Bill McCallum on Problem-Solving

Today’s commonplace quote is from from Mathematical Musings: Meaning and meaninglessness.

The logical connections between mathematical ideas make it possible for students to reason, to make sense of what they are learning, to figure things out for themselves.

But that same logical structure allows us to teach procedures without reasoning. Students can work through problems, follow the steps, and get a page full of correct answers without having any idea about the conceptual relations that underlie what they’re doing. Students can use math symbols like the playing pieces of a game, never bothering about what they mean but just following the rules.

That’s why mathematical narration is so important. When we ask students to explain what they are thinking as they solve a problem, we force them to realize that they are supposed to be thinking.

Because real math is always about reasoning and making sense of things. And when students wrangle their vague thoughts into specific words, it helps them see mathematical relationships more clearly.

• Math Game Monday: Symmetry Challenge

• Children Have Mathematical Ideas

• The Problem with Socratic Questions

• Thinking Thursday: Counting Puzzles

[If you’re reading this late, the “Math Game Monday” and “Thinking Thursday” posts have a time limit. Each one stays on my blog for about a month, and then falls away, replaced by new games and journaling prompts. You can always find the newest posts at DeniseGaskins.com/blog.]

I’m still catching up on all the details of running a one-person business, too many of which tend to fall aside during the final rush of getting a new book to print.

ALERT:

If you’re thinking about buying one of my ebooks or printable activity guides, now’s the time to shop.

I’ve decided to raise digital-book prices soon. Most of them have been the same since before the pandemic, and the pressure of inflation plus the hassle of dealing with spambots (who like to target low-priced items when testing stolen credit cards) combined to force my hand.

You have until the end of August to grab a back-to -school book at current prices.

(Print book prices will not change, at least not this year.)

CHARLOTTE MASON’S LIVING MATH:

If you ordered a book from the Kickstarter, all the rewards are now in the mail. If you haven’t received your book (s) or at least a tracking number for your package—please let me know!

For those who missed the Kickstarter, you can buy Charlotte Mason’s Living Math now on my store.

I hope to get retail store preorders uploaded soon, for those who prefer to shop at your favorite online store, but I’ve been having login issues with my distribution site. If it’s not one thing, it’s another.

But really, the full-color edition at my store is the same price as the black-and-white retail edition will be, and it’s ever so much nicer for reading.