VIEWS: 14 PAGES: 110 POSTED ON: 2/26/2012 Public Domain
1 Questioning To Expose and Evoke Thinking and Understanding in Mathematics Marian Small 2 Today’s Agenda How questions differ Questions to focus on the important mathematics Evoking reasoning by piquing curiosity Questions to support DI Questions to improve the practice element of lessons (if we have time) 3 How Questions Differ in Evoking Student Thinking 4 Let’s look at an example You are considering how to subtract linear and quadratic polynomials. Algebra tiles are available. You could ask: What is (-3x) – (-4x)? Or you could ask: Explain how you calculated (- 3x) – (-4x). Or you could ask: Why does it make sense that, for positive x, (-3x) – (-4x) is more than (-3x) – (-x)? …. 5 Let’s look at an example Or you could ask: How could you use a model to show what (-3x) – (-4x) should mean? Or you could ask: Choose two polynomials and subtract them. What is the difference? How do you know? 6 Differences in Intent Do I want them to be able to get an answer? [What is (-3x) – (-4x)?] Do I want them to be able to explain an answer? [Explain how you calculated (-3x) – (-4x).] Do I want them to be able to describe why a particular answer makes sense? [Why does it make sense that, for positive x, (-3x) – (-4x) is more than (- 3x) – (-x)? ] 7 Differences in Intent Do I want them to see how a particular aspect of mathematics fits into a bigger picture? [How could you use a model to show what (-3x) – (-4x) should mean?] Do I want to ensure that all students can answer? [Choose two polynomials and subtract them. What is the difference? How do you know?] Which of these types of questions are important to you? All of them? some of them? why? 8 An Important Message It is important that even struggling students meet questions with these various intents, including (maybe even especially) making sense of answers and relating to other math ideas. It is important that questions focus on the math that matters. 9 Are Intent Differences the Same as Process Differences? 10 The Processes Problem solving Reasoning and proving Reflecting Selecting tools and strategies Connecting Representing Communicating 11 Let’s Compare You could ask: What is (-3x) – (-4x)? [process elicited unclear] Or you could ask: Explain how you calculated (-3x) – (-4x). [communicating; selecting tools and strategies] Or you could ask: Why does it make sense that, for positive x, (-3x) – (-4x) is more than (-3) – (-x)? [reasoning; connecting] 12 Let’s Compare Or you could ask: How could you use a model to show what (-3x) – (-4x) should mean? [representing; communicating; connecting] Or you could ask: Choose two polynomials and subtract them. What is the difference? How do you know? [communicating; ???] 13 Another example A table of values begins as below. 1 2 3 4 200 180 160 140 Some possible questions: What comes next? How do you know? [explaining; communicating] Why does it make sense that (10,20 ) is an entry in the table? [making sense of an answer; reasoning]..... 14 Another example (cont’d) 1 2 3 4 200 180 160 140 Can you be sure what the other entries in the table would be? [relating to other mathematical ideas; connecting] How many entries would you need if you were told that the decreases in the term value are constant? [relating to other mathematical ideas; connecting] 15 And one more…. A grade 9 student says that the volume of a cylinder cannot be the same as the volume of a cone; it always has to be three times as much. Do you agree? Explain. [making sense; communicating, reasoning] Or one cylinder is twice as tall and twice as wide as another. Does it make sense that the volume of the bigger cylinder is twice as much? [making sense; communicating, reasoning] 16 And one more…. Or why is it useful to have a formula for the volume of a cone that is based on its radius? [bigger math idea; connecting] Or How is it possible for a shape like a cylinder or cone to have exactly the same volume as a prism shape? [bigger math idea; connecting] 17 Now you try Choose a different topic. Think about and be ready to share what is important in this topic. Then let’s focus on two “new types” of questions presented earlier. Create a question where the intent is that students are able to describe why a particular answer makes sense. OR where the intent is that students see how a particular aspect of mathematics fits into a bigger picture Write your questions on chart paper and post. 18 And so… It is important to identify the math that matters. It is important to develop questions that focus on students making sense of the math. It is important to practice