Reasoning and Proof
Urban Mathematics Leadership Network
Christine Avila
Mathematics Coordinator
February 11, 2014

What is Reasoning?
Reasoning: The lifeblood of mathematics.
The engine that drives problem solving.
Its absence is the root cause of teaching and learning by rote.
(Wu, 2011)

•   There is an intimate nature between our standards and our assessment.

Standard for Mathematical Practice #3
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies.
Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Claim 3: Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.

Targets for Claim 3
Target A : Test propositions or conjectures with specific examples.
(DOK 2).
Target B: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. (DOK 3, 4).
Target C: State logical assumptions being used. (DOK 2, 3)
Target D: Use the technique of breaking an argument into cases. (DOK
2, 3)
Target E: Distinguish correct logic or reasoning from that which is flawed and—if there is a flaw in the argument—explain what it is. (DOK 2, 3, 4)
Target F: Base arguments on concrete referents such as objects, drawings, diagrams, and actions. (DOK 2, 3)
Target G: At later grades, determine conditions under which an argument does and does not apply. (DOK 3, 4)
6

“ Children must adopt new rules for fractions that often conflict with wellestablished ideas about whole number ” (p.156)
Bezuk & Cramer, 1989

Reasoning starts in the classroom
•   The standards ask students to “ … extend previous understandings of operations with whole numbers.
”
Between which two consecutive whole numbers is the sum of 5/4 and 1/3?
Between which two consecutive whole numbers is the sum of 5/4 and 2/3?

4 +
5
2
= ?

4
5
2
= ?

How can K-2 work on operations with whole numbers and work on fractions in grade 3 support students ’ thinking about these problems in grades 4 & 5?
4 +
5
2
= ?
Even  before  learning  the   exact  sum,  can  students   tell  you  between  which   two  whole  numbers  the   answer  lies?
4
5
2
= ?
Even  before  learning  the   exact  product,  what  can   students  tell  you  about   the  value  of  the  product?
11

And all of that work pays off later when students can not only solve equations, but reason about them.
4 + x =
13
2
4 x = 10
12

Some progressions can be traced back pretty far in the standards …
What is the value of x in each of the following equations?
4x + 3x = 3x + 20
4x + xy = xy + 20
It is in grade 1 where students first learn the meaning of the equal sign:
1.OA.D.7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5 , 4 + 1 = 5 + 2.
No<ce  the  similarity  in  the   structure  of  the  grade  1  example.
13

Show the work a student might do
•   How would a student who has not been taught a procedure for solving this equation attempt to find a value for x that makes the equation true?
4x + 3x = 5x + 20
14

One potential way. Why might a student skip from x=5 to x=10? [MP.8?]
4x  +  3x  =  5x  +  20
If  x  =  3
If  x  =  4
If  x  =  5
If  x  =  10
21  =  35
28  =  40
35  =  45
70  =  70
15

MP.8 — Look For and Express Regularity in Repeated Reasoning
"Maintain oversight of the process, while attending to
the details.”
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Finding ways to measure important aspects of the standards
Which statement is correct about the value of x in the following equation?
4x + xy = xy + 20
(a) The equation is true for all real values of x.
(b) The value of x that makes the equation true depends on the value of y.
(c) There is exactly one value of x that makes the equation true.
(d) There is no value of x that makes the equation true.
17

Assessing “ Reasonableness ”

19

Explain the flaw in the chain of reasoning.
In the first group, 3/5 of the cats have spots. In the second group,
1/3 of the cats have spots. All together, 4/8 of the cats have spots.
Therefore, 3/5 + 1/3 = 4/8.

What do their futures look like?
Slide 21

Purposes of the Math Reasoning Project
•   Enhance the knowledge base regarding authentic evidence of mathematical reasoning.
•   Increase the validity of test scores by increasing the breadth of the Common Core standards that are measurable via computers.
•   Strengthen educator engagement and ownership of the assessments.
•   Enable students to incorporate graphical representations using natural user interfaces in their response to mathematics items.
•   Improve the efficiency of the scoring of mathematical reasoning by use of automated processes.

“ On the CMT [Connecticut Mastery Test],
I work and work and work on a problem.
Then I go to fill in my answer and it ’ s not there. But this test [Smarter Balanced pilot] let me tell how I solved it.
”
-- Grade 3 student from Wethersfield, CT, when asked by the district math coordinator to give feedback on the
Smarter Balanced math pilot

Attempts to “ Capture ” Student Work often
Eliminate the Autonomous Reasoning Called for in the Content Specifications
Imagine the same problem was posed as a series of questions in an attempt to “ capture ” reasoning:
•   What is the volume of ½ the candle?
•   What is the volume of ½ the vase?
•   How much sand in each vase?
•   How many sand for all vases?
•   How many bags of sand?
•   How much would that cost?

Gavin, M. K., Casa, T. M., Chapin, S., Copley, J. V., & Sheffield, L. J. (2008). Project M2: Using Everyday
Measures: Measuring with the Meerkats from Project M2: Mentoring Young Mathematicians series. Slide 29

Future Collaborations for Advancing the
Field
What relationship does Smarter Balanced have with other organizations working on efforts related to CCSS-M implementation?
–   CTB, AIR, MI, SCALE, & DRC
–   Khan Academy
–   Desmos
–   Illustrative Mathematics
30

Change is a given.
Progress is not …
You have decide that you are going to make progress!

Context of Lesson Planning
•   The LEVEL OF THINKING that goes into the preparation of the plan that matters most!
•   The outcome of the lesson is about understanding the mathematics embedded in the task or problem – not merely completing the task

Mathematical Learning Goals of Patio
Task
•   Students will understand three mathematical ideas regarding functions:
1.
Linear functions grow at a constant rate
2.
There are different ways of writing an explicit rule that defines the relationship between two variables
3.
The rate of change of a linear function can be highlighted in different representational forms: a.
As successive differences in a table of [x, y] values in which values for x increase by one b.
As the m value in the equation y=mx+b c.
As the slope of a function when graphed
Smith and Stein 2011