Every month inside the Build Math Minds Facebook group, we watch out for a hot topic. Something that sparked a lively debate, a lot of comments, some unsureness or anything that might be interesting to dive a little bit deeper into. Our hot topic this month inside of the group is about why multiplication is not just groups of.

I’m Christina Tondevold, The Recovering Traditionalist, and today I’d like to ask the question: Does Multiplication mean ‘Groups of’? in our quest to build our math minds so we can build the math minds  of our kiddos.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog.

Join The Build Math Minds Facebook Group

Putting Essential Understandings of Multiplication and Division Into Practice By John Lannin, Kathryn Chval, Dustin Jones, Barbara Dougherty

Valerie Faulkner’s post Multiplication and Arrays – Does Order Matter?

“Groups of” is not the way everyone says it.

In the U.S. it is most common to hear people say “4 groups of 6”, but some other countries say it as “4 taken 6 times.”  Both expressions are true and it is important for us as educators to understand that the way we phrase multiplication problems in the US, isn’t necessarily how the rest of the world does. 

That being the case, teaching our kids to understand multiplication as ‘groups of’ is an effective place to start. However, we should NOT demand that they only think of it in that way. In other words, we should be exposing them too many ways to visualize multiplication.

The book, Putting Essential Understanding of Multiplication and Division into Practice (by NCTM) dedicates an entire chapter to the meaning of multiplication where they start it off by saying; “multiplication is a scalar process involving two quantities, with one quantity -the multiplier – serving as a scaling factor and specifying how the operation resizes, or rescales, the other quantity – the multiplicative unit.  The rescaled result is the product of the multiplication.”

So yes, even with that ‘ scalar process’ definition in mind, the case can still be made that when you see 4 x 6, and think of it as “4 groups of 6”, the 6 is being scaled up 4 times. Just remember that this is not the only way to look at it.

Keep the progression of multiplication in mind.

In order to ensure that we teach multiplication as a ‘scalar process’ efficiently, we need to first make sure we understand the progression of multiplication.

As kids transition from multiplication with whole numbers, to learning how to multiply decimals, fractions, and other types of numbers, it’s important that we understand the progression of multiplication when we teach the ‘scalar process’.  If we only use ‘groups of’ as our method of phrasing, it can end up being understood as repeated addition. This visual works well with whole numbers, but once you get into those different types of numbers we mentioned before, the concept can become more confusing.

For example, with the equation 2/3 x 3/5, repeated addition doesn’t work here. However, scaling does work in this example.

Let’s look at 3/5.  If we scale 3/5, 1′ whole’ time, it would be the ‘entire 3/5’. In other words, 3/5 x 1.

But the problem is 2/3 x 3/5 not 3/5 x 1 so we only want to scale 3/5 by 2/3. Here’s how we can represent that visually.

This is 2/3 of a group of 3/5.

With this scaling process, we can show different types of numbers being multiplied as kids progress into fractions and other numbers. Lets use 2 x 3×5 as another example

This can also still be described as ‘2 groups of 3/5’. So it is true that there are many ways to look at multiplication problems but we always need to remember that we need to think about the visual that we’re helping kids create. We want to be able to give them ways of understanding that transfer as they work with more complicated numbers.

Context matters

Context is what determines which numbers mean what. It does not matter if a kid shows you ‘4 groups of 6’ or ‘6 groups of 4’ when all you give them is 4 x 6. The only time it matters is if those numbers are put into a specific context. Once that context is established, then one of those numbers is the multiplier and the other is the multiplicative unit. Without context, it should be up for interpretation which is which. For example:

“Christina has 3 bags of books with 15 books in each bag. How many books does she have?”

That establishes a very different visual than if we asked;

“Christina has 15 bags with 3 books in each bag. How many books does she have?”

Both have the same answer of 45, but the visual you get when you imagine Christina holding 15 bags of books versus holding 3 bags of books is very different.

In this case, the context matters because it creates the visual. Establishing this context can make it easier to build discussions about which number/thing is the group (multiplier) and which is the amount in each group (multiplicative unit). In the first example, 3 bags of 15 books each, the 3 would be considered the multiplier whereas 15 would be the multiplicative unit. In the second example of 15 bags and 3 books in each bag, it would be the exact opposite.

Ultimately, multiplication is a scaling process, it’s not just ‘groups of’.

A common complaint or frustration that we’ve seen with teaching multiplication in this way is that “when kids get into middle school and/or high school, the teachers will expect them to have learned it a certain way.” Valerie Faulkner has a post that goes into more detail explaining why the order of numbers in bare multiplication problems does not matter even when you’re using arrays and area models as it relates to that argument. She says,

“They did, however, mention that there is one decent argument for this position.  That is that when we get to Algebra there is a formal understanding that the first numeral/adjective/factor in a multiplication expression does represent the row in a matrix and the second numeral/adjective/factor represents the column.  But this is still not reason enough to make this a ‘rule’ in an elementary class.” She goes on to explain “When you formalize ideas into rules too early, you detract from students’ abilities to have discussions and describe their thinking. You also injure their ability to think about the difference between an abstract expression (3 x 4), a tool to represent it’s meaning (array – rows and columns), and the actual thing you might want it to represent (a patio)”

Multiplication can be seen as ‘groups of’ but it really is so much more than that. 

So moving forward, allow your students to share how they think about multiplication, keep the progression in mind and that multiplication really is a scaling process, and remember that the order of how kids say or model the problem really does NOT matter unless the problem is in a context that helps kids see which number is the multiplier and which is the multiplicative unit.

I hope this video has helped you build your math mind so you can build the math minds of your students.

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