How & Why (Understanding Math)

We’ve been looking at Deans for Impact’s white paper, “The Science of Learning,” which summarizes cognitive science research and its implications for teachers (and homeschoolers). The paper poses six key questions that help us to apply the principles of science in teaching our children.

This week, let’s explore how students learn and retain new information.

Because if we want our children to be able use what we teach them, the first step is to help them remember it.

Continue reading The Science of Learning 2: Practice

We all want our children to thrive, to grow, to learn, to succeed in school and in life.

Cognitive science focuses on the “learning” part of life, studying how people acquire and retain information and then incorporate it into their understanding of the world. Earlier this year, Deans for Impact published a new edition of their white paper, “The Science of Learning,” summarizing cognitive science research and its implications for teachers.

Download a copy and read it for yourself:

• The Science of Learning

In this series, we’ll go through Deans for Impact’s six key questions and learn how to apply the principles of science in teaching our children. First, let’s explore how students understand new ideas.

Because if we want our children’s learning to stick, our focus must always be on helping them make sense of things.

How Do Students Understand New Ideas?

Principles from The Science of Learning:

• Students learn by connecting new knowledge to what they already know.

• Students have limited working memory capacity that can be overwhelmed if given too much information at once or if tasks are cognitively too demanding.

• Cognitive development does not progress in a fixed sequence at age-related stages. Understanding does not happen all at once or just one time; the mastery of new concepts happens in fits and starts.

Tips for Teachers

• Encourage students to remember what they’ve already learned, and help them connect new knowledge to old.

• Use stories, activities, and visuals that clearly relate to the concept being studied. Avoid distracting clutter that is merely decorative.

• When working example problems, discuss the reasoning behind each step and help students make connections.

• Don’t limit students to “grade level” ideas, but wonder about big concepts.

• Don’t assume that learning something once is enough. Repetition and reinforcement make learning solid.

My Comments

We make sense of new ideas by connecting them to things we already know, by recognizing relationships.

I appreciate the reminder of how important background knowledge is to new learning. It’s easy for me to jump straight into the new material, thinking that’s the interesting part of a lesson, but I need to let my students orient themselves by remembering what we’ve done before.

Just as I begin a history read-aloud with, “Tell me what happened last time,” I can begin a math lesson with, “Remind me of what you know about division.” My goal is not to make connections for my students, but to open their eyes so they can see the relationships for themselves.

Try This Today: Number Webs

For each post in this Science of Learning series, I’ll share an activity idea that gives you and your children a way to put it into practice. Today, let’s play with relationships and see how many connections we can find between math ideas by creating Number Webs.

For best results, get a large whiteboard or a sheet of poster-size paper.

Write a number or math expression in the center of your space. Then take turns writing equivalent expressions, other ways to say the same thing. Draw lines to show how these new expressions branch out from the original to form families of related math.

For example, if you begin with the number five, branches might include:

• 1+4, 2+3, etc.
• 10÷2, 15÷3, and so on
• 20/4, 50/10, etc.
• What else can you think of?

As you build your number web, each transformation will spark new ideas. You can create branches off the branches. For example, you could shoot off from 15÷3 with an expression like (5+5+5)÷3, or (7+8)÷(9−6).

For each bit of math you add, be prepared to explain why it fits in the family you chose, or how it’s different from the others and needs to start a new branch. Often an expression could fit in several places, and players may argue for their own point of view. But the player whose turn it is has the final say where to place his or her creation.

Instead of (or in addition to) writing in a new expression, players may draw in a line that connects it to an expression on a different branch. Numbers relate to each other in various ways, and there is almost always more than one way to think about linking them. For example, you might join the division expression 15÷3 with the fraction 15/3 and the multiplication (⅓)×15.

Don’t worry about keeping your Number Web neat and systematic. Explore and let the connections get a bit wild. There’s always another relationship to discover.