The Dilemma of the Launch: Why is it so Difficult for Mathematics

Report

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
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
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
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
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).
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).
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).
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).
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 reexplain 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.
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
http://blog.mrmeyer.com/
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
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 ______?
1+1=2
1+3=4
3+5=8
7 + 11 = 18
13 + 19 = 32
201 + 305 = 506
30. Conjecture: The product of any two odd
numbers is ____?
1x1=1
1x3=3
3 x 5 = 15
7 x 11 = 77
13 x 19 = 247
201 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 ______?
1+1=2
1+3=4
3+5=8
7 + 11 = 18
13 + 19 = 32
201 + 305 = 506
30. Conjecture: The product of any two odd
numbers is ____?
1x1=1
1x3=3
3 x 5 = 15
7 x 11 = 77
13 x 19 = 247
201 x 305 = 61,305
Comparing Two Versions of a Task:
How are they the same and how are they different?
TASK B’
TASK B
Consider the construction below.
VISUAL REASONING Explain why the following
method of drawing a parallelogram works.
State a theorem to support your answer.
1.Use a ruler to
draw a segment
and its midpoint.
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.
2. Draw another
segment so the
midpoints
coincide.
3. Connect the
endpoints of the
segments.
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.
Comparing Two Versions of a Task:
How are they the same and how are the different?
Task C
Task C’
GEOMETRY For Exercises 45 and 46, use the
diagram below that shows the perimeter of the
pattern consisting of trapezoids.
45.
46.
Write a formula that can be used to find the
perimeter of a pattern containing n
trapezoids.
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.
What is the perimeter of the pattern
containing 12 trapezoids?
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.
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).
Do the differences matter?
YES!
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?
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.
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.
Donald’s Task (D and D’)
TASK D
a. Simplify each expression.
Task D’
1. Solve the following examples.
(-2)2
(-2)3
(-2)4
(-2)5
(-2)2
(-2)3
(-2)4
(-2)5
(-3)2
(-3)3
(-3)4
(-3)5
(-3)2
(-3)3
(-3)4
(-3)5
2.
b. Make a Conjecture Do you think a negative
number raised to an even power will be
positive or negative? Explain.
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?
c. Do you think a negative number raised to
an odd power will be positive or negative?
Explain.
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?
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.
t
1
2
3
4
P
4
6
8
10
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.
Elaine’s Task (E and E’)
TASK E
TASK E’
Patterns The table of values below describes the
Patterns The table of values below describes
the perimeter of each figure in the pattern of perimeter of each figure in the pattern of blue tiles. The
blue tiles. The perimeter P is a function of the perimeter P is a function of the number of tiles t.
number of tiles t.
a. Write down any observations you notice about the
pattern and how the pattern changes with each
t
1
2
3
4
figure added.
P
4
6
8
10
b. Consider the 10th figure in the pattern. Describe in
words how to draw the 10th figure and what the
perimeter would be. Explain how the perimeter
a. Choose a rule to describe the function in the
connects to the picture.
table.
c. What would the perimeter of the 100th figure be?
Explain how you know.
A. P = t + 3 B. P = 4t C. P = 2t + 2 D. P = 6t – 2
d. Write a rule to related the perimeter P to the
b. How many tiles are in the figure if the
number of tiles t. Explain how each part of your
perimeter is 20?
formula relates to the tile pattern.
e. Graph the function. Why does it make sense that
c. Graph the function.
the graph has the shape it does in relation to the
pattern?
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°.
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?
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?
George’s Task (G and G’)
Task G
Task G’
2. The number of dots in the figures below
are the first four rectangular numbers
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.
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.
2. What are the first four rectangular numbers?
3. How do the numeric values relate to the
picture?
4. Describe the picture of the 10th rectangular
d.
Predict the 7th and 8th rectangular
numbers.
e.
Write an equation for the nth rectangular
number r.
number.
5. Use words, diagrams, or symbols to
generalize the pattern. How do you know
your generalization is true?
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
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.
Reflecting on Your Own Practice
 What value (if any) do you see in modifying or “rebuilding”
tasks?
Thanks!
This work was made possible through the National Science
Foundation funded “Cases of Reasoning-and-Proving
(CORP) in Secondary Mathematics” Project (NSF DRL
#0732798)
Peg Smith
[email protected]
Comparing Tasks A and A’
Similar
Different
 Both ask students to
 Task A’ asks students to
complete a conjecture about
odd numbers based on a set
of finite examples that are
provided
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
Comparing Tasks B and B’
Similar
Different

 Task B: perform the construction
Same geometry content

Based on the same
construction

Relates to a specific
theorem about the
diagonals of a parallelogram
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
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.
Comparing Tasks D and D’
Similar
Different

Both tasks ask students to
calculate the answer when -2
and -3 are raised to odd and
even powers

Task D states the two options
for the conjecture while Task D’
asks students to make the
conjecture

Both tasks ask students to
consider the sign of the
products when odd and even
integers are raised odd and
even powers

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
Comparing Tasks E and E’
Similar
Different

 Task E’ first asks students to


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.
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
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
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


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
Both ask for a generalization
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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