Mathematics PCK
Workshop #1
Day 1
RAMALLAH, PALESTINE MARCH 22 & 24, 2013
Dr. Elizabeth (Betsy) McEneaney
[Mac-Uh-Ninny]
Dept. of Teacher Education and Curriculum Studies
UMass – Amherst
emcenean@educ.umass.edu
Introductions
YO U R N A M E
W H E R E YO U W O R K A N D W H AT YO U D O T H E R E
W H AT YO U H O P E TO L E A R N I N T H E S E T W O D AY S
O N E I N T E R E S T I N G T H I N G A B O U T YO U T H AT N OT M A N Y P E O P L E K N O W
Goal for this workshop:
Enhance your capacity to prepare in-service teachers
to use effective strategies for interpreting and
transforming mathematics content knowledge into
pedagogically appropriate teaching strategies and
learning environments in mathematics, grades 5 to
10.
Honoring your experience and intuition…
Honoring your experience and intuition…
Much of this you
already know…
We will review,
synthesize,
brainstorm,
adapt.
By the end of the day today, I hope you
will be able to:
Identify major math concepts and skills that students need;
Anticipate and diagnose some typical conceptual difficulties;
Identify and incorporate students' prior knowledge effectively;
’
You hope to:
Our Mission in Math
•Consolidate good “procedural knowledge”
•Cultivate good “conceptual knowledge”
through active learning
•For both “concrete operational” and “formal operational” learners
•And for those who don’t know that they should care
•All while making it FUN!
Awareness of self as learner
One aspect of METACOGNITIVE SKILLS
We want students to develop this awareness
So… teachers must have this awareness
So… teacher educators must have this awareness
Your Math Autobiography
Please take 5 MINUTES to write down some thoughts about your experiences
with math so far.
We will share later.
Such as:
Was there a time that you really ENJOYED doing math? Describe the context and the feelings
you had.
Was there a time that you really DISLIKED doing math? What was that like?
How do you study math? What works for you?
How do you use math in your life?
What are some
variations we see?
WHAT VARIATIONS WILL WE SEE AMONG TEACHERS AND STUDENTS?
DOES THIS AFFECT HOW WE DESIGN INSTRUCTION?
The learner-centered
classroom
FIRST, A “SILENT CONVERSATION”
SILENT CONVERSATION ACTIVITY
No talking, please.
In your group, write answers, comments, reflections on the chart paper.
Later, you may write comments to other people’s reflections.
We will summarize later.
The “look” of a great math class
Teacher:
Students:
Setting and sharing goals
Actively engaging
Giving “wait time”
Working together
Checking for understanding
Using what they know
Encourage mathematical thinking
◦ Represent in multiple ways
◦ Integrate technology wisely
◦ Gain from “mistakes”
/misconceptions
Persevering
Taking Risks
From: Moynihan, C. 2012. Math Sense: The Look, Sound and Feel of Effective Instruction. Stenhouse Publishers.
The “sound” of a great math class
Teacher:
Students:
Supporting math discourse/”math
talk”
Justifying and clarifying their thinking
Analyzing the thinking of others
Engaging ALL students
BOTH: Active listening, Using math vocabulary, encouraging risk taking
The “feel” of a great math class
Purpose
Creativity
Collaboration
Building
Confidence
Pride
Myths, counterproductive
beliefs and misconceptions
about math
Free Write (2 minutes)
If we ask people on the street what they think about:
◦ MATH in general
◦ SCHOOL MATH
What to you think they would say?
Which of those ideas are:
Category A: UNTRUE, and we need to change that misconception!!
Category B: UNTRUE, but it’s OK.
Category C: TRUE!
Category D: TRUE, unfortunately. We need to change the conditions
that make it true.
Thinking about student misconceptions
in specific math topics
An insight from science education:
Misconception = “Everyday” conception
A mistake makes “sense” in a particular context
Example: ½ + ¼ = ?
Some common misconceptions in Grades
5-10 mathematics (Betsy’s “Favorites”)
“Multiplication makes bigger, division makes smaller.”
4.442 is a decimal, it can’t be written as a fraction.
½ of 6, 0.5 of 6 and 50% of 6 are all different quantities.
