Examining Student Proficiency in Algebra

Examining Students’ Understandings and Misconceptions

Urban Mathematics Leaders Network

David Foster at Lakeway, June, 2005

And what does x equal today?

AlgebraAlgebra

Shift in policies and practice of algebra in grades K-12 National movement for algebra for all students. National call for algebraic thinking in all grades

K-12. 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?

Data from an Annual Algebra Performance Exam administered to ninth grade students each year since 1999. Approximately 42,000 students were assessed, drawn from 35 member school districts.

The Silicon Valley Mathematics Initiative

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.

Show all your work clearly.

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 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.

PATHWAYSThis problem gives you the chance to:• find the dimensions of the paving stones used to make a pathway.

Bob uses paving stones to make a pathway.

The paving stones are rectangles and they are allthe same size.

2.5 ft

The pathway is 2.5 feet wide.

Find the length and width of the paving stones.Explain how you figured it out.

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[5]

Findings from Pathways Students in geometry demonstrate similar levels of

algebraic understanding as algebra one students. 38% of geometry students demonstrated no success

with this task. 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.

Make Half

This problem gives you the chance to:

• solve a problem using fractions

5 ?

12 ?If you add 2 to the top and bottom of

5

12 you get a fraction equal to one half.

5 2

12 2

7

14

1. What number could you add to the top and bottom of these fractions tomake one half? Remember that you must add the same number to both topand bottom. Show your work.

2?5 ?

2?6 ?

2?7 ?

Algebra

Ninth

Grade

2. Use your answers in question 1 to find the answer to these:

2 ?

35 ?

2 ?

n ?

Show your work.

Findings from Make Half 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.

Algebra one students had difficulty completing the task other 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.

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

Performance in High Poverty High School Districts

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..

In 2003, MAC administered similar tasks to 4th and 7th 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.

Algebraic Thinking at 4th Grade

Algebraic Thinking at 7th Grade

Algebraic Thinking Across Grade Levels 78% of the fourth grade students demonstrated they could recognize, extend and graph

the linear pattern.

60% of the seventh grade students demonstrated they could recognize, and extend the same linear pattern.

30% of fourth graders met all the demands of their task that included finding and verifying the functional relationship for a given case.

35% of the seventh grade student met similar demands of the fourth grade task, finding and verifying the functional relationship for a given case.

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.

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.

TOOTHPICK SHAPESTom uses toothpicks to make the shapes in thediagram below.

shape 1 6 toothpicks

shape 2 9 toothpicks

shape 3 shape 4

1. How many toothpicks make shape 3?_________________

2. Draw shape 4 next to shape 3 in the diagramabove.

5. Tom says, “I need 36 toothpicks to make shape 12.”Tom is not correct. Explain why he is not correct.How many toothpicks are needed to make shape 12?

5th Grade Task

(Q. 3&4 Make a table and draw a graph)

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.

Toothpick Stairs

This problem gives you the chance to:

• extend and check a given pattern

• use values in a table to derive a formula

Jake makes staircases with toothpicks arranged in squares.He counts the number of toothpicks he uses and makes a table to show his findings.

Staircase 1 Staircase 2 Staircase 3 Staircase 4

4. Write down a rule or formula linking the number of toothpicks on the perimeter, P,

with the staircase number, S.

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5. Write down a rule or formula linking the number of toothpicks inside thestaircase, I, with the staircase number, S.

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__________________________________________________________Explain how you figured it out.

Findings from Toothpick Stairs

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 extending the quadratic pattern.

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 ProjectInstitute 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

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

Problem Solving and Variable – - Border Problem & Postage Stamp

Equality and Algebraic Representation – Banquet Tables

Linear and Non-Linear Functions – Window Frame Problem

Polynomial Functions - Spot Problem (regions of circles)

Slope and Graphical Representation – Lesson Story Graphs

Polynomials (Multiplication and Factoring) – Miles of Tiles

Rate of Change and Technology – Match the Graph

The Border Problem

QuickTime™ and aSorenson Video 3 decompressorare needed to see this picture.

QuickTime™ and aSorenson Video 3 decompressorare needed to see this picture.

The Border Problem

Sharmane: 4•10 - 4 = 36

Colin: 10+9+9+8 = 36

Joseph: 10+10+8+8 = 36

Melissa: 10•10 - 8•8 = 36

Tania: 4•9 = 36

Zachery: 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

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.

Teacher Reflections

Achievement on Algebra Performance Exam, 9th Grade

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