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Paradigm Shifts In Teaching Higher Level High School Mathematics iMATHination Conference 2014 Q-Center, Saint Charles, IL Saturday, January 25, 2014 Mr. Lamont Holifield, Presenter Agenda • Understanding the Urgency • What is Math? and Why Do We All Need It? • What’s Going Wrong In the Classroom? • Key Strategies and Ways of Working • Making a Difference on the Front Line Disturbing Statistics!! • • • In a recent international assessment of mathematics performance conducted in forty countries across the world, the US ranked a lowly twenty eight. When the level of spending was taken into account, the US sank to the bottom of the forty countries. Interest in mathematics is declining. Dr. Boaler, who was a professor at Stanford University, reported that the average number of math majors over the past ten years was sixteen out of approximately 1,470 students. Furthermore it had dropped by 19 percent at other four year institutions at the same time. Nonresident aliens receive approximately 44 percent of all master’s degrees and 35 percent of all bachelor’s degrees in engineering, mathematics and information science in the United States. -Taken from What’s Math Got to Do With It, Boaler Turn and Talk 1. 2. 3. What’s most surprising about this information? What are the ramifications? What can we as mathematics educators do to reverse these disturbing statistics? “ Lots of people think knowledge is what we want, and I don’t believe that, because knowledge is astonishingly transitory. We don’t employ people as knowledge bases, we employ people to actually do things . . . Knowledge bases comes out of books. So I want flexibility and continuous learning . . . And I need team work. Ray Peacock, Research Director at Phillips Laboratories • • • What does this appear to be saying about the mathematics of work and life? How does this compare with the mathematics of the classroom? What are the similarities and what are the differences? ” What is Mathematics? In pairs or alone, please prepare two definitions of mathematics. As you craft the definitions, do it from the following lenses. • • If we asked your students to define mathematics, what would they say? If we asked you for your definition, what would you say? What is Mathematics? Student’s Perspective • • Math is number. Mathematics is a lot of rules. Mathematician’s Perspective • • • Mathematics is a study of patterns. Mathematics is a set of connected ideas. Mathematics is a human activity, a social phenomena, a set of methods used to help illuminate the world. “ Computation involves going from a question to an answer. Mathematics involves going from an answer to a question. Such work requires creativity, original thinking and ingenuity. —Peter Hilton, an algebraic topologist ” “ Mathematics is a performance, a living act, a way of interpreting the world. Imagine music lessons in which students worked through hundreds of hours of sheet music, adjusting the notes on the page , receiving checks and crosses from the teachers, but never playing the music. ” • • How are mathematics and music related? What does this comparison seem to say about the way we approach mathematics? What’s Going Wrong in the Classroom? Sample Scenarios Topic: Variables Presentation 1 Solve the following: 1. 9 + 18 − 2 2. 2 (𝑏 + 2) Presentation 2 The Hickson household contains 3 people of different generations. The total age of the three family members is 90. Use variables and create an equation to: (a) (b) Find reasonable ages for the 3 Hicksons. Find another set of reasonable ages. One student in solving this problem wrote: 𝐶 + 𝐶 + 20 + 𝐶 + 40 = 90 What do you think C means here? How do you think the student got 20 and 40? What set of ages do you think the student came up with? Interview with a Student of Mathematics • Interviewer: What does it take to be successful in a mathematics class? • Student 1: Pay careful attention! Listen to the teacher! • Interviewer: How is mathematics similar to other subjects. • Student 1: It really isn’t. In math you have to remember, in other subjects you can think about it. It is ironic that math—a subject that should be all “ about inquiring, thinking, and reasoning—is one that students have come to believe requires NO THOUGHT! • • • How does this quote strike you? What can we do about it? What implications does this quote have on our practice? ” Mathematics Problem to Consider A woman is on a diet and goes into a shop to buy some turkey slices. She is 1 given three slices which together weigh of a pound, but her diet says that she 1 3 is only allowed to eat of a pound? How much of the three slices can she eat 4 while staying true to her diet? Which of these represent the correct way to think? Question: What is 18 𝑋 5? Student 1 Response Student 2 Response 18 + 2 = 20 10 𝑋 5 = 50 20 𝑋 5 = 100 8 𝑋 5 = 40 50 + 40 = 90 5 𝑋 2 = 10 100 − 10 = 90 Which of these represent the correct way to think? Question: What is 18 𝑋 5? Student 3 Response Student 4 Response 15 𝑋 5 = 75 5 𝑋 18 = 10 𝑋 9 3 𝑋 5 = 15 75 + 15 = 90 10 𝑋 9 = 90 “ The beauty of mathematical problems is that they can be seen or approached in different ways and, although many have one answer, they can be answered using different approaches. ” Ponder the Following: Suppose you were asked to present a lesson