Teaching Strategies: Worthwhile Mathematical Tasks Appropriate Use of Technology

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 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 transcript from teacher educator Fran Curcio In this problem, Janel has set the stage for further learning in mathematics because these tools - graphs, tables, and algebraic expressions... Read More

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

 Read transcript from teacher Tom Reardon Hopefully, I'm also bringing in good mathematical terminology and good mathematics to set up the problem and solve the problem... Read More

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:

 Read transcript from teacher educator Diane Briars One of the things that's notable in this lesson is the task itself, the problem with which Tom started... Read More

 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:

 Read transcript from teacher educator 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... Read More

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

 Read transcript from teacher educator 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 high-achieving countries... Read More

 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 transcript from teacher educator 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... Read More

 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?

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 teacherJanel Green I thought their presentations were great. We saw a few misconceptions that took place, which is great... Read More

Read what Fran Curcio says about Janel's careful use of language to communicate important mathematical ideas:

 Read transcript from teacher educator Fran Curcio Janel's attention to language helps students make the transition from equations to inequalities. Read More

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

 Read transcript from teacher educator Diane Briars How frequently do you need to go back and actually connect up to the real-world setting? Read More

 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 teacherTom 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."... Read More

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

 Read transcript from teacher Janel 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. Read More

 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?