MATHEMATICAL TASKS TO SKILLS Dr. Betti Kreye Virginia Tech Dr. Jean Mistele Radford University March 21, 2016 Southwest Virginia Higher Education Center Session 2C

Please complete the provided task individually without collaboration.

Why Focus on Tasks? Classroom instruction is generally organized and orchestrated around mathematical tasks The tasks with which students engage determines what they learn about mathematics and how they learn it The inability to enact challenging tasks well is what distinguishes teaching in the U.S. from teaching in other countries that had better student performance on TIMSS

The Importance of Mathematical Tasks 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

Differences Matter Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking. Stein, Smith, Henningsen, & Silver, 2000

Differences Matter The level and kind of thinking in which students engage determines what they will learn. Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

Categorizing Tasks Low-Level Tasks memorization procedures without connections High-Level Tasks procedures with connections doing mathematics The TASK ANALYSIS guide in your packet, provides what we think of as the hallmarks of tasks at each level of demand. The TAG can be a helpful tool as you consider the level of demand of tasks that you use. BUT I WANT TO ARGUE THAT WHILE COGNITIVE DEMANDS ARE A PRIMARY CONSIDERATION, THEY ARE NOT THE ONLY CONSIDERATION.

Characteristics of Rich Tasks High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008) Significant content (i.e., they have the potential to leave behind important residue) (Hiebert et. al, 1997) Require justification or explanation (Boaler & Staples, 2008) Make connections between two or more representations (Lesh, Post & Behr, 1988) Open-ended (Lotan, 2003; Borasi & Fonzi, 2002) Allow entry to students with a range of skills and abilities Multiple ways to show competence (Lotan, 2003) Fenciing task has these characterisitics Consider the Patio Tiles as another example of a rich task.

Factors Associated with the Maintenance and Decline of High-Level tasks Decline Maintenance Routinizing problematic aspects of the task Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior Engaging in high-level cognitive activities is prevented due to classroom management problems Selecting a task that is inappropriate for a given group of students Failing to hold students accountable for high-level products or processes Scaffolding of student thinking and reasoning Providing a means by which students can monitor their own progress Modeling of high-level performance by teacher or capable students Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback Selecting tasks that build on students’ prior knowledge Drawing frequent conceptual connections Providing sufficient time to explore

Group Activity Form two groups based on grade levels taught: one for teachers of grades K-7, the other teachers of grades 8 – high school. Work together with partners within the group to complete the following activity.

Differences Matter 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

Mathematical Tasks: A Critical Starting Point for Instruction …a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a task for, and some means of, independent thinking.” Polya, 1973/1945

An addition to the mathematical Instructional model When trying to ensure that students progress from their engagement in mathematical tasks to developing the required skills – make sure that the movement from the pictorial representation to the symbolic clearly develops the algorithm: concrete pictorial development of a generalization/procedure/algorithm algorithm / symbolic fluency (with understanding)

Discussion Prompt As a mathematics teacher leader within a building, without any release time from your classroom obligations, how might you be able to assist teachers within your building to implement effective instructional changes?

Take - Away Using one of the provided index cards, please record one new idea that you will take away from this session.

Dr. Betti Kreye Dr. Jean Mistele bkreye@vt.edu jmistele@radford.edu