Teaching Matters!
NCTM Tools to Support
Implementation of Effective
Mathematics Teaching Practices
Diane J. Briars, President
National Council of Teachers of Mathematics
dbriars@nctm.org
Cathy Martin, Director of Mathematics & Science
Denver Public Schools
cathy_martin@dpsk12.org
Jeff Ziegler, Curriculum Supervisor 6-12 Mathematics
Pittsburgh Public Schools
jziegler1@pghboe.net
Urban Mathematics Leadership Network
February 10, 2015
Th
heSPa
attern
nTask
k
1
2
3
1
4
5
1. Whatpatt
W
ernsdoy
younoticceinthesetoffiggures?
2.
Howmany
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hat
co
ouldbeu
usedtodeetermineetheshap
peofand
dtotalnu
umberoff
sq
quaretileesinfigure7.You
urdescriiptionshouldbecclear
en
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othatano
otherperrsoncoulldreadittanduseeittothin
nk
ab
boutanotherfigu
ure.
3.
Determineeanequaationforthetotallnumberrofsquarresinanyy
gure.Explainyou
urrulean
ndshowhowitreelatestothevisuaal
fig
diiagramofthefigu
ures.
4.
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quationtthatmatcchesthed
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5.
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f youknew
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Page 1
S-­‐Pattern Task – CLIP 1 Teacher: Jeff Ziegler District: Pittsburgh Public Schools Grades: 11-­‐12 Group 1 – First Interaction with Teacher 1 S: The top is your x but that’s constant. That’s always gonna stay the same. 2 S: Okay. 3 4 S: So, then you have to find the number that’s before x to get to that. You already found what you have to add to get to that. Now you have to multiply to get to that. 5 S: We figured this (pointing to the table)… 6 S: We found out a pattern. 7 S: I can’t do equations. I can’t factor. So...yeah, we figured that out first period. 8 9 T: Okay, so, you’re...okay so you’re going to start with a table and see if you can find the equation from the table? 10 S: Yeah. 11 T: Okay. 12 S: But we don’t know how...we don’t know… 13 S: I don’t even know how to start to find the equation there. 14 15 S: We know what the “b” is. We don’t know if it should be x + 3 or x + an odd. X + an odd ‘cause these are all odds on the bottom. 16 T: Right. 17 18 S: They’re always going to be odds. So it’s plus 2 between them. There’s a difference of two between them. 19 T: Okay. 20 S: We already know the top rows are x. One, 2, 3, 4...like that’s our x. 21 T: Right. 22 S: Our pattern. The growth is…I don’t know what the growth is.  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH Clip ID 2398 Page 2
23 S: You got something. 24 S: The growth is an odd. It’s like odd numbers. 25 T: Okay, so you have the table, you have these numbers written out. 26 S: Right, if we had the equation we should be . . . 27 S: How do we graph it with no equation? 28 29 T: Well no kidding. Oh yeah, if I gave you the equation, life would be great. What do you have? 30 S: This. 31 T: Which is a what? 32 S: S. 33 T: Okay. It has what? 34 S: Squares. 35 T: Okay. How many? 36 S: 26. 37 T: In number 5? 38 S: Yes. 39 40 T: Okay, that’s 26. There’s no other way you can come up with that number 26 than just counting? 41 S: You can go by, like ... 42 S: So x plus . . . 43 S: He’s leaving us. Group 2 – First Interaction with Teacher 44 S: I broke it down real easy, real simple to this. It obviously looks real simple. So… 45 T: Do you guys know what he’s doing?  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH 2
Page 3
46 S: Yeah. 47 S: Yeah. 48 S: We all helped. 49 T: Hold up. You did this. Tell me what you did. You don’t know? 50 S: I was working by myself. 51 T: Oh, okay. Do you know what he was doing? Okay what’s this? 52 S: This is a group effort. 53 S: I know. 54 55 56 57 58 S: Whatever the pattern number is, not even looking at...not even looking at this, just whatever the pattern number is, you take it and you times it by 2 because there’s 2, there’s obviously 2 rows and each...the top row and the bottom row both have the number...this number, the 2. And then times that by 2 and that will give you the top and the bottom and the middle is a square so… 59 T: Right. 