on expanding binomials. In other words, students must be able to expand the following binomial: (2𝑥 + 4)5 • What would be the layout of the lesson? • How would you introduce the lesson? • What was the rationale for the choices you made? Common Core State Standards (CCSS) “These standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build lessons from two decades of standards-based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep” -National Governor’s Association for Best Practices and Council of Chief State School Officers Eight Mathematical Practices and Their Intent Mathematical Practice 1: Make Sense of Problems and Persevere In Solving Them Intent of the Practice • • • • Students make conjectures about the meaning of a solution and plan a solution pathway. Students try special cases or simpler form to gain insight. (They hypothesize and test conjectures.) Students monitor and evaluate their progress and discuss with others. Students under multiple approaches and ask the question, “Does this solution make sense?” Eight Mathematical Practices and Their Intent • • • • Mathematical Practice 2: Reason Abstractly and Quantitatively Intent of the Practice Students can decontextualize a problem by representing the problem symbolically for a solution. Students can contextualize a problem by attending to the meaning of the quantities involved in the problem. Students can create a coherent representation of the task or problem presented. Student can attend to the meaning of the quantities involved in a problem as well as the units appropriate to problem. Eight Mathematical Practices and Their Intent Mathematical Practice 3: Construct Viable Arguments and Critique the Reasoning of Others Intent of the Practice • • • • Students make conjectures and can explore the truths of those conjectures. Students justify their conclusions and communicate them to others. Students compare the effectiveness of two plausible arguments. Students listen, read and respond to the arguments of others for sense making and clarity. Eight Mathematical Practices and Their Intent • • • • Mathematical Practice 4: Model with Mathematics Intent of the Practice Students represent mathematical concepts by using such tools as diagrams, tables, charts, graphs, calculators, graphing calculators, and various forms of area or volume models. Students use symbols and tools to represent real-world situations. Students routinely interpret their mathematical results in the context of the problem situation. Students are comfortable making assumptions and approximations to simplify a complicated real-life situation and are comfortable testing those assumptions. Eight Mathematical Practices and Their Intent Mathematical Practice 5: Use Appropriate Tools Strategically Intent of the Practice • • • • Students choose an appropriate tool for the problems and tasks presented in class and homework. Is the tool necessary to the mathematical task or problem? Students know the limits of each tool for providing accurate solutions to the problems. Students detect tool-generated errors by estimating reasonable solutions without the tool. Students use tools to explore and deepen their discovery and understanding of concepts. Eight Mathematical Practices and Their Intent Mathematical Practice 6: Attend to Precision Intent of the Practice • • • • • Student communicate precisely to others. Students use clear definitions of terms in discussing their reasoning. Students express numerical answers with a degree of precision appropriate for the problem context. Students calculate accurately and efficiently. Students are careful about specifying units of measures and using proper labels. Eight Mathematical Practices and Their Intent Mathematical Practice 7: Look for and Make Use of Structures Intent of the Practice • • • Students’ attention is consistently drawn to the structure of mathematics as it occurs. For example, the teacher consistently asks, “What do you notice is happening here? How does this compare to our previous experiences?” Students are engaged in exploring numerical and visual patterns that reveal structures. Students can use strategies that shift the perspective of a problem—like auxiliary line in geometry. Eight Mathematical Practices and Their Intent Mathematical Practice 8: Look For and Express Regularity in Repeated Reasoning Intent of the Practice • • Students notice and discuss if their results seem reasonable while solving the problem and when they solved the problem. Students notice and can articulate patterns in calculations that can become generalized properties or formulas. Valuables Ways of Working Lectures Student-Led Discussions Individual Work Types of Tasks Long Applied Project Short Questions Contextual and Abstract Investigations Five Practices for Orchestrating Productive Mathematics Discussions Anticipating Connecting Sequencing Monitoring Selecting Credits and Resources Used Common Core Mathematics in a PLC at Work, Timothy D. Kanold and Matthew Larson Five Practices for Orchestrating Productive Mathematics Discussion, Margaret S. Smith and Mary K. Stein What’s Math Got To Do With It?, Jo Boaler Contact Informati0n Mr. Lamont Holifield, M.A. Urban Prep High School – Bronzeville Campus 2710 S. Dearborn Street, Chicago, IL 60616 Mobile: 773-307-0734 Email: lholifield@urbanprep.org