Activity - Bringing The Practices to Action (Worthwhile Mathematical Tasks)


A. Relevant Research and Readings (Deepening our Common Knowledge)


NCTM Brief. (April 8, 2010) “Why is Teaching with Problem Solving Important to Student Learning” http://www.nctm.org/news/content.aspx?id=25713

Smith, M. S., Bill, V., & Hughes, E. K. (October 2008) “Thinking Through a Lesson: Successfully Implementing High Level Tasks”. Mathematics Teaching in the Middle School. Vol. 14 No. 3.

Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.

Driscoll, M. (2007). Fostering geometric thinking: a guide for teachers grades 5-10.
Heinemann.

Driscoll, M. (1999). Fostering algebraic thinking: a guide for teachers grades 6-10. Heinemann.

The following online resources are starting points for gathering math tasks. These tasks are good, but also consider how to modify or adapt:

NCTM Illuminations: http://illuminations.nctm.org/

Howard County Public Schools Common Core Curriculum -
https://secondarymathcommoncore.wikispaces.hcpss.org/

Annenberg Learner: http://www.learner.org/

Inside Mathematics: http://www.insidemathematics.org/

B. Designing Worthwhile Mathematical Tasks (Guiding Task Development)


The NCTM Brief (April 8, 2010) outlines the following criteria to guide the creation of a worthwhile mathematical task:

  1. The problem has important, useful mathematics embedded in it.
  2. The problem requires higher-level thinking and problem solving.
  3. The problem contributes to the conceptual development of students.
  4. The problem creates an opportunity for the teacher to assess what his/her students are learning and where they are experiencing difficulty.
  5. The problem can be approached by students in multiple ways using different solution strategies.
  6. The problem has various solutions or allows different decisions or positions to be taken and defended.
  7. The problem encourages student engagement and discourse.
  8. The problem connects to other important mathematical ideas.
  9. The problem promotes the skillful use of mathematics.
  10. The problem provides an opportunity to practice important skills.

While each math task may not address each of the 10 criteria, it is expected that criteria 1-4 are adequately addressed regardless of the math task content.

To guide your team through a worthwhile mathematical task design process, have the team:

  1. Start small. Sometimes the thought of developing a task can be overwhelming. Tasks can vary in length and complexity. Have the team consider tasks that are smaller in length in the beginning and gradually move to more demanding tasks.

  2. Begin by gathering math tasks that are currently available. Select a task that may be relevant to what the team may be teaching and complete the task. Having a team of teachers complete the task is essential to understand potential approaches/solution strategies, possible misconceptions, and cognitive demands of the task. As the team completes and discusses the task, have the team reflect upon the following:

    • What is the purpose of the task?
    • What understandings does this task help to develop?
    • How might your students do this activity or solve this problem?
    • What materials should be made available for students?
    • What difficulties might students encounter in solving the task? How might we scaffold without removing the cognitive demand?
    • How will you know if students have mastered the key understandings?

Determine when it would be appropriate to give the task to students. If possible, have all members of the content team give the same task so data collection can be consistent.

  1. Have the team evaluate a few math tasks. Gather tasks from a variety of sources (textbooks, Internet, professional books, NCTM resources, etc.). Have teachers score using the attached scoring tool. Have team members compare their evaluation scores and discuss why they gave their particular ratings. Next, challenge the team to select one of the tasks and generate a list of recommendations that will improve the quality and cognitive demand of the task. [Link to the Worthwhile Task Evaluation] Once the task has been revised, determine when it would be appropriate to give the task to students. If possible, have all members of the content team give the same task so data collection can be consistent.
  2. As a team, select a topic/concept for which you would like to create a task. Use the NCTM criteria and Smith (2008) “Thinking Through a Lesson Protocol” [link to the Peg Smith Thinking through a lesson MTMS article] as a guide as the team either revises a textbook/Internet activity or creates a new activity. Have other teachers complete the task and review for quality control and recommendations for revision. Once the task has been revised, determine when it would be appropriate to give the task to students. If possible, have all members of the content team give the same task so data collection can be consistent.




C. Standards for Implementation (Guiding our Work with Students and Informing Data Collection)


Once a task is written and revised, the next focus must be on the implementation of the task. Even the most worthwhile task can lose rigor and cognitive demand during implementation (Stein et al, 2000). Here are some guidelines to follow to help maintain the cognitive demand of the task during implementation:

Before the task:
  • Gather a variety of tools and/or manipulatives that could be used to solve the problem. Items may include chart paper, graph paper, rulers, Katie Cubes, color tiles, two-color counters, patty paper, etc.
  • Discuss with your team how you will measure understanding. What are the big ideas and how do you know students have gained that understanding. Make sure to not focus on one specific method/strategy; instead consider various ways you may gather information about student understanding (a BCR, exit ticket, group presentation, pencast, etc.)

During the task:
  • Speak less. The students should be the ones grappling with the task, not the teacher(s).
  • Avoid lengthy explanations of how to use these tools. Remember, these are just possible tools to use, but students need to select tools that will work for their strategy.
  • Avoid spending a lot of time in the beginning explaining the task. Rather have students take a few minutes to process and work on the task. You now have the option to pause the task and have groups share out what they are thinking or to simply allow students to continue with the task.
  • Be open to new ideas. It is likely that some students will devise a strategy that you might not have considered.
  • Be careful with the amount of support you provide. Scaffold only when necessary and avoid enabling the student or doing the task for him/her.
  • Make sure you are gathering necessary information or data to inform your reflection.

After the task:
  • Review student work and data.
  • Meet with your team to discuss the task implementation and reflect.


D. Tools for Reflection (Discussions for Improvement and Lessons Learned)


Once the team has had students complete the math task, the team should meet to reflect using focus questions to guide the reflection. The team should analyze student work and data collected as evidence of what students have been able to accomplish and understand. Here is a sampling of questions that could be used during the team reflection:
  • What about the task implementation went well?
  • What was students’ understanding of the key concepts upon completion of the activity?
  • What misconceptions (if any) did students encounter as they completed the task?
  • Did students approach the task in a way that was not previously considered? How so?
  • Based on the data your team gathered, what are next steps for instruction?
  • If you were to facilitate this task in the future, what changes (if any) would you make to the task and/or your implementation?

It may be helpful to revisit the Worthwhile Task Evaluation to analyze the task again now that the task has been implemented. Compare the new score to what score was previously given and discuss what may have caused a change in how you scored the task.



E. Sharing our Work


Collaboration is essential for the development and implementation of worthwhile mathematical tasks in all courses. We encourage teams to share tasks with one another and we will post some of the resources.