Denisse R. Thompson University of South Florida, USA 2011 Annual Mathematics Teachers Conference Singapore June 2, 2011 “Reasoning mathematically is a habit of mind, and like all habits, it must be developed through consistent use in many contexts.” (Principles and Standards for School Mathematics, p. 56) Recognize reasoning & proof as fundamental aspects of mathematics Make and investigate conjectures; Develop and evaluate mathematical arguments and proofs; Select and use various types of reasoning and methods of proof. (Principles and Standards for School Mathematics, p. 56) Singapore curriculum framework “… the possibility of proof is what makes mathematics what it is, what distinguishes it from other varieties of human thought” (Hersh, 2009, p. 17) “Students cannot be said to have learned mathematics, or even about mathematics, unless they have learned what a proof is” (Hanna, 2000, p. 24) Meaning or purpose of a proof Use of empirical examples as a proof Lack of knowledge of needed concepts Definitions and notation Unfamiliarity with proof strategies Knowing how to get started Monitoring one’s progress while attempting a proof How can we ensure that students have many opportunities to engage with reasoning, proof, and justification throughout their secondary curriculum? How can those opportunities provide teachers with insight into their students’ thinking that can help modify and enhance instruction? The textbook is a “variable that on the one hand we can manipulate and on the other hand does affect student learning.” (Begle, 1973, p. 209) Look for opportunities within the textbook, and when not present, consider how we might modify items or tasks to engage students in reasoning and explaining their thinking. finding counterexamples making conjectures investigating conjectures developing arguments evaluating arguments correcting mistakes in logical arguments (Johnson, Thompson, & Senk, 2010) The use of examples and non-examples is an important prerequisite to making and evaluating conjectures. One or several examples cannot prove a generalization true. But one counterexample can disprove a statement. Epp (1998) argues that finding counterexamples is easier than writing a proof – good first step. Give an example to show that m – n = n – m is not necessarily true. Find a counterexample to show that a2 > a is not always true. Give a counterexample to show that (x + y)2 = x2 + y2 is false. (Prentice Hall Algebra I, 2004) Notice that the directions tell students how to start. As students make a generalization, they may come to realize that a proof requires showing the statement is true in all cases. Notice that r2/r2 = 1, where r is not 0. Does this suggest a definition of a zero exponent? Explain. (Holt Algebra I, 2004) The quadratic formula provides solutions to ax2 + bx + c = 0. Make up some rules involving a, b, and c that determine the solutions are non-real. (Key Advanced Algebra, 2004) b b 4ac x 2a 2 Consider the equation y = 3x. Write a conjecture about the relationship between the value of the base and the value of the power if the exponent is greater than or less than 1. (Glencoe Advanced Math: Precalculus, 2004) Students do not necessarily know if the conjecture is true or false, so they have to bring other reasoning skills to bear. ◦ This is more aligned with the way that mathematicians work. Determine whether the pair of monomials (5m)2 and 5m2 is equivalent. Explain. (Glencoe Algebra I, 2004) ◦ There might be several ways that students could explore this conjecture – try some numbers, graph the two expressions, use an algebraic proof. ◦ Students with different learning styles have different ways to engage with the problem. If you use a calculator to graph y = x2 and y = x4 it may look as if x2 x4 for all values of x. Use the zoom feature on a graphing calculator and inspection of tables for each relation to test that conjecture. (Core Plus Course 3, 1999) Deductive arguments might occur for specific cases as a precursor for more general cases, what we typically consider as a proof. Explain how you could verify that the Product-ofPowers Property is true for 23 * 24. (Holt Algebra I, 2004) Write a convincing argument to show why 30 = 1 using the following pattern. 