LEARNING HOW TO LEARN
FROM TEACHING :
NOTICING STUDENTS’
M AT H E M AT I C A L R E A S O N I N G
ABOUT INTEGERS
W E S T E R N R E G I O N AL N OY C E C O N F E R E N C E
N OV E M B E R 1 4 , 2 0 1 5
R AN DY PH I L I P P
C AS E Y H AW T H O R N E
Basketball Challenge:
How many passes by the team dressed in white?
Basketball Challenge:
How many passes by the team dressed in white?
A. 16 passes
B. 17 passes
C. 18 passes
D. Other____
E. I lost track
Basketball Challenge:
Did you notice anything unusual?
A. There was a gorilla beating his chest.
B. There was a gorilla but he didn’t beat his chest.
C. What are A & B talking about? There was no gorilla!
About half the people do NOT notice the gorilla!
(Simons & Chabris)
Lessons from the Basketball Challenge:
• Classrooms are complex places!
A teacher is bombarded with a “blooming, buzzing, confusion” of sensory data.
-­‐ B. Sherin & Star, 2011, p. 69
• What we notice, that is, what we see and make sense of is related to
– …what we already know.
– … what we are looking for.
– … the goals we hold.
PROFESSIONAL NOTICING
At this point in my development as a teacher…
–What do I easily notice in my classroom?
–What do I not notice? What are my gorillas?
PROFESSIONAL NOTICING
What we notice, that is, what we see and how we
make sense of it, is related to what we already
know, what we are looking for, and what our
goals are.
In particular, the practice of professional noticing
takes time and specific knowledge to develop.
A student teacher asks you: What are your
goals for a unit on proportional reasoning?
How might you respond?
Problem: There are two banks. Money is invested in each, for the
same period of time. Which bank provided a better return, or were
they the same?
Bank #1: $200 à $ 800
Bank #2: $700 à $1,400
Person 1: “Bank #2 is better, because you have more money.”
Person 2: “Bank #2 is better, because you make $700 at Bank #2 and only $600 at Bank #1.”
Person 3: “Bank #1 is better, because look:
$200 à $ 800
$400 à $ 1,600
and I already have way more!
$600 à $ 2,400
Person 4: “Bank #2 doubled the money, but Bank #1 is 4 times as much. Bank #1 is better.”
–
Here is an important mathematical symbol.
What are ways you think about this?
Please write them down.
–
How many of you thought about
a) Subtraction?
b) Negative?
c) The Opposite Of?
Think about the role the “–” symbol plays in the
following sequence.
3–x=5
–3 + 3 – x = –3 + 5
0–x=2
–x=2
x = –2
A student teacher asks you: What are
your goals for a unit on integers?
How might you respond?
WHEN STUDENTS LEARN ONLY RULES
To answer 6 – -2, many students rewrite the
problem as 6 + +2 to get 8.
WHEN STUDENTS LEARN ONLY RULES
THE PROBLEM WITH RULES
What kind of view of mathematics do these
students have?
SOLVE THE FOLLOWING
a) -2 + 5 = 🀆
b) -3 + 🀆 = -5
c) -8 – –3 = 🀆
INTEGER REASONING FRAMEWORK
Ways of Integer Reasoning
Order-Based Reasoning
Descriptions
Using the sequential and ordered nature of numbers to reason about a
problem (e.g., counting strategies or a number line with motion).
Analogy-Based Reasoning Relating negative numbers to another idea and reasoning about
negative numbers on the basis of characteristics observed in this other
concept. This way of reasoning may be characterized by thinking of the
sign and magnitude of the number separately and interpreting the sign
as taking on the meaning based in the analogy.
Formal Reasoning
Negative numbers are treated as formal objects that exist in a system
and are subject to fundamental mathematical principles. Formal
strategies often involve comparisons to other, known problems so that
the logic of the approach remains consistent and underlying structural
principles are not violated.
Computational Reasoning Using a procedure, rule, property, or calculation to arrive at an answer.
ROSIE
🀆+5=3
ROSIE
5+🀆=2
ROSIE
Rosie can solve 🀆 + 5 = 3, but is unable
to solve 5 + 🀆 = 2.
Can you explain this?
JAMES
-7 – 🀆 = -5
JAMES
James can solve -7 – 🀆 = -5, but is unable
to solve 1 + -2 = 🀆.
Can you explain this?
RANK THE FOLLOWING
(1-EASIEST, 4 HARDEST)
Letter
Item
a)
6 + -3
b)
-3 + 6
c)
-8 – ___ = -2
d)
-5 + -1
Grade 4
Grade7
RANK THE FOLLOWING
(1-EASIEST, 4 HARDEST)
Letter
Item
Grade 4
Grade7
a)
6 + -3
4
2
b)
-3 + 6
3
1
c)
-8 – ___ = -2
2
4
d)
-5 + -1
1
3
RANK THE FOLLOWING
(1-EASIEST, 4 HARDEST)
Letter
Item
Grade 4
Grade7
a)
6 + -3
4
2
b)
-3 + 6
3
1
c)
-8 – ___ = -2
2 (61%)
4 (53%)
d)
-5 + -1
1
3
TEACHERS’ INTEGER REASONING
(a) c + -2 = -10
(d) 5 – c = 8
(b) -9 + c = -4
(e) -3 – c = 2
(c)
(f) 6 + c = 4
-3 + 6 = c
TEACHERS’ INTEGER REASONING
1. Did teachers demonstrate multiple ways of
non-computational reasoning?
Yes
2.Were teachers able to articulate goals other
than computational fluency?
No
3. Was there a correlation between teachers’
instructional goals and their own reasoning?
No
4.Were teachers able to identify and describe
students’ integer reasoning?
Some
TEACHERS’ INTEGER REASONING
Teachers use integer ways of reasoning in thoughtful and
rich ways, but do not hold these as their instructional
goals because they are not cognizant of this knowledge.
We hope to reconceptualize integer instruction around
ways of reasoning, providing teachers alternative
conceptions to focus on.
WHEN LOOKING AT STUDENT WORK,
WHAT DO YOU LOOK FOR?
a) to find out if students have correct or incorrect answers;
b) to find out if students have mastered the method you have
presented;
c) to find out if students have other ways of solving (correctly);
d) to find out how students are thinking, in what ways are these
correct or incorrect (and consider how to connect d to a, b, c).
PROFESSIONAL NOTICING
5 PRACTICES
Anticipating what students will do--what strategies they will use
in solving a problem
Monitoring their work as they approach the problem in class
Selecting students whose strategies are worth discussing in class
Sequencing those students' presentations to maximize their
potential to increase students' learning
Connecting the strategies and ideas in a way that helps students
understand the mathematics learned
INTEGER REASONING
PEDAGOGICAL IMPLICATIONS
DISCUSSION