60 61 S: You min...you do 2...you do the...ah, I’m going to call this x. (X – 1)2. That will give you the middle and you just add them together. 62 S: I understand him, I just can’t explain it. 63 S: Did you understand that, Zieg? 64 T: Yes. 65 66 67 S: And then I got another one, I got another one, though. I don’t know...if you take it and go this way, rectangle, you take x + 1 and then do x -­‐ 1 and that will give you, that will give you this dimension right here. 68 T: Go back to the first one. There. Look at what he’s doing. Tell me what he is doing. 69 S: What do you mean, like? 70 71 T: When he came up with it, when he was explaining the top row and the bottom row and the center, do you know what he was talking about? 72 S: Yeah. 73 T: What?  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH 3
Page 4
74 S: I don’t know how to say it, but I helped him do that, too, like it’s not just all him. 75 T: Okay. So tell me, show me. I mean, do you have it in your head? Is it on paper? 76 77 78 S: It’s in my head. He said that, okay, the middle, there’s one square in the middle and then there’s 2 on the top. Subtract 1 to get the number...subtract one and square it to get the number of the boxes in the middle. 79 T: Okay. 80 81 S: So x -­‐ 1...so 3 -­‐ 1 is 2 and then you square 2 to get 4 in the middle and then you multiply the whatever sequence you’re on times 2 ‘cause there’s a top and a bottom. 82 83 84 85 T: Okay. And that’s how you came up with the equation? So, okay, can you take his equation 2x + (x -­‐ 1)2 and can you put it to a picture? Can you put it to these pictures? Like let’s, let’s, let’s pull out, let’s say number 4, okay? If we take this, how does this picture right here relate to 2x + (x -­‐ 1)2 ? 86 S: So that’s simpler than... 87 S: ...and then you add 2. You see what I’m saying? You see what I am saying, Nick? 88 S: That’s simpler than… 89 S: That’s the easy way to break it down. You go from...you just take these 2…  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH 4
Page 5
S-­‐Pattern Task – CLIP 2 Teacher: Jeff Ziegler District: Pittsburgh Public Schools Grades: 11-­‐12 Group 2 -­‐ Second Interaction with Teacher 90 91 92 T: Okay, I’m back. All right, so when I left, I asked the group to come up and explain how we came up with 2x + (x -­‐ 1)2 and where that relates in the picture. So are you ready to tell me? 93 S: Yeah. 94 T: Okay. 95 S: Well, actually I think that she made a different equation. I think hers is better. 96 S: I thought we were just going to go with this one. 97 S: We have one. 98 S: Hers is x2 + 1. 99 S: Yeah, but that doesn’t explain the picture. 100 T: What I asked when I left was does 2x + (x -­‐ 1)2 fit the pattern, correct? 101 S: It fit the pattern. 102 S: Yeah. 103 T: Okay. What I wanted to know when I left was how does it relate to the tiles? 104 S: Um… 105 T: Where is 2x in these tiles? Where is the (x -­‐ 1)2 in these tiles? 106 S: Well x is that number right there. 107 T: Okay. 108 S: And 2, you just multiply 2 by that number… 109 T: Why? 110 S: Which gives you…  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH Clip ID 2404 Page 6
111 T: Why? 112 S: Because, um… 113 T: The tiles… 114 S: Because it gets bigger. It doubles. 115 S: Yeah, it doubles… 116 T: What doubles? 117 S: That…that…the tiles. Like for 1, it doubles and then for 2 it doubles, 3 it doubles. 118 S: That’s the top and the bottom. 119 S: Oh, it’s the top...oh it’s the...oh, yeah. Those 2. 120 S: The x is the top number and the bottom. 121 T: What’s the matter? 122 S: I don’t...I mean, I understand but I can’t really explain it. Like those 2… 123 124 T: If you want, see, my thoughts always were if you really, truly understood then explaining would be the easy part. 125 S: Well, I do understand but...2 is right here. That’s where they got the 2 from. 126 S: 2x. 127 S: Like 2x ‘cause you just take out those 2 and then use that. And then x -­‐ 1 is like 4 -­‐ 1. 128 S: Which is 3. 129 130 S: Three. Oh yeah. How much is...each row right there. Then you square it and that’s how much is in the middle. 131 T: Okay, so… 132 S: Has to be a square number. 133 134 T: Take the sheet of paper right now. Take number 4. Separate...I want you to actually manipulate those black tiles on here. Show me the 2x, show me the (x -­‐ 1)2. 135 S: All right. 136 T: Okay.  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH 2 Page 7