35 = 243, 34 = 81, 33 = 27, 32 = 9, … (Glencoe Algebra I, 2004) The following statements support the reasoning behind the definition of a0 for all positive values of a. For each step shown, supply a general property of number operations to support that step. 1=ax-x = a0 So, 1 = a0 (Core Plus Course 2, 1998) On one chemistry test, Amelia scored 97 when the class mean was 85 with a standard deviation of 4.8. On a second chemistry test, Amelia scored 82 when the class mean was 75 with a standard deviation of 2.7. On which test did Amelia score better in relation to the rest of the class? Explain your reasoning. Evaluating an argument is at a different level than writing one’s own argument. A teacher or peer may have used a different approach, and students need to be able to determine if these arguments are valid or not. An algebra class has this problem on a quiz: Find the value of 2x2 when x = 3. Two students reasoned differently. Student 1: Two times three is six. Six squares is thirty-six. Student 2: Three squared is nine. Two times nine is eighteen. Who was correct and why? (Key Discovering Algebra, 2007) Students are told there is a mistake and they have to find it. This type of task is similar to evaluating an argument, except that students know there is an error. Find the error. Nathan and Poloma are simplifying (52)(59). Nathan Poloma (52)(59) = (5 * 5)2+9 (52)(59) = 52+9 =2511 =511 Who is correct? Explain your reasoning. (Glencoe Algebra I, 2004) Find the error. x2 + 2x = 15 x(x + 2) = 15 x = 15 or x + 2 = 15 x = 15 or x = 13 (Glencoe Algebra II, 2004) The following statements appear to prove that 2 is equal to 1. Find the flaw in this "proof." Suppose a and b are real numbers such that a = b, a ≠ 0, b ≠ 0. a=b a2 = ab a2 - b2 = ab - b2 (a – b)(a + b) = b(a – b) a+b=b a+a=a 2a = 1 2=1 (Glencoe Algebra I, 2004) Use vocabulary to signal that proof-related reasoning is needed ◦ ◦ ◦ ◦ ◦ ◦ Explain Explain why Why Show Show that Prove Highlight concepts that you know are potential difficulties for students ◦ ◦ ◦ ◦ Through finding counterexamples Through investigating conjectures Through identifying common errors Through creating an argument and having students evaluate it Use examples of student work (anonymously) to generate tasks, particularly for evaluating arguments or correcting mistakes Consider using language that does not give away the answer ◦ Prove or disprove ◦ True or false ◦ Is the student correct? Why or why not? Replace 1 or 2 problems in each homework assignment with tasks in which students are expected to engage in reasoning ◦ Students need to be convinced that such tasks are not going away Name a decimal that estimates the value of point A. 0 A 1 Why did you give A that value? Response 1 0 .5 A = .75 1 It w as about half way between my .5 mark 1 and 3 /.75 the way betw een 0 1 4 Response 2 A = .9 0 1 I g ue ssed Response 3 (Chappell & Thompson, 1999) 0 A = 15 1 Be c a u se w he n I d ivid ed 01 in to 20 = pa r ts A = 17 Do .3 and .30 name the same amount? Explain your answer. Response 1 ◦ No, because .3 is three and .30 means thirty so they can’t be the same amount Response 2 ◦ Yes, zeros put on a decimal like 0.3 or .30 don’t matter. Zeros put on a decimal like .03 do matter Response 3 ◦ Yes, .3 = .30 because saying .3 instead of .30 is just reducing it. 3 30 The first one is 10 100 reduced Typical problem: ◦ An item normally costs $250 but is on sale for 20% off. What is the sale price, before tax? Possible revision to encourage reasoning: ◦ When an item is on sale at 20% off, you can always find the costs of the item (before tax) by multiplying its original price (non-sale) price by .80. True False If you marked True, explain why this works. If you marked False, explain why the statement is false. (Thompson et al., 2005) Pick a specific price and show that both ways work. Pick an arbitrary price, x, and use the distributive property to show that x – 0.2x = (1 - 0.2)x = 0.8x Responses to such tasks help us learn whether students have a conceptual understanding of the mathematical