1/8 = 0.8
1
𝑎
1
𝑏
+ =
1
𝑎+𝑏
1.25 > 1.4
𝑥3 + 𝑥2 = 𝑥5
Probability of flipping a coin and getting heads, then tails. ½+ ½ = 1
Adapted from: Misconceptions in Mathematics. www.counton.org
A story problem
Hassan is on the wrestling team. To be in class one, he can weigh at
most 48 kg, and to be in class two he can weigh more than 48 kg but
no more than 55 kg. Hassan is hoping to wrestle in class two. How
much is he allowed to weigh?
What common mistakes do students make?
Perhaps my all-time favorite:
(𝑎 + 𝑏)2 = ? ?
Maybe it’s a favorite of yours, too!
What mistake do students make, and what is the underlying
misconception?
Adapted from: Collins, A. and Dacey, L. 2011. The Xs and Whys of Algebra. Stenhouse Publishers.
And the list goes on…
THE TAKE AWAY? “MISTAKES” AREN’T USUALLY RANDOM.
TEACHERS NEED TO PAY AT TENTION TO MISTAKES AND LEARN FROM
THEM.
What should math
teachers know?
Shulman’s view of teacher knowledge
Shulman’s Pedagogical Content
Knowledge (PCK):
• Represent subject so that it is
comprehensible to others
• Understand student
conceptions/misconceptions
• Curricular knowledge
Mathematical Knowledge for Teaching
(MKT)
MKT: The deep knowledge of mathematics that allows for Shulman’s PCK
Where does
MKT fit on
the
diagram?
Thames and Ball article in your packet
Their approach: Look at real teaching, decide
what kind of math knowledge is needed.
Teachers need a LOT of math, even in the lower
grades
It isn’t the same kind of math that
mathematicians need.
Example: Adding and subtracting with nearten
45 – 19 =
USING THE HUNDREDS CHART
100
45 – 19 =
90
91
92
93
94
95
96
97
98
99
80
81
82
83
84
85
86
87
88
89
70
71
72
73
74
75
76
77
78
79
60
61
62
63
64
65
66
67
68
69
50
51
52
53
54
55
56
57
58
59
40
41
42
43
44
45
46
47
48
49
30
31
32
33
34
35
36
37
38
39
20
21
22
23
24
25
26
27
28
29
10
11
12
13
14
15
16
17
18
19
0
1
2
3
4
5
6
7
8
9
Math
knowledge
needed:
MKT is multi-faceted
Including the ability to:
Some questions to think about:
In small groups:
What do you think mathematicians think about Ball’s idea
of “Mathematical Knowledge for Teaching”? Does it
matter what they think?
If Ball is correct, what does this mean for how we train
and develop math teachers?
Extending Shulman’s theory: Technology
TPACK: Technological Pedagogical Content Knowledge
FREE WRITE:
How do you see
Technological
knowledge
influencing what
math teachers need
to know?
Instructional Models
How are your teachers currently
structure their math lessons?
Does that structure support deep
and active learning?
Two alternative models
5E AND “LAUNCH , EXPLORE, SUMMARIZE”
TAKE A FEW MINUTES TO READ THE DESCRIPTIONS OF THESE TWO
MODELS
Give feedback to a
teacher
VIDEO OF A 6 TH GRADE MATH CLASS
WOULD YOU WANT THIS TEACHER TO CHANGE INSTRUCTIONAL MODEL
AND APPROACH?
Designing Worthy Tasks in Math
WORTHY: Having great merit, character, or value.
(Worthy for whom??)
How can we increase the worthy tasks we ask students
to work on?
Is everything that happens in a great math class
worthy work?
Another Silent
Conversation
THE PROBLEMS WITH MATH “PROBLEMS”
A teacher talks
about making
“good math
problems”
Mr. Dan Meyer
7th grade teacher
TED Talk
Summarizing the day
We clarified the goals we share for these PCK workshops.
We developed greater awareness of ourselves as math learners.
We reviewed common misconceptions about math and math learning.
We addressed a handful of common math mistakes and their associated “misconceptions.”
We reviewed Shulman’s work on teacher knowledge and extensions such as Ball’s MKT and
TPACK.
We considered what makes a math problem “worthy” of student engagement and persistence.