## Insights Into Algebra 1: Teaching for Learning

# Linear Functions and Inequalities Teaching Strategies: Worthwhile Mathematical Tasks

The National Council of Teachers of Mathematics (NCTM) has defined worthwhile mathematical tasks as those that:

- Are based on sound and significant mathematics
- Use knowledge of students’ understandings, interests, and experiences
- Develop students’ mathematical understandings and skills
- Stimulate students to make connections and develop a coherent framework for mathematical ideas
- Promote communication about mathematics
- Promote the development of all students’ dispositions to do mathematics.

(Source: NCTM, *Professional Standards for Teaching Mathematics*, 1991, p. 25)

The Workshop 2 videos present examples of how teachers can use worthwhile tasks effectively in the algebra classroom, even with students who don’t consider themselves interested in mathematics. In* Principles and Standards for School Mathematics* (*PSSM*), NCTM states:

In effective teaching, worthwhile mathematical tasks are used to introduce important mathematical ideas and to engage and challenge students intellectually. Well-chosen tasks can pique students’ curiosity and draw them into mathematics … worthwhile tasks should be intriguing, with a level of challenge that invites speculation and hard work. Such tasks often can be approached in more than one way … Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge.

… When challenged with appropriately chosen tasks, students become confident in their ability to tackle difficult problems, eager to figure things out on their own, flexible in exploring mathematical ideas and trying alternative solution paths, and willing to persevere … When students work hard to solve a difficult problem or to understand a complex idea, they experience a very special feeling of accomplishment, which in turn leads to a willingness to continue and extend their engagement with mathematics.

(PSSM, 2000, p. 18)

### Sound and Significant Mathematics

Tom Reardon’s lesson demonstrated the use of sound and significant mathematics because he chose an example that required an in-depth investigation of linear functions. The data points in the problem were all part of a mathematical formula that AT&T used to determine the cost of a long distance phone call. Tom was able to have the students choose two points, write an equation, predict values using the equation, and understand the real-world meaning of the slope and y-intercept based on the data. Through the use of one problem, Tom was able to help students reinforce their understanding of all the important concepts of linear functions.

Janel Green’s lesson demonstrated the use of sound and significant mathematics because she chose a context with which the students were familiar to help them understand the important connection between solving equations and inequalities numerically, graphically, and algebraically. While Janel’s context seemed to be relatively simple, the mathematical ideas the students discovered were powerful, abstract, and quite difficult. Both teachers chose their lessons carefully and with their mathematical goals in mind.

Read what Fran Curcio has to say about this mathematical task and how this lesson can be used again and again in the students’ study of mathematics:

**Transcript from Fran Curcio**

In this problem, Janel has set the stage for further learning in mathematics because these tools – graphs, tables, and algebraic expressions – are not just limited to linear equations. Further in their studies of mathematics, [students] will encounter these forms of representation and be able to apply what they’ve learned in these lessons.

See what Tom Reardon has to say about selecting a task that contains sound and significant mathematics:

**Transcript from Tom Reardon **

Hopefully, I’m also bringing in good mathematical terminology and good mathematics to set up the problem and solve the problem. We say we’re solving equations, but sometimes just to set the problem up and translate it from words to symbols is a big step and that’s something we’re trying to do here. So the big ideas – in fact it was part of their assignment at the end to write down some of the big ideas – are to review those ideas of slope and how you find it, how do you find the y-intercept, and what does it mean? How do you graph, how do you look at a table, how can you solve equations using graph tables, function notations? Those things were included in today’s lesson. Multiple representations of a problem situation is probably the ultimate thing I tried to get across today.

Diane Briars adds more comments about the importance of the task Tom Reardon selected and how it can help students see the relevance and importance of mathematics:

**Transcript from Diane Briars**

One of the things that’s notable in this lesson is the task itself, the problem Tom started with. He brought in real-world data, so he was able to get some data analysis ideas into the lesson along with the fundamental algebraic goal of the lesson, which was how do we create a linear equation given two points? Students had examples of some real data points [and] got to talk about representing those points in different ways. They did multiple representations: They started out with the table, and then they did the graph. It’s also notable that later on in the lesson, the students were talking about the slope and they [understood that the increase of $0.24 per minute was the reason for the slope of the line.]

