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Creating Opportunities For Students to Engage in Reasoning and Proving: Modifying Existing Tasks Peg Smith University of Pittsburgh October 25, 2012 North Carolina Council of Teachers of Mathematics 2012 State Mathematics Conference

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Agenda Provide rationale for the importance of reasoning-and- proving Discuss the need and method for modifying textbook tasks Compare and Discuss Three Sets of Tasks Analyze Tasks Modified by Teachers Consider the value of engaging in this work

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What is Reasoning-and-Proving? The work in which mathematicians engage that culminates in a formal proof involves searching a mathematical phenomena for patterns, making conjectures about those patterns, and providing informal arguments demonstrating the viability of the conjectures. Lakatos, 1976

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What is Reasoning-and-Proving? By focusing primarily on the final product - that is, the proof - students are not afforded the same level of scaffolding used by professional users of mathematics to establish mathematical truth. Therefore, reasoning-and-proving should be defined to encompass the breadth of activity associated with: identifying patterns making conjectures providing proofs, and providing non-proof arguments. Stylianides, 2008

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Why Reasoning-and-Proving? There is a growing consensus that high school mathematics programs need to include a greater emphasis on reasoning and proof. Reasoning and proving are central to the mathematical practices identified in CCSS (2010). Practice 2: Reason abstractly and quantitatively Practice 3: Construct viable arguments and critique the reasoning of others Practice 7: Look for and make use of structure Practice 8: Look for and express regularity in repeated reasoning NCTM in their most recent policy document, argues that reasoning and sense making “should occur in every mathematics classroom everyday” (2009, p.5).

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Proof is Challenging for Students Student have difficulty constructing deductive arguments even after a high school geometry course (e.g., Senk 1985, 1989). Students have a difficult time understanding the role of proof and empirical evidence (e.g., Chazan, 1993). Many secondary school and university students believe that empirical evidence constitutes a proof (e.g., Chazan, 1993; Coe & Ruthven, 1994; Harel & Sowder, 1998). Students often believe that proofs are acceptable only if they possess a particular form, type, or representation (e.g., Healy & Hoyles, 2000).

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Proof is Challenging for Teachers Many teachers favor empirical arguments over proofs, finding them more convincing or easier to follow (e.g., Knuth 2002a). Secondary math teachers tend to view proof largely as a specific topic of study rather than as a tool for doing mathematics or as a stance toward mathematics in general (e.g., Knuth, 2002b). Many high school mathematics textbooks limit serious work on reasoning-and-proving to high school geometry, with a focus on verifying already-known results that are obvious to students (Johnson, Thompson, & Senk, 2010; Schoenfeld, 1994). Current teaching practices perpetuate the view that the teacher is the sole arbiter of mathematical knowledge – an authoritarian proof scheme (e.g., Harel & Rabin, 2010).

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Proof is Challenging for Teachers Many teachers favor empirical arguments over proofs, finding them more convincing or easier to follow (e.g., Knuth 2002a). Secondary math teachers tend to view proof largely as a specific topic of study rather than as a tool for doing mathematics or as a stance toward mathematics in general (e.g., Knuth, 2002b). Many high school mathematics textbooks limit serious work on reasoning-and-proving to high school geometry, with a focus on verifying already-known results that are obvious to students (Johnson, Thompson, & Senk, 2010; Schoenfeld, 1994). Current teaching practices perpetuate the view that the teacher is the sole arbiter of mathematical knowledge – an authoritarian proof scheme (e.g., Harel & Rabin, 2010).

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Math Class Needs a Makeover Dan Meyer You Are Doing Math Reasoning Wrong If... Students don't self-start. You finish your lecture block and immediately you have five hands going up asking you to re- explain the entire thing at their desk; Students lack perseverance; They lack retention; you find yourself re-explaining concepts three months later, wholesale; and There's an aversion to word problems, which describes 99 percent of my students. And then the other one percent are eagerly looking for the formula to apply in that situation. This is really destructive.

