Wednesday, November 17, 2010

9 x 7 = 63

I was teaching Billy how to add and multiply today. We use tokens as a manipulative, so the kids can show me what they are thinking when they do arithmetic.

I asked Billy to share with me how he multiplies. He then explained that he does a lot of flash cards at home, and that he knows how to use his fingers to multiply.

When talking to kids, my premise is talk less-listen more, so I asked him to explain. He told me about how he remembers what 9 times 7 is. I later Googled this trick and here's what he basically said:

This may be an easier way to do 9's with finger math. Choose the number you will multiply by nine. Count to that number beginning with the pinky finger of the left hand with palms facing down. Once you get to that number, fold that finger down. The numbers to the left of the folded finger are tens. The numbers to the right of the folded finger are ones. Example: 9 x 7 = 63 Count to seven starting with the left pinkie finger. That should put you to the pointer finger of the right hand. Fold that pointer finger under. To the left you have six digits or 60. To the right you have 3 fingers or 3. 63!
When I asked Billy what 9 times 7 was he responded with 63! I then asked him how he knew that. He felt content with just holding his fingers up and nodding at them - as if to say, "I just showed you, silly". So I asked him to prove the answer was 63. He thought about it and said:
Well, 9 times 5 is 45, and that's two groups of 9 short of 7, and I know that 9 plus 9 is 18, so 18 plus 45 is 63.
Someone might hear this and say: "Look, the trick worked. He understood what he was doing." My response: it is far more likely that Billy has developed this number sense not because of the trick but in spite of it. Also, if teachers explicitly say or just implicitly hint that the most important ability in math is quickly knowing the right answer, then kids will sacrifice thinking for precision at the cost of understanding. In other words, Billy was able to reason why 9 times 7 was 63, but I had to invite him to need to do it.

After experiencing all this, I noticed that an anonymous parent left the following comment on my post about Alberta's new math curriculum:

As a parent of a child in Grade 4, I have serious concerns about this "new" math. They are not learning basic foundational skills, such as multiplication. Without consistent practise of these skills, how can a child go on to apply their knowledge to various forms, like this math insists? I am heading into parent teacher interviews to discuss why my previously math-loving child now hates math, and we find that we are doing the activities she LOVES (like math facts) at home because she isn't learning them at school...
If a teacher is provided the appropriate professional development and they understand the theory behind Jean Piaget's constructivism, then this "new" math actually reflects the very essence of how people learned arithmetic before we had all these tricks and alogrithms - essentially making this "new" math a very "old" math.

Constance Kamii explains why this kind of learning is actually the best way to learn the basic foundational skills such as multiplication:

It took centuries for mathematicians to invent, or construct, “carrying” and “borrowing.” When we teach these algorithms to children without letting them go through a left-to-right process, we are requiring them to skip a step in their development.
Honeslty, if you want to immerse yourself in understanding this "new" constructivist math, there are three books by Constance Kamii that you really need to read:

Young Children Reinvent Arithmetic

Young Children Continue to Reinvent Arithmetic (2nd Grade)

Young Children Continue to Reinvent Arithmetic (3rd Grade)


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  2. I think it's interesting that you take issue with Billy's trick for 9x7, but you're fine with his reasoning. He begins with 9x5=45. How did he know that? What evidence is there that he didn't just "know" 9x5??? It's because he goes on talking and we see later that he understands the grouping process. I don't know how old Billy is, but as a high school teacher, I need my kids to multiply without starting at 45 and adding 18. At some point they have to "know" the answer without reasoning it through every time. Billy clearly learned the concept. If someone then taught him a trick to help him remember, I am all for it...

    You say, "it is far more likely that Billy has developed this number sense not because of the trick but in spite of it." What if Billy learned number sense and then learned a trick to help him remember the answers more quickly... They don't have to even be related. let's not confuse a good mnemonic with an undermining of educational methodologies...

  3. I've had a similar experience with multiplying by nine. It was interesting for me that the majority of kids I asked to tell me how took a lower number and then did the add-on, the same as Billy. I have had a couple do the multiply by ten and subtract -- which seems to me to be a higher level strategy -- so even here, you can't teach a 'standard algorithm'.

    It took me a couple of years of teach, but I have been convinced that "Show me how" always works better than "this is how" for learning!


  4. @Deb, Billy is 11 (grade 6). While he is showing signs of mathematical reasoning and problem solving skills with arithmetic, he is no where near the stage where I would feel comfortable abandoning this kind of number sense development.

    Also, traditional math curriculums and assessments tend to concern themselves far more with product at the exclusion of process. The harder = better mentality encourages the worst kinds of teaching and learning.

    When Billy is ready to move beyond making sense of computation, then I do agree these tricks and algorithms are important to provide students with the time and effort to do more than just arithmetic. But I don't believe we have anything that resembles a balance in our current system.

  5. Most kids are being taught these tricks and algorithms in grade 1, 2 and 3. Their understanding for place value and number sense tends to be out of whack for good reason - too often their development is being stunted by the system itself.

  6. After teaching middle school math for many years and coaching middle school math teachers, I've seen so many occasions where students are taught tricks for remembering without any conceptual development or number sense activities. As a proponent of Brain-Based Learning, I definitely believe that mnemonics are valuable memory tools. However, without ensuring that students understand a concept prior to learning mnemonics, we are hindering their desire to ever learn the "why" behind the mathematics. When students think they already know something, they tend to perceive number sense and conceptual development activities as going backwards.

