Number Bonds

Number_Bonds_1

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.

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

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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.

Teaching Time

rsz_img_runI started the race at 7:54 am and crossed the finish line at 10:27 am.  How long did it take me to run the race?

This problems, often labeled as elapsed time problems, are difficult for students to solve.  One reason is that students often have trouble coordinating between the two different units–hours and minutes.  In fact, in one study, only 58% of eighth-grade knew that 150 minutes is equal to 2 1/2 hours (Jones and Arbaugh 2004).

I recently witnessed this first hand in a third grade class. The students really struggled with these types of problems. Many students wanted to subtract 754 from 1027. I didn’t know what to do. So I went to the research and found some work by Juli Dixon that used open number lines to help students reason about these types of problems.

Open number lines can be used by students to count up, count down and find distances between numbers.

Here’s one way to use the open number line to solve the problem above.

rsz_time_line

Now, that’s not the first problem I had students begin with, but I wanted you to get a sense of how the open number line works.

Dixon also notes that you should allow students to use the number line in ways that make sense to them. You should not prescribe one way of using it to solve a problem, as there are multiple ways that work. Students can share the different strategies with the class.

As I’ve mentioned before, I’m all for diagrams that allow students to reason about problems. In this case, I love that the diagrams allow for students to record what is happening as it is very hard to keep track of all the steps. I’ll talk more about how I built up to this another time, but I’m curious to hear what you have done with your students.

Want to know more?  Check out the articles below:

Dixon, Juli K. “Tracking Time.” Teaching Children Mathematics, 19 (August 2008): 18-24

Jones, Dustin L., and Fran Arbaugh. “What Do Students Know about Time?” Mathematics Teaching in the Middle School 10 (September 2004): 82–84.

Monroe, Eula Ewing, Michelle P. Orme, and Lynnette B. Erickson. “Working Cotton: Toward an Understanding of Time.” Teaching Children Mathematics 8 (April 2002): 475–79.

Draw a Picture: Check out the webinar

Draw a Picture: Check out the webinar

In March, I did a session for Global Math Department on drawing diagrams. If you don’t know about Global Math Department, you should check it out. They present free video conferences and webinars every Tuesday night and you can join in and participate that night or you can watch the recording later if you can’t make it.  Here’s the recording of my talk.

It was an interesting experience for me. I find that whenever I have to write a blog post or prepare a presentation, I learn more about the topic. The process of synthesizing and preparing the message I want to deliver forces me to think about the content in different ways.

It was also a unique experience because it was a different type of interaction with the audience than I’m used to. When I do workshops, I can read the body language of the members and I can interact with them differently. My workshops are also much more interactive and I do far less talking than the participants. With the webinar I did, I could read the comments and interact that way but it was a different form of communication for me.

I’ve been thinking a lot about how to foster learning with teachers. What activities help them develop new ideas? When I look at my own learning, the experiences that help me are sometimes listening to talks or reading books or articles. But I think experiencing new ways of learning math and then trying these methods out with students help the most.

I’m really interested in how to create different learning experiences for teachers.  I’d love to hear any ideas you have or more about what learning experiences have been helpful for you.

Multiplying Fractions with Meaning

When I do workshops with parents, I often get complaints about why students don’t just memorize things.  It’s often followed by, “I learned math that way and I’m fine!”

I then ask them to solve a multiplication of fractions problems, say 4/5 x 2/3.  Inevitably, one person will tell me to draw “butterfly wings” and cross multiply. Another will tell me to find a common denominator, multiply the numerators and leave the denominators the same. Eventually, someone will say that you multiply the top and then multiply the bottom.  Don’t even ask about what happens when I ask them to explain why that procedure works.

I tell this story because most of us learned multiplication of fractions without any meaning. As a result, if we forget the memorized procedure, we don’t know how to reason about it.

I’ve been using the following type of problem to help learners begin to develop meaning for what it means to multiply a fraction by a fraction.

Ms. Placa made a tray of brownies.  She put icing on two-thirds of the pan.  She then put sprinkles on four-fifths of the brownies that had icing on them.  What fraction of the pan of brownies have sprinkles and icing on them?

Before students begin, I have them estimate whether the answer is going to be bigger or smaller than two-thirds. This leads to an interesting discussion and will help addresses a major misconception later on that students have about multiplication always making things bigger.

Then students draw pictures.  Here’s one possible sequence of drawings and student thinking:

1.  I’ll draw a pan and shade two-thirds with icing.

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2.  Now I’ll cut the iced brownies into fifths and put sprinkles on four-fifths of those.

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3. Hmmmm, I know I have 8 brownies with sprinkles and icing on them, but what size are they?  I can’t tell because all the brownies in the pan aren’t the same size. Oh, I have to make some more cuts to have equal sized brownies.   

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Now I know that the brownies are fifteenths and eight of them have sprinkles and icing on them so eight-fifteenths of the pan are brownies with sprinkles and icing. 