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Math Class Needs a Makeover “So 90 percent of what I do with my five hours of prep time per week is to take fairly compelling elements of problems …from my textbook and rebuild them in a way that supports math reasoning and patient problem solving.” Dan Meyer (March, 2010) Google dan meyer ted for video

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Comparing Two Versions of a Task Compare each task to its modified version (A to A’, B to B’, C to C’) Determine how each pair of task is the same and how it is different Look across the three sets of tasks and consider: what the modifications in the tasks were trying to accomplish what modification principles can be generalized whether the differences between a task and its adaptation matter

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Comparing Two Versions of a Task: How are they the same and how are they different? TASK A MAKING COJECTURES Complete the conjecture based on the pattern you observe in the specific cases. 29. Conjecture: The sum of any two odd numbers is ______? = = = = = = Conjecture: The product of any two odd numbers is ____? 1 x 1 = 17 x 11 = 77 1 x 3 = 313 x 19 = x 5 = x 305 = 61,305 TASK A’ For problems 29 and 30, complete the conjecture based on the pattern you observe in the examples. Then explain why the conjecture is always true or show a case in which it is not true. MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases. 29. Conjecture: The sum of any two odd numbers is ______? = = = = = = Conjecture: The product of any two odd numbers is ____? 1 x 1 = 17 x 11 = 77 1 x 3 = 313 x 19 = x 5 = x 305 = 61,305

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Comparing Two Versions of a Task: How are they the same and how are they different? TASK B VISUAL REASONING Explain why the following method of drawing a parallelogram works. State a theorem to support your answer. TASK B’ Consider the construction below. Use the construction with a variety of starting segments. 1. Make a conjecture about the type of figure that the construction produces. 2. Using the properties that you know about that figure, make a mathematical argument that explains why that figure is produced each time by the construction. 1.Use a ruler to draw a segment and its midpoint. 2. Draw another segment so the midpoints coincide. 3. Connect the endpoints of the segments. 1.Use a ruler to draw a segment and its midpoint. 2. Draw another segment so the midpoints coincide. 3. Connect the endpoints of the segments.

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Comparing Two Versions of a Task: How are they the same and how are the different? Task C GEOMETRY For Exercises 45 and 46, use the diagram below that shows the perimeter of the pattern consisting of trapezoids. 45. Write a formula that can be used to find the perimeter of a pattern containing n trapezoids. 46. What is the perimeter of the pattern containing 12 trapezoids? Task C’ 1. Make as many observations as you can about the trapezoid pattern. 2. Find the perimeter of the first four trapezoids shown above. 3. Find the perimeter of the pattern containing 12 trapezoids without drawing a picture. 4. Write a generalization that can be used to find the perimeter of a pattern containing any number of trapezoids. 5. Using words, numbers, and/or connections to the visual diagram, prove that the generalization you created in part 4 will always works.

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Ways to Modify Tasks 1. Engage students in investigation and conjecture instead of just giving answers. 2. Provide all students with access to a task by first making observations about a situation before moving on to more focused work. 3. Require students to provide a mathematical argument, proof, or explanation. 4. Take away unnecessary scaffolding. 5. Ask students to explore a situation by generating empirical examples and looking for patterns.

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What Were the Modifications Trying to Accomplish? Press students to do more reasoning and justifying than the original versions of the task. Give students more access to the task. Give students the opportunity to do more investigation and less of just giving answers. Engage students in PROOF (without actually saying PROOF).

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Do the differences matter? YES!

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Analyzing Teacher’s Modifications Consider the original task. How could you modify the task so as to increase it potential for engaging students in reasoning-and-proving? Why would you want to? Review the modification created by teacher (e.g., Donald (Task D became D’). What did the teacher accomplish with his or her modification?

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Ways to Modify Tasks 1. Engage students in investigation and conjecture instead of just giving answers. 2. Provide all students with access to a task by first making observations about a situation before moving on to more focused work. 3. Require students to provide a mathematical argument, proof, or explanation. 4. Take away unnecessary scaffolding. 5. Ask students to explore a situation by generating empirical examples and looking for patterns.

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Donald’s Task (D) TASK D a. Simplify each expression. (-2) 2 (-2) 3 (-2) 4 (-2) 5 (-3) 2 (-3) 3 (-3) 4 (-3) 5 b. Make a Conjecture Do you think a negative number raised to an even power will be positive or negative? Explain. c. Do you think a negative number raised to an odd power will be positive or negative? Explain.

