The Dangers of “New Math”: When Teaching Multiplication Becomes More Complicated Than Multiplication

Somewhere along the way, we seem to have confused understanding mathematics with performing elaborate rituals to prove we understand mathematics.

I recently watched a video of a young girl solving a multiplication problem.

It was a particularly challenging mathematical exercise, one that has tormented generations of students and undoubtedly pushed the limits of human intellectual achievement.

6 × 4

The answer, as anyone who has learned their multiplication tables knows, is 24.

But apparently, getting to 24 is no longer sufficient. There must be a process. A carefully prescribed series of steps, complete with diagrams, counting exercises, and enough unnecessary labor to make an accountant reconsider his career.

The girl wrote 6 × 4 at the top of her paper.

Then she drew four circles. Underneath them, she drew six horizontal lines.

She counted the circles and wrote 4 on the first line.

Then she counted them again and wrote 8 on the second.

Then 12.

Then 16.

Then 20.

Finally, after six rounds of counting, she arrived at 24.

Congratulations. After performing an elaborate sequence of additions, she had successfully rediscovered multiplication.

I wanted to bang my head against the table.

Vigorously.

But then something interesting happened. The more I thought about what I'd witnessed, the more I realized that the problem wasn't quite as simple as it appeared.

Because I had learned multiplication in almost exactly the same conceptual way.

And that realization led me down a rather fascinating rabbit hole about mathematics, education, memory, and the peculiar difficulty of remembering what it was like not to know something.

Before Multiplication, There Was Addition

I grew up in Brazil in the 1970s and early 1980s.

I remember learning how to count. Then came addition and subtraction.

Subtraction was introduced as the opposite of addition. We learned that if 5 + 3 = 8, then 8 − 3 = 5.

We were also taught that we couldn't subtract a larger number from a smaller one.

Of course, that wasn't universally true. We simply hadn't learned negative numbers yet. Within the natural numbers we were working with, the restriction made sense.

Then, in second grade, came multiplication.

And here's the part that complicates my initial outrage: I remember being taught that multiplication was repeated addition.

Six times four was six groups of four.

4 + 4 + 4 + 4 + 4 + 4 = 24.

Exactly the same mathematical principle demonstrated by the girl in the video.

But there was another important part of the lesson.

We learned that multiplication was a shortcut for repeated addition, and that it was much easier to memorize the multiplication tables—the tabuada, generally from 2 through 9—than to perform those additions every time.

In other words, the repeated additions were a means to an end.

They weren't supposed to become the end itself.

When division entered the picture, it was taught as the opposite of multiplication, just as subtraction was the opposite of addition.

The four basic operations weren't isolated tricks. They were relationships.

By third grade, we were solving problems such as:

□ + 3 = 8

We didn't call them first-degree equations, but that's precisely what they were.

Replace the square with x, and you have elementary algebra.

We already understood that the missing value could be found by reversing the operation.

That progression made sense then, and it makes sense to me now.

Which raises an uncomfortable question.

If I learned multiplication through repeated addition, why did watching that child do essentially the same thing frustrate me so much?

The Curse of Knowing

There's a cognitive phenomenon known as the curse of knowledge.

Once we know something, it becomes surprisingly difficult to imagine what it was like not to know it.

I can remember learning multiplication.

I can remember the multiplication tables.

I can remember the relationship between addition, subtraction, multiplication, and division.

But there's something I absolutely cannot remember.

What it felt like to look at 6 × 4 and not immediately know that the answer was 24.

Try it yourself.

Look at this:

6 × 4

Can you look at those symbols without thinking 24?

I can't.

I don't calculate the answer. I don't add four six times. I don't visualize groups of objects.

Twenty-four simply appears in my mind.

The association has become automatic.

And that is what makes the experience of a beginner so difficult to reconstruct.

A child who hasn't mastered multiplication sees a problem requiring conscious effort.

I see an answer.

We're looking at the same mathematical expression, but we're not having the same cognitive experience.

It's not that I've forgotten mathematics. Quite the opposite.

I know it so thoroughly that I can no longer experience the uncertainty that preceded knowing it.

Try Experiencing Illiteracy

There's a surprisingly effective way to approximate this phenomenon.

