```Examining Student
Proficiency in Algebra
Examining Students’ Understandings and
Misconceptions
David Foster at Lakeway, June, 2005
Algebra
And what does x equal today?
Shift in policies and practice of
National movement for algebra for all
students.
 National call for algebraic thinking in all
 State and District policies to make Algebra
One a middle school course.
 What evidence is there that students are
being more successful with these changes in
policy and practice?

The Silicon Valley Mathematics Initiative
Data from an Annual Algebra
Performance Exam
students each year since 1999.
Approximately 42,000 students
were assessed, drawn from 35
member school districts.
Erica is putting up lines of colored flags for a party.
The flags are all the same size and are spaced equally along the line.
1. Calculate the length of the sides of each flag, and the space between flags.
2. How long will a line of n flags be?
Write down a formula to show how long a line of n flags would be.
The Findings from Party Flags
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The task may be approached as a system of
simultaneous equations, almost no algebra
students used such an approach.
49% of algebra students had no success.
44% accurately found the two lengths (most
commonly by an estimation strategy only using
one constraint).
21% correctly used both constraints (the length of
three flags is 80 cm. and the length of 6 flags is
170 cm.).
7% of the students were able to develop a valid
generalization for n flags.
P ATHWA YS
T his p r o b le m g iv es y ou th e chanc e to :
• fin d t h e d im e n s io ns o f th e pavi n g st o n e s u s e d to ma ke a path w a y .
Bo b us e s pa vin g s t one s t o m a k e a pa th way.
Th e pa vin g s t one s a r e rec ta n g le s a nd t he y a re al l
t h e s a m e s iz e .
2.5 ft
Th e pa th way is 2.5 fee t w id e .
F in d the len g th a n d w id th of t he pa v ing s t o ne s.
Ex pla in h o w y o u f ig u r ed it o u t.
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[5 ]
Findings from Pathways
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Students in geometry demonstrate similar levels of
algebraic understanding as algebra one students.
38% of geometry students demonstrated no success
36% of the students considered only one constraint
and estimated the correct width and length.
25% used algebraic notation to describe the
relationships between the two unknown lengths.
10% of the students used the two sets of constraints in
making sense of the task and finding a solution set.
M a k e Ha lf
Algebra
T his p ro bl em g ive s yo u t h e cha n ce t o:
• s o lve a p ro b le m usi n g f ra ct ions
5?
Ninth
12  ?
If y o u ad d 2 t o th e to p a n d b o t t om of
5
12
y o u g e t a fr ac t ion e q u al t o one h al f.
52
12  2
1.

7
14
W h at num be r co ul d y o u a dd t o t h e t o p an d b ot t om of t he s e fr ac t ions t o
m ak e on e h a lf ? R e mem b er t h a t y o u m u s t a d d the s ame n u m be r t o b o th t o p
a nd b o t t o m . Sho w y o u r wo r k.
2 ?
5  ?
2 ?
6  ?
2 ?
7  ?
2.
U s e y o u r a n sw er s i n q u e s t ion 1 t o fin d the a n sw e r to t he s e :
2?
35  ?
2 ?
n ?
Sho w y o u r w or k.
Findings from Make Half
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The essential mathematics of this task is to find a pattern,
apply the pattern to a specific case, determine a generalized
rule and simplify an algebraic expression. The mathematics
of the task is core to algebra one standards.
than finding the simple number pattern (n: 1,2,3,..). Only
38% of the students were able to successfully complete part 1
and were able to apply that knowledge to part 2.
6% of algebra one students were successful in developing a
generalization in order to write a valid algebraic expression.
4% of the students were able accomplish the entire task
including simplifying the expression.
Performance in High Poverty High School Districts
One thing I notice is that in the high schools most of the
assessment is multiple choice so students have little
experience in using their algebra in any non-standard
ways. This is because the low performing schools have
set up units of study so students can continue on in the
course and "restart" a unit that they have failed and
retake exams on units they have failed. So they have
this whole system of multiple choice tests for each unit
of study. Needless to say multiple choice testing
prepares one for little except more multiple choice tests.
Dr. Joanne Rossi Becker
Building Rules to Represent Functions
One of the big ideas in algebra is building rules to represent
functions. The assessments we have examined over the past
five years reveal considerable information regarding student
thinking in regards to pattern and function development..
grade students. Students were challenged to find the linear
pattern of the perimeter of consecutively aligned hexagons.
