What is the difference between "old math" and "new math"?

 In the middle of my class this morning, a student asked a great—and very fair—question:

“What is the difference between ‘old math’ and ‘new math’?”

Instead of answering directly, I wrote a problem on the board:

29 + 38

These are adult learners. They have years and years of experience doing mathematics, so I asked them simply to solve the problem and share how they thought about it.

The first strategy I heard was:

“Twenty and thirty is fifty. Nine and eight is seventeen. Fifty and seventeen is sixty-seven.”

Then came the familiar strategy:

“Nine and eight is seventeen. Write the seven, carry the one. One, two, and three is six. So sixty-seven.”

And then a third student said:

“I took one from 38 and gave it to 29. That makes 30 + 37, which is 67.”

There was an audible gasp across the room.

So I stopped.

“Why the gasp?”

Several students responded with almost exactly the same words:

“I didn’t know you could do that!”

And that might be one of the best explanations of the difference between what people often call “old math” and “new math.”

For many of us, our experience with math was primarily about learning the way to solve a problem. We were shown a procedure, practiced that procedure repeatedly, and became proficient at carrying it out. Maybe we eventually understood why it worked. Maybe we didn’t.

And to be clear, there is nothing wrong with knowing an efficient procedure. The standard addition algorithm represented in the second strategy is powerful, efficient mathematics. I want students to know it.

But I don’t want that procedure to be the boundary of what they believe they are allowed to do mathematically.

What people often call “new math” is really an effort to help students make sense of mathematics—not simply perform it.

Look again at the three strategies.

The first student decomposed the numbers by place value:

29 + 38 = 20 + 30 + 9 + 8   (Strategy 1)

The second used the standard algorithm. (Strategy 2)

The third student used compensation. They recognized that moving one from 38 to 29 does not change the total:

29 + 38 = 30 + 37 (Strategy 3)

Same problem.

Same answer.

Three different ways of thinking.

The goal isn’t to make children learn twelve different ways to add just for the sake of learning twelve different ways.

The goal is to help them understand numbers well enough that they can choose a strategy that makes sense.

That is harder to teach.

Understanding is messier than following a set of steps. Children may think about a problem differently than we do. Their work may not fit neatly into a single algorithm. Teachers have to listen to their reasoning, make sense of it, and help connect one idea to another.

But there is something incredibly important gained in that messiness.

Students begin to see mathematics as something they can reason about, not simply something they have to remember.

And perhaps most importantly, we want fewer adults sitting in a mathematics classroom someday, watching someone turn 29 + 38 into 30 + 37, and gasping:

“I didn’t know you could do that!”

Because they should know.

They should know that mathematics is not a collection of rules handed down to us that we must obediently follow.

It is a connected system of ideas that we can understand, reason about, and use flexibly.

Maybe that is what “new math” is really trying to accomplish.

Unleashing Children's Multiplicative Brilliance

 

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.