**Reflection:**

Think about a mathematical task that your students engage in and reflect on how it helps them come to understand sound and significant mathematics.

### Knowledge of Students' Understandings, Interests, and Experiences

A worthwhile mathematical task builds on students’ understanding of concepts, interests them, feels familiar to them, and has mathematical significance. The familiar setting helps them focus on the underlying mathematical concepts. It’s very important to choose a task that directly relates to the concepts and procedures you want to teach. For example, Tom knew his students would be curious to see how phone bill charges are calculated and how the charges can vary depending on time and day. But his underlying goal for the problem was to build on what his students already knew about linear functions and help them deepen their understanding of slope, y-intercept, predicting unknown values, and understanding graphs and tables.

Janel asked a student to model the football jersey that the hot dog sales would help purchase for the team. She built on this familiarity and interest to lead the students to the understanding that problems can be solved in multiple ways, using tables, graphs, and algebra.

Read what Fran Curcio has to say about how Janel used her knowledge of her students to present a problem that they could successfully explore:

**Transcript from Fran Curcio**

In this particular setting, where there are students of a variety of learning abilities and levels of preparedness, Janel has provided a problem that allows for multiple entries and multiple exits. Multiple entries allow students to use what they know, build on their experiences, and apply it to the problem. How they mix all of this together and work together allows them to come up with a solution to the problem and walk away from the problem knowing something that they didn’t know before.

Diane Briars talks about the importance of embedding skill-building in lessons that focus on conceptual understanding:

**Transcript from Diane Briars**

I think what’s interesting about the new video study [Third International Mathematics and Science Study Repeat (TIMSS-R)] is finding the range of pedagogical models that are used in other highachieving countries. Japan was the only country that really had students engage in one big problem for most of the lesson. Other countries had effective pedagogical styles in which students were engaged in multiple problems. Sometimes they were realworld settings, sometimes they weren’t. What seemed to make the real difference, or what set the U.S. apart from high-achieving countries, is that regardless of the set-up of the problem at the beginning, U.S. teachers tended to focus on the procedures in the lesson. Even if the lesson seemed like it was going to be conceptual, the U.S. teachers tended to turn lessons into procedure lessons: the big focus was [on] how to do it. In the other countries, there was much more emphasis on the conceptual underpinnings. What I think you see in this video of Tom’s lesson is someone who is trying to get at the concepts, so it was not strictly a procedural lesson and didn’t turn into a procedural lesson. It really was to get at the concepts as well as the procedures.

**Reflection:**

Both Tom and Janel chose tasks that they thought would build on students’ understanding, interests, and experiences. Describe how a task that you use in your classroom achieves this goal.

### Stimulating Students to Make Connections

Students develop a framework for mathematical ideas when they model a situation in a variety of ways and then make connections between the different methods. Teachers should deliberately select tasks that provide windows into student thinking so they can see whether this is happening, especially in areas where students tend to have misconceptions. Tom’s lesson exemplified this when he chose a set of five different data points that all lie on the same line. He was able to engage his students in a discussion about which data points to select and whether or not it mattered if they chose different data points. This exploration allowed students to conclude that they could select any two points on the line and calculate the same equation.

Read what Diane Briars has to say about this aspect of Tom’s lesson:

**Transcript from Diane Briars**

[The students] weren’t quite sure that if they had picked two different data points they would actually get the same equation, even though those two data points were on the same line. That the students were unsure surprised me, but they were able to go through and try it out, say, “Oh yeah, of course, they are on the same line, it would make sense,” and really get that it’s two points that determine a line, and any other points that you have on that same line are going to give you the same equation. I thought that was a good question for him to ask, and, clearly, it was productive. It may seem obvious to those of us who are very familiar with it, but the students really weren’t sure. They had to do that kind of exploration themselves.

**Reflection:**

Describe a mathematical task that you use that helps students make connections and develop a framework for their understanding of mathematical ideas. Why is it effective?

