The SERP-SFUSD partnership work in mathematics was made possible by the generous support of the S.D. Bechtel Jr. Foundation.

The development of this website was made possible by a contribution from the Intel Foundation.

### The Team Considers the Use of Problems in Middle School Mathematics

Over the years of our work together, we noticed that while we all use problems in our mathematics teaching, we did not share a vision as to why. So together we decided that there were three important questions to consider as we use word problems.

1. Is the intention of the word problem to develop general problem-solving proficiency, as with a puzzle? If so, how might sense-making and multiple representations be made public in the class setting?

2. Is the problem designed intended to shed light on a specific concept? If so, what might we do if students use a simpler approach to get the correct answer instead of more advanced grade-level mathematics we were eager for them to use?

3. Does the problem include specific vocabulary that has a specific mathematical function (such as rate)? And equally important, is the general academic language in the problem (such a "evaluate" or "partial") understood by the students?

With these questions in mind, we have assembled resources that we think would have been most helpful for us in tackling related teaching challenges. We hope that you benefit from the information as well.

• Moving Beyond
• Multiple Representations
to Generalize Mathematics
• Strategic Use of
Diagrams

There is nothing "wrong" with focusing on the solution of a math problem. But we found that we shared a concern about the truly haphazard approaches among students we often witness. One colleague referred to this as "mashing numbers." Typically, students would try to quickly find the numbers embedded in the word problem and then run through various familiar operations to see what they get. They don't take time to think about the premise of the problem, to consider the question being asked of them, and to thoughtfully decide on an approach to solve.

In order to address this shared concern, we began think of problems in three parts: stem, question, and solution.

What we found especially helpful about looking at word problems as three-part entities was that it enabled us to parse them so that students could be challenged to conceptualize the question themselves. For instance:

What we found especially helpful about looking at word problems as three-part entities is that it enabled us to parse them so that students could be challenged to conceptualize the question themselves. For instance:

Another approach we tried was having students examine solutions to get at what is the mathematical question at hand might be:

 Go deeper with video: See a student work through a "dragonfly problem" similar to the one above. Visit Alison Oliver's San Francisco classroom to see two students work through the approach described below.

### One Problem Many Ways

What is the purpose of this approach?
Students learn how to make sense of word problems by matching or generating various representations of problems. The different representations of a problem might be an equation, table, diagram, or a graph. By systematically making connections between these different forms throughout the year, students will better understand the situations of word problems and will continue to develop the underlying mathematics. As students learn to write word problems they will encounter the related structure, language, and text, thus making such problems easier to solve.

How will the students achieve this?
Students are given one to several different representations (word problem, equation, table, graph/diagram) and they have to produce the rest. The number of representations provided by the teacher will vary based on the readiness of the students.

So, you guessed it. Back to our friend the dragonfly. Examine the various ways that this problem can be represented by students. (Click on any portion to enlarge)

 Now consider the many options we have as teachers to have students reason through the meanings of these representations by providing some, but not all, the representations above and asking them to provide the rest. The marked up versions below show the versatility of the worksheet. Our group found that asking students to use the familiar format for a challenging thinking task (such as composing a stem and question from an equation) helped them a great deal. Finally, here is an example of making a specific mathematical point when using multiple representations. For example, imagine you are teaching the concept of unit rate to your class. From the same activity as shown above, you have four helpful examples with which the students are already very familiar.

### Strategic Use of Diagrams

A diagram of a problem situation in mathematics shows where the numbers come from. Numbers can be known, unknown, or variables (sets of numbers). For example:

We are pouring water into a water tank.  5/6 liter of water is being poured every 2/3 minute.  How many liters of water will have been poured after one minute?

Where are the numbers going to come from? Not from “water tanks.” You change to gas tanks, swimming pools, or catfish ponds without changing the meaning of the word problem. All of the numbers referred to (given, implied or asked about) come from:

• The number of liters poured
• The number of minutes spent pouring
• The rate of pouring (which relates liters to minutes)

A diagram should show where each of these numbers comes from. We need a way to show liters and a way to show minutes. Examples of diagrams for this situation are below.

The examples range in abstractness. The least abstract strategy is not a good reasoning tool because it fails to show where the numbers come from. The more abstract examples are easier to reason with, if the student can make sense of them. Our goal is to teach students to make sense of, produce and reason with abstract diagrams that show all the numbers, their relationships. A good practice is to first make a more concrete diagram in early sense-making, then revise it to a more abstract diagram for reasoning purposes. Click images to enlarge.

Level 1:
A picture that shows the objects of the context without showing where the numbers come from (thus, this is not useful diagram for a mathematics class). A clock is a poor representation of elapsed time, as it only shows current time. The problem is about elapsed time.

Level 2:
A diagram that shows the numbers as properties of pictures. The relationship between time and liters is not shown. A timeline is much better than a clock for elapsed time.

Level 3:
A diagram that lines up the liters with the minutes in a valid way. The tank still looks like a tank. Students might draw a diagram like this and invalidly line up the 1 mark on the liters with the 1 mark on the timeline. If they don’t line anything up, they are not ‘relating’ the variables in their reasoning. The picture of the tank makes it hard to see "0" liters and so hard to realize they line up.

Level 4:
A diagram that abstracts both variables to a single number line. This makes reasoning about the relationship much easier. Another version of this is two parallel lines lined up correctly (see number strips from Singapore). In either version, these are number lines.

Why the "1s" don't line up:

Each number line has a "1", 1 liter and 1 minute. It is valuable for students to understand why the 1 on the liter line the "1" on the second line. Why are the 1s where they are? The two lines are constructed based on the given relationship: 5/6 of 1 liter in 2/3 of 1 minute. 5/6 liter lines up with 2/3 minutes. "0" liters lines up with "0" minutes. The point on the liter line that lines up with the 1 minute shows how many liters per minute, the "unit rate".

Level 5:
Coordinate graph shows even more about the relationship between the variables. It shows the rate as the slope of the graph. This is the most mature diagram and a set of mathematical proficiencies and knowledge in its own right. We need to teach kids how to reason with graphs, not just interpret them. Use the graph to answer questions like: How many liters per minute? How long to pour 10 liters?