The Mathematics Problem Solving Model A Professional Development Program Presented by Sarah Enoch September 10, 2011

The Mathematics Problem Solving Model Designed by Education Northwest, an educational research lab in Portland, OR Research-based framework Central themes of the model: Teaching mathematics through problem-solving Formative assessment as a tool to enhance instruction

Assumptions about Student Learning Two important influences on how children learn mathematics are the tasks and problems they engage in and the interactions they have about them. Teachers’ ability to understand students’ mathematical development is enhanced by their ability to notice and describe what students say and do.

Two important influences on how children learn mathematics are the tasks and problems they engage in… Effective instructional practice revolves around the use of problem solving tasks and student-centered discussions about these tasks (See, for example, Boaler & Humphreys, 2005; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, …& Human, 1997; Kilpatrick, Swaffor, & Findell, 2001). it was found that higher learning gains were achieved when teachers implemented tasks that were cognitively demanding (Stein and Lane, 1996) Students that engage in cognitively demanding problem solving tasks have opportunities to learn problem solving, reasoning skills, and higher order thinking (Wood and Turner-Vorbeck, 2001).

…and the interactions they have about them It is necessary to create opportunities for students to clarify their solutions, providing their reasoning and justification for the approaches they take and connecting it to relevant mathematical content (Hiebert et al., 1997; Kazemi & Stipek, 2001) Also, students should have opportunities to look for patterns and make generalizations around problem solving tasks, attempting to defend and justify their conjectures, with reasoning playing a central role in the process (Yackel & Hanna, 2003).

Teachers’ ability to understand students’ mathematical development is enhanced by their ability to notice and describe what students say and do Analysis of student work is a valuable tool for creating effective mathematical instruction. An awareness of students’ understanding and thinking around mathematical concepts and tasks makes it possible to appropriately choose tasks that are of high cognitive demand, but still within the students’ reach (Stein, Grover, et al., 1996) Knowledge of student work also supports appropriate teacher questions useful for building upon student thinking to develop and make connections between mathematical ideas (Grouws, 2003).

Teacher’s knowledge about students’ thinking can be improved in the following ways: Posing problems specifically designed to access students’ thinking (Lesh, Hoover, Hole, Kelly, & Post, 2000) Asking good questions during instruction that access students’ thinking (Wiliam, 2007) and listening attentively to their students’ thinking (Schifter, 1998) Examining student work to make a connection between their idiosyncratic ways of thinking and more conventional mathematics (Mewborn, 2003)

Mathematical Learning Objectives Teachers’ instructional move that lead towards building conceptual understanding should begin with and be subsequently guided by the identification of an instructional learning goal (Sherin, 2002) the implementation of a problem solving task should take place concurrently with the mathematical concepts and procedures in the classroom’s curriculum (NCTM 2000; Lesh & Zawojewski, 2007) Implementing a problem-solving tasks is NOT “Problem-Solving Friday”.

Mathematical Learning Objectives Teachers are supported in this element of the MPSM by: Examining the mathematical content and skills required for implementation of the MPSM tasks Looking at their own curriculum to see where these tasks would best fit.

Mathematical Tasks Problem-solving tasks are open-ended have multiple solution methods lead to multiple possible solutions the focus is not on finding the answer, but on the processes that students use to arrive at their solutions. These problems should be non-routine and challenging, but not inaccessible (Becker & Shimada, 1997).

Mathematical Tasks: Cognitive Demand Low-level tasks require little to no cognitive demand to complete are focused on producing correct answers have no connections to related concepts require little explanations. Memorization tasks are straight-forward tasks that require students to recall previously learned information to solve and do not afford opportunities to use procedures. Procedures without connections tasks are algorithmic in nature, requiring the use of previously learned procedures.

Mathematical Tasks: Cognitive Demand High-level tasks: require students to explore mathematical relationships, processes, and concepts demand self-monitoring of one’s thinking require students to access relevant knowledge from past experiences Procedures with connections tasks focus students on the use of broad procedural pathways for the purpose of deepening understanding of the concepts underlying the procedures, requiring students to engage with conceptual ideas in order to successfully complete the task. Doing mathematics tasks require complex, non-algorithmic thinking, and require considerable cognitive effort.

Mathematical Tasks: Cognitive Demand Teachers are supported in this element of the MPSM by: Being supplied with tasks which were identified by the professional developers as doing mathematics tasks. analyzing problem solving tasks for the purpose of categorizing tasks by their level of cognitive demand taking low-level tasks from their own curricula and adapting them to raise the cognitive demand of the tasks

Implementing Tasks It was found that several factors contributed to the decline of cognitive demand during implementation: 1) tasks became non-problematic as the teacher either reduced the requirements of the task or completed the challenging steps for the students; 2) the task was inappropriate for the students, indicating a lack of knowledge about students on the part of the teacher; 3) the focus shifted to finding the correct answer; 4) too much or too little time was allotted to complete the task; 5) a lack of accountability on the part of the students; 6) classroom management issues.

