Mathematics Teaching: Learning, Extending, Following, and Leading (not necessarily those, and not necessarily in that order) Cynthia Lanius June 11, 2008 Recognize Anne

Teaching <=> Learning Math What do we mean by “learning” math? It means different things to different people. Conceptual Understanding? Skills and Processes? Mathematical Thinking? In Texas, it means scoring well.

When in Texas, Do as the Texans Do TAKS State-Wide Competitors

Learning - Mathematical Proficiency Conceptual understanding—comprehension of mathematical concepts, operations, and relations Procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic competence—ability to formulate, represent, and solve mathematical problems Adaptive reasoning—capacity for logical thought, reflection, explanation, and justification Productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. Which of these does the TAKS measure? Kilpatrick, Jeremy; Swafford, Jane; and Findell, Bradford (Editors) (2001). Adding It Up: Helping Children Learn Mathematics [Online]: http://www.nap.edu/catalog/9822.html.

Belief in Diligence and One’s Own Efficacy Means – If I work at it, I can learn math. It might be a struggle, but I can do it. It’s not something magical that takes some special powers to learn.

The Shrug, Blank Look, “I-Don’t-Know” Syndrome How do students approach mathematics that they haven’t seen? Helpless? - Wait for the teacher to guide them? “When are you going to teach me? Why aren’t you teaching me?” Don’t have sense of “diligence and one’s own efficacy”

Developing Diligence and Efficacy This is a whole session Fluid: Learners make judgments about their capabilities based on comparisons of performance with peers successful and unsuccessful outcomes on standardized and other measures, and feedback from others such as teachers

Mathematics Learning http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf

To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem solving skills. Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive, each facilitating learning of the others. Teachers should emphasize these interrelations; taken together, conceptual understanding of mathematical operations, fluent execution of procedures, and fast access to number combinations jointly support effective and efficient problem solving.

Problem Solving “A problem-centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion” Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc., 2000.

Learn, Extend, Follow, Lead Explore, Observe, Develop, Wonder Problem Solving Schoenfeld calls problems “starting points for serious explorations, rather than tasks to be completed” As opposed to most students’ approach—find the answer and move to the next problem as quickly as possible. Learn, Extend, Follow, Lead Explore, Observe, Develop, Wonder Schoenfeld, A.H. "Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics." Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. Ed. D.A. Grouws. New York, NY: Macmillan, 1992. 334–370. Title page

Diligence (Struggle) Misunderstanding, confusion, and being wrong are natural parts of problem solving, and being able to recognize that you are wrong can be just as valuable as knowing that you are right. But the process of reaching that answer, struggling, being wrong, trying something else, is what is called learning. Antithesis of math as remembering. Make Math Easy. Shouldn’t traumatize the kids by having them struggle.

math.rice.edu/~lanius/Lessons

Problem Exploration – “starting point for serious explorations” Corina has a large piece of paper. She tears it into two equal pieces, and hands one to Mark. She continues to do this; tearing the paper she has left into two equal pieces, handing one to Mark. a) After two tears, how much of the original paper does Corina have left? How much does Mark have? b) After three tears? c) After four tears? d) After twenty tears? e) After infinitely many tears?

Changing the Problem I want to make this problem richer by adding some geometry to it. This provides more potential for rich explorations. I look for a geometric figure so that when I tear it, I will have two pieces that are similar to the original figure. Ahhh, an Isosceles Right Triangle.

Sharing a Triangle Cynthia has a large isosceles right triangle. She tears it into two congruent triangles that are similar to the big one, and hands one to Anne. She continues to do this; tearing the triangle she has left into two congruent triangles, handing one to Anne. a) After two tears, how much of the original triangle does Cynthia have left? How much does Anne have? b) After three tears? c) After four tears? d) After twenty tears? e) After infinitely many tears?

Anne’s Amount of Triangle 1/2 =.50 = 50% 1/2 +1/4 = 3/4 =.75 = 75% 3/4 +1/8 = 7/8 =.875 = 87.5% 7/8 +1/16 = 15/16 =.9375 = 93.75% 15/16 +1/32 = 31/32 =.96875 = .96.875% 31/32 +1/64 = 63/64 =.984375 = 98.4375% 63/64 +1/128 = 127/128 =.9921875=99.21875% S=(2n -1)/2n or 127/128 +1/256 = 255/256 =.99609375= 99.609375% If we kept doing this forever, would she ever have more than the whole triangle? 1/2 S=1-1/2n Wht about perimeter at each stage? 1/8 Cynthia’s Part - 1/2n goes to zero 1/4 1/32 1/16 1/128 1/64 1/256 That means the sum converges to a finite number.

