"There is no other decision that teachers make that has a greater impact on students' opportunity to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages the students in studying mathematics."
(Lappan and Briars 1995, p. 138)

Designing or selecting worthwhile mathematical tasks is essential to creating an effective mathematics classroom. It is through tasks that the curriculum and the discipline of mathematics comes alive. Starting with students' prior knowledge and then creating rich, mathematical tasks, teachers help "students to proceed gradually from their informal knowledge of the ideas in the domain to more formal notions" (Romberg and Kaput 1999, p. 9).

When designing tasks, researchers Romberg and Kaput ask teachers to consider five questions:

• Do the tasks lead anywhere?
• Do the tasks lead to model building?
• Do the tasks lead to inquiry and justification?
• Do the tasks involve flexible use of technology?
• Are the tasks relevant to students?
  (pp. 9–12)

Reproduced from Thomas A. Romberg and James J. Kaput, "Mathematics Worth Teaching, Mathematics Worth Understanding," in Elizabeth Fennema and Thomas A. Romberg (eds.), Mathematics Classrooms that Promote Understanding (Mahwah, NJ: Lawrence Erlbaum Associates, 1999), pp. 9, 10, 11, 12. 

Through tasks, students come to see that mathematics is a network of intriguing ideas. This means tasks should be created or selected that encourage students to "wonder why things are, to inquire, to search for solutions and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas and questions for students" (Hiebert et al. 1996, p. 12).

Provide Time for Students
Teaching through tasks requires time. Students must have the time to develop understanding of worthy mathematical ideas. They must be provided time to:

• construct relationships
• extend and apply mathematical knowledge
• explore how ideas are connected
• find alternate solutions
• justify their thinking
• extend the task or problem
• create generalizations
• reflect on the experience
• communicate what they have done and what they understand to others
• make the mathematical knowledge their own.

It also means that students are provided with "penetrating techniques of thought that [can be used] to solve problems, analyze situations and sharpen the way [they] look at [the] world" (Burger and Starbird 2005, p. xi). When students work in this way with mathematics, new tasks or problems arise from their inquiry, helping them to see mathematics as a dynamic, living discipline that requires them as much as they require it. This is the heart of an effective mathematics classroom.

Setting a Context
Mathematical tasks and problems should be situated within a context that both engages the learners and yields insights into important, worthwhile mathematics. A story, current controversy, compelling mathematical terrain or historical account could be used to set the context.

Personal Stories
The Grade 2 students arrived back at Red Deer Lake School after a snow day and wanted to know how the school fan-out system worked. When did the principal get phoned in the morning? Who did he phone? How long did it take to phone every student in the school? Were there different ways to organize a fan-out system for a school? Which fan-out system took the least amount of time to reach all students?

Seizing the opportunity, the teacher turned students' questions and wonderings back to them. They spent a glorious morning exploring various fan-out systems, inventing their own, debating the merits of each and deciding on the most effective and efficient fan-out system for a school.

Moments like this happen in every classroom. These opportunities to turn students' questions and wonderings into a worthy, mathematical exploration await teachers who are attuned to the curriculum, the subject discipline and their students.

Traditional and cultural personal stories can also provide a context for mathematical inquiry. Sticks is an example of this.

Literature
If the World Were a Village: A Book about the World's People by David J. Smith provides a wonderful opportunity for students to grasp various features of the world's population. By shrinking the world's population to a village of 100, students have the opportunity to understand better who we are, how we live and how fast we are growing. It provides an opening to explore various number concepts, such as fractions, decimal fractions, percent, proportion, scale and ratio, as they come to understand the world's people through number.

• Anno, Mitsumasa and Tsuyoshi Mori. Anno's Three Little Pigs. London, UK: The Bodley Head, 1986.
• Burns, Marilyn. The Greedy Triangle. New York, NY: Scholastic, 1994.
• Clement, Rod. Counting on Frank. Milwaukee, WI: G. Stevens Children's Books, 1991.
• Enzensberger, Hans Magnus. The Number Devil: A Mathematical Adventure.New York, NY: Henry Holt, 1998.
• Schwartz, David M. How Much Is a Million? New York, NY: Lothrop, Lee & Shepard Books, 1985.
• Scieszka, Jon and Lane Smith. Math Curse. New York, NY: Viking, 1995.
• Tahan, Malba. The Man Who Counted: A Collection of Mathematical Adventures. New York, NY: Norton, 1993.

Using literature, historical accounts, personal stories and current controversies to open a robust inquiry into a worthy mathematical topic provides a common context and gives all students a common experience.

Current Controversies
From the UN millennium goals, what would it take to reduce by half the proportion of people living on less than a dollar a day?

How might you conduct a count of a species like the polar bear (or another animal that is always moving) to determine whether its population is decreasing?

Is the quality of your local water improving? Collect evidence over time to determine whether this is the case.

Compelling Mathematical Terrain
What do leaf arrangements, pine cones, sunflowers, daisies, drone bees and rabbits all have in common? Fibonacci numbers. An exploration into numbers and nature opens a rich mathematical terrain.

Order and pattern have a particular soothing effect on humans. We are drawn to the symmetrical. From patterns in floor tiles and wall coverings to First Nations beadwork and butterfly wings, symmetry can be found almost everywhere. An inquiry into symmetry through decomposing figures into their congruent parts yields fascinating insight into symmetry.

Historical Accounts
Some intriguing mathematical investigations begin with a historical account.

•	Queen Dido Uses Her Wits Forced to leave her home in Italy, Queen Dido arrived in Northern Africa in an area that became known as Carthage or modern-day Tunisia. She needed to purchase some land for herself and her servants. She went to King Jambas and convinced him that all she needed was the amount of land she could enclose with a bull's hide. The land she enclosed became the city of Carthage. How is this possible?
•	Descartes' Fly Folklore has it that one day as René Descartes lay in bed, he watched a fly crawl along the ceiling. As he lay there, a flash on analytic geometry struck him. He wondered whether he might be able to relay the position and the path of the fly to someone who wasn't there to watch the fly. This would require that he be able to determine the relation connecting the fly's distances from two adjacent walls. Descartes is the creator of the Cartesian plane and analytic geometry.
•	Penalty for Divulging Pythagoras was born around 560 B.C.E. During that time, it was fashionable for Greek men to wear long robes; however, Pythagoras preferred to wear the Persian fashion of trousers. He is known for forming a secret community of mathematicians. One day while on a ship, one of Pythagoras’s followers revealed to an outsider that  was not a rational number. As penalty, he was thrown overboard.

A helpful guideline for designing worthwhile mathematical tasks entitled Standards and Scoring
Criteria for Mathematics Task, is located at http://qhs.humble.k12.tx.us/download/5.2.1_resource_r
ubrics/Mathematics Tasks Criteria.doc. Teachers can use this rubric to rate tasks they have previously created, or thoughtfully consider the criteria as they begin to design new ones.

Reference Materials

• Sources of Problems
• Bibliography