developing questions that focus on building connections- how new math ideas are related to and built on older ones 19 Differences in Cognitive Complexity 20 Bloom’s taxonomy Knowledge Comprehension (info in another form) Application Analysis Synthesis (original, creative thinking) Evaluation (how a concept is consistent with values) Gallagher and Ascher’s 21 Taxonomy Cognitive- memory Convergent thinking Divergent thinking Evaluative thinking How would you categorize these 22 Grade 9 text questions? Substitute the given values and evaluate each expression: e.g. 3x + 5 when x = 2 On a multiple choice test- you earn 2 points for each correct answer and lose 1 point for each incorrect answer. a) Write an expression for a student’s total score. b) Maria answered 15 questions correctly and 3 incorrectly. Find her total score. How would you categorize these 23 Grade 9 text questions? Describe a situation that can be modelled by an algebraic expression. Illustrate your expressions using algebra tiles or a diagram. A student is confused about the difference between a term and a polynomial. Explain so that she could understand the difference. Consider the expression 3(x + 5). Why do you need to use the distributive property? Why can you not just add the terms in the brackets first? Relating processes to cognitive 24 complexity Problem solving Reasoning and proving Reflecting Selecting tools and strategies Connecting Representing Communicating 25 For example… Representing: Notice the different levels: What does -5x look like if you represent it with algebra tiles? Which representation of x2 + 2x + 1 helps you see that it is a perfect square? When could the representation of a repeating decimal as a fraction be helpful? 26 For example… Communicating: Notice the different levels: What is a quadratic relation? Why does it make sense that a polygon with more sides has a greater total angle degree? How do you know that if you cut up a pentagon in different ways that the total angle sum will still be the same? 27 Look at a resource Use one lesson from a resource you have with you. Where do most of the “questions” fall? Are they low level, mid-level, high level? In the course of a lesson, what is the sequencing? <For example, in one resource I examined, in a Grade 9 lesson on equation solving, the questions went like this: L, L, H, L,L, L, L,L,L,L,L,L,L,L,L,L,M,L,L,M,H,H> 28 Some examples Can we hear some examples of the mid level questions you met? The high level questions you met? Cognitive complexity categories 29 don’t… Tell us if the question matters Tell us if the question is appropriate for the student to whom it is addressed Tell us how inclusive the question is 30 And so… It is important that struggling students need to meet questions in the higher categories. It is important to understand that it may be necessary to “scaffold” them up, but there are concerns about how much to do this; we want students to become independent and self-scaffold, rather than waiting for our scaffolding all the time. (see next slide) Mosenthal taxonomy 32 a periodic table of learning Type of requested info (6 zones from naming a person, thing, etc. to an indeterminate question) 8 types of processing (from identify to persuade) 4 types of match (from locate to generate) 33 Keep in mind… Keep in mind that cognitive complexity is a different thing than interest-generating. We’ll talk a bit more about this later. 34 Compare these I could ask: You had $1500 in the bank. You earned 4% simple interest. How much would you earn in 3 months? Or I could ask: You had some money in the bank. You earned $15 interest in one year. What might the rate have been and how much might you have had in the bank? 35 Possible answers to question 2 10% on $150 1% on $1500 5% on $300 2% on $750 50% on $30 36 Differences in Open-ness Think about this-- Are my questions useful for the whole range of students or only : Average? Average and above average? Below average and average? The most below average students? 37 Another example Compare the question: Which difference (i.e. 1st difference, 2nd difference, 3rd difference, …) is constant for a quadratic function? to How are linear and quadratic functions alike? How are they different? 38 We’ll look more at open-ness later. 