Report

Math for America San Diego Holistic Problems and the Common Core Standards Susie Amoroso, Yekaterina Milvidskaia, Ovie Soto Outline 1. Background and Purpose 2. A Student’s Perspective 3. A 2nd Year Teacher’s Perspective 4. A 15th year Teacher/Support Provider Perspective 5. Q&A – All Purpose 1. To introduce attendees to holistic problems (HP’s). 2. To discuss the implementation of HP’s. 3. To demonstrate the ability of HP’s to support the Common Core Standards’ eight mathematical practices. BACKGROUND: Common Core’s 8 Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. BACKGROUND: Connecting Common Core Math Practices with Proof Schemes Taxonomy Common Core: • Reason abstractly and quantitatively • Use appropriate tools strategically • Look for and make use of structure • Look for and express regularity in repeated reasoning Deductive (Proof Scheme): Harel and Sowder (1998) • Ability to pause and probe into the meaning of the symbols • Make changes to expressions/equations/geometric objects, anticipating their effects and compensating for them in a goal-oriented fashion • Attend to the generality aspect of conjectures • Attend to regularity in a process using it as a reason results generalize BACKGROUND: Synthesis of Common Core’s 8 Mathematical Practices 1. ARE YOU NUTS??!! 2. Does counting on fingers qualify as “Using appropriate tools strategically”? BACKGROUND: Synthesis of Common Core’s Mathematical Practices 1. Students come to see mathematics is a human activity. • Make sense of problems and persevere in solving them. 2. Teachers extend the locus of authority. • Construct viable arguments and critique the reasoning of others. 3. Teachers help students learn to reason deductively. • Reason abstractly and quantitatively. • Look for and make use of structure. • Look for and express regularity in repeated reasoning. • Attention to precision. BACKGROUND: A Teaching Dilema Algebra A polygon has n sides. An interior angle of the polygon and an exterior angle form a straight angle. a. What is the sum of the measures of the n straight angles? b. What is the sum of the measures of the n interior angles? c. Using your answers above, what is the sum of the measures of the n exterior angles? d. What theorem do the steps above lead to? California Geometry (Prentice Hall): #46, p. 162 BACKGROUND: A Teaching Dilemma If the problem were changed (see below), what potential benefits and obstacles might a high school Geometry teacher encounter? What might students try? A polygon has n sides. An interior angle of the polygon and an exterior angle form a straight angle. What is the sum of the measures of the n exterior angles? Adapted from California Geometry (Prentice Hall): #46, p. 162 BACKGROUND: A Teaching Dilemma Students’ Mathematics Formal Mathematics Teacher’s Knowledge Enacted Curriculum BACKGROUND: Teaching Practices Problems constrain teachers and teachers constrain problems. A teaching action refers to what teachers in a particular community or culture typically do in the classroom. A teaching behavior is a typical characteristic of a teaching action. Teaching Practices Teaching Actions Teaching Behaviors BACKGROUND: Holistic vs. Non-Holistic Problems A holistic problem refers to a problem where one must figure out from the problem statement the elements needed for its solution—it does not include hints or cues as to what is needed to solve it. A non-holistic problem, on the other hand, is one which is broken down into small parts, each of which attends to one or two isolated elements. Often each of such parts is a onestep problem. BACKGROUND: Purpose Questions we hope to address: 1. What are some examples of holistic problems (HP’s)? 2. What are some potential benefits and obstacles in using HP’s for teachers? 3. How do HP’s connect to the common core mathematical practices? 4. What teaching practices do we see as closely connected with the successful implementation of a holistic problem? A Student’s Perspective • BA in Mathematics (UCSD). • Math 121A and B • Finishing a Masters in Education (UCSD). • Completed 1 year of student teaching. A Student’s Perspective • Connections between different areas of Mathematics o Algebra, Geometry, and Calculus • Multiple solutions to a single problem o Multiple approaches and ways of thinking • Problems Necessitate Conjectures and Theorems Find the Center of a given Circle Geometric Approach #1 Step 1: Pick any 3 points on a circle and connect them to make A2 a triangle. A1 A3 Step 2: Construct Perpendicular Bisectors of the Triangle. This can be done using a Compass Continued Step 3: The perpendicular bisectors intersect in a point, call this point O. Claim: Point O is equidistant from the vertices, and hence is the center of the Circle. Step 4: The Claim is proved using The Perpendicular Bisector Theorem: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment Algebraic Approach Step 4: Set the two expressions equal to each other to find the point of intersection. This point will be the center of the Circle. Geometric Approach # 2 Step 1: Construct a chord that is a diameter (proved by Thale’s Theorem). Step 2: Repeat. Claim: The point of intersection of two diameters is the Center. This can be proved by definition of a diameter and intersection of lines. The Tactile Approach • Step 1: Fold paper to create