# Dots!

The New York Times published an article about Common Core homework this week.

As is going to be the case with a news article (in contrast to say, a post on a blog dedicated to children’s mathematical ideas), one can’t really learn any mathematics from the piece. One critique got hit twice, though—that children are being forced to draw lots of dots.

Here near the beginning of the piece:

Ms. Nelams said she did not recognize the approaches her children, ages 7 to 10, were being asked to use on math work sheets. They were frustrated by the pictures, dots and sheer number of steps needed to solve some problems.

And a bit later:

Her daughter, Anna Grace, 9, said she grew frustrated “having to draw all those little tiny dots.”

“Sometimes I had to draw 42 or 32 little dots, sometimes more.”

I have no interest in picking up political issues surrounding the Common Core State Standards on this blog.

But I do think a parent frustrated by all those dots deserves an explanation of what all those dots are for.

Before we begin, please be assured that there is absolutely no mention of dots in the Common Core. What is mentioned is the array. An array is a collection of things arranged in rows and columns. We have discussed arrays before here at Talking Math with Your Kids. They are very useful tools for representing an important meaning of multiplication—that multiplication is about some number of same sized-groups.

Arrays (with dots or other things) are useful tools for making these groups visible, either actually visible or visible in the mind.

So I asked Tabitha (7 years old) to draw some dots for me.

Me: Tabitha, [neighbor girl and best friend] wants to play. Before you go outside, can you draw that picture for me? Three rows of five dots.

Tabitha (7 years old): That’s easy! Fifteen.

She is probably counting by fives here. She completes her picture for me.

I know that neighbor girl is waiting. I decide to press my luck.

Me: What if it had been 3 rows of 6?

There is a long, thoughtful pause.

T: Eighteen!

Me: How did you know that?

She shrugs her shoulders. Now is not the time to force things. Neighbor girl is waiting. So I offer a strategy.

Me: Let me tell you how I think you might know it.

T: OK.

Me: Six is one more than five. So each row would have an extra dot. That’s 15 for the 3 rows of 5, and then 16, 17, 18.

T: [smiles] Yeah.

We share a high five and she is out the door for a morning of clubhouse shenanigans in the backyard.

Quick note: Tabitha does not let me get away with stating her strategies incorrectly. I have done this before—summarized how I think she is thinking—and when I get it wrong, she objects. I am glad about this.

## So what do we learn?

This is what those dots are for. They give us something we can talk about. Without those rows and columns, the conversation is so much more abstract. We were picturing those dots in our minds as we talked about counting them.

The three rows of five she drew gave us a jumping off point for imagining the three rows of six we discussed. Three groups of five now has a relationship for her to three groups of six.

More importantly, the strategy of finding new facts based on old facts (here that 3 groups of 6 is 18 based on knowing that 3 groups of 5 is 15), has been introduced explicitly. It is something we will talk about in the future, and something she will know to consider.

Without the array, it is not at all clear to me that she would have been able to know what 3 groups of 6 is. She could have drawn 3 unorganized groups of 6, I suppose, and counted them individually. But this is a much less sophisticated strategy, and she is ready for more than counting individual objects.

## Starting the conversation

Many children do not naturally see rows and columns. Given an array, they may haphazardly count the objects around the edge, then in the middle. This often leads to double counting and skipping things.

But even children who are very good at keeping track of their haphazard counting—and who can get correct counts every time—may not see the row and column structure of an array.

So put 15 pennies in 3 rows of 5. Have your child count them and notice whether she counts in rows and columns, or whether she counts in some less structured way. Model the counting yourself so that she can see an example of the rows and columns at work. Don’t worry if she doesn’t see the structure yet, but do make a note to do more of this kind of counting in the future—seeing the structure of an array is an important stepping stone to multiplication and to the measurement of area and perimeter.

Or just have her draw dots.

## 18 thoughts on “Dots!”

1. Monika says:

Thank you for this explanation. I read that article yesterday and was confused about what the exact complaint was. As a child I “discovered” dots on my own and was often told to stop using them, so I had to hide my dots! I’m now a successful business owner but I still secretly depend on dots to do math in my head. This is the first time I heard that dots might actually be OK! Thank you

2. I do think that having to draw 42 (or more!) dots to illustrate a simple calculation is over the top. Some kids won’t be able to keep the rows and columns neat enough to see when they get to 42 (or more!!!)’. At this point it might be more sensible, and still meaningful to draw six little boxes and write a 7 in each one.