39 Teachers need to know that.. Right now, most questions (and there are a lot of them) are low-level and convergent and not open. Although questions in a math class are usually used to assess understanding, it is not as much to elicit what the student understands about the concept, but whether they follow the details and/or the instructions for a task. 40 Rapid-fire What is the slope of the line that goes through (1,3) and (2,5)? What does slope mean? How do you calculate it? What does change in y mean? etc. 41 Exposing student thinking What do they know? How do you find out what 42 students know? First you think about what you mean by the word “know”. Suppose, for example, the issue is the Pythagorean theorem. What do you want students “to know”? 43 Is it… Is it what the theorem says numerically? geometrically? Is it how to solve for one side length if you know the other two? Is it how to use it to solve a problem when the use of the theorem is obvious? Is it to how to use it to solve a problem when its usefulness is less obvious? 44 45 46 Let’s try another topic Consider the topic of quadratic relations. What might we want students to know? For example 47 (cont’d) - How to write it in factored form - What information factored form provides - What the graph will look like - The effect of coefficient/constant changes on the graph - How many points define the quadratic uniquely - What situations quadratic relations describe 48 (cont’d) - Write 3x2 – 11x -4 in factored form. - How does writing a quadratic relation in factored form make it easier to graph it? - Is the graph of y = 3x2 + 4x – 5 more like the graph of y = 3x2 -5 or more like the graph of y = 4x – 5? Explain. 49 (cont’d) - You graph y = 6x2 + 5x + 1. Does the graph change more if you increase the 6 to 7, the 5 to 1, or the 1 to 2? - Jane said that you need to know 3 points on a parabola to graph it; Aaron said that you need 2 and Amanda said that you need 4. Who is right? How do you know? - Describe a real-life situation that could be described by a quadratic relation. How did you know that a quadratic would make sense? 50 You try Try this in a group. You want to find out what students know about linear equations with integer coefficients and/or solutions. What kind of middle to higher level question might you ask? What will a student response tell you about what the student knows? Play act to figure out what “scaffolding” questions you might need if kids get stuck and how you decide when and/or whether to scaffold. Write your questions on chart paper and post. 51 Some possibilities Why is it useful to isolate the term involving the variable to solve an equation? Why does it not change the solution to the equation? If you have an equation like -3x – 2 = 4x + 12, how do you know that the answer is probably going to be negative? How do you know that there are a lot of equations with a solution of x = -3? How is solving -3x – 2 = 4x + 12 like solving 3x – 2 = 4x? How is it different? 52 And so… You need to think about what it means to know. The curriculum expectation helps, but it is limited. A more detailed analysis is required. This is something for teachers to do with colleagues. 53 How do you evoke reasoning by piquing curiosity? Investigation of Surprising 54 Phenomena Try to explain these 55 Example 1 56 One way to subtract is like this: 5 11 - 3 86 275 = 200 – 70 – 5 = 125 57 How could we explain? Using models 58 How could we explain? Using integer arithmetic 500 + 10 + 1 – (300 + 80 + 6) = (500 – 300) + (10 – 80) + (1 – 6) = (500 – 300) – (80 – 10) – (6 – 1) 59 Example 2 Multiply numbers between 5 60 and 10 On one hand, raise the number of fingers that tells how much more than 5 the first value is. On the other hand, raise the number of fingers that tells how much more than 5 the second value is. The fingers up are the tens. Multiply the fingers down for the ones. 61 An explanation Call the first number (5+a) and the second one (5+b). The fingers raised, respectively, are a and b. The fingers down, respectively, are (5 – a) and (5 – b). The result, based on the trick is 10(a+b) + (5 – a)(5 – b). • Simplifying the expression above gives 10a + 10b + 25 – 5a – 5b + ab. [This happens to be 25 + 5a + 5b + ab) 62 Example 3 63 Choose three consecutive numbers, square them, and add the squares. Divide by 3 and record the whole number remainder. What happened? Why? 64 So why was the remainder 2 (n-1)2 + n2 + (n+1)2 = 3n2 +2 But it’s also true that n2 + (n+1)2 + (n+2)2 = 3n2 + 6n + 5 = 3n2 + 6n + 3 + 2 65 Or use a model 66 Example 4 67 Comparing fractions To get a fraction between two other ones, add the tops and add the bottoms, e.g. Between 3/5 and 8/9 is 11/14. Between 1/3 and 4/5 is 5/8. Is this always true? 68 An algebraic approach If a/b < c/d, then To compare a/b (a+c)/(b+d) and c/d, then ad(b+d) < bd(a+c) < bc(b+d) = abd + ad2 < abd +bcd < b2c + bcd (since ad <bc) (since ad <bc) 69 A visual approach 70 Compare these You are