principles or whether they are just following a set of procedures rotely. For all numbers x and y, is it true that x2 + y2 = (x + y)2? ◦ Yes No Imagine that someone does not know the answer to the question. Explain how you would convince that person that your answer is correct. Student Response 1 Student Response 2 ◦ Well, just take, for example, x = 8 and y = 6 ◦ So 82 + 62 = 100 and (8 + 6)2 = 196. So it’s wrong to say “all numbers” ◦ ◦ ◦ ◦ Show any two in here 52 + 62 = (5 + 6)2 25 + 36 = (25 + 36)2 61 612 ◦ 42 + 82 = ( 4 + 8)2 ◦ 16 + 64 122 = 24 Is (x + 4)2 = x2 + 16? Explain why or why not. Sample Responses with graphing calculators ◦ No, (x + 4)2 = 49 and x2 + 4 = 13 ◦ Yes, (x + 4)2 = 16 and x2 + 4 = 16 What caused the difference? Another variation: Is (x + 4)2 = x2 + 16 always true, sometimes true, or never true? Explain. ◦ Students failed to realize that the calculator evaluated the expression for the value that is stored in x. Typical problem: Possible revision: Approaches: ◦ Write y = 4x2 + 24x + 31 in vertex form. ◦ On a test, one student found an equation for a parabola to be y – 5 = 4(x + 3)2. For the same parabola, a second student found the equation y = 4x2 + 24x + 31. Can both students be right? Explain your answer. Graph both equations Expand the first one Rewrite the second into vertex form Substitute a value for x into both equations – if two different y-values result the two equations are not equal ◦ It is possible that neither is correct. ◦ ◦ ◦ ◦ With someone near you, take one of the following problems and write 2 modifications to engage in proof-related reasoning. Grade 7: Solve 4x – 7x < -24 Grade 8: Find the mean and median of a set of data. Grade 9: The product of two consecutive integers is 552. Find the integers. Grade 10: Given that vector a = (6, 8) and vector b = (r, 0), where r is positive, find the value of r such that |a| = |b|. Original item: ◦ Solve 4x – 7x < -24 Possible revisions: ◦ Correct the mistake in the following solution: 4x – 7x < - 24 – 3x < -24 x<8 Find a counterexample to show that x < 8 is not the solution to 4x – 7x < -24. Original item: ◦ Find the mean and median of a set of data. Possible revisions: ◦ True or false. Explain. In any data set, the mean is always greater than the median. ◦ Show that when 5 is added to every value in a data set, the mean and median both increase by 5. ◦ Find a set of 10 values so that the mean is 25 and the median is 18. Original item: ◦ The product of two consecutive integers is 552. Find the integers. Possible revisions: ◦ To find two consecutive integers whose product is 552, Balpreet first took the square root of 552. She got 23.49468025. So, she decided the numbers were 23 and 24. Will her method always work? Justify your solution. ◦ Jericho found the product of 12 and 46 to be 552. Do his numbers satisfy the problem? Why or why not? Original item: Possible revisions: ◦ Given that vector a = (6, 8) and vector b = (r, 0), where r is positive, find the value of r such that |a| = |b|. ◦ Under what conditions would the two vectors a = (6, 8) and b = (r, 0) have congruent magnitudes? Explain. ◦ Marshall wanted to find the value of r so that vector b = (r, 0) and vector a = (6, 8) have equal magnitudes. He submitted the following work: Sqrt (r + 0) = sqrt (62 + 82), so r = 100. Evaluate his reasoning and correct any errors. Thank you! [email protected] Begle, E. (1973). Lessons learned from SMSG. Mathematics Teacher, 66, 207-214. Bellman, A. E., Bragg, S. C., Charles, R. I., Handlin, Sr., W. G., & Kennedy, D. (2004). Algebra 1 Florida Teacher’s Edition. Needham, MA: Pearson Prentice Hall. Chappell, M. F., & Thompson, D. R. (1999). Modifying our questions to assess students’ thinking. Mathematics Teaching in the Middle School, 4, 470-474. Coxford, A. E., Fey, J. T., Hirsch, C. R., Schoen, H. L., Burrill, G., Hart, E. W., Watkins, A. E., Messenger, M. J., & Ritsema, B. E. (1998b). Contemporary mathematics in context: A unified approach Course 2. Chicago, IL: Everyday Learning. Coxford, A. E., Fey, J. T., Hirsch, C. R., Schoen, H. L., Burrill, G., Hart, E. 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