Questioning by fjzhangxiaoquan

VIEWS: 14 PAGES: 110

									                                                    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




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 www.onetwoinfinity.ca (quick links: Upper Canada
  DSB)

								
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