Implementing Tasks Several factors contribute to the maintenance of cognitive demand: 1) the task built on students’ prior knowledge; 2) an appropriate amount of time was allotted for students to work on the task; 3) a high level of performance was modeled; 4) there was sustained pressure for explanation and meaning; 5) scaffolding was provided without taking away from the complexity of the task; 6) students were encouraged to self-monitor their work; 7) the teacher helped the students to draw conceptual connections

Implementing Tasks: The Use of Teacher Questioning Maintaining the cognitive demand of a problem-solving task is particularly challenging for teachers when they see their students are struggling with a task and they want to relieve that anxiety for their students (Henningsen & Stein, 1997). Questions that support students in moving their thinking forward should encourage students to provide mathematical argumentation and make mathematical connections (Kazemi & Stipeck, 2001).

Implementing Tasks Teachers were supported in this element of the MPSM through Discussion of questioning frameworks to distinguish different types of questions (Boaler & Humphrys, 2005; NCTM, 2000) Observation of video recordings followed by discussion of the questioning practices of teachers that successfully maintained the cognitive demand of problem solving tasks in their classrooms.

Feedback Guide as Lens Five problem solving traits : Conceptual Understanding Strategies & Reasoning Communication Computation & Execution Insights These traits are distinct from, but are useful for supporting, the Standards for Mathematical Practice (see handout)

Analyzing Student Work The analysis of student work allows for teachers to gain insight into how their students think about and understand mathematics, allowing teachers to… better understand how they can support their students in moving forward their thinking and shape subsequent instruction (Sowder, 2007; Franke et al., 2007).

Plan Next Step Two possible avenues: Classroom Group “Feedback” (discourse around the problem solving task) Follow up on written feedback

Written Feedback The primary purpose of formative feedback is to increase the knowledge, skills, or understanding of a student within a certain content area (Shute, 2008). An important characteristic of good feedback is that the nature of feedback provided should be guided by the teacher’s instructional goals and the teacher’s knowledge of the student (or students) in question (Narciss & Huth, 2004). Research shows that the most learning gains are associated with feedback in which the student is given information about correct results, some explanation, and suggestions for specific activities to undertake in order to improve (Nyquist, 2003).

Formative Feedback Guide Organized by the five traits of problems solving: Conceptual Understanding, Strategies & Reasoning, Communication, Computation & Execution, and Insights. Designed for teachers to give more detailed information when providing written feedback suggests possible avenues for improving the issues evident in the student’s work Example of suggested feedback from the guide: I’m not sure you used __(mathematical term)__ correctly. Check the definition and see if there is a better word to use.

Classroom Use of Feedback Good feedback is not useful when students do not reflect upon it. Ways that feedback can be put to use in the classroom: Revise and resubmit peer feedback student reflection on written feedback Particularly these last two processes encourage student autonomy through metacognitive behavior (Shute, 2008)

Classroom Group “Feedback”: Discourse around Problem Solving Tasks Students sharing their problem-solving strategies is a valuable way for students to build new mathematical thinking (Hiebert, 2003) Students randomly volunteering to share can lead to limited opportunities for students to engage in mathematical thinking around a problem solving task (Chazan & Ball, 2001; Nathan & Knuth, 2003; Stein, Engle, Smith, & Hughes, 2008) The MPSM proposes planning for discourse as follows: Identify which students’ work will be discussed Determine the order in which the work will be presented plan appropriate questions to make the mathematics salient (Stein et al, 2008; Stein et al, 2009).

Identification of Student Work Identification of student work for classroom group “feedback” may be based upon the mathematical ideas present in the solution representations used misconceptions that are evident in the work sharing a particularly well-written solution making sure that a student who hasn’t presented in a while gets an opportunity to share

Sequencing of Student Work Strategies for sequencing student work include, but are not limited to starting with the least sophisticated solutions and progressively building a sophisticated solution (Groves & Doig, 2004). sharing incorrect solutions up front to eliminate misconceptions sharing two conceptually similar solutions together to create opportunities for students to make connections between the strategies sequencing in such a way to allow a mathematical lesson to emerge (Stein et al., 2008).

Planning Questions for Discourse Questions a teacher might ask include requests for students to provide justification for the strategies they used (Hiebert et al., 1997; Kazemi & Stipek, 2001) questions that lead students to make sense of the mathematical ideas used to solve the task (Boaler & Humphreys, 2005; Sherin, 2002) questions that prompt students to make connections between strategies (Hiebert & Wearne, 1993; Kazemi & Stipek, 2001) questions that encourage students to formulate and prove conjectures and generalizations around the mathematics in the task (Fraivillig, Murphy, & Fuson., 1999; Hiebert, & Wearne, 2003; Yackel & Hanna, 2003).

Questions? Please feel free to me any requests for further information (including a reference list) at