teaching how to create the equation of a line knowing its slope and a point on the line. You have not yet taught the rule, but students know the concept of what the slope is. You could ask: A line goes through (2,5) and its slope is 3. What is the equation of the line? Or you could ask: A line goes through (2,m). Its slope is m. What has to be true about the equation of the line? 71 Lines through (2,m) and slope m 72 Multiplying binomials Compare these: Use your algebra tiles to show what (x + 2)(3x + 4) looks like. OR Tara said that if you multiply (ax + b) (cx +d) and a, b, c, and d are consecutive integers, the number of algebra tiles needed to show the product is odd. Decide if you agree or not and why. 73 Try… (2x + 3)( 4x + 5) (x + 2) (3x + 4) (5x + 6)(7x + 8) Does the number of tiles being odd or even change if one or both of the + signs become a – sign? 74 And so… Teachers need to recognize that engagement means interesting mathematical situations, not just questions about “skateboarding”. Kids, for the most part, are curious. 75 The Power of Open Questions To find out student thinking, but include all students 76 A graph goes through the point (1,0). What could it be? 77 Possible responses x=1 y=0 y = x- 1 y = x2 - 1 y = x3 - 1 y = 3x2 -2x -1 78 A sequence question An arithmetic sequence begins with 8 and goes up by 3. What is the 20th term? The number 48 appears in an arithmetic sequence. What’s the sequence and where does it appear? 79 Possibilities 8, 16, 24, 32,… <6th term> 48, 50, 52, 54,… <1st term> 1, 2, 3,…. <48th term> 80 Using powers Write 88 as the sum of powers. 81 Possibilities 12 + 12 + …. + 12 (88 of them) 22 + 22 + … + 22 (22 of them) 52 + 52+ 52+ 22 + 22 + 22 + 12 52 + 62+ 33 82 Trig question Instead of: A ladder leans against a wall at an angle. The angle it makes with the ground is 25°. If the ladder rests 2.8 m up the wall, how long is the ladder to the nearest tenth of a metre? You could ask: • A ladder leans against a wall at an angle. If the ladder rests 2.8 m up the wall, how long could the ladder be and what angle does it make with the ground? 83 Or this.. A farmer wants to build enclosures for his animals. He has 40 m of fencing to build three, identical adjacent square pens. Use an equation to represent the amount of fencing you need and solve it to solve the problem. You could not force the use of an equation, could allow the students to choose the amount of fencing and/or choose the number of pens. 84 Some “opening up strategies” Start with the answer instead of the question. Ask for similarities and differences. Leave the values in the problem somewhat open. 85 Start with the answer. The solution to the equation is x = 2. What is the equation? The difference of two rational numbers is -3/5. What are the rational numbers? The slope of the line is ¾. What points does the line go through? The value of a power is 6561. What could the power be? 86 Try it Open your resource to a lesson. Choose a convergent question and open it up by starting with the answer. 87 Similarities and differences. How are quadratic equations like linear ones? How are they different? How are arithmetic sequences like geometric ones? How are they different? How is dividing rational numbers like dividing integers? How is it different? 88 Try it Use the resource you brought. Find a convergent question or situation. Describe how to open it up using the similarities and differences strategy. 89 Leaving values open Choose values for the height and radius of a cone. Calculate its surface area. 90 You try Choose three questions from a text you use (or some other source you use) and open them up. OR open up three of these: Determine the sum of the interior angles of a pentagon. Write the equation of the line goes through (2,6) and a slope of -3. Graph y = 2(3x-4)2 + 8. Solve the equation x – 4 = 13. A cube shaped box has sides 10 cm long. What is the surface area of the largest cone that fits in the box? Write your questions on chart paper and post. 91 The Power of Parallel Questions 92 The idea is to use two similar tasks that meet different students’ needs, but make sense to discuss together. 