From Confusion to Clarity: How Complex Instruction Supports Students’ Mathematical Proficiency

1. From Confusion to Clarity: How Complex Instruction Supports Students’ Mathematical Proficiency Frances K. Harper Michigan State University INTRODUCTION FINDINGS Theme 2: Tasks and instructional tactics must vary to best suit the specific content and the needs of the students. • The task - Introducing linear functions through non-linear patterns: • Groups of 4. • Each student receives a different pattern. • One pattern is linear; three are non-linear. • Multiple representations: pile patterns, tables, graphs and rules, with connections between these representations and across the 4 different patterns. • “I’ve designed a whole bunch of things where people feel like it’s backwards. And I want it backwards on purpose. This is a task about…non-linear patterns that lets me transition to the much simpler linear patterns….The completed posters that I’ve kept are the ones where kids realize, ‘Oh, we have these things in common. We’re going to color-code them the same. Or set up our poster where it’s matching from one section to another.’ They’re looking on this meta-level that makes the actual looking-for-patterns and color-coding almost trivial in comparison, but that’s the key to the linear component. • Three themes related to Guillermo’s beliefs about promoting mathematical proficiency through CI emerged from the data. Complex Instruction (CI) provides teachers with a powerful tool for promoting rigorous and equitable learning. The research question for this project asks: How is mathematical proficiency developed within a classroom environment where the teacher places a strong emphasis on equitable learning? This research project considers one teacher’s beliefs about and goals for his mathematics teaching. In particular, this case study looks closely at the resulting practices that give more students access to challenging and rigorous math tasks. • “Kids need to be protected or challenged in all sorts of ways. You just have to try different things.” • In an example from classroom observations, Guillermo: • Enlarges a section of one student’s graph • Helps students identify the growth pattern • Connects to concept of slope • Changes the scale of the graph • Presents a challenge for students • Relies on the high achieving students to interpret the graph • Provides more scaffolding for low achieving students than for high achieving students • All students have access to the challenging task DATA • Teacher: High school mathematics teacher recognized for his effective use of CI, Guillermo Reyes • Students: 4 male students from ethnic minority backgrounds at an urban high school • Data Collection – 3 phases • Observations: observed Guillermo leading discussion section of course designed to introduce pre-service teachers to CI – used to develop interview protocol • Interviews: one interview conducted by Jo Boaler in 2004; one interview conducted by Frances Harper in 2012 – used to develop frameworks and coding scheme • Classroom Observations & Task: 4 video excerpts (total time: 15 minutes) selected from 6 hours of classroom observation; assignment details for corresponding video selection Theme 3: Math is complex in nature; therefore, instruction should focus on making math meaningful to students • “(A student) latched onto this meta-agenda of sense making and showing off knowledge…showing off connections, showing off relationships. And that’s what I want kids to remember. That they had this experience with mathematics where they felt smart and knew it was grounded it something…I just opened the doors for some actual performance and learning.” • Guillermo emphasizes: • Loving students • Breaking down barriers to mathematical understanding • All students have the ability to succeed • Setting up all students for success • Assigning competence • An example: Guillermo assigns competence by noting the clear connections between a low-achieving student’s table and pile pattern and offers this student’s work as a model for the whole group. Theme 1: Struggle and confusion are necessary for learning meaningful mathematics. • “Watching kids wrestle with all this affirms for me that it’s the right sequence. That asking them to do something much, much harder then makes the subsequent stuff more obvious to them.” • Complex & rigorous math tasks: • Allow for a smoother transition to topics where fluency is the goal. • Give more students access to deeper conceptual understanding. • Push the limits of reasoning and understanding. FRAMEWORKS • Math Proficiency • Demonstrate procedural fluency & conceptual understanding • Create representations of problems and solutions • Make connections by identifying & explaining patterns • Justify solutions & generalize • Apply procedures, rules or understanding to new contexts • Reverse mathematical processes (check answers) • Complex Instruction • Multiple ability task • Modified authority structure • Treatment of status issues Contact: harperfr@msu.edu