Math Triangle of Meaning

I've been on sabbatical this semester researching ways to help students who are English Learners learn math in US classrooms. Part of the project has involved interviewing both adults and children who have settled in the US having grown  up in other countries, just like I did. This week I have interviewed a couple of third graders who were born in Burma and came to the US via Thailand.

Both students spoke English well so communication was not an issue. In the first interview the student did something quite remarkable in that it was so exactly in line with a particular theory I use in my math education courses. It is always so neat when this happens because it reconfirms the value of a theory-based education. When we can make an observation and relate it directly to a particular theory, or piece of a theory, as teachers we are in such a better position to know where to go next in the instructional process.

The particular observation happened as I was interviewing the student on his understanding of basic numeracy; more specifically being able to read, write and understand 3 and 4 digit numbers. The triangle of meaning is a piece of theory that identifies the  three component of a piece of math understanding; the idea, the word and the symbol. For example 5 is "five" and *****. Put one of these at each  corner of a triangle and there are 6 relationships (e.g hold up five fingers and ask how many, hold up 5 fingers and ask student to show the correct numeral). Students may  frequently have some of the relationships but not all 6.

Yesterday the student kept reading 4 digit number incorrectly. For 4,582 he would say "forty-five thousand and eighty two". So I asked him to write numbers such as 4,275. He did this correctly every time. I then asked him to read back to me the numbers he had correctly written and he read them incorrectly using tens of thousands as in the example above.

He could model the numbers using base ten blocks and he could write them when given them orally  but he could not read them even when he had written them. The important lesson here is to remember that just because a student can write a number doesn't mean to say she/he can read it, or knows what it means.