Eventually we can get to writing number sentences and to imagining what would happen with larger numbers. We can then start to generalize what rule would work for multiplying any two fractions. But I think starting with a picture and context provides a nice foundation for starting to think about multiplying fractions.

What are your thoughts?  How do you usually teach students to multiply fractions?

Want to know more?  The study below gives a more detailed progression of how this type of thinking was fostered and some of the background knowledge it requires.

Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267.

Using pictures as a tool to justify

One of the things that is particularly interesting to me right now is how we help young students develop the ability to justify why things work in math. Often, kids know the correct rule but have no idea why the rule works. When I ask them to explain why a rule works, they wind up just listing the steps of the procedure.

Part of developing a conceptual understanding in math is being able to anticipate what procedure to use and why that procedure works.  For example, when I am trying to convert a mixed number into an improper fraction, I don’t simply need to know that I multiply the denominator by the number of wholes and add the numerator.  I also need to know why that rule works and why it will give me the correct number of parts in the mixed number.

I’ve found that children (and even adults) can have a really hard time writing or even explaining out loud their justification for why something works.  They often say “I know why but I can’t explain it.”

Recently, I have noticed that drawing diagrams or pictures is one way to help them begin to justify.  It’s as if the diagram allows them a way to make explicit what they are doing when they perform a calculation.

For example, the other day I was working with a group of elementary school teachers on fractions.  They worked in groups to draw the pictures like the one below to show why the calculations they performed worked.

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It was an interesting experience for them because they had to think deeply about what it really means to say, multiply fractions, as opposed to just remembering the formula.  It also gave them something to refer to when explaining to the rest of the class.

I’m currently digging through the research on this, but I’m curious to hear what your experience has been. What do you do to help student justify in math?

Using Diagrams to Make Sense of Fraction Word Problems

I’ve been spending a lot of time in schools and classrooms these past two weeks using diagrams with students and teachers. As a result, I’ve been neglecting this blog but I’ve been learning so much. Hopefully, I’ll start making more time to write about what I’m doing.

Here’s the latest problem I’ve been tackling with students:

Each lap around the lake is 3 ¾ miles long. Nicora runs two-thirds of a lap and then stops.  How many miles did she run?

Some students have no idea how to get started. Others add the two fractions. Others convert the mixed number to an improper fraction and then multiply.  Of the ones who chose to multiply, many make mistakes–they convert the mixed number to an improper fraction incorrectly, they forget the rule for multiplying fractions or they forget how to convert the product to a mixed number.

Students with a strong understanding of fractions and good number sense could do the problem in their head. They thought about decomposing 3 ¾ into 3 parts and then taking two of them. The million dollar question is “why don’t the other students see the problem this way?”

In an effort to help them see the problem differently, this is how we worked on it together.

First, I asked them to tell me about the problem in their own words.  Was it about cupcakes?  Unicorns?  What is going on in the story?  I’ve found that many students need to be forced to think about what the problem is asking them to do because they are so used to circling numbers and randomly picking an operation to solve them.

I then asked them to draw a bar that shows how far a lap is.

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Next, I asked them to show how far two-thirds is.

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Finally, I asked them to figure out how many miles two-thirds of the lap was.  Some decomposed the three and then the three-fourths.  Others used guess and check.

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After we did a series of similar problems, I asked students to solve the problem without actually drawing the diagram. When they had to picture the diagram in their head, they started to develop shortcuts. One student told me that she could just divide the whole number into three and divide the fraction into three and then add them together to get the amount for one-third.  To find two-thirds, she thought about doubling the amount for one-third.

I wanted to share this example with you all because I think it shows how representations can help make explicit what may be obvious to other students. I also think it shows a nice progression of how students can use a representation to build a strategy to solve the problem.  How is this different than telling them the procedure?  Because the students build a process to solve the problem that makes sense to them.  Furthermore, they can always go back to the diagram if they forget the procedure.  So often, I find that students don’t forgot things because they are lazy, but rather because the procedure never made sense to them to begin with. My goal is to help students build on what makes sense to them, not to show them an algorithm that makes sense to me, but may not make sense to them yet.

Using diagrams to ease the transition to algebra

People sometimes feel compelled to tell me how much they hate math. When I ask when this started, many point to algebra (the rest say fractions, but we can talk about that another day).

We know that many students struggle when they get to algebra. Researchers have been studying how to ease this transition through “early algebra.” They found that simply introducing traditional algebra concepts at an early age isn’t the answer.

Instead, Blanton & Kaput (2005) talk about “algebrafying” the elementary school curriculum, in which elementary school concepts are developed in a way that allows generalizations about properties and relationships to become more explicit.  For example, students are asked to identify and generalize about patterns, relationships and structure in mathematics. They are given tasks that require them to reason about unknown quantities. At early ages, they develop the ability to identify, describe and analyze how quantities vary in relation to each other.

