In Principles to Actions, the National Council of Teachers of Mathematics (NCTM) calls for the implementation of tasks that “promote reasoning and problem solving” (p. 10) as one of eight “research informed teaching practices” (2014, p. 7). We argue that the implementation of high-level tasks affords opportunities for teachers to then engage in the other research informed teaching practices recommended in Principles to Actions. These teaching practices include using and connecting mathematical representations, facilitating meaningful mathematical discourse, posing purposeful questions, building procedural fluency from conceptual understanding, supporting productive struggle, and eliciting and using evidence of student thinking (NCTM, 2014). For instance, students cannot be prompted to make connections between mathematical representations if the task does not afford the opportunity to do so. Further, teachers and students cannot make connections between mathematical representations if they all create the same representation.
Project TASK supports approximately 80 elementary teachers in their identification, implementation, and reflection of high-level tasks. In grade-level groups (K-5), six instructional leaders comprising faculty from Drake University, math consultants from Heartland AEA, and teacher leaders in Des Moines Public Schools are providing sustained professional development around the implementation of high-level tasks.
Rich Mathematical Tasks
To get started, we provided teachers with three print resources: Teaching Student-Centered Mathematics (Van De Walle, Lovin, Karp, & Bay-Williams, 2014), Problem Solving for All Seasons (Markworth, McCool, & Koziak, 2015) and Mine the Gap (SanGiovanni, 2016). We used Teaching Student-Centered Mathematics to provide teachers with rationale and a foundation for teaching through problem solving. Problem Solving for All Seasons offers rich tasks organized around cultural activities that occur during the four seasons. Iowa Core standards and Standards for Mathematical Practice are identified for each task. Mine the Gap includes hundreds of tasks that promote thinking and reasoning aligned to the Iowa Core State standards. In Mine the Gap, there are subsequent tasks that follow up or add to the primary tasks, which promote continued mathematical opportunities for students. After teachers pose tasks, we ask them to reflect on task implementation and student work.
We chose one task per month for all teachers to pose to their students. For example, we asked the fourth and fifth grade teachers to pose Task 17A from Mine the Gap, Grades 3-5 (see below).
We found Task 17A to be particularly productive for our students because their experiences, background knowledge, and perspectives on fractions differed immensely. Because of this, the conversation and deliberation between the students was tremendous!
Through our project work, we are learning more about how students respond to tasks and how we can adapt tasks to make them more productive for students. We asked our third grade teachers to pose The July Fireworks Task from Problem Solving for all Seasons (Markworth, McCool, & Kosiak, 2016) below:
The city of San Francisco is planning a very special fireworks show for the Fourth of July. In addition to their standard fireworks, the organizers are including special fireworks that will go off at specific times, as shown in the table:
|White Chrysanthemum||Every 2 minutes|
|Blue Bloom||Every 3 minutes|
|Red Explosion||Every 5 minutes|
If the organizers want to start and end the fireworks show with all three of these special fireworks going off together, how long could the fireworks show be? Explain your reasoning.
Teachers found that the task was inaccessible to many students. Thus, we generated a list of adaptations that could be used to make the July Fireworks Task more accessible:
- Change the task to include just two fireworks
- Change the number choices to 2, 3, and 4
- Change the order of the fireworks to have Red Explosion first. When students were listing multiples, they did not go very far in their counting. If Red Explosion is first, students might go farther in their listing of multiples.
An adapted Fireworks Task could be given before the original to provide students with an entry point.
Teacher Julie Zimmer wrote the following reflection about the benefits she sees from her involvement with Project Task:
I feel very fortunate to be a part of Project Task and have enjoyed seeing my students develop confidence as mathematicians. I have a culturally diverse class with varying mathematical background, who have grown from the weekly exposure to rich tasks related to the Iowa Core standards. It is amazing to see their self-esteem increase regarding their ability to work through meaningful problems when given an opportunity to make connections to the problem and converse with others. One of the most important things I have learned is to incorporate visuals at the beginning of every problem. This allows all students the ability to formulate thoughts and ideas related to the mathematical task in a non-threatening way. In addition, it incorporates the arts, which is a building goal as a Turnaround Arts school.
Once they have engaged their mind through visuals and conversations, students get to work on their own to grapple with the given tasks. They are more willing to try their best after making these connections to the problem. Then the real magic begins – when they share their thinking with others. They construct and defend their positions in a way that never would have come to life without these tasks. I have seen my students get excited about tasks and able to persevere through problems that they may have initially said were too difficult. Most recently, it came to life when defending why one part shaded out of four cannot be one fourth if the pieces are not equal. The opposing view was sure that it was one fourth because there was one shaded out of four, bringing to play the importance of size of fraction pieces.
Problem solving in mathematics supports real-world thinking and problem solving. The greatest challenge seems to be, “Where do I start?” What we have learned and observed over the course of this professional development is that you simply start. Provide students with a rich mathematical task, that’s it! Watch, listen, and observe as students find and use strategies that make sense to them, reason through errors, collaborate with peers, and much more. Where will you start? What rich mathematical tasks make sense for you and your students?
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