# Questioning by fjzhangxiaoquan

VIEWS: 14 PAGES: 110

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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
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

 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

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

 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
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

 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.

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

 10% on \$150

 1% on \$1500

 5% on \$300

 2% on \$750

 50% on \$30
36

Differences in Open-ness
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

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
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
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
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

 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

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?
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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?

• 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”

 Ask for similarities and differences.

 Leave the values in the problem somewhat open.
85

 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
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

• 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

Determine the slopes of the tangents to f(x) = 2x3 +
3x2 algebraically.

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

 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?

1          2          3           4
8          11         14          17

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

 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

 A teacher must analyze, and not superficially, how
students might differ to decide how to differentiate
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
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
- 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