# Rules that Expire

Why can’t we just teach them the rule? At what point do we stop this “exploration” and just give them the rule? When I was in school, the teacher just taught us the rule–we didn’t need to understand it.

I’ve talked before about why teaching math as a bunch of rules to memorize is a problem–even if you teach the rules in a really creative way or with a catchy song. If students don’t have any understanding of why rules work, they begin to think of math as this mysterious thing that doesn’t make sense. They mix rules up and constantly need a teacher or answer key to tell them if they picked the right rule. They lose the ability to make sense of things on their own. I often talk with students who can reason about a problem and get the right answer, but get it wrong because they are following a rule that they incorrectly memorized.

I recently read a great article “13 rules that expire” that talks about another problem—the fact that many of the rules we teach kids in elementary school don’t work anymore when students move to middle and high school math. For example, when I taught middle school, my students were often confused about multiplication of fractions. They kept telling me that “multiplication makes numbers bigger.”

I recommend taking the time to read the article. It’s is an important reminder to be precise with language when working with young students. It’s also a good read for middle and high school teachers. It gives some insight into the struggles students face when they try to reconcile the rules they may have learned with new experiences that break those rules.

I’m interested to hear what you think of it.

# Three things I’m loving this week.

September is off to a great start.  Here are some highlights from my week:

1. This activity:

I love that this task is accessible and challenging to different level learners.  Students who didn’t know their multiplication facts used drawing and square tiles to help them figure out the dimensions. Higher level students were able to tackle part C.

After watching Annie Fetter’s webinar about Math Forum’s Problems of the Week, I am hooked. Not only is it a great source of interesting tasks, the teacher resources for engaging students in solving problems are awesome!

3. Mystery number

Kids love solving puzzles.  At the start of the week, each student in the class picked a favorite number. Every day when the students entered the class, they listed different facts about their number. The day I visited they were working on factors. A student would share the factors for his or her mystery number and the class would have to guess the mystery number. I’ve never seen kids so excited about factors.

Unrelated to work, I am loving training in this fall weather for my next half-marathon. I am also loving that is pumpkin spice latte season (even though I’m a little sad to see iced coffee season leave)  What about you? What are you loving this week?

# Setting Routines that Build Number Sense

This was the first full week of school in New York City! As a result, I’ve spent a lot of time talking with teachers about routines. Veteran teachers know the importance of setting routines and new teachers quickly learn that without routines, the best planned lessons quickly fall apart.

While routines for classroom management are super important, they aren’t what I want to talk about. I’m more interested in setting routines that help develop mathematical thinking– especially number sense.

Counting circles are a perfect example of such a routine. Once you set the routine, the counting circle can be used in many different ways and in many different grades to develop number sense. I learned all about them from the one and only Sadie. Definitely read her post to learn more. I also highly recommend reading more about them in Number Sense Routines even if you don’t teach K-3.

The other book I’ve been loving is High Yield Routines. It’s an easy read and I got a lot of great ideas from it. Here’s one I’ve been playing with.

Today’s Number  Students are given a carefully chosen number and asked to list everything they know about it. Students then generate different representations of the number–drawings, equations, examples. Through discussion about the different representations, student can be pushed to think about different ways to decompose and recompose today’s number.

What I love about this is that it can be easily adapted to different grades. Today’s Number can be a whole number, a fraction, decimal, a negative number, an irrational number and so on… In addition, students can access the task at whatever level they are at. Some may have only one representation, others may have many that they can begin to compare and contrast.

For me, this would work well as a Do Now. Maybe I would do it once a week or maybe more at the beginning or the year. It can also be used as part of your homework and the discussion can take place at the start of class. I also think it might be neat to share with parents as an activity they can do at home with their kids. However you use it, I think it’s a great routine to start at the beginning of the year and track how students’ number sense builds throughout the year.

I’d love to know what you think.

What routines do you use that help students build number sense?

# Welcome back

I’m back. It’s been a while.

I took a break this summer–from blogging, from twitter, from workshops. I did a little writing, some for my dissertation and some for academic papers. I went to Vancouver and gave a talk at a conference. I visited Puerto Rico and drank pina coladas by the pool. I spent some time in the mountains in Utah, and at the beach in Cape Cod. It’s been a great couple of months.

And now it’s back to school. I always have mixed feelings when summer ends. It’s by far my favorite time of year. For many reasons. The beach and the pina coladas being a major one. Another is that it gives me the time and space to reflect on the past school year. What went well? What didn’t? What are my goals for this year? How do I do things better? Do I still want to do what I am doing?

Here’s what I realized this summer. The thing I love most about what I do is building relationships–with students and with teachers. It was hard for me to do that well last year because I was bouncing around to different schools all the time. So this year, I’ll only be working at three schools. I’ll be at each school at least once a week supporting the math teams. I’m excited to really be part of the teams instead of popping in for a visit here and there.

• Meaningful Feedback.  I had the chance to finally meet Michael Pershan and pick his brain about math mistakes over coffee. We talked about a lot about meaningful student feedback. What does it look like? How does written feedback differ from verbal feedback? What types of tasks or questions promote learning from one’s mistakes?
• Guided Math Groups and/or Centers. Most classrooms I work with contain children with a wide range of abilities. While there are a lot of great ways to use open ended tasks that students of all abilities can access, we also need to work with students at their level in small groups. I’m trying to figure out the best ways to make that work in both elementary school and middle school.

I’m excited to get back on twitter and read what all of you have been up to. It’s good to be back.

# 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.

# Teaching Time

I 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.

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

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.

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

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.

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.

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?