two semicircles • Step 2: Fold paper again to create another two semicircles What Makes the Problem Holistic? No hints or cues as to how to approach the problem No “Apply this Theorem, or Use ____ Formula” Multiple ways of solving the problem Geometric, Algebraic, & Logical Multiple ways of thinking Perceptual, Empirical, & Deductive Necessitates a Particular Mathematical Concept Why must the bisectors of the chords intersect at the center of a circle? A Teacher’s Perspective Promotes academic debate among students. CC: Construct viable arguments and critique the reasoning of others. Real-life thinking - Students need to figure out an approach before applying tools they have. CC: Use appropriate tools strategically. Holistic Problems afford different learning modalities. This give students better opportunity to find an approach that works for them. Who am I? • Susan Amoroso • In 2008, earned B.S. in Mathematics and was selected as a Math for America Fellow • Just completed 2nd year of teaching • Low-income, high-minority high school in San Diego County, CA • Attended first MfA San Diego Summer Institute in July 2009 My Intro to Holistic Problems At MfA San Diego Summer Institute in July 2009. Knew right away this was different: • No lecture • No explicit discussion of teaching strategies • Given holistic problems to solve • Worked in small groups • Every group shared its solution #1: At what time after 4:00 will the minute hand of a clock overtake the hour hand? Step 1: Started with a picture of such a clock. Step 2: Divided the clock face into 360° Thus, the starting locations are 0° for the minute hand and 120° for the hour hand. In one minute, the minute hand moves 6° and the hour hand moves 0.5°. Step 3: Create a Table Time (in Location of Minute minutes) Hand (in degrees) 0 0° 1 6° 2 12° 3 18° t 6t Location of Hour Hand (in degrees) 120° 120.5° 121° 121.5° 120 + 0.5t Step 4: Write and Solve an Equation 6t = 120 + 0.5t 5.5t = 120 t ≈ 21.82 ≈ 21 minutes 48 seconds The minute hand will overtake the hour hand at approximately 21.82 minutes, or 21 minutes and 48 seconds. Follow Up Problems #2 – At what time after 7:30 will the hands of a clock be perpendicular? #3 – Between 3:00 and 4:00 Noreen looked at her watch and noticed that the minute hand was between 5 and 6. Later, Noreen looked again and noticed that the hour hand and the minute hand had exchanged places. What time was it in the second case? What I Learned About Learning Mathematics Power of images (and alternatives) • e.g., Degrees vs. proportions Okay for students to see that math is messy Mathematical procedures are tools; quantitative reasoning is the goal What I Learned About Teaching Mathematics Intellectually intriguing Multiple points of entry • Some students solved proportionally • Some solved algebraically Lend themselves well to differentiation Connection to Common Core Standards Reflect real-life nature of problem-solving Require students to make sense of the problem Often provide multiple points of entry – students develop their own problemsolving strategies based on their strengths Connection to Common Core Standards (Cont’d) Require students to reason quantitatively (in context) and attend to the meaning of quantities Students must communicate effectively by making clear mathematical arguments, explaining their reasoning, justifying conclusions and critiquing others’ solutions Fast Forward Two Years Knew this was the type of learning environment I wanted to create in my classroom Currently, I teach very traditionally • Homework review • Toolbox Notes • Independent Practice Effective use of holistic problems is a complex process Systems of Equations #1 – Tree A is 2 feet tall and growing at a rate of 1 foot each year. Tree B is 6 feet tall and growing at a rate of 0.5 feet each year. In how many years will the trees be the same height and what will that height be? Students: Represented their work in a table… My Goal: Solving by Graphing Systems of Equations #2 – Tree C is 4 feet tall and grows 0.5 feet each year. Tree D is 7.3 feet tall and grows 0.25 feet each year. In how many years will the trees be the same height and what will that height be? Students: Represented their work in a table… Goal: Solving by Substitution or Elimination Challenges for Students Not being able to get started Poor organization Addition errors, particularly on #2 Lack of perseverance Calculating growth for fractions of a year Partial Table of Values for #2 Years of Growth Height of Tree C Height of Tree D 0 4 7.3 1 4.5 7.55 2 5 7.8 3 5.5 8.05 12 10 10.3 13 10.5 10.55 14 11 10.8 15 11.5 11.05 16 12 11.3 17 12.5 11.55 Challenges for Teacher How to help students get started Getting students to share solutions with the class Time issues Moving from contextualized to de-contextualized The Big Picture The value and benefits of using holistic problems Understanding that change takes time A Dual Perspective: Teacher/Support Provider Personal Background: 1. 15 years teaching high school mathematics. 