Category Archives: Number Sense

Number Bonds


These past few weeks have been a bit busy for me. So much great stuff has been going on, but I haven’t had a chance to write about any of it. This week, I forced myself to sit down and write a little bit so I don’t forget all of it.

One of the things I’ve been thinking about is how to develop number sense with young students. Building strong number sense is one of the most important things elementary school teachers can do. There are many ways to do it and I’ve been trying to figure out the advantages and disadvantages of different strategies.

The activities that are particularly interesting to me help develop students’ ability to decompose and recompose numbers. Understanding that a whole can be composed from different parts is a big idea for students. They don’t see a connection between all the different facts that for example, add to 8.

Students who develop these connections will have an easier time developing the big idea of decomposing numbers. When faced with a fact like 9 + 6, they can think about decomposing either of the numbers into parts that may make the calculation easier. For example, by decomposing the 6 into 1+ 5, they can think about adding the 1 to the 9 to make 10 and then adding the 5 to the 10 to get 15.

So how do we help students decompose and recompose numbers?

Number bonds are one way. I like them because they can help students visualize the different parts that can be used to create a certain number.

You can start by having students use hands-on materials, like cubes and counters, and asking them to find all the different ways they can break a number into parts. They can then represent what they are doing with a picture. Here are some different number bond diagrams for the number 8.



After students have had lots of practice with hands-on materials and drawing diagrams, they can then move to using numbers.



I like number bonds because they provide a nice visual for students to use to think about decomposing and recomposing numbers. Of course, this is just the beginning. Students then need to think about strategies for decomposing and recomposing numbers in ways that help make computations easier. However, I think it’s important to make sure students have this foundation.

What do you think? Do you use number bonds with your students? Are they helpful?

Want to know more? Check out: Baroody, A. J. (2006). Mastering the Basic Number Combinations. Teaching Children Mathematics, 23.

Building Fluency and Number Sense

I think the most important thing elementary school teachers can do in math class is to build students’ fluency with rational numbers.  The research shows that weak number sense and fluency underlies many difficulties students have with math (Geary, Bow-Thomas, & Yao, 1992).

Just to be clear, I don’t mean having students memorize their multiplication tables or race to answer questions the fastest.  I mean having them be fluent with numbers similar to the way we think about being fluent in a language.

Here’s a definition of fluency from NCTM’s Principles and Standards for School Mathematics: “Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently” (p. 152).

To me, that means I need to ask:

  • Can they work flexibly with numbers?  Can they decompose and recompose different numbers easily and in a variety of ways?
  • Can they use mental math to solve problems or do they always need to resort to pencil and paper and a traditional algorithm?
  • Do they have efficient ways to solve problems?

I’ve heard a lot of complaints recently about attempts to teach fluency to students–mostly related to complaints about the common core. Parents don’t understand why we are teaching students these new ways to add or subtract instead of the just showing them the traditional algorithm they learned in school.

The thing is that many of these strategies aren’t new. Students who have strong number sense and fluency have been developing these strategies on their own. What’s new is that we are now explicitly teaching all students these strategies. A parent who attended one of my workshops explained it nicely:

  • This makes more sense to me than to me than how I learned math.  I am an Engineer with 5+ years of calculus and I find the thought process to solve the problems the kids are working on is much closer to how I think, but I had to figure it out on my own.

I’ll talk more about some of the ways we can develop fluency in students, but if you want to read about it now, check out:

O’Loughlin, T. (2007). Using Research to Develop Computational Fluency in Young Mathematicians. Teaching Children Mathematics14(3), 132-138.

“But I learned it that way”: The case for student-invented strategies

Recently, I have been receiving a lot of push back from parents and teaches who don’t think it is worthwhile to expose students to methods other than traditional algorithms. They tell me how they memorized the procedures in math class and they don’t understand why their students or children can’t do the same. They complain about all these diagrams I encourage students to use and don’t understand why students should invent their own strategies to solve problems.

I held this belief when I first started teaching as well. However, I soon realized that my students weren’t learning math by memorizing procedures. Maybe they could do the problems in class that day, but ask them to do a mixed set of problems a couple weeks later and they were lost. When I began to encourage students to invent their own strategies, two things happened:  1. They got the problems right more often. 2. They started to like math better and become more confident.

Last week, I was working with 4th graders on the following problem:

Ms. Placa bought 15 packets of pencils.  Each packet had 12 pencils in it.  How many pencils did she buy?

Students were able to represent the problem with a diagram or explain why they knew they needed to multiply 15 times 12.  Awesome, right? Yes. Until they tried to multiply 15 times 12 using the traditional algorithm. Over half the class got the wrong answer.

I asked them whether 30 was a reasonable answer. They knew it wasn’t, but kept going back to the algorithm. Then, one student pointed to the picture she drew–15 boxes with the number 12 written in each one. She suggested counting by 2’s 15 times and then counting by 10’s 15 times and then adding them together. She could even explain to the class why that worked. Later, I learned she was one of the “weaker” math students in the class.

