### Meaning of factors in multiplication: four groups of 2, or 4 taken two times?

Someone asked me about the interpretation or meaning of the two numbers in multiplication, such as 4 × 2. Do you view it as four groups of 2 , or as four, taken two times ? Here's their question: "...I noticed that your explanation about how to interpret a math phrase, such as 4 × 2, is different from how I have understood it and taught it to my older daughter who was using RightStart Math at this level. Let me explain: I see that you are using the first factor to denote number of groups and the second factor tells how many are in each group (4 × 2 would be interpreted 4 groups of 2). On the other hand, I have taught and thought of it reversed, with the first factor telling how many are in each group and the second factor telling you how many groups (4 × 2 would be 4 taken 2 times). I know that the difference in explanations does not ultimately change the product, but I am wondering what you thoughts are. When I conceptually picture what I'm doing I see it as 4