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Week 5: Blogosphere and multiplication (February 13-19)

Next week, the Math Future network is holding an online event with John Mason, which our course participant Julia Brodsky will co-host. John is the author of some of my favorite books on education. He invites us to discuss several topics dear to many of us in this course - early learning, multiplication patterns, hands-on activities, and a controversy that sparked a lot of blog conversations and flames over the last five years. I hope some of you can attend (it's at 2pm ET on Wednesday January 22), and this task, among other things, will help make the conversation more interesting.

John writes: "It is often said that 'multiplication is repeated addition' when what is meant is that 'repeated addition is an instance of multiplication'. I have been developing some tasks which present 'scaling as multiplication' based around familiarity with elastic bands."

1. Find some blogs discussing whether multiplication is repeated addition or not, and comment.
2. If you want to have higher chances of a reply, use "last month" option in the Google Blog Search tool http://www.google.com/blogsearch (see the screenshot).

3. Post Comment to this task, linking blog posts you found, a copy of your comments, and your thoughts on on what blogs do for these distributed conversations around a deep topic.

• This topic came up today on a homeschooling forum I frequent. It began as the related question, "Do you think teaching Division as "repeated subtraction" is confusing?" but the discussion quickly grew to include multiplication. I added a very long comment ...

The problem with slogans in teaching: Whether we are the teacher or the student, once we accept a slogan as dogma, we stop thinking.

"____________ is simply ____________ ."
Fill in the blanks however you want, and you have a BAD statement for a teacher to make.

"Division is simply repeated subtraction?"

As to the original question: I agree with the other commenters that "repeated subtraction" is a poor way to introduce division. On the other hand, it IS a great mental math technique to have in your toolbox for solving certain math problems. And the slogan does encapsulate ONE way of looking at division.

Let's consider the types of story problem (or real life) situations our student might meet which require division. First, we might have some amount of stuff that must be shared evenly among a certain number of whatevers, and we need to find out how much stuff each whatever will receive. Second, we may have some amount of stuff that must be measured out in chunks of a certain size, and we need to find out how many chunks we can make.

The latter situation looks much like subtracting the size of the chunk over and over until we run out of stuff. For instance, we might need 3/4 yard of fabric to make a certain type of pillow cover -- so how many pillows could we cover with 6 yards of fabric?

Also, as another commenter has pointed out, this understanding of division is at the heart of the standard long-division process.

Conclusion: Don't teach with the slogan. But do, as a teacher, think about what might have inspired the slogan and how it might help you develop a deeper, more flexible understanding of division.

This slogan has the SAME problem as the statement about division (it encapsulates ONE way of looking at one very limited application of multiplication), but because the multiplication slogan is so familiar to us, we teachers don't recognize the problem. We have a familiar, comfy slogan, and we don't think deeply enough to realize the problem this can cause for our students.

If we train our students to think "multiplication is repeated addition", then we have no cause to complain when those same students can't solve story problems or when they get confused trying to remember the fraction rules.

Consider:
(2/3) x (5/6) = (2 x 5) / (3 x 6)
but
(2/3) + (5/6) is NOT = (2 + 5) / (3 + 6)
Why not?
Isn't multiplication just a special type of addition? So WHY are the rules so different?

The Fibonacci Series is created by repeated addition of the two previous numbers. Is that multiplication? We can form the square numbers by adding up the odd numbers: 32=1+3+5, and 42=1+3+5+7, and 52=1+3+5+7+9. That's definitely repeated addition, and squaring a number is a sort of multiplication...

The problem with the definition "multiplication is repeated addition" is that it leave unstated the MOST IMPORTANT difference between the two operations. That's why so many students are reduced to staring blankly at a story problem, asking, "Do I add or multiply?" We haven't given them any way to recognize the difference.

For more examples of how not understanding the difference between addition and multiplication makes learning fraction rules difficult:

For a more thorough exploration of the "repeated addition" debate:

Then how SHOULD we teach multiplication?

If we accept this argument, if we agree to no longer define basic multiplication as "repeated addition", then what? How does that affect the way we teach?

Mainly, we need to change our focus from how to why.

We can teach multiplication in much the same way that we do now, using manipulatives arranged in groups or rows, pictures of multiplication situations, and rectangular arrays of dots or blocks. But instead of drawing our student’s attention to the process of adding up the answer, we want to focus on the fact that the items are arranged in equal sized groups.