Look at the following word:

გამარჯობა

What do you see?

If you can read Georgian, you see a perfectly ordinary greeting: gamarjoba, meaning “hello.”

If you're like me, however, you see something resembling two or three lowercase d's, several u's, a couple of m's, and a particularly elaborate x.

My brain tries to interpret the unfamiliar characters using the alphabet it already knows.

But the actual meaning?  Nothing.

Now consider Arabic:

مرحبًا

Or Hindi:

नमस्ते

To someone familiar with these writing systems, the words are immediately recognizable.

To someone unfamiliar with them, they might as well be decorative patterns.

Within due proportions, it's rather like looking at ancient cuneiform.

You know the marks represent language. You understand that information is encoded in them. You might even recognize recurring patterns.

But you cannot read them.

And here's the fascinating part: try looking at an ordinary English word without reading it.

ELEPHANT

Don't think about the animal. Don't pronounce the word in your mind. Just examine the shapes.

Difficult, isn't it?

Your brain has spent years developing an automatic association between those symbols and their meaning.

You cannot simply decide to become illiterate for thirty seconds.

You can remember learning to read, but you can't easily recreate the experience of looking at familiar letters and finding them meaningless.

And that's almost exactly what happens with multiplication.

Once 6 × 4 becomes 24 in your mind, it stays that way.

So What's Actually Wrong With “New Math”?

First, a clarification.

What Americans often call “New Math” today is not necessarily the historical New Math movement of the 1950s through the 1970s, which emphasized topics such as sets, mathematical structures, and abstraction.

Modern approaches emphasizing number sense, visual models, decomposition, and multiple solution strategies are not all manifestations of that historical movement.

But in everyday conversation, “New Math” has become a convenient umbrella term for arithmetic instruction that seems unnecessarily complicated to adults who learned different procedures.

And there's a legitimate criticism buried underneath the jokes.

A teaching method can be mathematically correct and still be pedagogically inefficient when used at the wrong stage of learning.

There's nothing inherently wrong with showing children that multiplication represents repeated addition.

There's nothing wrong with drawing circles, arranging objects into groups, or visually demonstrating the distributive property.

For beginners, such representations can be extremely valuable.

Consider:

6 × 4 = (5 × 4) + (1 × 4)

20 + 4 = 24.

That's not pointless complexity. It's an introduction to a mathematical property that will become immensely useful in algebra.

But there's a difference between understanding an operation and being required to reconstruct it from first principles every time you use it.

Imagine requiring an experienced driver to explain the mechanics of internal combustion before starting the engine.

Or demanding that a programmer manually demonstrate binary addition before writing a + b.

Understanding the underlying process is valuable.

Having to perform that explanation every single time is not.

The same applies to multiplication.

Once a student understands what multiplication means, memorizing multiplication facts is not a retreat from understanding.

It's a way of making that understanding useful.

Memorization Is Not the Enemy of Understanding

Some educational debates frame memorization and conceptual understanding as competing philosophies.

One side wants children to memorize answers.

The other wants them to understand why those answers are correct.

But that's a false choice.

We need both.

Understanding that multiplication is repeated addition provides the foundation.

Memorizing the multiplication tables makes arithmetic efficient.

That efficiency matters because human working memory is limited.

A student solving a complicated algebraic expression shouldn't have to dedicate significant mental effort to calculating 6 × 4.

Those elementary operations should be available almost instantly, leaving cognitive resources for the actual problem.

Think about reading.

When you first learned, identifying individual letters required attention.

Then came syllables, words, and sentences.

Eventually, decoding became sufficiently automatic that you could concentrate on meaning rather than the mechanics of recognizing letters.

Imagine having to sound out every letter of every word for the rest of your life.

You could still read.

But reading would become exhausting.

Arithmetic fluency performs a similar function.

When basic calculations become automatic, mathematics becomes less about executing elementary procedures and more about reasoning.

And that should be the objective.

Not to eliminate understanding.

Not to glorify memorization.

But to use both in the service of increasingly sophisticated thought.

When the Method Becomes More Important Than the Answer

There is, however, a particularly objectionable practice sometimes associated with modern mathematics instruction.

A student arrives at the correct answer but loses marks because they didn't use the prescribed method.