Group 1 -­‐– Second Interaction with Teacher 137 S: You add 1. 138 T: Are you listening? (Teacher directs this to the student he is sitting next to.) 139 S: I am. 140 141 S: All right, you have 2 on the bottom, 2 on the top (referring to the second figure in the sequence). 142 S: I get it. 143 144 S: You got 1 in the middle that’s an extra. So you add that. That’s plus 1. So it’s 2 and 2 is 4. That’s 2 squared is 4 then you add this extra 1 in the middle. That’s 5. 145 T: Okay. What is, what is he telling me here? Where (x + 1)2 came from. 146 S: Are you asking me? 147 T: Yeah, I’m asking you. 148 149 S: Oh, I get it. It’s like, because you start off with 1 and then you times it by itself and then you add 1. 150 T: Okay. 151 S: And then you just keep going...you want me to keep going? 152 153 154 T: No, what I want you to do is...I want you to take these, these black tiles that are sitting right here (referring to the figures of tiles) and I want you to show me, I want you to show me, where do you see 2 squared? And then where’s the plus 1 at? 155 S: Like… 156 T: Where’s the 2 squared? 157 S: Right here. 158 T: What’s that? 159 S: And right here. These are 2. 160 T: Okay. 161 S: Then the 1 is the middle.  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH 3 Page 8
162 163 164 165 166 167 T: Okay. So, for number 2, for pattern 2, top row and the bottom row, you’re putting those 2 together, making a square and adding 1 to it. So, if I did the same thing in pattern number 3, I took the top row and the bottom row and I put them together, is that 3 times 3? Is there 1 left over? (Students acknowledge that this doesn’t seem to work.) So, what I’m telling you is, how do you manipulate these tiles for your x2 + 1. If it obviously works… 168 S: I mean, ‘cause look, there’s 1, 2, 3; 1, 2, 3; 1, 2, 3, we’re just doing it like that. 169 T: Where? 170 S: And there’s the 1 left over. 171 S: You can do that. 172 S: Come on, I can do this. 173 S: And this is the way to do 4, 4, 4. 174 175 T: Okay, you have…How many tiles do you have in pattern 3? How many tiles do you have in pattern 3? 176 S: 10. 177 178 T: Okay. I’m giving you 10 individual tiles on this piece of paper. Okay? They’re not touching. 179 S: They’re a new pattern? 180 181 T: I want you to take those 10 tiles and I want you to show me how you put them together to get x2 + 1. That’s what I want you to show me. 182 S: We can go like this. Look. 183 T: I’ll be back. 184 S: No, just stay with us. 185 S: “I’ll be back” (mimicking the teacher).  2007, 2009, 2010, 2011, 2014, 2015 UNIVERSITY OF PITTSBURGH Page 9
4 Acce
ess and Eq
quity Beliiefs Surve
ey
SD = Stro
ongly Disagre
ee
D = Disa
agree
A = Agree
SA = Strongly A
Agree
Belief
SD
D
A
SA
1. Equ
uity is the sa
ame as equa
ality. All stud
dents need tto receive tthe
sam
me learning opportunitie
es so that th
hey can achiieve the sam
me
aca
ademic outcomes.
2. Matthematics ab
bility is a function of op
pportunity, e
experience,
and
d effort—nott of innate in
ntelligence. Mathematiccs teaching and
learning cultiva
ate mathem
matics abilitie
es. All stude
ents are
cap
pable of partticipating an
nd achieving
g in mathem
matics, and a
all
desserve supporrt to achieve
e at the high
hest levels.
3. Students who are
a not fluen
nt in the Eng
glish languagge are less a
able
to learn
l
mathe
ematics and therefore must
m
be in a separate tra
ack
for English lang
guage learne
ers (ELLs ).
4. Matthematics le
earning is ind
dependent of
o students’ culture,
con
nditions, and
d language, and teacherrs do not ne
eed to consid
der
any
y of these fa
actors to be effective.
5. Placing studentts in classess and groups of studentss with simila
ar
ability/achieve
ement promo
otes studentts’ achievem
ment and
ena
ables them to
t make the greatest lea
arning gainss.