Students were asked questions regarding the functional
relationship between the number of hexagons in a sequence
(adjacent hexagon share a side) and the figures outside
perimeter.

78% of the fourth grade students demonstrated they could recognize, extend and graph
the linear pattern.
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60% of the seventh grade students demonstrated they could recognize, and extend the
same linear pattern.
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30% of fourth graders met all the demands of their task that included finding and
verifying the functional relationship for a given case.
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and verifying the functional relationship for a given case.
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Only 18% of the seventh grade students were able to successfully meet all the demands
of their task which required students to write a functional rule between the number of
hexagons and the perimeter of the figure.
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Most students show little understanding of the functional relationship between the
domain and range. Students tend to focus merely on the consecutive terms in the range
set.
T OO TH PI CK S HA PES
Tom us es t oo thpicks to ma k e th e sh a p e s in th e
di agram b el ow .
shape 1
6 toothpicks
shape 2
9 toothpicks
1 . H o w m an y too thpicks
shape 3
shape 4
ma k e sh a p e 3 ?_________________
2 . D ra w sh a p e 4 n ex t t o sh a p e 3 in th e di agram
a b ove .
(Q. 3&4 Make a table and draw a graph)
5 . Tom s ay s , “I n eed 36 toothp ic k s to m ak e s hap e 12.”
Tom is n o t corre ct . E x pl a in wh y h e is n o t corre ct.
H o w m an y too thpicks are n ee d e d to ma k e sh a p e 12 ?
Findings from Toothpick Shapes
 The
fifth grade students demonstrated the ability to identify
the pattern, extend it, complete a t-table of values, graph the
function and find a specific functional value for a given
sequence number.
 Over 50% of the 5th grade students were to successfully
complete all aspects of the task.
 70% demonstrated competence in all 5 parts of the task.
 89% of fifth graders met the core standards of the task.
 This is just one of many assessments that show students
coming out of elementary grades with basic algebraic thinking
skills for attacking linear function problems.
T o oth p ic k S ta irs
T his p ro bl em g ive s yo u t h e cha n ce t o:
•
•
ex te n d a n d check a g
use va
iven p a tter n
lue s in a t a bl e to d er ive a fo rmu la
Jake m akes s ta irc ases w ith too thpi cks arra nged in squ are s.
H e c ount s th e nu m be r of too thpi cks h e us es and m akes a table to sho w hi s f ind ings.
Staircase 1
Staircase 2
Staircase 3
Staircase 4
4. Wr ite down a r ul e or formu la link ing th e nu m be r of too thpi cks on the pe rim eter, P,
w it h th e sta irca se numb er , S.
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5. W r ite do w n a r ul e o r form ula link in g the n u m b e r of t o oth pi ck s in sid e th e
stai rc as e , I, w ith th e sta ir ca se n u mb er , S.
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Expl ai n how you
fi gu red it out.
Findings from Toothpick Stairs
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Algebra one students demonstrated similar levels of
understanding functions as did fifth grade students.
65% of ninth graders could merely extend a linear
pattern and and complete the t-table.
38% more algebra one students were successful in
19% of algebra students were successful at writing an
algebraic rule for the linear relationship (success on
question 4)
4% were successful at writing the quadratic rule
(question 5).
Conclusions
In general, the findings from five years of performance
assessing of students enrolled in an algebra one course shows
that students have minimal utility with algebraic language and
constructs when presented with non-routine exercises.
Most algebra students demonstrated minimal skills in
interpreting problems, formulating successful strategies,
understanding linear systems or building rules to represent
function.
Students demonstrate lack of skills in generalizing even linear
relationships. In fact, students rarely show growth in depth of
understanding functional relationships between 4th grade and
completion of algebra 1.
Lessons to be Learned
Algebra One instruction must be taught in
problems solving context. The valuable
mathematical tools learned in Algebra One must
be useful to students and experienced in a manner
where students learn to apply algebraic reasoning
and language to problem situations. The
assessments are evidence of whatever algebraic
knowledge is acquired in class is fleeting and
obscured from use when students are presented
with non-routine problems.
Call to Action
There needs to be a dramatic shift in instruction for algebra. The
data is clear, algebra students by and large can not use algebra to
solve problems. Algebra instruction must involve problem
solving, applications to real problem situations, and taught in a
manner that promotes thinking and making meaning. The data
also shows that students in elementary grades are using algebraic
thinking to make sense of situations. Secondary teachers must
build on these foundations to further students’ knowledge.