Now, to be fair, there are circumstances where this makes perfect sense.

If the lesson is specifically about demonstrating the distributive property, then writing 24 without showing the decomposition doesn't demonstrate mastery of that particular skill.

A teacher is entitled to assess the skill being taught.

But that's different from treating a mathematically correct solution as though it were mathematically incorrect simply because the student used another valid method.

If the question is “What is 6 × 4?”, then 24 is the answer.

If the question is “Demonstrate 6 × 4 using repeated addition,” then repeated addition is part of the assignment.

Those are different questions.

And educators should be careful not to confuse them.

Because when students learn that following a prescribed procedure matters more than mathematical reasoning, they're learning something rather unfortunate.

They're learning compliance, not necessarily mathematics.

The Problem With Judging a Lesson From a Video

Now comes the part where I have to acknowledge the limitation of my own reaction.

I saw a short video of a child solving a multiplication problem through repeated counting.

I don't know whether she was encountering multiplication for the first time.

I don't know whether the teacher had specifically requested that method.

I don't know whether the exercise was designed to demonstrate the concept of multiplication rather than to practice multiplication fluency.

And those details matter.

If this was an introductory lesson, the circles and lines might have been entirely appropriate.

If the child had already mastered multiplication and was nevertheless required to perform six rounds of counting to calculate 6 × 4, the exercise becomes much harder to justify.

But the video alone doesn't establish which situation we're looking at.

That distinction is important, because criticizing a teaching method without understanding its instructional objective risks committing the very error we're accusing the educational system of making: judging the procedure without considering its purpose.

My immediate frustration was real.

But frustration isn't evidence.

What We Should Actually Be Teaching

The goal of elementary mathematics should not be to produce children who can mechanically execute one approved algorithm.

Nor should it be to produce children who can recite multiplication tables without understanding what multiplication means.

It should be to develop students who understand mathematical relationships, can perform basic operations fluently, recognize when different methods are useful, and eventually solve unfamiliar problems independently.

A student should understand why 6 × 4 = 24.

They should also eventually know that 6 × 4 = 24 without having to count anything.

And when they encounter something more complicated, they should have enough understanding to choose an appropriate strategy rather than blindly following a memorized procedure.

The teaching aids should support that progression.

The circles are scaffolding.

The repeated additions are scaffolding.

Even the multiplication tables, in a sense, are scaffolding for more advanced mathematics.

But scaffolding serves a purpose: helping construct something that can eventually stand without it.

If a student must continue drawing circles to solve basic multiplication problems indefinitely, something has gone wrong.

Not because drawing circles is inherently bad.

But because the student hasn't yet achieved the fluency that multiplication is supposed to provide.

The Real Danger Isn't New Math

I began this discussion wanting to bang my head against a table because a child needed six lines and four circles to solve a problem whose answer I knew instantly.

Then I remembered that I had learned multiplication as repeated addition myself.

And eventually I realized something more significant.

The ease with which I solve 6 × 4 today makes me a remarkably unreliable witness to the difficulty of learning it for the first time.

I can remember the lessons.

I can remember the sequence of concepts.

But I cannot remember the actual experience of not knowing that 6 × 4 = 24.

That knowledge has become inseparable from how I perceive the problem.

And this is precisely why educational debates require more than adults declaring that the way they learned was obviously superior.

Our memories are incomplete. Our expertise changes our perception. And what looks like an absurdly complicated procedure to someone who mastered a skill decades ago may be a valuable intermediate step for someone learning it.

That doesn't mean every new teaching method is good.

It doesn't mean every criticism of modern education is misplaced.

And it certainly doesn't mean we should celebrate unnecessary complexity.

It means we should evaluate teaching methods according to what they actually accomplish.

Does the method help children understand mathematics? Does it help them become fluent? Does it prepare them to reason independently?

Those are the questions that matter.

Because there's nothing wrong with taking six steps to teach a child something new.

The problem begins when we forget that the point of those six steps is to help the child eventually take one.

And if, after years of mathematics instruction, a student still has to count four circles six times to calculate 6 × 4, then perhaps we should stop congratulating ourselves on how sophisticated the teaching method is.

And start asking whether we've actually taught multiplication.