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.

mult1

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

mult2

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.   

mult3

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.

IMG_0629

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.

Slide1

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

Slide2

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.

Slide3

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.

Slide1

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.

div1

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.

div2

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.

div3

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.

model1

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

model2

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.

“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

Solving Multi-Step Word Problems

Recently, I was working with fourth-graders on the following word problem from the 2013 New York State test:

Ms. Turner drove 825 miles in March. She drove 3 times as many miles in March as she did in January. She drove 4 times as many miles in February as she did in January. What was the total number of miles Ms. Turner drove in February?

Let’s put aside for a minute what we think about all of this testing or the common core state standards. Let’s ignore the issue of whether or not this is a “real-world” problem or whether solving these types of problems will make students career or college ready.

Instead, let’s pretend our goal is to help students solve these types of multi-step word problems. What would you do?

When I pose that question to teachers, I often hear them tell me how they would break down the steps and explain each one to the student. That’s one strategy, but I’m going to guess that if the student sees a problem that is a little bit different, they are going to be at a loss as to what to do.

One thing I try to do when I am approaching problems like this with students is to make the numbers a little friendlier at first so I can see if the students are getting stuck because they don’t know how to think about the problem or because they are struggling with the computation. In this case, I changed 825 to 900.

Another thing I do is to encourage them to use diagrams. This problem is a great example of why I think bar models are a good tool for students.

Here’s how a student used a bar model to solve this problem.

rsz_1img_0585 barmodel2

I think that’s a pretty neat way to solve the problem. Of course, drawing and using this type of diagram is not spontaneous. Students need to have exposure to it and understand how to use it. But once they have it as a tool, they can use it to model a variety of situations.

We can talk more about how to introduce these types of models at another time and we can also talk about what the limitations are, but first I’d love to know how you would approach this problem with your students.

One Strategy for Attacking Word Problems

They know the math but they can’t do word problems. 

They can do all the problems on the page correctly, but they skip the word problems at the end. 

It’s a reading comprehension problem. They don’t understand what they are reading. 

The dreaded word problem. In many classes I taught, students weren’t used to being asked to read in math class and they certainly weren’t used to thinking about what they were supposed to do. Math was about following a set of procedures and getting the right answer. Math wasn’t about thinking about what types of calculations might help you make sense of a situation.

Early in my career, I fell into the trap of teaching them to circle the key words.  That became a problem very quickly.

Then I tried teaching them Polya’s four step model: 1. Understand the problem 2. Make a plan 3. Carry out the plan 4. Look back

This didn’t help either.  If students managed to understand the problem and explain it in their own words, they didn’t know how to make a plan.

What finally worked, particularly with my struggling students, was encouraging them to use pictures. At first, students created elaborate drawings of the situations that often focused on superficial aspects of the story. Then, I guided them to use simple diagrams to represent the story– diagrams that helped organize the information given and what was missing. I found that after a bit of practice, students were able to solve a variety of word problems, including complicated multi-step ones, with a picture.

Recently, I’ve been working with teachers and students on drawing diagrams to solve word problem and I figured it would make sense to go to the research and see if there was evidence that supported my experiences.  It turns out there is.

The benefits of using diagrams for story problems include:

  • Reducing the memory demands
  • Assisting in unpacking the situation
  • Helping identify important information
  • Focusing students on the quantities involved in the situation and the relationships between them
  •  Flexibility  Diagrams can be used across grade levels and for solving both routine and non-routine problems

It is important to mention that these studies don’t encourage the use of any pictorial representation, but rather ones that highlight the structure of the problem. Bar models or part-whole diagrams are some of the representations I’ve found successful.

I’ll talk more about these particular models later, but I’d love to know about your experience with word problems. What have you found to be successful?

Want to know more? Check out:

Diezmann, C., and L. English. (2001). Promoting the use of diagrams as tools for thinking. In A.A. Cuoco and F. R. Curcio (Eds.), The role of representation in school mathematics. Reston, VA: National Council of Teachers of Mathematics, 77-89.

Yancey, A. V., C. S. Thompson, and J. S. Yancey. (1989). Children must learn to draw diagrams. Arithmetic Teacher, 36 (7), 15–23.

Teaching Mathematics to Middle School Students with Learning Difficulties by Marjorie Montague and Asha K. Jitendra  

Asking questions that help students think algebraically

Over the past two weeks, I’ve been working with both students and teachers on algebra tasks. One of the things we have been doing is writing rules for different geometric patterns.

For me, identifying key questions that will guide students towards meeting the learning target for the day is one of the most important things I do when planning a lesson.

I find that it can be easy to do really cool activities that get students engaged and have great potential but if the questions we ask don’t pull out the mathematics, kids get excited about math class but don’t learn much. The questions asked as the students work need to guide them to see the connections and pull the mathematics out of the situation (which may be obvious to us but not to them).

I think sometimes the resistance to inquiry based approaches is because if students aren’t carefully guided to see the mathematical connections, it’s no better (and maybe even worse) than simply telling them a procedure and having them practice a bunch of problems.

When working on my algebra lesson, I went to researcher Mark Driscoll’s book, Fostering Algebraic Thinking: A guide for teachers, Grades 6-10. In it, there are various questions that can be used to foster different types of algebraic reasoning. Below are some of the questions he suggests using when students are engaged in writing rules to represent functions:

Questions for building rules to represent functions

  • How are things changing?
  • Is there information here that lets me predict what’s going to happen?
  • What steps am I doing over and over?
  • How can I describe the steps without using specific inputs?
  • What if I do the same thing with different numbers? What still holds true?  What changes?
  • Can I write down a rule that will do this job once and for all?
  • Why does the rule work the way it does?
  • Does my rule work for all cases?
  • Now that I have an equation, how do the numbers in the equation relate to the problem context?

I used these questions to help assist small groups when they were stuck as well as to guide the whole class discussion. They helped students see the connection between the pattern they were drawing and the algebraic rule that could be written to describe it.

Want to know more?  Check out: Fostering Algebraic Thinking: A guide for teachers, Grades 6-10 by Mark Driscoll