    Another issue with student's learning mnemonics without prior conceptual knowledge is that many times they don't even remember the mnemonic correctly. At that point, the damage is compounded because they can't even properly apply the mnemonic that is supposed to help them get the "right answer". For example, many of my 7th graders would have learned the following:
    a good thing and a good thing equals a good thing
    a bad thing and a bad thing equals a good thing
    a good thing and a bad thing equals a bad thing

    This was supposed to help them remember the rules for multiplying and dividing integers. Unfortunately, they usually couldn't remember which integer operations this applied to. So, not only did they have no conceptual understanding of multiplying and dividing integers, they couldn't even remember the rules. If they had solid conceptual understanding of integer operations, they would have known why the rules work and therefore would have had better retention of the integer rules.

    Joe, I also recently discovered your blog and enjoy it very much. I found it because the name is the same as the blog I just started. Thanks for being a proponent for positive change!

  7. Thanks for your thoughtful response. I think part of our problem here in the States is two-fold. First, we really group kids together for the most part by age, in no way taking into account that kids learn various topics at vastly different speeds at various points in their lives and have vastly different abilities. Therefore when it's time to move on because most of the kids get it, we teach a mnemonic to get the answers faster, regardless of whether they understand the "whys" or not. The second problem is of course standardized testing. We teach to the test and care nothing for understanding. Just pass... here's the tricks.

    I think to really teach math the way you suggest we in the States would need a complete overhaul of the entire system as well as a restructuring of what we are actually trying to teach. In that vein I offer a couple of videos that you might find interesting. You may have seen them as they have points in common with your ideas.

    Thank you for the conversation.

  8. Hi Joe,
    There is so much I want to comment on here so I'm going to start at the top:

    1. The parent who wrote on your blog, "They are not learning basic foundational skills, such as multiplication." Needs to realize that we are doing so much more than teaching these "basic facts." We are teaching them to think and reason. This is so difficult to get across to parents and teachers. Basic facts are not the end all and be all to understanding math. I know many adults who do not know there basic facts but have strategies on how to solve them. We want kids to have strategies as well.
    2. This comment is for Deb. I would be thrilled to have a student know to start at 9x5 is 45 then to know that 9+9 was 18 and to add them together. This is mathematical reasoning. Yes, perhaps he could have did 2x9 instead but in the grand scheme of things I'd be happy with 9+9. This student obviously has a benchmark and an understanding of what to do with the numbers in this equation. Please point me to the area of the curriculum where it says, "At some point they have to "know" the answer without reasoning it through every time." When you find it for me, I will be satisfied that you know the curriculum you teach. With teachers who continually challenge Billy's thinking and continue to ask him how he got the answer, he'll discover and become more proficient with his answer.

    So I guess my husband who solved the exact same question with saying, 6X7 is 42 and 3x7 is 21 and 21 and 42 is 63 is also wrong because he didn't "just" know the answer and shouldn't have reasoned his way to the answer. He is a perfect example of someone who has number sense because he has strategies and mathematical reasoning to solve problems.

    This student should be applauded for his reasoning because he has a strategy like the one mentioned. Perhaps he's playing around with the mnemonic understanding that a parent or friend taught him. He's coming to understanding it by experimenting with it.

    3. I know you didn't mean to insult some of us who don't teach tricks and algorithms but you did. Did you ever consider that some of the primary teachers in this province actually do teach for understanding and that when they head to the higher grades like 4, 5 and 6 that these teachers unteach what we've tried to do? I say this because a very wise man named Grayson Wheatley told me that if teaching for understanding/reasoning doesn't continue throughout a child's educational experience they will loose what they've been taught. I pride myself on teaching my grade two's how to think and reason based on their ability and level. I don't teach tricks and algorithms.

    Teachers in the province are working to understand this new curriculum. Yes, there are those who still teach traditional algorithms and tricks but please don't place blame on all of us or "most" of us.

    I applaud you on your journey this year with improving your understanding of the math curriculum and teaching students to understand numbers. Constance Kamii's articles are relevant not just to primary teachers but to high school teachers are well.

    Thanks for posting such an interesting blog.


  9. In Ontario, we have been using a "constructivist" math curriculum for years. No "tricks" or "algorithms" are taught in the early grades, yet there has been very little improvement in conceptual mathematical understanding. (Ontario's scores on international tests have been steadily declining, while Alberta's have not. I'm wondering if that's about to change.) Most kids find "constructivist" teaching methods frustrating and confusing, and parents who think their kids have a knack for math externally train them in truly challenging programs like Spirit of Math.

    For a *progressive* alternative to "constructivist" math (I use the quotation marks because many people--mathematicians included--have argued that this type of math pedagogy is based on a fundamental misunderstanding and misapplication of constructivist theory), google Jump math. Or read my recent blog post about my daughters' struggles with the Ontario math curriculum:

  10. This sounds like an example of a conundrum I've heard more than a few math teachers discuss... is it more important that students understand the underlying meaning of what multiplication is, or that they can immediately give the answer to a "basic math fact". It's easy to say they should have both... but that still doesn't parcel out where emphasis will be placed. The 'fingers trick' or any other 'trick' (which isn't really a trick, but an application of the underlying theory of numbers, without any accompanying explanation) is just a replacement for not letting kids use a calculator. The 'trick' doesn't require the student to understand anything about multiplication, but because it doesn't use a calculator, it "must be okay" because "Billy got the right answer on his own". Pffft.