2. BA – Math, MA – Math, PhD – Math Ed (May 2010) 3. Continue to teach 2 classes at Patrick Henry High School 4. MfA Summer Institute Facilitator 5. MfA Program Associate Master Teaching Fellows and Field Support 6. Graduate of a previous summer institute Where do they meet? Prepare a poster to be presented to your classmates answering the following question. Will these two lines meet on the left side of the page, the right side of the page, both sides of the page, or never? Explain your reasoning with the following rules: 1. You must find something to measure and say something about how to measure it. 2. You must use your measurements in some way to defend your answer. Where do they meet? Interesting student thinking that surfaced: 1. Parallel did not mean the same thing to all students even though it had been defined. 2. Alternative conception: (Jasmine) near and far perceptual intersection. Where do they meet? 1. Jasmine’s distinction helped authenticate the task. 2. Students came up with several productive ways to tell if lines are parallel: a. If the “distance” between them ever shrinks, the lines are not parallel. “Distance” can be measured with a ruler. b. If you have the Geometer’s sketchpad, you can measure the slopes of the lines. Equal slopes parallelism. c. If the lines head in the same direction, then the lines are parallel. d. Draw a perpendicular line to one. If it is perpendicular to the other, then the lines are parallel. e. Draw a transversal. Check for congruent angles in the “same position of both groups of angles formed”. Where do they meet? Given the information below, will lines AB and DF meet on: A. The half-plane containing point C. B. The half-plane containing point D. C. Neither A nor B. D. Both A and B. Where do they meet? Given the information below, will lines AB and DF meet on: A. The half-plane containing point C. B. The half-plane containing point D. C. Neither A nor B. D. Both A and B. Where do they meet? Given the information below, will lines AB and DF meet on: A. The half-plane containing point C. B. The half-plane containing point D. C. Neither A nor B. D. Both A and B. Where do they meet? Given the information below, will lines AB and DF meet on: A. The half-plane containing point C. B. The half-plane containing point D. C. Neither A nor B. D. Both A and B. Where do they meet? Given the information below, will lines AB and DF meet on: A. The half-plane containing point C. B. The half-plane containing point D. C. Neither A nor B. D. Both A and B. Follow-Up: In a previous class discussion, it was established that lines AB and DF would meet in the halfplane containing point D. Assuming this is the case, what would be the angle formed at the point of intersection? BACKGROUND: Teaching Practices Problems constrain teachers and teachers constrain problems. A teaching action refers to what teachers in a particular community or culture typically do in the classroom. A teaching behavior is a typical characteristic of a teaching action. Teaching Practices Teaching Actions Teaching Behaviors Teaching Practices with Potential to Foster Deductive Reasoning Teaching Actions Teaching Behaviors Dealing with student thinking Ensure that debate/discussion is truly public Eliciting student responses Ask students to communicate her solution process Direct the class to similarities/differences between solution processes and underlying mental imagery. Ask for alternatives in the presence of correct solutions Devolve responsibility to students Encouraging them to make and prove conjectures Allow errors to persist Introduce new tools/concepts Through attention to students’ natural capacity to be puzzled (intellectual need) Goals! Nobody’s Perfect. Working with MfA Fellows 1. Try NOT to invent holistic problems… use the textbooks’ problems whenever possible. 2. Think about the inception of knowledge. Where does it come from? 3. 1 week model. 4. Encourage collaboration. Cell Phone Tower Problem A cell phone tower sends out a signal that reaches a fixed distance in every direction. Some can carry calls up to 5 miles from the tower while less powerful towers can only carry calls within a mile of the tower. If there is a cell tower at Point A that reaches 5 miles, how can we determine what space is in the calling area? If there is a cell tower at Point B that reaches 3 miles, how can we determine what space is in the calling area? Where does the calling area of the towers overlap? Vanessa Davis, MfA Fellow Fancy Cell Phone Tower Problem Verizon has come out with a new tower plan that will provide faster internet service and clearer calls. However, this new service requires two cell towers. The cell phone user must be within 10 miles of both cell towers combined. (So if a person is 7 miles from one tower, she can only be 3 miles away from the other.) Verizon plans to put the towers 4 miles apart. If A and B are the locations of the two towers, what is the calling area for this new service? Vanessa Davis, MfA Fellow Acknowledgements We would like to thank NSF, The Noyce Foundation, and Math for America for this opportunity. www.mathforamerica.org/sandiego BACKGROUND: Language Conjecture, Proving & Proof Schemes A conjecture is an observation made by a person who has doubts about its truth. A person’s observation ceases to be a conjecture and becomes a fact in her or his view once the person becomes certain of its truth. Proving is the process employed by an individual to remove or create doubts about the truth of an observation. A person’s proof scheme consists of what constitutes ascertaining and persuading for that person. - Harel & Sowder, 1998, p. 244