What makes Algebra Difficult for Some Students to learn?

Steve Rhine Pacific University

Rachel Harrington Western Oregon Univ

NCTM (2008)

“Knowing algebra opens doors and expands opportunities,

instilling a broad range of mathematical ideas that are useful

in many professions and careers. All students should have access to algebra

and support for learning it.”

The Gatekeeper

• Algebra serves as a filter to higher education (Ladson-Billings, 1998 and Moses & Cobb, 2001)

• “Algebra can mean the difference between menial work and high level careers” (Helfand, 2006)

• Over the past 17 years a significant gap has remained or increased for students of low socio-economic status and minorities on standardized tests (NAEP, 2008)

Failure

• Los Angeles: 48,000 students took Algebra in 2004. 44% failed. 17% finished with D’s. Three-fourths failed when they repeated Algebra.

• Chicago: Failure in Algebra is significantly increasing.

• Oregon: 31% of students failed the state test in math at 11th grade (2012-13)

• “It triggers drop-outs more than any other subject.” (Helfand, 2006)

Response

• “Algebra for all” programs across the country.

• Double periods of Algebra.

• Move Algebra into the 8th grade

• Consider algebra foundation in elementary

• Reform programs: Connected Mathematics and College Preparatory Mathematics

• Milgram: Go back to traditional teaching.

An error is not merely a failure by a student

but rather a symptom

of the nature of the conceptions

which underlie his/her mathematical activity.

(Balachef, 1984)

• Read 800+ articles on how students think in algebra

• Encyclopedia of Algebraic Thinking

• Formative Assessment Database

• Video Database

• 18 iOS apps

algebraicthinking.org

Transition from Arithmetic to Algebra

• from what is numeric (digital) or verbal to symbolic

• from what is specific to what is general

• from work with what is known to work with unknowns

• from intuitive processes to school rigorous (algebraic) processes

What is different about algebra?

Student Thinking

Student Thinking

• x + 5• Acceptance of Lack of Closure• Process-Product Dilemma

• expression is a procedure and answer simultaneously• Add 5 to the variable x• The number that is 5 more than x

• 8 + 4 = ? + 5• 100% of 145 6th grade students answered 12 or 17

Variables• Students struggles in algebra can often be traced

back to their understanding of variables.

• Students struggle with • the name of an object (m is Michael), • the name of an attribute (m is Michael’s height), • or a measurement or quantity (m is meters).

• Fruit Salad Algebra• 3a + 5b + 2a - 8b

VideoWhich is larger 2n or n + 2 ?

VideoWhich is larger 2n or 2 + n ?

• When can what variables represent change?

• What variables represent can change

• across context (3x - 2 = 3 vs. 5x + 34 = 4)

• or not (i.e. which is larger, 2n or n + 2?)

• but stay the same within a context (4x + 4 = 5x - 8)

Variables

Is  the  following  always,  sometimes,  or  never  true?    Why?                                    h  +  2  =  h  

• Response:  sometimes,  because  the  /irst  h  could  be  5  and  then  the  answer  would  be  7.

Formative Assessment Problems

Formative Assessment Problems

What  value  or  values  could  “x”  be  in  the  expression?                                                                                x  +  x  +  x  =  12  

• Response:  (2,5,5)  and  (10,1,1)  could  work  because  “x” is  unknown  so  it  could  be  anything.

Formative Assessment Problems

If  c  +  d  =  10  and  c  <  d,    what  does  “c”  equal?  

• One number (e.g., c = 4, d = 6) (39%)

• Only integer responses (e.g., 1, 2, 3, 4)(42.9%)

VideoWhat can ’n’ stand for?

Can “n” stand for 4? Yes (56%, 77%, 87%)*

Can “n” stand for 37? Yes (30%, 67%, 81%)*

Can “n” stand for 3r + 2? Yes (26%, 30%, 47%)*

Representation

*Percent of 373 6th, 7th, and 8th grade students (respectively) in Weinberg, et. al’s (2004) study with that answer.

