141_Focusing on the Development of Children’s Mathematical Thinking: CGI

Focusing on the Development of
Children’s Mathematical
Thinking: CGI
Megan Loef Franke
UCLA


Algebra as focal point


“Algebra for All”
(Edwards, 1990; Silver, 1997)



“gatekeeper for
citizenship”
(Moses & Cobb, 2001)



Difficult transition from
arithmetic



Not move high school
curriculum to elementary
school



Engages teachers in a new
way, new content


Algebra as generalized arithmetic
and the study of relations


Viewing the equal sign as a
relation
57 + 36 =  + 34



Using number relations to
simplify calculations
5 x 499 = 



Making explicit general
relations based on fundamental
properties of arithmetic
768 + 39 = 39 + 


Equality

8+4= +5

Equality Data (8+4= 
+5)
Student Responses1

1

Grade

7

12

17

12 & 17

1st & 2nd

5%

58

13

8

3rd & 4th

9

49

25

10

5th & 6th

2

76

21

2

Falkner, K., Levi, L., & Carpenter, T. (1999). Children’s understanding of
equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-6.



True/false number sentences:
from worksheets to index cards






Shift from a focus
on answer to a
focus on reasoning
Shift from a focus
on a single problem
to a sequence
Shift from sharing a
single strategy to a
conversation
around the
reasoning


Sequence of Number
Sentences
3+4=7
5 + 5 = 8*
7=3+4
6=6+0
6=6
6=3+3
4+2=3+3
* denotes false number sentence


Mathematical Content
Equality
Number Facts
Place Value

Number Sense
Mathematical Properties
Multiplication
Equivalence

7=7
5+5=4+6
250 + 150 =  +100
45 = 100 + 20 + 
5+6=6+
3× 7=7+7+7
½=¼+¼


Relational Thinking

24 + 17 – 17 = 34 + 

1,000 – 395 = ___
999 – 395 + 1


Relational Thinking


Solve:



576 + 199 = □





1,000 – 637 = □




576 + 200 - 1

999 – 637 + 1

4 x 24 + 5 x 24 = □


10 x 24 - 24


Generating Conjectures
Making relational thinking explicit

Representing
Conjectures
 
b+0 = b
 
c+d = d+c

Variables
k + k + 13 = k + 20


Experimental Study Design


Volunteer, urban, low
performing elementary schools
in one district (19)



District working to improve
opportunities in mathematics



Schools randomly assigned to
year 1 or year 2 professional
development work



School site based PD monthly



On site support



End of one year assessed
teachers (180) and students
(3735)


Teacher Findings
Generating strategies for 8 + 4 =  + 5






No differences in
teachers’ perceptions on
time spent on algebraic
thinking tasks in
classrooms
No differences on
knowledge of algebra
Differences in teachers’
knowledge of student
thinking- strategies and
relational thinking

Numb
er of
strate

gies

Participat Noning
Participati
Teachers ng
Teachers

1

6%

44%

2

38%

41%

3

25%

12%

4 or
more

31%

4%


Student Findings


Students in algebraic
thinking classrooms
scored significantly
better on the
equality written
assessment.



Students in 3rd and
5th grades were twice
as likely to use
relational thinking



Publications


Book for teachers:

Carpenter, T., Franke, M., & Levi, L. (2003).
Thinking mathematically Integrating
arithmetic and algebra in elementary school.

Portsmouth, NH: Heinemann.


Research article:

Jacobs, V., Franke, M., Carpenter, T., Levi, L. &
Battey, D. (in press). Exploring the impact of
large scale professional development focused
on children’s algebraic reasoning. Journal for
Research in Mathematics Education.


Conjectures


Is a focus on children’s thinking
enough?


Show what students are capable of



Counter narratives



Change what we consider basic skills



Create ways in schools to make room for
understanding



Watch for how the status quo limits
opportunities…find ways to challenge it