6. Effe
ective teach
hing practice
es (e.g., eng
gaging stude
ents with
cha
allenging tassks, discoursse, and open
n-ended pro blem solvingg)
havve the poten
ntial to open
n up greaterr opportunitiies for highe
erord
der thinking and for raising the mathematics acchievement of
all students, in
ncluding poo
or and low-in
ncome stude
ents.
7. Students posse
ess different innate leve
els of ability in
mathematics, and
a these ca
annot be cha
anged by insstruction.
als have it while
w
others do not.
Cerrtain groups or individua
8. Equ
uity is prima
arily an issue
e for schoolss with raciall and ethnic
diversity or significant num
mbers of low
w-income stu
udents.
Adapted from
f
Principles to Actionss: Ensuring Ma
athematical SSuccess for A
All, NCTM, 201
14, p. 11.
Page 10
Mathematics Identities
 Mathematics identity includes:
1. beliefs about one’s self as a mathematics learner;
2. one’s perceptions of how others perceive him as a mathematics learner,
3. beliefs about the nature of mathematics,
4. engagement in mathematics, and
5. perception of self as a potential participant in mathematics (Solomon, 2009).
Identities and Motivation
 Understanding the strengths and motivations that serve to develop students’ identities should be
embedded in the daily work of all teachers.
 Mathematics teaching involves not only helping students develop mathematical skills but also
empowering students to seeing themselves as capable of participating in and being doers of
mathematics.
o When students identify themselves as participatory and doers of mathematics, they make
positive connections and are motivate to achieve at high levels.
o This understanding of students’ identities gives teachers insights to how and why some
students might make positive connections with mathematics and others do not.
 Teachers can use this understanding to provide opportunities for students to use mathematics to
examine personal, communal, and social contexts.
o In providing these opportunities, students may find the motivation and connections with
mathematics to see the relevance for their future thus developing a mathematics identity.
Identity Affirming Behaviors
 Identity-affirming behaviors influence the ways in which students participate in mathematics and
how they see themselves as doers of mathematics.
o A student who identifies himself as being good at mathematics might exhibit behaviors and
participate to maintain his status as a person who is “smart” or good at mathematics.
 In mathematics teaching and learning we see identity-affirming criteria emerging as learners are
labeled as “smart,” “gifted,” “proficient,” “at-risk,” or “on grade-level”
 Teachers affirm mathematics identities by providing opportunities for students to make sense of and
persevere in challenging mathematics.
o Students should be engaged with mathematics that requires active participation, asking
questions, problem posing, and reasoning.
o This kind of teaching values all students’ thinking and uses pedagogical practices, such as
differentiated tasks, mixed ability groupings, and publicly praising contributions and
perseverance, to cultivate and affirm mathematical participation and behaviors (NCTM
2014).
o Policies and Practices impacting Access and Equity. What kinds of
work/practices/strategies impact the differentials for Access and Equity?
Identities: Supporting Mathematics Teaching and Learning
 Mathematics teaching should leverage students’ culture, contexts, and identities to support and
enhance mathematics learning (NCTM, 2014).
 By understanding the significance and relevance of the alternative identities and contexts,
mathematics teachers can draw on community resources to understand how they can use contexts,
culture, conditions, and language to support mathematics teaching and learning.
Robert Berry Q. III, 2015
Page 11
Vignette 2: Caroline and Craig (Adapted from Chval & Davis, 2008)
Caroline
Caroline is a gifted seventh grader who has access to challenging mathematics in both her
gifted pull-out program and in her mathematics class. Caroline participates in the pull-out
program two days a week with other gifted students. Thus, she and the other students are
with their teacher only three days a week. However, her teacher recognizes the
importance of differentiating instruction for her students every day. She realizes that she
must give careful consideration to this instruction because gifted children require
different and more flexible educational experiences. As a result, Caroline’s teacher
provides thought-provoking problems and structures them in ways that provide multiple
entry points for the whole class. She also encourages her students to demonstrate what
they know during small-group and whole-group discussions, creating a safe and
respectful environment where all students can solve problems in different ways. This
classroom environment makes Caroline truly enjoy her mathematics class because she
feels respected, engaged, challenged, and creative. All these elements will allow her to
excel in mathematics.
