The Common Core State Standard’s Mathematical Practices, place a strong emphasis on higher-level mathematical thinking and reasoning. Students at all levels are expected to make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning.  In order for students to meet these standards, they will need to engage with mathematical tasks that are cognitively challenging.  The Common Core State Standards are suggesting teachers engage students with cognitively challenging tasks because research has consistently indicated that the greatest student learning gains occur in classrooms in which cognitively challenging tasks are used and the demands are consistently maintained throughout instruction (Boaler & Staples, 2008; Hiebert & Wearne, 1993; Stein & Lane, 1996).

The Mathematical Task Framework (MTF) (Stien, Smith, Henningsen, & Silver, 2009) served as the guiding framework for analysis in this study and models the progression of a mathematical task from its original form throughout a lesson. The MTF provides a guide for differentiating mathematical tasks based on the level of cognitive demand of the task (memorization, procedures without connections, procedures with connections, and doing mathematics).  The first phase of the MTF identifies the initial step for the teacher to select cognitively challenging mathematical tasks as the basis of instruction in their classrooms.  The second and third phase of the MTF indicate that as the task is set up and implemented in the classroom, there is the opportunity for the cognitive demands of the task to be maintained or decline (Stein, Grover & Henningsen, 1996).

This study analyzed high school mathematics teachers’ selection and implementation of instructional tasks before and after their participation in a professional development initiative that focused on selecting and enacting cognitively challenging mathematical tasks.  Specifically, the research questions that framed this study are the following:

1) What is the cognitive demand level of mathematical tasks presented in high school math classrooms?

2) During implementation is the cognitive demand of High-Level tasks maintained or decreased?

3) Are there changes in teachers’ selection and implementation of instructional tasks after participating in a professional development initiative?

Methods

 Data was collected at the beginning of a professional development initiative (ARCK, described later) and again after the initiative was completed.  Data sources consisted of the following from thirteen high school mathematics teachers:

• Five consecutive days of lesson plans including all mathematical tasks used for instruction
• A video taped lesson from one of the five consecutive lessons

All teachers were participants in the ARCK (Assessment, Reflection, Community, and Knowledge) project. ARCK provided a 2-year professional development initiative for high school mathematics teachers. Through the purposeful creation of a Professional Learning Community amongst the teachers and employing self-study methodologies, a continuing professional development program was developed that could be sustained once the project was completed. The emphasis was on using cognitively challenging instructional mathematical tasks, mathematically productive classroom discourse, and formative assessment.

The mathematical tasks from the five consecutive lessons were analyzed for the potential cognitive demand level using the MTF Task Analysis Guide (Stein, Smith, Henningsen, Silver, 2009).  We then analyzed the videos of high-level tasks using the Instructional Quality Assessment rubrics for Academic Rigor  (Matsumura, Garnier, Slater, & Boston, 2008).  The IQA rubrics for Academic Rigor result in a score of 4-1, as described briefly below:

4: Students engaged in exploring and understanding the nature of mathematical concepts; there is explicit evidence of students’ reasoning and understanding.

3: Students engaged in complex thinking or in creating meaning for mathematical concepts; but there is no explicit evidence of students’ reasoning and understanding.

2: Students engaged in using a procedure however, students did not make connections to the concepts or meaning underlying the procedure being used.

1: Students engaged in memorizing or reproducing facts, rules, formulae, or definitions.

Results

At the beginning of the professional development initiative:

36% of mathematical tasks presented in the study participants’ high school mathematics classrooms had a high-level of cognitive demand. Of the high-level tasks presented 72% consisted of tasks at the “Procedures with Connections” level, while 28% were at the highest level of “Doing Math.” The remaining 61% of mathematical tasks analyzed had low-level cognitive demand, 95% of which were at the “Procedures without Connections” level.

Of the thirteen teachers’ lesson videos, six had high-level tasks (procedures with connections or doing math) as their main instructional task.  Analysis of these six videos found that only two of the teachers maintained the cognitive demand levels during implementation of the task.

We found that most teachers selected instructional tasks from their textbooks, and these textbooks did not support them in selecting cognitively challenging tasks.  Additionally, teachers struggled to maintain the cognitive demand levels of instructional tasks during implementation.  Thus teachers needed opportunities to learn about these issues in their professional development initiative.

After the professional development initiative was completed:

Preliminary analysis of the mathematical tasks used in five consecutive lessons indicates teachers are selecting more cognitively challenging tasks as their instructional tasks than they were at the beginning of the professional development initiative.

References

Boaler, J., & Staples, M. (2008).  Creating mathematical futures through an equitable teaching approach: The case of Railside School.  Teachers College Record, 110, 608-645.

Hiebert, J., & Wearne, D. (1993).  Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic.  American Educational Research Journal, 30, 393-425.

Matsumura, L.C., Garnier, H.E., Slater, S.C., & Boston, M.D. (2008).  Toward measuring instructional interactions “at-scale.”  Educational Assessment, 13(4), 267-300.

Stein, M.K., Grover, B., & Henningsen, M. (1996).  Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms.  American Educational Research Journal, 33, 455-488.

Stein, M.K., & Lane, S. (1996).  Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project.  Education Research and Evaluation, 2, 50-80.

Stein, M.K., Smith, M.S., Henninsen, M., & Silver, E.A. (2009).  Implementing standards-based mathematics instruction: A casebook for professional development (2^nd Edition).  New York: Teachers College Press.