I: Could “n” stand for 15 + 27?S: I think so. Well, actually, I don’t think so really because variables just stand for one number. You could have “n” plus another letter or variable and “n” could be 15 and the other variable could be 27. n = 15 and p can equal 27 so then if you did n + p it would equal 42.I: Can “n” stand for (3r + 2)?S: I don’t think so because it couldn’t really stand for, well, actually yes it could because it’s all in parentheses so the r could stand for another number and then it would all be one number. Inside the parentheses you’d have to do come up with an answer.I: What about (3r) + 2? Could n stand for that?S: No. Because what is in the parentheses stands for a separate number and it’s basically what you did up here (points to 15 + 27). It’s as if you were doing 10 plus 5 in parentheses and then plus 27 out of the parentheses.I: Could “n” stand for “r”?S: I don’t think it could because n, variables stand for numbers and if n were a variable then it would have to stand for a number, not another letter.I: Would it make a difference if I put parentheses around it? (“r”)S: I think it might because then “n” is standing for what “r” equals, not just “r” but I’m not quite sure.

Student ThinkingI: (Simplify e + 2 + 6) Why did you put 13?

S: I added 2 and 6 to get 8, but I didn’t really know what to do next so

I guessed that because “e” is the fifth letter of the alphabet, the answer

must be 13.

I: What is x? (given 10 – x, when x = 6)

S1: 9. Because x is just like 1. Like having one number. And so you take

one of the x’s out of the tens and you get 9.

S2: Well, x equals 1. By itself it is 1, the x.

S3: x is just one single thing, so like x times x is just like 1 times 1.

Variables

• A = LW

• 40 = 5x

• sin x = (cos x)(tan x)

• 1 = n (1/n)

• y = kx

What is a variable?

Variables• A = LW

• a formula indicating a relationship with letters that are connected to a physical reality

• 40 = 5x• x is an unknown, solved by algebraic manipulation

• sin x = (cos x)(tan x)• abstract, not necessarily connected to a solution

• 1 = n (1/n)• a property, to be used for manipulation

• y = kx• y & x vary in a dependent/independent relationship and

k is a constant

I: (h + 10) What is 10h (or h10)?

S: “h” added to 10.

I: If x = 6, what is the value of 2x?

S: 26. You put the 6 in for the x because it is supposed to

something more than 20.

SymbolizationStudents often have logical thinking

but struggle with symbolizing

Students’ Thinking

• Letters represent whole numbers (75%)

• Different letters represent different numbers (74%)

• The letters chosen and the numbers are related• x, y, z and 3, 4, 5 or 10, 20, 30 (10%)• letters higher in the alphabet are bigger (10%)• alphabet value: e is the 5th letter = 5 (13%)

Strategies for Teaching Variable

• Use prior knowledge; i.e. numbers only

• 3 = 2 + 1 or 2 + 1 = 3 x = 3 or 3 = x• a sense of balance• a variable as a container that can hold anything

• x = 3, x is another name for 3

• What are other names for 3? What can x hold?• 2 + 1 = 3 so x = 2 + 1• 12/4 = 3 so x = 12/4• (14 + 4)/6 so x = (14 + 4)/6

• Money bags and coins

Strategies for teaching Like terms

• 3(4) + 2(4) = 5(4)

• 6(7 - 2) + 3(7 - 2) = 9(7 - 2)

• 8(15/3) - 2(15/3) = 6(15/3)

• 4x + 5x = 9x 8d - 5d = 3d

• 5(6 - x) + 8(6 - x) = 13(6 - x)

• 2(7) + 6(8) + 4(7) + 3(8) = 6(7) + 9(8)

• 5x + 2y + 3x + 4y = ?

• 5m + 2p - 3m + 4p = ?

Algebraic Relations

• Understanding Features• Equals Sign and Inequalities• Negative Numbers• Ration Numbers

• Understanding Processes• Variables on one side• Variables on both sides• Solution Strategies

Mathematical Issues

Two approaches• "Repeatedly doing the same thing to both sides

of the equation" until you arrive at the answer.• The "change sides, change signs" approach.

Students’ selection of a solution strategy depends extensively on the strategies employed

in lessons by their teachers.

Student Interviews

Find numbers for a, b, and c such that

• all numbers will work• there are no numbers that will work

[a=0 and b, c can be anything except zero]

Teaching Strategies

Estimate

• 1• ½ • 0• don’t know

(x-3) + 4 = (x-3) + x

Teaching Strategies

1. Identities: 2. Non algebraic equations:3. Equations with more than one unknown:

3x+4y = a4. Trivial equations: x=25. Functions: f(x)=2x+16. Inequalities and expressions: 3x+2 > 4x +5

algebraicthinking.org

Steve Rhine, Ed. D.Pacific University

steverhine@pacificu.edu

Rachel Harrington, Ph. D.Western Oregon University

harringr@wou.edu