We regularly receive emails with great questions about various approaches to math education. Dr. Maria answers them by email. You can ask your question here. We edit some of the answers to share at this blog. Names and personal details are removed to protect anonymity.

*Q: I am curious about your thoughts on Jo Boaler’s work. She does not believe in memorizing formulas or even the times table. I think it was very helpful for my child to memorize the times table. I am unsure of how sound her theories are, especially moving to higher math. Many of my homeschooling friends love her work though. *

A: I like the parts of Jo Boaler’s work that I’ve seen so far. I’ve read “Experiencing School Mathematics,” the book that came from her dissertation research in the 1990s, and followed some of her further developments. Dr. Boaler works with school systems. Since the school systems are, unfortunately, caught in political turmoil, her work is constantly subject to disinformation and other attacks. This means we can’t trust all sources about Dr. Boaler’s work.

So it’s better to read Jo Boaler’s directly to know where she stands. To quote her: “It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact.” Dr. Boaler emphasizes that memorizing does not equal understanding. She opposes timed tests and memorizing instead of understanding. Don’t only memorize times tables and call that “learning multiplication.” You can read more of her words on the topic here: https://www.youcubed.org/evidence/fluency-without-fear/

For my part, I also believe it’s a good idea to be fluent in times tables. There are multiple (ha!) paths to fluency, and memorizing is a good one. In the essay linked above, Jo Boaler says she never memorized all times tables. I did, but I never memorized addition facts; 8 plus 4 still goes, *8 => 10 and 2 extra => 12* in my mind, albeit lightning-fast. (I used to play and win math and science Olympiads as a kid; I needed speed.) **Different learners achieve fluency differently**. My math friends and I developed a system for memorizing times tables efficiently, and with an eye on supporting future concepts, including algebra. Our system offers choices that cover different kinds of learners. It is a part of this course: https://naturalmath.com/multiplication-explorers/

I celebrate when I see a fluent teen with a home-field advantage on the multiplicative conceptual field: multiplication, division, factorization, proportions, and so on. But here is something else to keep in mind: memorizing may not be the best START for learning multiplication. Most students first need to learn what multiplication is and what it means, where in life you multiply, and how to see patterns within times tables. Based on that connected understanding, students can *then* memorize the times tables well. This way, multiplication becomes a cornerstone for algebra.

Together with my colleagues, I developed an at-your-own-pace course called Multiplication Explorers that supports both deep understanding and efficient memorizing. The page has three “math sparks,” sample activities from this course that you can try with your child. Two are aimed at understanding what multiplication is, and the third helps you to see patterns in the multiplication tables. https://naturalmath.com/multiplication-explorers/

We also have a multiplication poster with 12 examples of where in life you multiply, such as computing areas, using symmetry, or counting the number of combinations. You can view it online or get a printed copy. https://naturalmath.com/multiplication-models-poster/

Here are some other Natural Math multiplication resources that you might find helpful.

Download a card that teaches an ancient merchant’s multiplication trick first recorded around the 15th century. Back then, merchants used finger reckoning to calculate prices and profits. A calculating device took half a room, while finger math was easy to take on a trip. Merchants twiddled their fingers away from the prying eyes of their competitors: that’s where the phrase “under the table” came from – https://naturalmath.com/s/wp-content/uploads/2019/05/AncientMultiplicationTrick_NaturalMath.pdf

From Buttons to Multiplication Blind Spots, a blog post about a no-prep activity that explorers multiplication – https://naturalmath.com/2015/05/from-buttons-to-the-multiplication-blind-spots/

Explore the commutative property of multiplication and add your ideas on whether 2×3 is really the same as 3×2 – https://naturalmath.com/2013/04/what-would-you-rather-have-commutative-game/

*Editing: Yelena McManaman *

*Proofreading: Emilie Desmarais*

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