Anne’s Amount of Triangle If we did it infinitely many times ( we couldn’t but if we could) how much of the triangle would Anne have? Exactly 100%, yes the whole triangle. 1/2 What about perimeter at each stage? More areas for exploration. 1/8 1/4 1/32 1/16 1/128 1/64 1/256

because Another Way of Saying It .999 = 1 .333 = 1/3 + .666 = 2/3 EXACTLY!!! .333 = 1/3 because We believe … .3333 3 1.000 + .666 = 2/3 .999 = 1 Remember, we’re extending that original problem. Using it for a springboard for other math.

More Proof .999 = 1

And More                                        It will work for any two fractions that add to 1 and have a repeating decimal representation. Nice Problem: Ask students to find more!!! .999 = ?

An Algebraic Proof .999 = ? So 1 = .999… Let x = .999… 10x = 9.999…                                        So 1 = .999… Let x = .999… 10x = 9.999… Subtract 9x = 9 X = 1 X = .999… X = 1

What would happen if… Cynthia has a large piece of paper. She tears it into four equal pieces and hands one piece to Anne, one to Richard, and one to Jackie. She continues to do this; tearing the paper she has left into four equal pieces, handing three pieces…. a) After two tears, how much of the paper does Cynthia have left? How much does Anne have? b) After three tears? c) After four tears? d) After six tears? e) After infinitely many tears?

Exploration Rough sketch Is this? Is this a trapezoid? What kind of geometric figure is this? 8 Is this? Rough sketch 8 8

Trapezoid – Exactly one pair of opposite sides parallel Quadrilaterals Parallelograms Trapezoids S q u a r e s Rhombus Rectangles

Trapezoid - AT LEAST one pair of opposite sides parallel Quadrilaterals Trapezoids Parallelograms S q u a r e s Rhombus Rectangles

This special isosceles trapezoid, where the legs and a base are the same length, has the characteristic, that its interior region can be divided into four congurent figures that are similar to the original figure.

Form a large trapezoid from the four small trapezoids.

1/4 1/4 1/4 1/4 ¼ of a ¼ Anne’s part ¼ = .25 ¼ + 1/16 = 5/16 = .3125 ¼ = .25 ¼ + 1/16 = 5/16 = .3125 5/16 + 1/64 = 21/64=.328125 21/64+ 1/256 = 85/256=..33203125 85/256 + 1/1024 = 341/1024=..3330078125 1/4 1/4 1/4 1/4 ¼ of a ¼ 1/16 1/64 1/256

I Notice ½ + ¼ + 1/8 + 1/16 + 1/32 + 1/ 64 + 1/128 + … + 1/2n + … = 1/1 and ¼ + 1/16 + 1/64 + 1/256 + 1/1024 + 1/4096 + …+1/4n + … = 1/3 I Wonder if 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + …+1/3n + … = or 1/5 + 1/25 + 1/125 + 1/625 + 1/3125 + …+1/5n + …= or 1/c + 1/c2 + 1/c3 + 1/c4 + 1/c5 + 1/c6 + …+1/cn + …=

1/3 1/9 1 27 1 0.333333 2 0.111111 0.444444 3 0.037037 0.481481 4 0.012346 0.493827 5 0.004115 0.497942 6 0.001372 0.499314 7 0.000457 0.499771 8 0.000152 0.499924 9 5.08E-05 0.499975 10 1.69E-05 0.499992 ?

An Algebraic Proof

And if a/c + a/c2 + a/c3 + a/c4 + a/c5 + a/c6 + …+a/cn + …=

Preventing Wrong Impressions Not all series of fractions will converge on a finite number. One that won’t 1/1 + 1/2 + 1/3 + 1/4 + 1/5 +…+ 1/n + … One that will 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …+ 1/n2 + …

Problem Solving=>Problem Exploring Anti-Inch-Deep Syndrome; Connects to some students that aren’t great problem solvers -- I’m a much better problem explorer than problem solver; An important approach to mathematics that students don’t usually develop until graduate school; and Demands creativity, leading, following.

http://www.teacherthankyoucard.org/ In Conclusion:

http://www.teacherthankyoucard.org/ Sen. Kent Conrad (D-ND) “…Each of these teachers commanded respect, and any one of them could have been a United States Senator. I am grateful for the impact they have had on my life."