I used to conduct math (teacher) trainings with "Julie." She was (is!) a phenomenal math teacher. But one day, while working with a group of 4th/5th grade teachers, the topic of multiplication table memorization came up. After a number of folks shared their ideas and perspectives--including their frustration with

*, Julie spoke, close to tears. She told us how she struggled throughout childhood with the fact that no matter what she did, she could not memorize multiplication facts. It became a huge point of anxiety. Yet mathematical thinking and strategies came naturally to her. She explained how at 40something, she still does not know her facts. BUT, her knowledge of math strategies is so great that she can compute quickly and easily.*

**kids who just don't get it**As she spoke, I reflected on my own experience with memorizing facts. As a 4th grader, I was scholastically competitive. Surprising, I know. When the class multiplication chart went up, the race was on! I was eager to conquer the chart (and beat my classmates.) I memorized it quickly. To this day, if I need to recall a fact, I do this sing-songy thing in my head to find it. ("Nine times nine is eighty-one.") But here's the irony... I didn't have a clue what I was doing--what multiplication meant. I was great at memorizing, but it was really no different than memorizing a random list of words, letters, or numbers. It carried absolutely no meaning. Yeah, I could use them to do higher level multiplication and division, but my knowledge of the gamut was pretty limited because I'd done nothing more than memorize, first the facts and later the larger algorithms.

My friend Julie, in the meantime, fully understood the mathematical thinking behind multiplication. She had a wide range of strategies to pull from in order to figure out facts. I meanwhile, had none.

Which brings us to the concept of fluency. The definition of fluency is "good command." I would wager that as kids, Julie had excellent command of multiplication facts while I had relatively little. Today, Julie uses strategies to compute facts just about as quickly as I can locate them with my little sing-songy memory.

When I teach multiplication, my goal is fluency, not instant recall. I want kids to command multiplication--by thoroughly understanding what it is and being able to express their understanding through a wide range of strategies.

That sing-songy voice occasionally fails me. Julie's strategies never do. Her toolbox is loaded; mine has one unreliable wrench. ;)

In Salman Khan's book, The One World School House: Education Reimagined, he quotes from Nobel Prize-winning Eric Kandel, a neuroscientist:

"For a memory to persist, the incoming information must be thoroughly and deeply processed. This is accomplished by attending to the information and associating it meaningfully and systematically with knowledge already well established in memory."In order for something to be memorized [well], it has to make contextual SENSE!

For more thoughts on fluency, read "Teaching for Mastery of Multiplication."

P.S. I'm on the edge of my seat, waiting for the online version of Number Pieces to come out. When it's available, I'll show you some spectacular strategies for multiplication. (Even this 40something can add a few tools to the ole' toolbox!)

Cindy, would you give us an example (or a few) of what Julie would do to find the answer to a particular problem? Did she know the 2's and 5's? (What was her basis to reason from?)

ReplyDeleteHi Sue,

DeleteHer strategies were pretty endless. Yes, I'm pretty sure she knew the 2s because she could double a number, and the 5s because she could count by 5s. It's been long enough ago that I couldn't tell you her exact set of strategies, but honestly, I'm not at all sure that she had a particular set. She could just think mathematically. So 4 x 6 could be (2x6) + (2x6) or 3, 6s (18, pretty easy to skip count), plus one more. I'm sure she also did things like figure 9x9 (which I sing-song to this day!) by saying that 10, 9s is 90 so one less 9 is 81. That kind of thing.

I never memorized my 12s very well (sing-song fail!) so I'm forced to use strategies for them. 10, 12s is 12, so 11, 12s is 132. You could also use a clock...5, 12s is 60 (thinking of a clock face), and twice that is 120 and one more 12 is 132.

All of this is the sort of thing that I've had to learn as an adult. I would have hugely benefited from it had I learned in 4th grade! :)

I also could never memorize my multiplication table (along with a lot of memorization facts). I do know my 2's and 5's just because we use them so much in our daily lives, but I still do not know 6, 7, 8, or 9 table facts. I never will, I am sure as I am 51 years old. I was held back in math because I couldn't memorize things such as this, but I did a lot of the tricks that Cindy spoke of in order to get the answers I need. When my children didn't seem to be able to pick it up, I did not sweat it, but instead taught them to think mathematically and I taught them tricks to get to the answers to the facts. It is much better to do this than make the child feel stupid by trying to make them memorize when they just can't. That is what happened to me.

ReplyDeletePhyllis, you sound just like my teaching partner. And from reading your blog, I know you are a whiz at math. What's very cool is that you've been able to shed the past and show your kids another way to think about the topic. Very cool.

DeleteThoughtful post. What is your advice for a mom teaching math but has her own set of fears regarding the subject?

ReplyDeleteHi Tiffany,

DeleteYou are in the same boat as a whole lot of (classroom) teachers. Don't know if that makes you feel better or worse! :) I had a huge set of fears myself. The fears diminish as I learn more and more strategies. Visual models and manipulatives help me to see and experience it...baby steps, really...to help it become less threatening. My dream? I'd love to develop a series of videos to give teachers and parents a chance to see all the incredible math strategies.