I tell the story because it is amazing what students can do when we allow them to invent strategies that make sense to them.  

The research supports my experience.  The benefits of student-invented strategies include:

1. Students make fewer errors than when using standard algorithms that they do not understand.

2. There is less of a need for remediation later on if students understand what they are doing and make connections on their own.

3. In many cases, using an invented strategy can be faster than using a standard algorithm.

Standard algorithms that are taught with meaning can have their place as well, but I think the case for allowing students to make sense of math needs to be made as well.

Want to know more? Check out:

Fuson, K.C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, WG Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 68-94). Reston, VA: NCTM

Gravemeijer, K., & van Galen, F. (2003). Facts and algorithms as products of students’ own
mathematical activity. In J. Kilpatrick, WG Martin, & D. Schifter (Eds.), A research companion to
principles and standards for school mathematics (pp. 114-122). Reston, VA: NCTM

Number Talks

In my last post, I talked about the importance of fostering mental math strategies in class.

A great resource for how to think about doing this in your class is Sherry Parrish’s book Number Talks: Helping Students Build Mental Math and Computation Strategies, Grades K-5.   I highly recommend taking some time to read the book over the summer.   It even includes a DVD with videos of model number talks.

So what is a number talk?

It’s a 10-15 minute conversation about a computation problem that is purposefully designed to allow students to communicate and justify their thinking.

What might it look like?

In the early grades, it might involve asking students for different pairs of numbers that add up to 10.

In the later grades, it might involve asking students for different ways to solve a multiplication problem like 25 x 16.

After sufficient wait time, students share their strategies, including justification for why a particular strategy works.

What I love about number talks is that they can easily be incorporated into a class.  Often, as teachers, we are bound to a particular curriculum that is beyond our control.  However, most of us can find 10 or 15 minutes to add in a number talk, even if it’s not every day.

I also love that they challenge different students.  Higher-level students can think of more complex solutions or multiple solutions while lower-level students can access the problem with a simpler solution.  And all students can benefit from hearing alternative methods.

Finally, I love that they allow for rich discussions to take place.  Students can talk about which solutions are mathematically correct or which may be more efficient.

Have you tried number talks in your class?  How have they worked?

Want to know more?

Check out this article:

Parrish, S. D. (2011). Number Talks Build Numerical Reasoning. Teaching Children’s Mathematics, 18(3), 198-206.

Mental Math

My sister gets nervous sometimes when we are at the store together.   She is afraid (rightfully so) that she may have to witness me give a math lesson at the cash register.   I can see why it might be embarrassing to her and I can also see why it infuriates people behind me in line, but sometimes I just can’t help myself.     As a math teacher, it frustrates me that the cashier can’t figure out the change when I give her a twenty, a one and a nickel for a bill that is $11.05.

Now, one could argue that the reason why the cashier can’t do this is because he or she did not receive enough instruction on the traditional algorithm for subtraction.  Maybe he or she used calculators too much in class instead of doing lots of “drill and kill” exercises.

But I don’t think that having lots and lots of practice with the traditional subtraction algorithm would help in this case.

The truth is, I don’t often use traditional algorithms when solving math problems in real life.  I do use a lot of mental math.

I count up or I count down when adding or subtracting.  I break numbers apart when multiplying.  I rarely think about multiplying 25 times 6 in my head using a traditional algorithm. Instead, I think of 20 x 6 plus 5 x 6 or I think about 25 x 4 plus 25 x 2.  It is just easier.  To figure out a 15% tip, I usually find 10% and then half it and add the two together.   Or most of the time I double it because waiters and waitresses really deserve that 20%.

I bet a lot of you operate the same way.  You don’t take out a pen and paper in the store or restaurant.

Yet as I mentioned in a previous post, some people still think that traditional algorithms are the only way to teach math.

As you may expect, I disagree.  How much more valuable is it to work flexibly with numbers? To be able to calculate things in your head quickly.  To have a sense of whether an answer is way off base.  I think this is the way we should approach math in the early grades.  It certainly would make my experiences at the grocery store less traumatic.

So what does the research say? Two researchers (Fosnot and Dolk, 2001) reported on a program called “Mathematics in the City” in which teachers fostered mental math computation strategies in children between the ages of 4 and 8.  They showed how these students developed a deep understanding of number and the operations of addition and subtraction.

Now is there a place for learning traditional algorithms?  I think there is.  But I think it’s often more valuable to do after students have had experience working flexibly with numbers.

So how do we begin to do this in our classrooms?    That I’ll talk about in my next post.  But feel free to write any suggestions you have in the comments below.

Want to know more?

Check out the Mathematics in the City site here.

Check out Fosnot and Dolk’s book: Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction.