What is the capital of Kansas?
If you are like me, you probably memorized in school that the answer is Topeka. But what else do you know about Topeka?
Unlike many of you, I grew up about 40 minutes away, so Topeka is more than just a fact I memorized. I remember field trips to the zoo from my tiny hometown of 700 people. I remember eating lunch at a park with a giant wooden slide made of rolling logs that practically guaranteed you would leave with splinters. As I grew older, I learned about the famous Brown v. Board of Education case connected to Topeka. And I still remember standing beneath John Steuart Curry’s Tragic Prelude mural in the state capitol rotunda.
In other words, I do not just know Topeka is the capital of Kansas—I have experiences and connections tied to it.
I think we want the same thing for children when they encounter a multiplication fact like 8×6. It is one thing to know the answer is 48. It is another thing entirely to experience the relationship.
I am not opposed to flash cards. But initially, those flash cards should help students see the mathematics, not just recite it. Students need opportunities to visualize the multiplicative relationship and talk about what they notice.
When students see 8×6, I want them to see 8 groups of 6. But I also want them to see something deeper: the reorganization of those groups into our base-ten system. Forty-eight is not just “the answer.” It is 4 tens and 8 ones. Multiplication is about reorganizing groups of units into decimal units.
Just as importantly, I want students to describe what they see using conceptually consistent language such as “groups of.” That language matches the mathematics students are visualizing.
We often default to the word “times,” but I have found that “groups of” language is more productive for many learners. “Times” tends to sound comparative, while “groups of” emphasizes the actual structure of multiplication. With both children and adult learners, I consistently notice richer reasoning when students describe multiplication as groups of units rather than simply “times.”
A few weeks ago, I asked an adult learner to solve 8×6. They responded by skip counting:
6, 12, 18, 24, 30, 36, 42, 48.
That is certainly a conception of multiplication, but it is largely additive reasoning disguised as multiplication. The learner must keep track of both the running total and the number of counts. Many elementary students—especially those who struggle in mathematics—approach multiplication this same way.
The problem is not that repeated addition is “wrong.” The problem is that students often never move beyond it. When multiplication remains rooted only in additive reasoning, students struggle to develop the multiplicative reasoning needed later for proportional reasoning, fractions, and algebra.
My goal is that when students see 8×6, they connect the fact to a network of ideas. Of course I want them to know the answer fluently, but I also want flexibility.
I want students to know that 8 groups of 6 and 6 groups of 8 produce the same product. I want them to understand that rectangular arrays can be viewed as rows or columns depending on how we define the unit.
And I want them to see how multiplication facts are connected to the properties of algebra. In fact, they can use these strategies to move from known facts to unknown facts.
For example:
• 5 groups of 6 plus 3 groups of 6 gives us 48. Welcome to the distributive property.
• 2 groups of 4 groups of 6 also gives us 48. Welcome to the associative property.
• 10 groups of 6 minus 2 groups of 6 gives us 48. Another use of the distributive property.
I also want students to see how halving and doubling preserve products:
4×12=48
16×3=48
These are not isolated tricks. They are connected relationships within multiplication.
And with a strong understanding of place value, students can extend these same ideas naturally to problems such as:
8×600
8×0.6
8×6/5
In every case, students are still working with 48 of a particular unit:
48 hundreds
48 tenths
48 fifths
The first two problems involving decimal units “place” nicely, 4 thousands and 8 hundreds, and 4 ones and 8 tenths because the regrouping happens at 10 of a particular unit (e.g., 10 hundreds = 1 thousand). The third problem, 8 x 6 fifths, can also be expressed as 48 of a unit, fifths, but since fifths regroup every 5 fifths, 48 fifths can also be expressed a 9 ones and 3 fifths.
That is the kind of multiplication understanding I want for children—not isolated answers but connected mathematical relationships.
Every place we travel gives us opportunities to make connections and build understanding. Learning state capitals becomes more meaningful when we have experiences tied to those places. The same is true for multiplication facts.
Students deserve more than memorized answers. They deserve opportunities to explore the rich mathematical connections hidden inside those facts.
To learn more ways to connect mathematics ideas that may seem disconnected check out my book, See It, Say It, Symbolize It: Teaching the Big Ideas in Elementary Mathematics.
To learn more about the author's work and contact him about professional development opportunities email him at patrick_e2e@outlook.com or check out his website www.elevate-2-excellence.com.