As you may have noticed, I’ve been obsessed with tape diagrams recently.  In fact, I’m doing a Global Math Department session about them tomorrow night if you want to hear more. One of the benefits I haven’t talked about yet is their potential for developing algebraic thinking.

Take a look at this problem.

Nicora wants to buy herself a new bicycle that costs $240.  She has already saved $32, but needs to make a plan so she can save the rest of the money she needs.  She decides to save the same amount of money each month for the next four months.    How much money must she save each month to meet her monthly goal of buying a bicycle?

Many of you probably solved it by setting up the equation $240= $32+ 4x and solving for x.

However, younger students could also solve it by setting up a tape diagram like the one below.

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After creating the diagram and using it to solve the problem, students can talk about writing a number sentence to represent the situation. This allows the to give meaning to each part of the equation. Students can discuss what the $240 represents, what $32 represents and what the missing number, x, represents. They can also discuss how they solved for the missing number. In this way, the tape diagram can be used to build developing algebraic thinking.

Want to know more?

Blanton, M., & Kaput, J. (2005). Characterizing a classroom practice that promotes         algebraic reasoning. Journal for Research in Mathematics Education, 412-446.

Using Tape Diagrams to Solve Division Problems

I’ve been working with K-5 teachers on how to use tape diagrams to solve problems. During one of these workshops, a first grader peeked in and asked, “Teachers have to go to school too?”  It got quite a laugh, but it’s true.  We need time to sit with our peers and try new ways to solve problems. I also think it reminds us of what it feels like to be a student and see things through a different lens.

One of the problems we worked on was:

Ms. Placa spends a total of $42 for 3 sweaters.  Each sweater costs the same amount. How much does each sweater cost?

Most of the groups solved this by using long division.

But let’s say you have students who don’t remember the long division algorithm (most likely because it never it held any meaning to them).  If they have been working with a tape diagram or bar models, they can solve it using that.

First they can draw a bar and split it into three equal boxes to represent the 3 sweaters.

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Then they can try guessing in order to determine what goes in the box. You want to work with them on using efficient strategies for guess and check. For example, 10′s are easy to count by, so they can start with 10s.

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They can see that they still need 12 more.  At this point, some students may know that they can divide 12 by 3 and it will give them 4.  If not, they can count by 2′s and then by 2′s again.

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When they add what is in each box, they get to $14 for each sweater.

The tape diagrams may seem strange to us at first if we haven’t used them before.  We need to learn how to use them just like students need to learn how to use them in a meaningful way. However, once they do, the bar models begin to make sense to them. Later, the models can be used to build to the algorithm so that the algorithm will have meaning for them.

Research shows that students can do well when they use diagrams to solve problems. However, the diagram alone does not hold magical powers.  While simply putting a tape diagram alongside a word problem aided higher ability middle school students, it did not seem to aid lower level students who received no instruction on how to use the diagram. As with any model or representation, students need guidance and opportunities to foster their understanding of it.

Want to know more? Check out:

Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14(1), 42-46.

Booth, J. L., & Koedinger, K. R. (2012). Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving. British Journal of Educational Psychology, 82(3), 492-511.

Yancey, A. V., Thompson, C. S., & Yancey, J. S. (1989). Children must learn to draw diagrams. The Arithmetic Teacher, 36(7), 15-19.

Using tape diagrams to solve fraction problems

Read the following problem and think about how a student might solve it.

There are 250 students in the fifth grade.  Three-fifths of them speak two languages.  How many speak two languages?

As I’ve mentioned before, I’ve been introducing tape diagrams, or bar models, to students and teachers recently as a way to solve word problems. These models have been popular in Singapore and Japan and are slowly working their way into some US programs. Essentially, the models use bars of different lengths to represent the size and relationships between different quantities.

Students who worked with tape diagrams previously solved the problem in the following way:

First, they drew a bar to represent the 250 students.

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Next, they partitioned it into 5 equal parts to represent the fifths.

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Then, they tried to figure out what number would go in each box. What was interesting at this stage was that students used different strategies depending on their abilities. Students who were fluent with division immediately divided 250 by 5. Students who were less comfortable with division tried to find what number times 5 would equal 250. They used guess and check until they arrived at 50. Other students used repeated addition and tried to find what number could be added five times to equal 250.

They then added the number to the diagram:

Slide1

Finally they either multiplied 50 times 3 or added 50+50+50 until they arrived at the answer.

What’s neat about using this model is that students can approach the problem at different levels. Lower level students can use guess and check or repeated addition until they build up their multiplication and division skills.

Furthermore, I think it helps students make sense of the problem and the mathematical operations they are using to solve it. After enough practice drawing the models, students can be asked to visualize what they would draw in their head.  This will help them to focus on what operations they are performing. It can also help them generalize about what they are doing to solve these types of problems.

Eventually, students will be able to solve the problems without drawing the models. However, starting with the models allows them to develop meaning for why they divide the set by the denominator and multiply by the numerator.

I’ll talk more about what the research says about these models next time, but I’d love to hear what your experience has been.