Craig
Craig, a gifted seventh- grade middle school student, is not engaged during his
mathematics lessons. The content is not difficult for Craig, and his participation is not
encouraged. For example, his teacher often says, "Craig, I know you know the answer. I
want to see if anyone else knows." This statement and similar comments have taught
Craig not to raise his hand in class. In addition, his teacher frequently tells him that he
cannot use his mathematics knowledge to reach an answer because some of the other
students have not yet learned it. For example, when his class was studying circles, Craig
was told not to use pi or his algebra skills to calculate area and circumference. This and
similar situations have frustrated Craig. As a result, he has learned not to initiate
questions or alternative approaches to solving problems. Later, during the school year,
Craig approached his teacher to request some challenging problems to work on
independently during class. Although the teacher took additional time to find
mathematics problems that would challenge Craig, he asked that Craig solve them
outside of class. This gesture helped challenge Craig but did not improve his classroom
experience. Craig disliked his middle school mathematics class because he felt that he
was not respected, engaged, or challenged. He was also prohibited from solving problems
using different methods than those used by his peers. Craig’s role in his mathematics
class- room had been reduced to observing or tutoring his classmates, rather than
learning.
Chval, K.B., & Davis, J. (2008). The gifted student. Mathematics Teaching in the Middle School, 14(5), 267-274.
Robert Q. Berry, III, 2015
12
Geoboard Dot Paper
5 x 5 (10 mm)
Name
42–Mastergrids for Mathematics
Date
www.worldteacherspress.com © World Teachers Press®
www.didax.com/2-5195
Page 13
Shifts'in'Classroom'Practice'
Shift'1:'From!same!instruction!toward!differentiated!instruction.''
Differentiated+instruction+but+
+ Same+instruction+for+all+
students.++
same+learning+outcomes+for+all+
+
students.++
'
Shift'2:'From!students!working!individually!toward!community!of!learners.!!''
+ Students+work+individually+on+
+
tasks+and+seek+feedback+from+
teacher+on+reasonableness+of+
strategies+and+solutions.+
Community+of+learners+where+
students+hear,+share,+and+judge+
reasonableness+of+strategies+and+
solutions.++
Shift'3:'From!mathematical!authority!coming!from!the!teacher!or!textbook!toward!
mathematical!authority!coming!from!sound!student!reasoning.!
+
'
Correctness+of+solutions+is+
determined+by+seeking+input+
from+teacher+or+textbook.+
Correctness+of+solution+is+based+
on+reasoning+about+the+accuracy+
of+the+solution+strategy.+
'
Shift'4:'From!teacher!demonstrating!‘how!to’!toward!teacher!communicating!
‘expectations’!for!learning.!'
+
+
+
Teacher+demonstrates+the+way+
in+which+to+solve+a+problem+and+
helps+students+in+solving+the+
problem+in+that+way.+
Teacher+facilitates+high$level+
performance+by+sharing+learning+goals+
and+expectations+for+products+that+
demonstrate+learning.+
Shift'5:'From!content!taught!in!isolation!toward!content!connected!to!prior!knowledge.!
+
Content+presented+in+ways+where+
+ Content+presented+independent+
of+its+connections+to+what+has+
been+previously+learned.+
explicit+attention+is+given+to+
making+connections+among+
mathematical+ideas.+
+
'
Shift'6:'From!focus!on!correct!answer!toward!focus!on!explanation!and!understanding.'
+
+
Discussions+and+classroom+
routines+focus+on+student+
explanation+of+how+they+solved+
a+task+and+if+it+is+correct.+
Discussions+and+classroom+
routines+focus+on+student+
explanations+that+address+why+
an+answer+is+(or+isn’t)+correct.+
Shift'7:'From!mathematics<made<easy!for!students!toward!engaging!students!in!
productive!struggle.'
+
Mathematics+is+presented+in+
small+chunks+and+help+is+
provided+so+that+students+reach+
solutions+quickly+and+without+
higher+level+thinking.+
Teacher+poses+tasks+and+
challenges+students+to+persevere+
and+attempt+multiple+approaches+
to+solving+problems.++
Bay$Williams,+J.+M.,+McGatha,+M.,+Kobett,+B.,+&+Wray,+J.+(in+press).+Mathematics*Coaching*Toolkit.+New+York,+NY:+
Pearson.++
Page 14