Teaching – Page 28 – Denise Gaskins' Let's Play Math

Quotations XIII: Mathematics Education Is Much More Complicated than You Expected

Posted on July 23, 2007September 29, 2023 by Denise Gaskins

Registrations have been rolling in for our homeschool co-op, and the most popular classes are full already. Math doesn’t seem to be a “most popular” class. I can’t imagine why! Still, many of my students from last year are coming back for another go, and I am getting spill-over from the science class waiting list.

Anyway, I have started planning in earnest for our fall session. As usual, I look to those wiser than myself for inspiration…

Many teachers are concerned about the amount of material they must cover in a course. One cynic suggested a formula: since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick.

— Paul Halmos

Continue reading Quotations XIII: Mathematics Education Is Much More Complicated than You Expected

Quotations XII: Mathematicians at Play

Posted on July 10, 2007May 13, 2022 by Denise Gaskins

This week’s quotes for teachers:

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

— Paul Halmos

There are many things you can do with problems besides solving them. First you must define them, pose them. But then, of course, you can also refine them, depose them, or expose them, even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found new goal they do lead to. It’s called play.

Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings.

— David Hawkins
The Spirit of Play [pdf, 1.4MB]
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians

See You Later! (Quotations XI)

Posted on June 19, 2007May 13, 2022 by Denise Gaskins

We will be heading out soon on vacation, so I will not be blogging for awhile. The rest of this week is devoted to packing. (I hate packing!) But before I leave, here is a longish quote on teaching math from the book I am reading this week: Ian Stewart’s Letters to a Young Mathematician.

A second reason why few students ever realize that there is mathematics outside the textbook is that no one ever tells them that.

I don’t blame the teachers… If your students are having problems remembering how to solve quadratic equations, the wise teacher will stay well clear of cubic equations, which are even more difficult….

Continue reading See You Later! (Quotations XI)

How To Start a Homeschool Math Club

Posted on June 18, 2007May 29, 2025 by Denise Gaskins

From a recent e-mail:

“Hello! I am on the board of a homeschool co-op. We have had requests for a math club and wondered if you have any tips for starting one. We service children from K-10th and would need to try to meet the needs of as many ages as possible.”

There are several ways you might organize a homeschool math club, depending on the students you have and on your goals. I think you would have to split the students by age groups — it is very hard to keep that wide of a range of students interested. Then decide whether you want an activity-oriented club or a more academic focus.

When I started my first math club, I raided the math shelves in the children’s section at my library (510-519) for anything that interested me. I figured that if an activity didn’t interest me, I couldn’t make it fun for the kids. Over the years we have done a variety of games, puzzles, craft projects, and more — always looking for something that was NOT like whatever the kids would be doing in their textbooks at home.

Continue reading How To Start a Homeschool Math Club

Quotations X: The Royal Road

Posted on June 15, 2007May 13, 2022 by Denise Gaskins

More quotations especially for teachers:

There is no Royal Road to Geometry.

— Euclid

Teaching is the royal road to learning.

— Jessamyn West

The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it.

— Marshall Cohen

Solving Complex Story Problems

Posted on May 29, 2007December 30, 2024 by Denise Gaskins

[Dragon photo above by monkeywingand treasure chest by Tom Praison via flickr.]

Let’s play around with a middle-school/junior high word problem:

Cimorene spent an afternoon cleaning and organizing the dragon’s treasure. One fourth of the items she sorted was jewelry. 60% of the remainder were potions, and the rest were magic swords. If there were 48 magic swords, how many pieces of treasure did she sort in all?

[Problem set in the world of Patricia Wrede’s Enchanted Forest Chronicles. Modified from a story problem in Singapore Primary Math 6B. Think about how you would solve it before reading further.]

How can we teach our students to solve complex, multi-step story problems? Depending on how one counts, the above problem would take four or five steps to solve, and it is relatively easy for a Singapore math word problem. One might approach it with algebra, writing an equation like:

$x - \left[\frac{1}{4}x + 0.6\left(\frac{3}{4} \right)x \right] = 48$

…or something of that sort. But this problem is for students who have not learned algebra yet. Instead, Singapore math teaches students to draw pictures (called bar models or math models or bar diagrams) that make the solution appear almost like magic. It is a trick well worth learning, no matter what math program you use.

Continue reading Solving Complex Story Problems

How Can We Teach Problem Solving?

Posted on May 28, 2007September 9, 2023 by Denise Gaskins

We continue to plan our co-op courses for next fall. Some of the classes I had hoped for will not happen, and my children are going to have to make some tough choices between the remaining topics. Unfortunately, they have not yet mastered the ability to be in two classrooms at once.

I have three math courses to plan, and I think I will focus as much as I can on teaching math through problems, even at the elementary level. These are once-a-week enrichment classes for homeschooled students, so I assume they have a “normal” math program at home. I want to introduce a few topics they might not otherwise see, to deepen their understanding of the topics they have studied, and to give them a taste of that “Aha!” feeling that comes from conquering a challenging problem. Has anybody done something like this, and can you recommend some good resources?

Continue reading How Can We Teach Problem Solving?

Quotations IX: A Good Student Is…

Posted on May 17, 2007May 13, 2022 by Denise Gaskins

Aaaargh! My Internet service is on the brink again. But since I had to run into my husband’s office to check email, I’ll take a few minutes to post a quick note. Here are a couple of quotes especially for teachers:

The only teaching that a professor can give, in my opinion, is that of thinking in front of his students.

— Henri Léon Lebesgue

A good student is one who will teach you something.

— Irving Kaplansky

In Honor of the Standardized Testing Season…

Posted on March 8, 2007September 9, 2020 by Denise Gaskins

[Feature photo above by Alberto G. (CC-BY-SA-2.0) via flickr.]

The school experience makes a tremendous difference in a child’s learning. Which of the following students would you rather be?

I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.

— Agatha Christie
An Autobiography

…or…

“Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”

“She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction? Take nine from eight.”

“Nine from eight I can’t, you know,” Alice replied very readily: “but—”

“She can’t do Subtraction,” said the White Queen. “Can you do Division? Divide a loaf by a knife — what’s the answer to that?”

“I suppose—” Alice was beginning, but the Red Queen answered for her. “Bread-and-butter, of course.”

“She can’t do sums a bit!” the Queens said together, with great emphasis.

— Lewis Carroll
Through the Looking Glass

…in other words…

If you could lead through testing, the U.S. would lead the world in all education categories. When are people going to understand you don’t fatten your lambs by weighing them?

— Jonathan Kozol
at Westfield State College’s 157th Commencement

Continue reading In Honor of the Standardized Testing Season…

Mastering Miquon: Top Ten Tips

Posted on February 20, 2007September 9, 2023 by Denise Gaskins

[Rescued from my old blog.]

I love Miquon math, but the program does feel odd to many homeschoolers, especially at first. It is so different from the math most of us grew up with that it takes time for the teacher to adjust. DJ asked for Miquon advice at a forum I frequent, but I thought enough people might find these tips useful to justify an expanded repost. If you have more advice on teaching Miquon, please chime in!

Continue reading Mastering Miquon: Top Ten Tips