## But Honey, You do know the Multiplication Table!

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 7-year-old Autumn realizes that she does know the complete multiplication table!  Here’s the story:

Autumn and I have been talking about multiplication and ways to think about multiplication since she was very young (see our other multiplication videos).  However, we never tied all the facts altogether into a single table.  So when one morning she confidently announced that she knew the multiplication facts up to multiplying by 5, I told her that she actually knew the multiplication facts up to 10.  She refused to believe me!  So we headed over to the studio and taped this video, partially to have her prove to herself she did know the facts, and partially to summarize all the great strategies we have been working on (in real life and in previous videos).

The video is awesome in that it captures on tape Autumn’s own disbelief in her own knowledge (education researchers take note).  Start watching at 6:00 and see her reaction at 6:09–priceless.

For teachers and parents, this video makes three points to think about:

• Assessments such as the one in this video are important part of education because they help students recognize for themselves that they actually know the material.  Autumn filled in every entry in the table, and it was only after she saw what she had accomplish did she realize she knew the facts.
• There are only 9(!) entries in the table that are tricky to memorize (watch starting at the 5:50 mark), not 121 as students sometimes think.  The 9 entries reduce to just 6 after applying the commutative property ($6\times 7 = 7\times 6$).  They are:

$6\times 6, 6 \times 7, 7\times 7, 7\times 8, 7\times 9, \mbox{and} \ 8\times 8$

These 6 facts can be effectively handled by linking them together by memorizing one fact, $6\times 7 = 42$, and building to the other facts using the distributive property.  The first fact can be given a story to help the child memorize it.  (In our case, I told Autumn the story of “What is the answer to the ultimate question of life, the universe, and everything?” from Douglas Adam’s Hitchhiker’s Guide to the Galaxy. BTW: The answer is 42, which literally begs, “What is the question?”) The remaining entries are found using unit math, i.e., counting in different units, via skip counting.  We have a lot of nice videos that introduce how to do this: Part I, Part II, Part III.

• Parents: This video once again shows that learning multiplication facts should not be done the old Textbook School Mathematics way where parents buy 121 flash cards and drill their child on all of them until she or he cries.  That’s too many facts too quickly.  There are better ways to learn the multiplication facts, and they all start with being selective about which groups of numbers are learned together.  For Autumn, she learned them in different batches: grouping 2s in one batch, grouping 3s and 4s in a different batch, grouping 1s, 5s, and 10s in a third batch, the 9s by themselves, and the tricky facts, $6\times 6$ to $8\times 8$, in a fifth batch.  The batches can be learned simultaneously over a year or two (after the first batch of 2s is used to explain multiplication).  The key is to remember to stay within the same batch during a learning session. Follow this advice, and learning the multiplication table can be a very pleasurable experience for you and your child!

A fun fact about the video: Autumn “hisses” like a cat at mark 7:45.  We have three cats and enjoy imitating their silly behaviors as a family.  If you have cats, you will probably recognize what Autumn was saying.  If you have dogs, then the equivalent would be if your dog picked his ears up, tilted his head slightly, and stared at you with the look of, “Stupid human, don’t you get it?”

CHANNEL: Growing up with Eureka
© 2016 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.
Video | This entry was posted in Growing Up With Eureka and tagged , , , , , , , , , . Bookmark the permalink.

### 9 Responses to But Honey, You do know the Multiplication Table!

1. Cassyt says:

So impressed that Autumn know the answer to the ultimate question of life, the universe and everything! Great Video!

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2. Hi Teachers: Autumn summarizes a bunch of different strategies for calculating 1×1 to 10×10. However, some of the real gems for teachers in the video include: (1) the importance of the right assessment in helping students recognize that they actually know the material, and (2) you can see in realtime some of the multiplication facts that slowdown Autumn’s calculation speed. Do you see other gems? Leave a comment!

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3. Interesting video. He continually reinforces that he believes the child knows the facts and more subtlety reinforces strategies for multiplying two numbers. For instances, the child didn’t know 7 X 4, but the adult provided the clue that 7 X 4 is 6 X 4 + 4. He didn’t make a big deal about it, he just said (I’m paraphrasing here) “Sure you know it…four more than 24.” Some might say that he is giving her the answer, but I would say that he is giving her a strategy for determining the answer. It isn’t so much that this child has memorized all 100 multiplication facts. I don’t think that she has–and when she says “I don’t know all of the multiplication facts.” I think she means that she hasn’t memorized all of the multiplication facts. The point the adult is making is a good one…You don’t have to memorize all of these multiplication facts. All you really need to do is to have a way of determining the correct answer.

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• Peter, these are very good observations. Thank you! Yes, one of the underlying themes of all of our videos is to show that there are positive, healthy ways to work with our children to pleasurably learn mathematics: Many of us recall with horror the flashcards, the drilling, the crying, with our own parents. So when I don’t make a big deal about providing a strategy in the video, you are right-on to point out that that’s a big deal!

You bring up an interesting point about memorization versus having a strategy. When Autumn said to me she didn’t know all the facts, she really meant that she didn’t think she knew how to derive all the facts. The reason why I can say this is because “memorizing” isn’t a word that is a main focus of any of our conversations about mathematics. It was also reasonable for her to say that to me since we had never looked at the entire multiplication table all together as a single “thing” (we do most mathematics verbally without paper-and-pencil). When she made her assertion, I realized that she didn’t know she knew the extent of her own knowledge, and hence one of the main reasons for the video: to show teachers the power of assessments in learning.

I do want to say that learning all the multiplication facts to fluency is very, very important. Of the thousands of engineers and scientists I know, I do not know a single one who isn’t fluent in the basics—fluency is the grease that greatly smooths out later learning. My expectation as her father is for her to know 7×8=56 by heart very soon.* It is also the expectations of Eureka Math that students become fluent by certain ages, see https://scottbaldridge.net/2015/02/23/fluency-without-equivocation. With that said, I definitely agree with the intent of your statement about determining the correct answer: one of the main points of our videos is to show by example that “how she gets to that point of fluency” is as important as becoming fluent itself.

* BTW: The very real tension between her knowing the strategies and knowing the facts is what makes this video so interesting to watch as a parent or teacher. A video of a child just regurgitating all 100 facts would be very boring!

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• Scott, I appreciate your extensive response to my reply. Thank you. I stand corrected regarding Autumn’s meaning of the phrase “I don’t know all of the math facts.” Thanks for the clarification.
I also appreciate the idea that automaticity of the 100 math facts is necessary (or very, very helpful) in later mathematics learning because it can enable students to expend more mental energy on the new knowledge and not because it is important to be fast in producing answers in math class. Cathy Seeley told us that Faster Isn’t Smarter and Jo Boaler is certainly an advocate for encouraging understanding over encouraging speed.
Who would have thought that so much could be discussed over a video of learning the multiplication facts [!]?
Take care.

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4. Jerry Lou says:

knowing how to get then can not replace memorizing them to achieve automaticity, actually, Autumn had trouble to come up with 4X6, and knew to add under father’s hint. Not achieving automaticity will use too much working memory as solving problems.

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5. Great video, reminds me of the days where she would yell out 3 4’s or 2 6’s. that’s how they do it in the Caribbean.

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