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Using Mathematical Modeling to Engage All Learners Doug Burge, Holmen School District [email protected] Dave Ebert, Oregon School District [email protected] Why Modeling? • Common Core State Standards for Mathematics – Standard for Mathematical Practice and Standard for Mathematical Content • Next Generation Science Standards “These standards are not intended to be new names for old ways of doing business. They are a call to take the next step.” -CCSSM p.5 Why Modeling? • Modeling helps build deeper understanding and application of content. • Modeling gives mathematics relevance – answers the question “why do we need to learn this?” • Modeling engages all learners – involving all students can be a Tier 1 Intervention. • Modeling sustains and enhances student interest. Mathematics is, after all, a human activity…. The pupil himself should reinvent mathematics. During this process, the learner is engaged in an activity where experience is described, organized, and interpreted by mathematical means. - Hans Freudenthal Steve Rasmussen at NCSM IGNITE The Wave What do students gain from this? • Students have a meaning of slope as a rate of change. • Students understand the meaning of y = mx + b It isn’t just an arbitrary formula. • Students are engaged in their learning. Pisa Example At a rock concert, a rectangular field 100m x 50m was reserved for fans to stand. The concert was sold out. Approximately how many fans were in attendance? a. 2000 b. 5000 c. 20,000 d. 50,000 e. 100,000 There is a gap between research and practice • Research: Modeling is valuable • Practice: Teaching modeling is difficult - assessment - planning - classroom control Which is less expensive, a t-shirt at the local store for $15, or a t-shirt at the mall for $12? Things necessary for effective modeling 1. Effective classroom management - task-oriented work, not just social group work 2. Learners are actively engaged - balance between independence and teacher guidance 3. Learners are engaged meta-cognitively - students reflect on their work Things necessary for effective modeling 4. A broad variety of examples - variation of real-world and mathematical contexts 5. Multiple solutions are encouraged - teacher’s solution is not the only, or even the best, answer 6. Repeat and practice - learning modeling is a long-term goal Things necessary for effective modeling 7. Assessment needs to reflect modeling 8. Positive beliefs and attitudes must be developed - students’ belief that math problems take more than 3 minutes to solve 9. Technology can be used as a tool 10. Modeling can be learned if there is quality teaching If a lighthouse is 30 m tall, how far away can a ship see the light? Steps to teach modeling 1. Understand the task – what does it mean? 2. Search the mathematics – what do you know and what do you need to know? 3. Use the appropriate mathematics. 4. Explain the result – does the answer make sense? If Beethoven’s 6th Symphony takes an orchestra 40 minutes to play, how long will it take to play th Beethoven’s 9 Symphony? High Performing Countries In high-performing countries, teachers place greater cognitive demands on students by encouraging them to focus on concepts and connections among those concepts in their problem solving. In Our Classrooms Teachers make mathematical tasks more explicit by breaking them down into smaller steps, specifying exact procedures to be followed, or even doing parts of the tasks. This robs students of the opportunity to develop meaningful mathematical understandings. Productive Struggle is Necessary Struggle means that students expend effort to make sense of mathematics. Struggle does not mean being presented information to be memorized or being asked only to practice what has been demonstrated. What Not to Do Don’t use word problems in which the premise is false. Don’t ask questions that only a math teacher would ask. (Glencoe McGraw-Hill) A better question… How close must the cheetah get to the gazelle in order to catch it? What information do we need to know to solve this problem? How are we going to find this information? (Glencoe McGraw-Hill) (Glencoe McGraw-Hill) U is for Undertow by Sue Grafton “In both junior high and high school, I had trouble staying focused in classes where I was doing poorly, math being my weakest subject.” “A train leaves Chicago for Boston traveling sixty miles an hour, while a second train leaves Boston, speeding toward Chicago at eighty miles an hour. A bird flies back and forth between the two… and that’s as far as I’d get. I’d start wondering why the bird was behaving so erratically, positing a virus affecting the bird’s internal gyroscope. I’d daydream about who was on the train and why they were going from Chicago to Boston. Then I’d fret about what was happening in Boston that residents had crowded into the fastest train out. I’d never been to Boston, and now I was forced to scratch it off my list.” Teachers matter most!