Developing and Using Meaningful Math Tasks

Developing and Using Meaningful

Math TasksThe Key to Math Common Core

Take a moment to record on a sticky: What is ameaningful Math Task?

NormsCourtesy Be on time Cell phones on silent,

vibrate, or off Be mindful of side-bar

conversations Focus on the task at

hand

Collaborative Promote a sense of

inquiry Frame meaningful

questions Pay attention of self and

others Assume positive

intentions Be reflective

Today’s OutcomesParticipants will have a better understanding of

what they need to expect from their students in math.

Participants will have a better understanding how to select and set up a challenging math task.

Participants will have a better understanding of how to analyze a problem or task.

Participants will have a better understanding of how to increase the cognitive demands of a math task.

Believe!The power to believe in your students is

the cornerstone for change.

The greater danger for most of us lies not in setting our aim too high and falling short

but in setting our aim too low and achieving our mark.

Michelangelo

3 Beliefs• Students are capable of

brilliance.• Understanding takes time.• There is more than one way.

Remember the learning community is a culture, not a structure!

What are we asking our students to:

Think about?Talk about?Understand?“Never memorize something you

can look up”Einstein

Challenging TasksThey are about understanding, not coverage and

should be our goals as teachers.Challenging tasks generate and demonstrate

understanding.We need to devote our class time to making

meaning of those juicy tasks to cultivate comprehension.

Selecting a Math TaskWhat are your goals for this lesson?What mathematical content and processes do

you hope students will learn from their work on this task?

In what ways does this task build on students’ previous knowledge?

What definitions, concepts, or ideas do students need to know in order to begin to work on the task?

Setting Up a Math TaskWhat are all the ways the task can be solved?How will you ensure that students remain

engaged in the task?What are your expectations for students as they

work on and complete this task?How will you introduce students to the activity so

as not to reduce the demands of the task?What will you hear that lets you know students

understand the task?

Supporting Students’ Exploration

What questions will you ask to focus their thinking?

What will you see or hear that lets you know how students are thinking about mathematical ideas?

What questions will you ask to assess students’ understanding?

Comparing Two Math Tasks

Martha’s Carpeting Task The Fencing Task Think privately about how you would go about solving each task (solve them).

(See the TASK handout.)

Comparing Two Math Tasks

How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?

Mathematical Tasks:A Critical Starting Point for

Instruction

Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.

Stein, Smith, Henningsen, & Silver, 2000

Mathematical Tasks:A Critical Starting Point for

Instruction There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995

Mathematical Tasks:A Critical Starting Point for

Instruction If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks. Stein & Lane, 1996

High Level vs. Low Level Tasks:Different Opportunities for

ThinkingRead Vignettes 1 (Mr. Patrick) and 2 (Mrs. Fox) (see VIGNETTE handout)

• Consider the following question: What opportunities did students have to think and reason in each of the two classes?

High Level vs. Low Level Tasks:Different Opportunities for

ThinkingLow-Level Tasks

– memorization (e.g., formulas for area and perimeter) – procedures without connections (e.g., Martha’s Carpeting Task)

High-Level Tasks– procedures with connections (e.g., use a drawing to explain why area = length x width)– doing mathematics (e.g., The Fencing Task)

Task AnalysisUse the Task Analysis Guide: think of how high-

and low-level tasks differ and why the differences matter.

What are your own experiences with using high-level tasks ?

What extent does your textbook support using high- level tasks?

What are other possible sources of such tasks?

Do High Level Tasks Decline?

Read Vignette 3 (Mrs. Jones) (see VIGNETTE handout) • Consider the following question: What factors might account for the decline of the task in Mrs. Jones classroom?

Conclusion Not all tasks are created equal—they provide different

opportunities for students to learn mathematics. High level tasks are the most difficult to carry out in a

consistent manner. Engagement in cognitively challenging mathematical

tasks leads to the greatest learning gains for students. Professional development is needed to help teachers

build the capacity to enact high level tasks in ways that maintain the rigor of the task.

Increase the Cognitive Demand of the Task

Increase complexityIntroduce ambiguitySynthesize strand of mathematicsInvite conceptual connectionsRequire explanation and justificationPropose solutions that reveal misconceptions or

common errors

Does the task have all of the following?

Multiple entry pointsVarious possible approachesA need for higher order thinkingOpportunities to synthesize, justify,

and explain

What is 6 + 4How can you make this into a

challenging task?Ten children went to the movie.

How many were girls? How many were boys? Explain your answer. Could there be other answers?

Change this into a Challenging Task.

A rectangle has a length of 4 units and a width of 3 units. What is the area?

Create rectangles with an area of 24 square units. How are they alike and different? Are their perimeters the same? Explain your observations.

Change this into a Challenging Task.

Three children shared a pizza. They each had the same amount. What fraction did each child have?

Three children are sharing a pizza. How might they share it? What fraction of the pizza could each child get? Justify your answer.

Make a task more challenging by changing the questions you ask.

Low- Three children fairly shared a box of 15 chocolates. How many chocolates are in each child’s share?

Middle- Same problem. How many chocolates are in each child’s fair share? Explain how you answered the question.

High- Same problem. Same first question. How much of the box of chocolates is one child’s share? How many times larger than one child’s share is the box of chocolates? Explain how you answered the questions.

InvestigationsFind a task in your Investigations manual.How can you make it more cognitively

challenging for students?Is the rigor of the tasks you are doing with

students high enough?

Invite students to:Describe their processReflect on their decisionsExplain their vigilanceConfirm their thinkingMake connectionsPromote discourse

Questions or Concerns?

Fill out exit slip