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Donald’s Task (D and D’) TASK D a. Simplify each expression. (-2) 2 (-2) 3 (-2) 4 (-2) 5 (-3) 2 (-3) 3 (-3) 4 (-3) 5 b. Make a Conjecture Do you think a negative number raised to an even power will be positive or negative? Explain. c. Do you think a negative number raised to an odd power will be positive or negative? Explain. Task D’ 1. Solve the following examples. (-2) 2 (-2) 3 (-2) 4 (-2) 5 (-3) 2 (-3) 3 (-3) 4 (-3) 5 2. Make some observations about any patterns that you notice. 3a.Using what you notice about the examples above, make a conjecture about negative numbers to an even power. 3b.How do you know that this will be true for all negative numbers? 4a. Using what you notice about the examples above, make a conjecture about negative numbers to an odd power. 4b. How do you know that this will be true for all negative numbers?

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Elaine’s Task (E and E’) TASK E Patterns The table of values below describes the perimeter of each figure in the pattern of blue tiles. The perimeter P is a function of the number of tiles t. a. Choose a rule to describe the function in the table. A. P = t + 3 B. P = 4t C. P = 2t + 2 D. P = 6t – 2 b. How many tiles are in the figure if the perimeter is 20? c. Graph the function. t1234 P46810

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Elaine’s Task (E and E’) TASK E Patterns The table of values below describes the perimeter of each figure in the pattern of blue tiles. The perimeter P is a function of the number of tiles t. a. Choose a rule to describe the function in the table. A. P = t + 3 B. P = 4t C. P = 2t + 2 D. P = 6t – 2 b. How many tiles are in the figure if the perimeter is 20? c. Graph the function. TASK E’ Patterns The table of values below describes the perimeter of each figure in the pattern of blue tiles. The perimeter P is a function of the number of tiles t. a.Write down any observations you notice about the pattern and how the pattern changes with each figure added. b.Consider the 10 th figure in the pattern. Describe in words how to draw the 10 th figure and what the perimeter would be. Explain how the perimeter connects to the picture. c.What would the perimeter of the 100 th figure be? Explain how you know. d.Write a rule to related the perimeter P to the number of tiles t. Explain how each part of your formula relates to the tile pattern. e.Graph the function. Why does it make sense that the graph has the shape it does in relation to the pattern? t1234 P46810

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Fiona’s Task (F and F’) Task F The measure of each interior angle of a regular n-side polygon is 180(n – 2)/n. For example, the interior angle measure of a regular (equilateral) triangle is 180(3 – 2)/3=60°. 77. Find the interior angle measure of a square. 78.Find the interior angle measure of a regular pentagon. 79.Find the interior angle measure of a regular hexagon. 80. Does the interior angle measure of a regular n-sided polygon increase or decrease as the n increase? Task F’ 1. The sum of the measure of the interior angles of any triangle is 180°. What is the interior angle measure of a equilateral (regular) triangle? 2. A square is another regular polygon. What is the interior angle measure of a square? 3. What about other regular polygons? Explore the interior angle measure of regular pentagons and hexagons. (If you’re stuck, try using what you know about triangles and squares.) 4. Can you come up with a general formula for finding the interior angle measure of any regular n-sided polygon? Will this formula always work? How do you know ?

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George’s Task (G and G’) Task G 2. The number of dots in the figures below are the first four rectangular numbers a. What are the first four rectangular numbers? b. Find the next two rectangular numbers? c. Describe the pattern of change form one rectangular number to the next. d. Predict the 7 th and 8 th rectangular numbers. e. Write an equation for the nth rectangular number r. Task G’ The numbers of dots in the figures below are the first four rectangular numbers. Assume that the pattern continues. 1. Write down everything you observe about the pattern. 2. What are the first four rectangular numbers? 3. How do the numeric values relate to the picture? 4. Describe the picture of the 10 th rectangular number. 5. Use words, diagrams, or symbols to generalize the pattern. How do you know your generalization is true?