### Promoting Communication About Mathematics

Both Janel and Tom worked hard to show their students that they valued their ideas and expected their students to communicate clearly about the mathematics. Janel built her lesson around groups discussing and coming to agreement on how to solve unfamiliar problems. She talked to her students about different methods for solving equations but wanted to see if they could reason about using that information to help solve inequalities. Her high expectations of her students were rewarded when the students were able to talk to each other and figure out how inequalities could be solved using tables, graphs, and algebra. The students also demonstrated their ability to communicate mathematical ideas when they presented their ideas to the class. They were comfortable talking in front of their peers and sharing the ideas that they learned from the activity.

Listen to Janel’s reflection on her students’ ability to communicate the mathematical ideas they were learning:

Listen to audio clip of teacher Janel Green |

**Transcript from Jenel Green**

I thought their presentations were great. We saw a few misconceptions that took place, which is great. Any time you see a mistake, it’s an opportunity to correct it and to help someone who doesn’t understand mathematics understand it better … It’s very important for students to summarize key concepts, because they have to understand what they learned. If they can’t summarize, they didn’t learn anything. It’s good when they are able to sum it all up for themselves and verbalize it, it has to be verbalized because when you are able to verbalize, you understand a little bit better.

Read Diane Briars‘ thoughts about different ways to make sure students make meaning of the mathematics while they’re working through a problem:

**Transcript from Diane Briars**

How frequently do you need to go back and actually connect up to the real-world setting? In [Tom Reardon’s] lesson, there was an initial real-world launch, then there was work with the mathematics. At the beginning, it was in pure mathematics, and then about 15 minutes into the lesson, there was another connection back to the real-world setting … [But] every time we talk about the slope of 24/100

^{ths}, should we have said that was 24 cents? Should we have taken the y-intercept of 85/^{ths}and reminded students that it was 85 cents? How often do you need to make those connections? Should students be working in the mathematics for a while and then come back and interpret in the real-world setting?

**Reflection:**

Share some tasks that you use in your classroom that promote mathematical communication by your students. Why do you think this is important?

### Improving Students' Disposition Toward Mathematics

Selecting worthwhile mathematical tasks should also convey messages about what mathematics is, and what doing mathematics entails. Tasks that require students to reason and to communicate mathematically are more likely to promote their ability to solve problems and to make connections. Such tasks can illuminate mathematics as an intriguing and worthwhile domain of inquiry. A central responsibility of teachers is to select and develop worthwhile tasks and materials that create opportunities for students to develop these kinds of mathematical understandings, competence, interests, and dispositions.

(Professional Standards for Teaching Mathematics, 1991, p. 24)

Listen to what Tom says about how the Phone Bill Problem can improve students’ disposition toward mathematics:

Listen to audio clip of teacher Tom Reardon |

**Transcript from Tom Reardon**

One of the other things I think helps the students see that I have high-level expectations is the problems we do are a little more substantial than just “solve this equation,” “simplify this expression.” My goal as a teacher is to find really, really good, what I call, cumulative problems or problems that you can take at the end and say, “This brings together all these ideas of mathematics.” If I can get those problems, that’s great, and if I can get those kids to do them, that’s great. I communicate to them that I think you can do them. I think you’re worth my trying to get you to learn how to do this. And I think it’s important that you can do this mathematics, so that maybe you’ll think it’s important to do this mathematics. And some of them buy into it, and some of them are just kind of along for the ride, but that doesn’t mean you stop trying.

Improving student disposition toward mathematics is also an important goal for Janel. Read what she has to say:

**Transcript from Jenel Green **

This was the first time they have ever made a connection between the three methods, and I think they really appreciated the power of mathematics today. So often, we hear people talk about how they hate math, and kids come in constantly with that negative attitude. But there’s nothing better than having a student come in my class in the beginning of the year hating it and then leaving thanking me and having a great time and saying, “You know what, Mrs. Green, math is not that bad.” There’s nothing more inspirational than that.

**Reflection:**

Describe how you help improve your students’ disposition to do mathematics. What types of tasks help students make sense of mathematics and value it as important to their lives?