In other words, we teach our students to recognize the multiplicand:

• Teach children the useful word “per” and how to recognize a “this per that” unit.
• Have them label the quantities in their workbook: 3 cookies per student, 5 flowers per vase, 1 eye per alien, or whatever.
• If your story problem has a "this per that" quantity, then it must be a multiplication or division problem. You may be able to solve it with an addition or subtraction approach (especially if the numbers are small), but the heart of the problem is multiplicative.
• I personally think that multiplication is repeated addition. I grew up learning it that way. Eventually I did have to memorize it but when you first start doing multiplication looking at it as repeated addition makes it easier to learn and understand. I think having blogs to discuss topics like these not only gives you a chance to express your opinion but also hear other sides of the topic. It allows you to be introduced to something you probably never thought of. It will give you a better understanding of where people are coming from, the sources they used, and the hands on experiences they might of did to get their information. Here are some of the blogs I found and commented on.

Comment: I think it's a good idea to introduce repeated addition when doing multiplcation with younger kids. Telling them that it will become difficult to do it as the numbers get bigger should be addressed so it's no surprise to them when they try to solve a problem like 12x22.

Comment: You bring up a good point and I agree with you that it will get confusing for kids, but some kids find it hard to just simply memorize answers to problems. So repeated addition is a different strategy for them until they are able to work their brain up to memorization.

• Keisha, thank you for the comment. You make a good point about modeling multiplication as repeated addition, rather than just memorizing answers. Many people here mentioned they like this model because it's simple and accessible to kids.

In yesterday's webinar, John Mason was showing us how to play with some rubber bands for multiplication. When you stretch the band, the length is multiplied. It's very hands-on too (not just memorizing answers). But, being hands-on and accessible to young kids, this is more like scaling than repeated addition. http://mathfuture.wikispaces.com/JohnMason

Just building on helping kids see where answers come from...

• This blog relates to our converstaion because it is great evidence that repeated addition is a great way for children to learn multiplication. The child said that it made learning easy and wondered why others didn't teach like this. She even later used this in a real life situatuion while helping her mom buy something on the internet.

Comment: This is really great! I am taking an online class about mathematics for younger children and this week  we were talking about debating whether multiplication is simply repeated addition. Your understanding of multiplication just happened to be that! You taught your daughter repeated addition and she found it so easy and actually used it in a real life situation. This is great evidence that multiplication is repeated addition and works well. Thanks for sharing!

This blog related to our conversation because it presented arguments for scaling rather than repeated addition. It was a good example of why we shouldn't think of multiplication as either one or the other necessarily, but that scaling tends to be more accurate of a description and can be applied to fractions instead of just integers.  Using examples such as stretching a line by a certain amount to demonstrate scaling made the example intuitive and is something I would include in a lesson, even if I was teaching about multiplication as addition for the most part.

Comment: As an elementary education major, I found this post to be interesting in presenting the arguments for scaling rather than repeated addition, while acknowledging that there is no one "correct" way to go about teaching or understanding multiplication.  My main concern with teaching multiplication as addition is that students may not fundamentally understand the difference between the two.  On the other hand, it is an easier concept to grasp at first. Some sort of combined lesson would introduce the concept best, as in daily life I tend to see multiplication in either way, depending on the context (i.e., scaling in order to determine the tip on a check vs. repetitive addition to determine the amount of total items for packages of multiple items.

• After searching through some blogs I have started to reflect on my experience with repeated addition and multiplication. As a student I was taught multiplication through repeated addition. Multiplication to me was just a shorter way of adding the same number. However I had the unfortunate event of acquiring Everyday Mathematics in the third grade, when I was learning multiplication. I learned all sorts of ways to multiply but none of them ever stuck and they always took forever.The lattice method is the one method I did use and still can to multiply. To this day I still cannot multiple the "old fashion" way with two double digit numbers, I can multiply the old fashion way if I mulptiply a number by a single digit. At some point I realize I will have to learn how to multiply the old fashion way but in school I could always get by with lattice.Moreover, many times on standardized tests, the only time I really needed to multiply by hand, I use repeated addition even if it was only to check my multiplication.