In the meantime, I'd give yourself a chance to try lessons before you do them with your child. Look at the upcoming lesson. If it's a multiplication fact, challenge yourself to work out the problem with manipulatives or a sketch. See it, move it, experience it. Learn what the math is behind the memorization.

And read Jo Boaler's book, What's Math Got to Do With It? Or Marilyn Burns' book, Math An American Phobia.

You can change your own math education. Unfortunately, a lot of us have to. Change it now, and your own kids won't need to! :)

Thanks for posting and kudos to you for facing fears!

I'm like you - I had no problem memorizing, but no real idea of what was going on with the numbers. When my oldest had trouble memorizing, I taught him with the story method, so he would be able to test - but really focused on fluency - it makes so much more sense at the end of the day.

ReplyDeleteHe is so lucky that you thought to do that! How long ago was it and how does he do now?

DeleteHave you read The Best of Times by Greg Tang? Highly recommend!

ReplyDeletehttp://www.amazon.com/gp/aw/d/0439210445/ref=mp_s_a_1?qid=1366037476&sr=8-4&pi=SL75

Oooh, yes, excellent suggestion!!!

DeleteThis is exactly why I don't have my children even *try* to memorize anything until about 4th grade. I want them to spend plenty of time practicing the thinking strategies. And even when we do work on memorization, it's primarily a mop-up operation of memorizing any strategies the kids haven't mastered, rather than memorizing the math facts themselves.

ReplyDeleteI like that..."mop-up operation of memorizing any strategies they haven't mastered" !! :)

DeleteThanks for the reflection!

Sing-songy voice occasionally fails me. Strategies never do.

ReplyDeleteGreat post!!

And a sing-songy reply, "Thank you!" ;)

DeleteI want to teach my students strategies to find the answer but I have at least 4 students who can't memorize. They struggle and try but it usually ends in tears. I don't think memorizing is important as long as they have the mental tools to find the answers:) thanks for sharing! I'm glad I'm not the only one who shares this opinion. www.missthirdgrade.com

ReplyDeleteI think your #s are probably pretty accurate...I don't think it's uncommon to find kiddos (or adults!) who struggle with memorization. Glad you're able to give them alternative tools!

DeleteTHANK YOU so much Cindy for this incredible post! I will be sharing it with my friends. It is such an important topic that often gets overlooked. What a blessing it is to have your advice!

ReplyDeleteThank YOU...it was your awesome question that set me to thinking about the topic. Number Pieces should be out later this week, so hopefully I'll be able to show some examples then that will help your friends. Hugs! :)

DeleteThanks for the great thought intriguing post...it has me thinking and I am excited to discuss it with my team tomorrow. With my first graders this same thing applies with their addition and subtraction facts. We want them to have tools in their toolbox to solve...and we want fluency not recall.

ReplyDeleteThanks for touching on a great topic and to get so many of us thinking deeper about this subject and how we teach it to our little ones!

Amanda

Learning to the CoreAmanda, you don't know how big a compliment this is! :) Everything that you are saying is absolutely true...and so important for our young (and old!) mathematicians. I hope you report back on your team discussion!

DeleteIn Asia country, multiplication fact is something that young child must know instantly at age of 9/10 by way of rote memory...but ever since I hv my daughter, I make her doing the sing song style to 36x36 since young by listening to an audio files...Does it work? In a sense it's. She could answer 4 when I ask 2x2 is....etc...that time she was only 2. And interestingly, ever after listening to the sing-song syle, the multiplication math music rhythm seems to have registered in her and it never fails her whenever she is asked to do her timetable even at age of 4. Of course, that's still not enough, hence I taught her grouping method to do multiplication solving skill for word problem such as the meaning of 2x2 is 2+2, or or ask how many 12 here, 12+12+12+12? As long as she could answer 4 group of 12, then I think it has reached my objective to teach her understanding in multiplication fact and then make her to write down as 4x12 ( 4 groups of 12)....

ReplyDeleteMy 10 and 7 y.o. do not have their math facts (subtraction, addition or multiplication) memorized. They can calculate and some of those facts they know just because they use them all the time. But like your friend, they use the knowledge they have to build on. I don't make them memorize. I want them to enjoy math and understand it.

ReplyDeleteI was taught to memorize multiplication. I had no clue what I was memorizing about! So I made sure my son didn't go through the same way. I made sure he understood the concept. It didnt matter if he had to take a longer time to calculate, to me, as long as he grasp the concept- it's all good. Here's how I did it http://toddlercanread.blogspot.sg/2011/04/multiply-them.html :)

ReplyDeleteI learned the old fashioned way but I've noticed that by working on number sense with K, she intuitively understood multiplication. Her ability to add quickly in her head led to quick retrival of times facts so that rote memorization wasn't necessary. It was painless. I think being able to see relationships in numbers is a much more useful tool than simple memorization. If I show her my way of coming up with an answer, she likes to tell me her way. She says she prefers her strategies. :)

ReplyDelete