93 A proportion example You used 240 g of rice. What was the total mass of the rice if… Task A: It was 1/3 of the total mass of the rice. Task B: It was 2/3 of the total mass of the rice. Task C: It was 40% of the total mass of the rice. • How do you know the mass is more than 240 g? • Is the mass more than double 240? • How did you figure out the mass? 94 Another one.. Task A: One electrician charges $45 an hour in addition to an automatic service call fee of $35. Another electrician charges $85 an hour and no additional fees. What would each company charge for a 40 minute service call? Task B: An electrician charges $75 an hour and no other fees. How much would she charge for a 40 minute service call? • How do you know the charge would be more than $40? • How did you figure out the fee? 95 Transforming graphs Task A: Compare the graphs of y = 3x2 + a for different values of a. Task B: Compare the graphs of y = 3[a(x-4)2] + 3 for different values of a. • What shape graph will it be? How do you know? • If a increases, does the graph get wider or narrower or neither? • If a increases, does the graph move up or down or neither? • How do your various graphs compare? 96 Systems of linear equations Task 1: Task 2 • How did you use the first piece of information? The second piece? • How did you know the numbers could not both be negative? • How did you know that their difference was -18? 97 Slopes of tangents Task 1: Determine the slopes of the tangents to f(x) = 2x3 + 3x2 algebraically. Task 2 Graph f(x) = 2x3 + 3x2 . Draw tangent lines to it at various points. How are the slopes changing? • What is the slope at 0? How do you know? • Where are the slopes negative? Where are they positive? How do you know? • How can you calculate the slopes? 98 Volume Determine the volume of one of these shapes. 15 cm 15 cm 20 cm 20 cm 15 cm 15 cm • Did you need all three measurements? Why? • What formula did you use? • How did you use the formula? 99 You Try Choose a task you typically assign. Come up with a meaningful way to set a parallel task so that the questions you ask after the completion of the task pertain to both tasks. Write your questions on chart paper and post. 100 Possibility You create a graph based on the table of values you choose. What is the y-coordinate of the point on the graph where the x-coordinate is 10? Task 1: 1 2 3 4 8 11 14 17 Task 2: 1 2 3 4 280 584 891 1187 101 Be aware Open and parallel questions are based on first deciding what the underlying big idea is so that the question is about that idea. Open questions are a bit vague, but carefully so, and focusing on the big idea which must be brought out. Parallel questions are focused on the same idea and they have to be close enough that the same “debrief” questions can be used. 102 Be aware Parallel tasks are used when more specificity is needed than with open tasks. A teacher must analyze, and not superficially, how students might differ to decide how to differentiate the tasks. 103 Better practice questions are also valuable They should include not just doing, but reasoning, too. 104 Place the digits 0-9 into the right spots. A line with slope []/[] goes through (9,[]) and ([],1) A line with slope 3/4 goes through ([],2) and ([],[]) A line with slope 5/7 goes through ([],6) and ([],[]) 105 Place the digits 0-9 into the right spots. A line with slope 7/3 goes through (9,8) and (6,1) A line with slope 3/4 goes through (0,2) and (4,5) A line with slope 5/7 goes through (9,6) and (2,1) 106 Solving right triangles Create 5 triangles using 2 pieces of information from the list below. Do not use the same information each time. If you use two side lengths, solve for an angle in the triangle. If you use an angle and a side length, solve for another side length in the triangle. 61° 52 cm 32 cm 20 cm 49° 35 cm 37° 30 cm 12 cm 40° 107 Underlying Theme We looked at the notions today that: Questions should focus on more meaningful mathematics than they sometimes do. At least a fair proportion of questions should be higher level and divergent more than they sometimes are. Questions should engage students. 108 Underlying Theme Questions should address the needs of as broad a spectrum of students as possible. We should encourage students to try to respond to the question without our scaffolding if at all possible, rather than making them reliant on our scaffolding. 109 In Summary You need to choose where to focus first, but then consider gradually working on - Figuring out what it means for students to know, e.g. Having students describe why an answer makes sense - Having students consider how an idea fits in the bigger mathematical scheme of things (the big ideas) - Using questions that pique curiosity - Considering what are appropriate follow-up or scaffolding questions, but ones that do not eliminate the need for thinking 110 In Summary - Making questions more inclusive by using more open questions - When questions are not as inclusive, considering offering choices by using parallel questions 111 You can download at… www.onetwoinfinity.ca (quick links: Upper Canada DSB)