• Christopher says:

I am going to concur with Atlas Educational below here, howardat58. Let’s leave critiques of teacher practice to Twitter and my other blog. Here, I would like to maintain a focus on the mathematics under study. Besides, we really don’t have any evidence of what the teacher was asking for in that article. If you look at the graphic with the article, which is supposedly illustrating the over-the-top dot making, you will notice that there is no call for students to draw dots. They are asked to show their work and to write an equation.

The point here is to help people understand that dot-drawing is not just a silly exercise in tedium; there is real mathematics that can be represented and done with dots.

I have no idea whether the child quoted was in a classroom where that was true.

3. Right now it seems that some teachers are teetering between following directions using more conceptual learning and using common sense. If teachers weren’t so used to living in fear of doing something wrong and getting fired, there’d probably be more learning going on. In the meantime, there’s nothing wrong with parents questioning. Just please, tread lightly as the teachers are just as nervous as the kids these days.

The point is to demonstrate thinking processes and encourage greater reflection, but common sense has to also come into the equation more often. How about using a good old fashioned stamp pad and stamps to show your rows of items?

4. Another thoughtful post Christopher thanks for sharing!
The structure and use of arrays make numbers accessible to students. When students see arrays they begin to think flexibly about number and begin to see that quantities can be manipulated and decomposed in multiple ways. Your example of using 3×5 to solve 3×6 is a great example.
Understanding number conservation is a critical piece to increasing a student’s number sense. Without the use of arrays we can never gain insight into students counting strategies. If this is the case, we miss the opportunity to assess how efficient they are when counting.
I think it’s extremely important to note that the expectation of CCSS does not ask that students draw an array for 63 x 32, but it does expect that students apply their foundational understanding of single-digit arrays to multi-digit.
Understanding the progression of learning and how concepts build on one another is often mistaken as tedious, when in fact it is a critical piece of conceptual understanding.

5. lora says:

Our problem with the “dots” was that my kid is a great memorizer. So she had all her facts down cold very early, and all this dot-drawing was not discovery but just a “picture” of something automatically available to her. Imagine if you, as an adult, were asked to answer 5 x 6 and draw an array. I guess you could argue that she should have been taught it earlier? I don’t know.

• Christopher says:

The point of the post was definitely not to defend particular homework assignments. I don’t know enough about the classroom practice of the teachers who assign them.

I definitely agree that if a kid knows 5 x 6 is 30 that drawing dots to demonstrate that it is probably wastes everyone’s time.

But again, I don’t know enough about what a kid knows to say whether there might be some benefit to it. Some kids know that 5 x 6 and 6 x 5 are the same, but they can’t say why. Arrays help with that. Some kids know that 5 x 6 and 10 x 3 are both equal to 30, but they don’t know that these facts are related to each other. Arrays can help with that, too.

Not every array needs to be drawn by the child. There is a lot of value in looking at and discussing pre-existing arrays, or those produced by other people.

The point of my response to the New York Times’s piece was simply that having kids draw dots is not grounded in wasting time on silly activities; there is real mathematics under there. Whether that mathematics is always brought out by teachers when they have kids draw arrays is another story, of course.

6. Another thing you can do with that array of dots? Play dots-and-boxes!
Here’s a nice online version, if you don’t know the game I mean: http://www.math.ucla.edu/~tom/Games/dots&boxes.html

Another technique is to use other objects to make your arrays. We use trio blocks which have the nice feature that you can use long segments to speed up the array and the holes in the top still give you the dots imagery at the end.

7. Why is learning times tables not acceptable in public education anymore? The “process skills” sound wonderful and certainly lend to greater ease in later mathematics courses; however, rote memorization of times tables at an early age proves extremely efficient and useful in tasks such as grocery shopping and budget balancing without the need for rows and columns of dots. Please feel free to respond, but try to keep the “new standards” advocacy jargon to a minimum. What are the real-world, applicable advantages for children to compute using an array over “old-fashioned” times tables? My “old-school education” enables me to calculate my estimated grocery bill including sales tax in my head before reaching the checkout counter, without the need for a pencil and paper to make rows and column of dots, because I know my times tables. In this day and age of soaring family debt and a less than stable economy, that is a necessary skill that sadly our public school children are simply not getting. If you need proof ask a 5th grader to add up your grocery basket in his or her head, calculate sales tax, and give you an estimated total. Please share your success stories.

