If Not Methods: Fraction Multiplication

Posted on by Denise Gaskins
Father and son doing math homework together

This is the last post (for now, at least) in our If Not Methods series about how to help children figure out tough calculations.

By the time students reach the topic of multiplying fractions, they have become well-practiced at following rules. After some of the complex procedures they’ve learned, a simple rule like “tops times tops, and bottoms times bottoms” comes as a relief.

But we know that relying on rules like that weakens understanding, just as relying on crutches weakens physical muscles.

If we want our students to think, to make sense of math, to figure things out, what can we do with a problem like 5/6 × 21 ?

Continue reading If Not Methods: Fraction Multiplication

If Not Methods: Mixed Numbers

Posted on by Denise Gaskins
A family doing math homework together

Continuing our series on teaching the tough topics of arithmetic…

Our own school math experiences led many of us to think that math is all about memorizing and following specific procedures to get right answers. But that kind of math is obsolete in our modern world.

The math that matters today is our ability to recognize and reason about numbers, shapes, and patterns, and to use the relationships we know to figure out something new.

But what if our children get stumped on a mixed-number calculation like 2 5/12 + 1 3/4?

Continue reading If Not Methods: Mixed Numbers

If Not Methods – Subtracting Fractions

Posted on by Denise Gaskins
Father and daughter doing math homework

We’re continuing our series of posts on how to build robust thinking skills instead of forcing our children to walk with crutches.

When we say, “Use this method, follow these steps,” we teach kids to be mathematical cripples.

If your student’s reasoning is, “I followed the teacher’s or textbook’s steps and out popped this answer,” then they’re not doing real math. Real mathematical thinking says, “I know this and that are both true, and when I put them together, I can figure out the answer.”

But what if our kids get stumped on a fraction calculation like 7/8 − 1/6?

Continue reading If Not Methods – Subtracting Fractions

If Not Methods: Scary Division

Posted on by Denise Gaskins
Father and son working on math homework

We’ve been exploring the many ways to help children reason about tough math problems, without giving them rules to follow.

As always, real math is not about the answers but the thinking.

But what about division with scary, big numbers? What if our kids get stumped on a calculation like 3840 ÷ 16?

When kids say, “I don’t know how”

We can teach without crippling children’s understanding if we follow the Notice-Wonder-Create cycle:

  • Notice everything about the problem.
  • Wonder about the possibilities.
  • Create something new: perhaps a solution or a math journal entry, or perhaps just a deeper level of understanding.

“Notice, Wonder, Create” is not a three-step method for solving math problems. It’s the natural, spiraling cycle by which our minds learn anything.

Continue reading If Not Methods: Scary Division

If Not Methods: Multi-Digit Multiplication

Posted on by Denise Gaskins
Mother helping her daughter with math homework

As we’ve seen in earlier posts, there are more ways to solve any math problem than most people realize. Teaching children to follow memorized steps and procedures actually cripples their understanding of number relationships and patterns.

But what if our children get stumped on a multi-digit multiplication calculation like 36 × 15?

Continue reading If Not Methods: Multi-Digit Multiplication

If Not Methods: Dividing Fractions

Posted on by Denise Gaskins
Mother and daughter working together on math homewrok

As I said in an earlier post, we don’t want to give our children a method because that acts as a crutch to keep them from making sense of math.

But what if our children get stumped on a tough fraction calculation like 1 1/2 ÷ 3/8?

Continue reading If Not Methods: Dividing Fractions

If Not Methods: Reasoning About Subtraction

Posted on by Denise Gaskins
Father and son reasoning about subtraction

We’ve been examining the fact that, while there may be only one right answer to a math problem, but there’s never only one right way to get that answer.

What matters in math is the journey. How do your children make sense of the problem and reason their way to that answer?

As always, real math is not about the answers but the thinking.

But if we don’t want to give our children a method, how can we teach? What if we pose a problem and the child doesn’t know how to solve it?

What if our children get stumped on a subtraction calculation like 431 – 86?

Continue reading If Not Methods: Reasoning About Subtraction

Musings: If Not Methods, Then What?

Posted on by Denise Gaskins

Last week, I quoted Pam Harris calling out a foundational myth of math education, the idea that we need to teach kids the methods that work on even the most difficult math problems.

“We have a misconception in math education that we think we need to teach methods so that kids can answer the craziest kind of a particular problem.

    “We would be far better served to teach kids to think about the most common kinds of questions WELL, and let technology handle the crankiest.”

    —Pam Harris

    Since many of us grew up in schools that taught these methods, they may feel like the only sensible approach to math. Without the standard procedures, how will our kids learn to do math?

    If we don’t teach subtraction with borrowing/renaming, how can students figure out calculations like 431 − 86? If we don’t teach fraction rules, how will they handle problems like 1 1/2 ÷ 3/8?

    Continue reading Musings: If Not Methods, Then What?

    Musings: A Common Misconception

    Posted on by Denise Gaskins
    Father and son thinking together about a math problem

    One of my favorite podcasts to listen to is Pam Harris’s Math Is Figure-Out-Able because she puts so many of my thoughts into words.

    For example:

    “We have a misconception in math education that we think we need to teach methods so that kids can answer the craziest kind of a particular problem.

      “We would be far better served to teach kids to think about the most common kinds of questions WELL, and let the cranky ones go to ChatGPT. Because they’ll recognize the sense of the answer.

        “Let technology handle the crankiest, and REASON about the rest of them.”

        —Pam Harris,
        the Math is Figure-out-able Fractions Challenge

        Well, I do think she’s wrong about the AI chatbot, because ChatGPT comes up with the strangest bald-faced nonsense about math problems. Wolfram Alpha is a much more reliable resource.

        But Harris’s main point stands. This misconception, this math-education myth, drives much of what happens in our classrooms and home schools today.

        Continue reading Musings: A Common Misconception