Unfortunately the data indicates that not only are secondary
teachers not expanding students’ algebraic thinking skills, that in
fact students’ problem solving and thinking skills may be eroding.
This is a disturbing trend has been documented repeatedly. It is
time to take dramatic action to change the practice of how algebra
is taught in secondary schools.
Algebra for All Project
Institute for algebra teachers
Sponsored by the SCVMP and SVMI
Dr. Joanne Rossi Becker
Cathy Humphreys and David Foster
Goals for the Algebra for All Project
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Provide ideas and materials to teach algebra
Work on problems with a group of teacher to learn
from each other
Promote higher-level thinking while still focusing
on the essential knowledge
Come away with a better understanding of how to
structure a class to be more open to alternative
thinking.
Reaching populations of students that are
unsuccessful
Build a better community of learners
Habits of Mind for
Algebraic Thinking
Doing – Undoing. Effective algebraic
thinking sometimes involves reversibility
 Building Rules to Represent Functions.
Input is related to output by well-defined
rules
 Abstracting from Computation. Abstracting
system regularities from computation

Fostering Algebraic Thinking, Driscoll, 1999,
Educational Development Center, Heinemann
AAP - Big Ideas
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Problem Solving and Variable – - Border Problem & Postage Stamp
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Equality and Algebraic Representation – Banquet Tables
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Linear and Non-Linear Functions – Window Frame Problem
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Polynomial Functions - Spot Problem (regions of circles)
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Slope and Graphical Representation – Lesson Story Graphs
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Polynomials (Multiplication and Factoring) – Miles of Tiles
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Rate of Change and Technology – Match the Graph
The Border Problem
QuickTime™ and a
Sorenson QuickTime™
Video 3 decompressor
and a
Sorenson
Video
3 decompressor
are needed
to see
this picture.
are needed to see this picture.
The Border Problem
Sharmane:
Colin:
Joseph:
Melissa:
Tania:
Zachery:
4•10 - 4 = 36
10+9+9+8 = 36
10+10+8+8 = 36
10•10 - 8•8 = 36
4•9 = 36
4•8 + 4 = 36
The Teacher’s Strategy
The teacher used the experience of the 10 by 10 border
problem to built algebraic understanding. She asked the
students to think about a smaller square, 6 by 6, and asked
the students to determine a set of equations of the 6 by 6
that matched the ways the students thought about the 10 by
10 square. They had to write new equations in the same
manner that Sharmane, Colin and the others had in the first
problem. Next the teacher asked the students to color a
picture of the border problem, to match each equation and
also write the process to find each total in a paragraph.
Now she felt the students were ready to use algebraic
notation to generalize each equivalent equation.
Student Equations
Generalizing For Any Size Square
 10+10+8+8=36
 Let
x be the number of unit square along
the side of the square.
x
+ x + m + m = total
x
+ x + (x-2) + (x-2) = total
Introducing Algebraic Notation
Moving from the specific to the general case.
Developing an understanding of variable and its uses.
Tying abstract ideas to concrete situations.
Fostering meaning to notation.
Developing the concept of equivalent expressions.
Encouraging efficiency and brevity in notation
Border Problem Lesson 3
Students conjectured that the following expressions
were equivalent to the original. The class was
challenged to verify their conjectures.
b2 - (b - 2)2 =
?
b2 - b - 22
?
b2 - b2 - 2
?
b2 - 2 - b2
? (b2 - 2)2
Teacher Reflections
The Border Problem was probably most significant for
me. This problem incorporates several aspects of
emerging algebraic thinking that are crucial to the
Algebra student. The problem ties in nicely with
existing “text book” curriculum and can be used as a
late sequel to more advanced concepts. Those who
teach out of Dolciani Algebra: Structure & Method
book 1 (Red Book) should consider using this problem
before/in conjunction with chapter 4 more specifically
the section on area problems. (4-9).
Teacher Reflections
The Border Problem allowed for most (if not all) students to
develop an algebraic expression, which would calculate the
square units in the border of a square frame. What I found is
that many of the students did not naturally use a variable in
their expression. In the future, I would require students to
work with several different size square borders; then have
them present their expressions while I compiled a list of
correct ones. We would then look for similarities and as a
Part II, I would have the expectation that generalizations be
made, and that a variable represent the same “part” of
different sized frames.
Achievement on Algebra Performance Exam, 9th Grade
P e r c e n t M e e ti n g S ta n d a r d s
100
90
80
70
60
50
40
30
20
10
0
1999
2000
2001
2002
Years of the exam
2003
2004
2005
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