I found this blog that led me to the second one on the definition of multiplication and how by definition multiplication is not repeated addition. Many other blogs that I came across referred to Keith Devlin's blog.

http://devlinsangle.blogspot.com/2011/11/how-multiplication-is-really-defined-in.html

I have come to the conclusion that repeated addition can be used as a strategy for learning multiplication but is not multiplication. I believe that students will understand how repeated addition only works with small numbers. If you think about it who would ven try and multiple 3/4 and 5/6? I am sure now I will run into a student that will try it but I will correct it at that time. I still see nothing wrong with using repeated addition to explain basic multiplication.

• Starting off, I knew very little on the debate of multiplication vs. repeated addition. Growing up, I don't remember teachers using that phrase, but I remember the concept. Like Kathy, it was easier for me to imagine multiplying as repeated addition. My multiplication lessons began by looking at two piles of M&Ms and realizing that multiplying them meant adding both together.

As I recall, I did not have trouble later on figuring out the difference when multiplying negatives or fractions. I believe that, while the terminology may not be the most appropriate in a very technical term, it's a helpful way to teach multiplication. I say "helpful" rather than "easy" because there really isn't an easy way to teach a more complex math topic. Yet, I believe that teaching children that it's repeated addition isn't necessarily harmful. Perhaps that exact phrase is wrong, but the concepts are far more simple to understand. For me, it doesn't make sense to complicate an already difficult topic for elementary students, simply because there's some debate. It feels like a better idea just to tell students that it's essentially repeated addition, it just gets harder later on with non-counting numbers. After all, a good portion of elementary school is focused on work with counting numbers.

I think what I disliked most about Devlin's argument was how condescending it seemed. I think that our comments on how we were unaware of this shows that it's not a very major debate just yet. Devon seemed almost frustrated that so many teachers were unaware of the difference, which feels unfair. I'll agree that he brings up very fascinating points, but I found myself on the defensive through most of his post.

http://www.maa.org/devlin/devlin_06_08.html

I did, however, find one blogger who went through the concepts of addition to prove that multiplication of natural numbers is repeated addition. I mist admit though, it feels almost bizarre that people can debate so heatedly about numbers, which I always believed to be objective, in a sense.

Comment: Thank you for your breakdown. I have just been researching the debate over multiplication vs. repeated addition, and found this very helpful in understanding this new topic. I believe that, because children mostly work with natural numbers, it should be acceptable to teach them multiplication through repeated addition, at least to some extent.

I also found the following blog useful, as it discussed the importance of teaching children multiplication through addition. It's hard for them to conceptualizer multiplication without having some sort of foundation. This site provides some activities for children to do which connects multiplication and repeated addition. The post is a bit older, but I felt like the material is still relevant.

Comment: This is a great resource for teachers just starting multiplication. I feel like these examples are engaging and would really help a child visualize multiplication. Beginning to multiply through activities like these may help children realize that more complicated topics aren't so intimidating!
• I think as a visual learner the explanation of multiplication as repeated addition really helped me to make sense of the process.  When I envision 5 groups of five being added together I'm able to see that this is 5X5 and that it equals 25.  I found some bloggers in a heated debate on the subject.  I especially liked what the blogger, JC had to say on the subject.  He or she said that with young children building an understanding on their prior knowledge of addition is a good way of starting out toward an understanding of multiplication.  With older students we might build a more abstract understanding. Here is the link.

• When I first read this week’s task, I had no idea what this controversy was about.  It seemed to me that multiplication IS repeated addition; I couldn’t see why that would be a problem.  I started with a simple Google search of blogs and went to The Number Warrior.  Here, http://numberwarrior.wordpress.com/ he discusses the original posts by Keith Devlin and responds with his thoughts.  His summation: “It is fine to expose students to multiplication as repeated addition if great care is taken to avoid the problem of identical mental models“ and “The research I’ve seen does not indicate a significant problem with multiplication-as-repeated-addition at the immediate point of learning”.  Honestly, that is my initial reaction.  To me, and perhaps because I am not a math person, I think that however a student reaches a correct answer is ok.  He used the example of a pan of brownies.  A student could multiple the rows (3 x 5) to get the number of brownies or they could count 5 for the first row + 5 for the second row + 5 for the third = 15 total, and done. To me, I’m not sure I see the problem in using either method.  Though one takes more time, the answer the student would give would be the same.  For some special education students, moving to a higher-level operation (multiplication) beyond the counting of rows may not be an option, and that’s ok.  They still arrived at 15.  I guess the problem is that you won’t always get the right answer by counting, and he didn’t provide an example of this so I still don’t fully understand the problem.  As a side note,  I saw Professor Droujkova’s website mentioned as a resource!

