Reflecting on Practice: Using Inquiry to Build Thinking Classrooms Unit 1, Session 2 2017 Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Farmer Jack Farmer Jack harvested 30,000 bushels of corn over a ten-year period. He wanted to make a table showing that he was a good farmer and that his harvest had increased by the same amount each year. Create Farmer Jack’s table for the ten year period. Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Worthwhile tasks… On the next page is the wordle from yesterday: “A worthwhile task is one that ….” At your tables discuss how the Farmer Jack task compares with what you see in the wordle. Reflecting on Practice Park City Mathematics Institute

Worthwhile Tasks Wordle

Park City Mathematics Institute Another Problem Sketch and Investigate Construct triangle ABC. Construct the midpoint of segment AB. Label it D. Construct the midpoint of segment BC. Label it E. Construct the midpoint of segment CA. Label it F. Construct a line perpendicular to segment AB through D (the perpendicular bisector of AB). Construct a line perpendicular to segment BC through E (the perpendicular bisector of BC). Construct a line perpendicular to segment CA through F (the perpendicular bisector of CA). The point where the perpendicular bisectors meet is called the circumcenter of triangle ABC. Reflecting on Practice Park City Mathematics Institute Harper & Edwards, 2011

Park City Mathematics Institute Wordle Part 2 Discuss how this task compares with the words in the wordle we associated with worthwhile tasks. Reflecting on Practice Park City Mathematics Institute

Worthwhile Tasks Wordle

No More Cookbook Lessons This problem (and its revisions) comes from an article in the Mathematics Teacher called “No More Cookbook Lessons”. Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Bucket List Look at the wordle and find a way to make 3 or 4 clusters of words that belong together as big ideas for what makes a task worthwhile. Write these clusters on your board. Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Gallery Walk Visit the board to your right CW and look at their clusters. ✔️ Clusters that you also identified ? Clusters that you are wondering about ! Clusters that surprised you Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Cognitive Demand Higher-level demands (Doing Mathematics): Require complex and nonalgorithmic thinking—a predictable, well- rehearsed approach or pathway is not explicitly suggested by the task, task instructions, or a worked-out example. Require students to explore and understand the nature of mathematical concepts, processes, or relationships. Demand self-monitoring or self-regulation of one’s own cognitive processes. Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task. Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions. Require considerable cognitive effort and may involve some level of anxiety for the student because of the unpredictable nature of the solution process required. Reflecting on Practice Park City Mathematics Institute Smith & Stein, 1998

Tasks and Cognitive Demand Over the last two days we’ve done four tasks: the penny problem, the circle-square problem, Farmer Jack, and the triangle center problem. How do these tasks match the characteristic of higher-level cognitive demand described by Stein and colleagues? Reflecting on Practice Park City Mathematics Institute

Thinking More About Tasks Reflecting on Practice Park City Mathematics Institute Principles to Actions, 2014

Park City Mathematics Institute A Reminder… Please don’t forget to read the Mathematics Teacher article called “No More Cookbook Lessons” for tomorrow with the lens of what makes a worthwhile task. You can find the article as a handout. Please bring this with you to the next session. Have a great day! Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute References Boaler, J., & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside School. The Teachers College Record, 110(3), 608–645. Harper S., & Edwards, M. (2011). A New Recipe: No More Cookbook Lessons. The Mathematics Teacher, 105(3), pp. 180-188. Reston VA: National Council of Teachers of Mathematics http://www.jstor.org/stable/10.5951/mathteacher.105.3.0180 . Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann. Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second- grade arithmetic. American Educational Research Journal, 30(2), 393–425. Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Liljedahl, P. (under review). On the edges of flow: Student problem solving behavior. In S. Carreira, N. Amado, & K. Jones (eds.), Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect. New York, NY: Springer. Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection. Mathematics Teaching in the Middle School, 3, 268-275. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development (Second Edition). New York, NY: Teachers College Press. 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. Educational Research and Evaluation, 2(1), 50-80. Stein, M., Grover, & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks in reform classrooms. American Educational Research Journal, 33, pp. 455 -488 Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Reflecting on Practice Park City Mathematics Institute

Park City Mathematics Institute Reflecting on Practice Park City Mathematics Institute

“Let d be the yearly increase and an be the amount harvested in year n “Let d be the yearly increase and an be the amount harvested in year n. Then an+1 = an+d and an = a1 + (n-1)d. The condition is that the 10 year total harvest is 30000 bushels, thus, S10 = ∑an = 30000 where S10 is the total number of bushels after 10 years. Now, Sn = (n/2)(a1+an), so S10 = (10/2)(a1+a10) = 5(a1 + a1+ 9d) = 30000. So 2a1+9d = 6000. Any pair (a,d) where a and d are both greater than 0 will produce a suitable table. There are an infinite number of tables if you do not restrict the values to be positive integers.” Secondary Mathematics Major MSU, Burrill, 2004