• Children today have little conceptual understanding of math due to the extensive list of standards that must be covered by such and such dates so they can be tested, sorted, and ranked ad nauseum. When I started teaching, we taught multiplication for several months, yet my last year of teaching (if you still call it that), our “team” scheduled 5 days for basic multiplication. And we wonder why education is disastrous?

I do admit to loving understanding over rote memorization in math though! Of course, I was that annoying child who always asked, “Why?” Insteas I was frustrated often and directed to follow algorithms with little understanding instead of being allowed time to discover how mathematics worked. Until the pressure cooker disappears, little genuine learning is happening in classrooms these days.

• Patrick Bureau says:

I Agree, one can calculate much faster in tables that it may be multiplication or by reverse division tables, than try to use “dots” to create the imagery of a total amount, the method is not a bad one, I recall using lines with circles in the 70’s and it did visually help, but was dropped early after 3rd grade for mental calculation of tables from 1 to 144 by sixth grade, and to this day the repetition of these table are far lasting in any work I do daily. My Fear as I see it in my 21 year old daughter is that when I ask for a simple surface calculation of a room of 11 x 14 she starts drawing lines on a paper and takes 5 mins to answer a simple table computation that took me 5 seconds to complete. That is what I believe people are gripping about, were no longer teaching methods that insight ” brain retention” but ” drawing diagrams” and in the real world, I hate point it as well as other have, but in ” this world” mental calculations are done daily by millions of older adults while the younger ones (millenials) are struggling to keep up .. developing a brain to understand rows and columns is a good thing, but retention of basic math is far more important than teaching to make dots to retain 3×5. sorry I don’t buy that argument one second. I have sat in a great many meetings with sales, engineers, tech support and Ceo’s to know when one of them ask you on the spot to evaluate a basic cost of products, your not pulling out a calculator or a piece of paper, you use your brains and spit the answer they want to hear….

8. DK says:

I have a 10-yr-old who is comfortable with the times table. Like a previous poster said, he already knows a lot of the facts and voices that the other methods are tedious. I think it’s great that they are teaching other methods to encourage conceptual understanding. However, if a child already gets it, choice should be preserved. Let him do the traditional algorithm even if the other kids are still drawing dots. Don’t mark his answer wrong or tell him to show his work in an array if he showed his thinking another way. Don’t make him write three sentences on his process. The problem isn’t the content or the various methods. Rather, it is that some teachers are scared as Atlas said, and they think kids MUST do math computation the way it is demonstrated in that unit of study accompanied by a written explanation.

9. DK and Patrick, I only pose this question to you: How do you know your child understands? Giving an answer does not indicate understanding. I memorized all of my multiplication tables in 2nd grade having to stand up and recite them in front of the class, but honestly, I had no idea what they meant and by high school my admission to an honors course was discovered to be a mistake. I was lost. I had no way to apply my knowledge having been able to memorize only.

If your child knows their multiplication tables well and you’re confident, instead of having them draw, ask them to solve extended problems. Surely, they can understand 200 X 6, 50 X 20, or 3/4 x 12. As an example, I used to teach my students the conceptual understand of fractions prior to algorithms. Some would naturally discover algorithms (which meant they owned the learning) and some needed more time that I could give (unfortunately).

Asking a child what is 1/2 x 12 is a good example of how rote procedure and conceptual understanding clearly part ways. Geez….it’s half of 12, but most children will multiply through the steps instead of actually thinking through naturally. THAT is what we’re trying to regain. Kids are so preoccupied with getting their work done that if I had a penny for every time a child said to me, “You make us think too much!” I’d be rich. It also breaks my heart to hear it because they’re missing out on the most precious gift of the love of learning.

3/4 x 12 is really……….half of 12=6 and 1/4 of 12 or 1/2 of the 6 which is 3; 6+3=9

Instead most teachers and students will do 3/4 x 12 = 36/4=9 While the first solution may take a bit longer, it lends to understanding. The second solution does not. The first solution allows for understanding so that when the equation is not available, students can more readily understand how to solve the problem. Without the equation, most children who solve using the second approach are lost without an equation.

In more than 2 decades of teaching, I can honestly say that some children will stare at you if you ask them what’s 1/2 x 30 and will go through the procedure. That’s not learning.

10. leon says:

Proponents of Common Core are missing the obvious: If extensive explanations of the method are neccessary, it is NOT an improvement.

• Jennifer says:

Explanations are included because it allows the students to demonstrate their conceptual understanding. I have students who “explain” their thinking by listing the process used to solve. This is not thinking. It is rote.