, from the use of it to "explain" greater than and less than to its use in teaching place value. The most common analogy I see used by teachers and their books is that "

**a fraction is part of a whole**".This analogy has devastating results. I routinely (in 100% of classrooms not using Singapore math, in more than 50% of the students) hear:

- 1. "
**there's no such thing as ten ninths.**" that's the majority response in classroom after classroom. Why? Because a fraction is**PART of a whole.**How can a part of a whole be bigger than the whole? What's the whole then? - 1b. therefore, they believe no fraction can be bigger than 1.
- 2. "You can't divide 6 things among 7 people." 6 things isn't one whole. It's 6.
- 3. "three thirds is
**A Whole.**" Not one. - 3b. Therefore, they don't know 3 divided by 3, written as a fraction, is 1. I often hear of students who ask "is this a division problem or a fraction problem?"

These problems are so severe because these students have teachers who manage not to notice these errors. No problems in their books, no lesson script in the teachers guides illuminates this to the teacher. They only see the most trivial of problems. 10/9 is beyond the pale.

The correct explanation is that

**What number? A number defined on the number line as follows:**

*a fraction is a number.*1/3 is the point on the number line when you break the unit length into 3 equal length parts, and take 1 part. the endpoint of that part is 1/3.

4/3 is the point on the number line when you break each unit length into 3 equal length parts, and take 4 parts. the endpoint of those parts is 4/3.

Yes, teachers will need to build up to this. They should do so.