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Challenges Being clear about the goal for the lesson and how the modification helps accomplish it Making sure the content of the modification matches or exceeds the original content Making sure that the modification is not just a surface level change that doesn’t really increase opportunities to engage in reasoning-and-proof Ensuring that there is sufficient scaffolding for students to actually do what is being suggested

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Impact In the final interview, 7 of the 9 teachers indicated that modifying task influenced their understanding of reasoning-and-proving or their understanding of teaching reasoning-and-proving. A few comments from the teachers: …the one thing that I really like just for practical reasons was modifying a task. Modifying several tasks of our own, it’s real useable things that you can take with you…Not everybody is lucky enough to teach from a curriculum that promotes it on its own, so sometimes you’re given a traditional book, and now we have some ways in which we can modify a task to help. …I had a crappy curriculum this year and it’s nice to know that we don’t have to completely reinvent something. That we can take something and just kind of change the words around have it be a good helluva task. So I guess this course has perhaps shown me some useful examples, as well as given me some feedback from the professor and other classmates about ways that I can create or modify tasks that I’ve already created to better apply to the aspects of reasoning and proving that are discussed in the course.

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Reflecting on Your Own Practice What value (if any) do you see in modifying or “rebuilding” tasks?

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Thanks! This work was made possible through the National Science Foundation funded “Cases of Reasoning-and-Proving (CORP) in Secondary Mathematics” Project (NSF DRL # ) Peg Smith

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Comparing Tasks A and A’ Similar Both ask students to complete a conjecture about odd numbers based on a set of finite examples that are provided Different Task A’ asks students to develop an argument that explains why the conjecture is always true (or not) Task A can be completed with limited effort; Task A’ requires considerable effort – students need to figure out why this conjecture holds up

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Comparing Tasks B and B’ Similar Same geometry content Based on the same construction Relates to a specific theorem about the diagonals of a parallelogram Different Task B: perform the construction once, then explain why the figure is a parallelogram Task B’: perform the construction several times, then conjecture about the type of figure produced Task B: state a theorem Task B’: make an argument using known properties to explain why the same figure is produced each time

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Comparing Task C and C’ Same Both tasks ask students to find the perimeter of the pattern containing 12 trapezoids. Both tasks ask students to find a rule that can be used to find the perimeters of a pattern containing any number of trapezoids. Both tasks require students to identify the pattern of growth of the perimeter. Different Task C’ provides more scaffolding by first asking students to make observations and then to find the perimeter of the first four trapezoid patterns. In Task C finding the perimeter of the 12 trapezoid pattern is after the general rule; in Task C’ finding the perimeter of the 12 trapezoid pattern is before the general rule. In Task C’ students are asked to create a generalization to explain why it always works.

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Comparing Tasks D and D’ Similar Both tasks ask students to calculate the answer when -2 and -3 are raised to odd and even powers Both tasks ask students to consider the sign of the products when odd and even integers are raised odd and even powers Different Task D states the two options for the conjecture while Task D’ asks students to make the conjecture Task D’ invites students to first make observations about what it occurring Task D’ moves beyond the examples by asking how they will know that the conjecture will always be true

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Comparing Tasks E and E’ Similar Both tasks use the same table and figures. Both tasks ask students to write a rule to describe the relationship between the perimeter and the number of tiles Both tasks ask students to graph the function. Different Task E’ first asks students to make observations about how the pattern changes. Task E’ asks students to consider figures 10 and 100 before writing the rule Task E’ asks students to relate the rule to the picture Task E’ asks students to relate the shape of the graph to the pattern

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Comparing F and F’ Both tasks involve the sum of the measures of the interior angles of regular polygons Both begin with triangles then move to squares, hexagons, and pentagons F gives students the formula; F’ engages students in an exploration and coming up with the formula F involves “plugging” values into the formula and getting answers. Only question 80 requires any level of thinking and it isn’t much; F’ is very open and students need to look for a pattern, use the pattern to generalize, and then to show that it always works

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Comparing G and G’ Both tasks involve the same pattern of growth Both ask students to determine the values of the first four rectangular numbers Both ask for a generalization G asks for an equation; G’ allows students to use different representations to express their generalization G begins with calculating; G’ begins with making observations G concludes with writing and equation; G’ asks students how they know the generalization will always work

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Task Modification Identify one task from a textbook that could be modified to enhance students’ opportunities to reason-and-prove. Modify the task with one or more strategies previously discussed and attach a copy of the original task and modified version of the task. Explain: How is the modified task similar to and different from the original task? In what way(s) does the modified task enhance students’ opportunities to reason-and-prove? What specific strategy or strategies did you use to make the modification?

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