I then navigated to A Mindful Madness at http://hilbertthm90.wordpress.com/.  This blog post was completely lost on me!  Perhaps you really do have to have some understanding of math to fully appreciate this argument, and I understand that.  For me, it must be similar to my field’s constant bickering about ending a sentence with a preposition.  I couldn’t tell if hilberthm 90 agreed with the idea or not.  He states, “You can make a definition of anything you want in math, but that doesn’t mean it exists.”  I’m guessing he thinks Devlin is flawed in his thinking.

Finally I went to Drat These Geeks! at http://myrtlehocklemeier.blogspot.com/.  This person I could understand!  Succinctly put the blogger posted:  “I don't know what else multiplication could be. Doesn't my computer base everything it does on repeated addition of ones and zeros? If Devlin comes up with a new paradigm wouldn't that be the greatest revolution in math in the latest 150 years?”  This, I understand!  (And for what it’s worth, I agree.)

I’m looking forward to hearing what the “math-ies” in this course have to say about this topic in which I admittedly can’t quite understand the controversy

• The multiplication/addition controversy can be difficult to fathom. I laid out my responses to Devlin's argument in two posts:

I found the book Children Doing Mathematics to be tremendously helpful in understanding the difference between multiplicative reasoning and the reasoning involved in addition situations -- and how very important it is for us as teachers to help children understand these differences.

Let me quote a few excerpts from chapter 7...

Children Doing Mathematics
Terezinha Nunes and Peter Bryant, c1996
Blackwell Publishers, Inc., Cambridge, MA

"It would be wrong to treat multiplication as just another, rather complicated, form of addition, or division as just another form of subtraction. The reason for this is that there is much more to understanding multiplication and division than computing sums. The child must learn about and understand an entirely new set of number meanings and a new set of invariants, all of which are related to multiplication and division but not to addition and subtraction.
...
"Additive reasoning is about situations in which objects (or sets of objects) are put together or separated. All the number meanings in additive situations are directly related to set size and to the actions of joining or separating objects and sets.
...
"Situations which give rise to multiplicative reasoning are different because they do not involve the actions of joining and separating. We will distinguish three main kinds of multiplicative situations: (1) one-to-many correspondence situations; (2) situations which involve relationships between variables; and (3) situations which involve sharing, division, and splitting.
...
(1) "One-to-many correspondence situations involve the development of two new number meanings: *ratio*, which is expressed by a pair of numbers that remains invariant in a situation even if the set size varies, and the *scalar factor*, that refers to the number of replications applied to both sets maintaining the ratio constant. It should be clear that neither of these meanings relates to the set size: the ratio and scalar factor remain constant even when the set sizes vary.
...
(2) "Even if children understand one-to-many correspondence, they probably need to build some different concepts in order to deal with situations where a relationship between two variables is involved. The novelties in dealing with relationships between variables relate to *fractions of units of measurement*, which appear in these situations because variables, unlike sets, are continuous quantities, and to a new type of number meaning that expresses the relation between the two variables, *a factor, a function, or an intensive quantity*.
...
(3) "Sharing involves a new view of part-whole relations, which differs from such relations in additive situations. In sharing, there are three values to be considered: the total, the number of recipients, and the *quota* (or the size of the share). The quota and the number of recipients are in *inverse relation* to each other: as one grows, the other decreases. Sharing also involves a new type of number, fractions."

• Sandra, you are bringing in a very interesting point about computers using discrete math and rational numbers for everything. It is in all senses correct that for a computer, multiplication is repeated addition! I've never thought about it this way. We need to tell Keith Devlin!

The paradigms where multiplication isn't repeated addition have to do with continuous life and irrational numbers. For example, multiplying the number Pi by itself is impossible to represent as repeated addition. But this is very technical and computational!

Let's look at some kiddie hands-on work in multiplication, though. A beautiful model is array, like an egg cartoon. It can definitely work as repeated addition by rows or columns. Another cool model is shadows. For example, shadow as you walk near a street light can be stretching or shrinking your height. Or the Batman Light projecting on the clouds. This is multiplication, but not repeated addition. Does it make sense?

• One of my favorite movies about social media - "The New Dork":

One of my favorite articles about its significance for education - "Bloom's Taxonomy blooms digitally":