Steve Rhine, Ed. D.
Willamette University
Variables

•“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.”

• 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 socioeconomic status and minorities on standardized tests (NAEP, 2008)

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

• “Algebra for all” programs across the country.
• Double periods of Algebra.
• Algebra is increasingly being pushed into the 8th grade.
• Reform programs: Connected
Mathematics and College Preparatory
Mathematics
• Milgram: Go back to traditional teaching.

• There are two kinds of subject-matter understanding that teachers need to have— knowing that (something is so) and knowing why (it is so)…
• Knowing that refers to research-based and experience-based knowledge about students’ common conceptions and ways of thinking about subject matter.
• Knowing why refers to knowledge about possible sources of these conceptions.
• ( Tirosh, Even, & Robinson, 1998)

• Seeing misconceptions as mistakes that impede learning is not a productive orientation for teachers.
• “The freedom to err is at the heart of developing mathematical knowledge.”
(Movshovitz-Hadar & Hadass, 1990)
• Errors are opportunities for learning when students are given the time to recognize and successfully reflect on their errors and teachers have strategies for supporting students in recognizing and reconciling their errors.

What is different about 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

• 8 + 4 = ? + 5
• 100% of 145 6th grade students answered 12 or 17
• x + 5
• Acceptance of Lack of Closure
• Process-Product Dilemma
• An expression can represent a procedure and an answer simultaneously
• Add 5 to the variable x
• The number that is 5 more than x
• Your experience?

What is a variable?
• Students struggles in algebra can often be traced back to their understanding of variables.
• Students struggle with the difference between the name of an object (e.g., the person
Michael), the name of an attribute (e.g.,
Michael’s height), or a measurement or quantity (h units).

What is a variable?
• A = LW
• 40 = 5x
• sin x = (cos x)(tan x)
• 1 = n (1/n)
• y = kx

• A = LW
• a formula indicating a relationship with
• 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 relationshiop and k is a constant

• What does the word “represent” mean
• in different contexts?

•
• 1. Is the following always, sometimes, or never true? Why? h + 2 = h
•
• 2. What value or values could “a” be in the expression?
7 + a + a + a + 10
• 3. If c + d = 10 and c < d, what does “c” equal?
• 4. Can you simplify this expression? If so, how?

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%)*
*Percent of 373 6 th , 7 th , and 8 th 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 (3 r + 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 (3 r ) + 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.

I: When is t + t larger than t + 4?
S: t + t is 2t and t + 4 is 4t so t + 4 is bigger.
I: (5a + 2a) Why did you put 7a?
S:
Anything can be an “a”, so you can put two apples and five apples, so then you can add the five and the two, which is seven, and then you can put seven “a” equals seven apples.
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.

I: (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.

• 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%)
• S’s ignore meaning of letters in abstract examples
• Add up numbers, put down each letter:
2x+8y+3x=13xy (56%)
• Add up numbers, put down letter that occurs most: 2x+8y+3 x
=13x (30%)

• S’s often have solid thinking but difficulty symbolizing
• They might be able to explain generally: “Add whatever number x is to whatever y is” (83%)
• They might be able to explain in an example: “If x were 2 and y were 3 I’d add, so it could be x + y ” (17%)
• Procedures students use might be tied to a specific context and not generalizable
• S’s believe in concrete, numerical solutions

How many black and white tiles in the nth term?
1, 4, 7, 10, ...

STAGES
• seeing (recognizing the pattern)
• saying (verbalizing a description)
• recording (creating a written representation)
ACTIVITIES
• identifying commonality across cases
• extending one’s reasoning beyond
• deriving broader results from particular cases

0 No justification Students are ignorant of the need for, or existence of justification or proof.
1
Appeal to an external authority
Reference is made to the correctness stated by some other individual or reference material.
2
Empirical evidence
Justification is provided through the correctness of particular examples. Students are aware of the idea of a proof, but consider checking a few cases as sufficient.
3
4
5
6
Strategic evidence
Students are aware that checking a few cases is not sufficient but are satisfied that checking extreme cases or random cases is proof.
Generic example
Deductive justification is expressed in a particular instance. Students believe that use of a generic example forms a proof for a class of objects.
Limited
Deductive
Justification
Students are aware of the need for a general argument, are able to understand the generation of such an argument, and are able to produce such arguments themselves in a limited number of familiar contexts.
Deductive justification
Students are aware of the need for a general argument, are able to understand the generation of such an argument (including more formal arguments), and are able to produce such arguments themselves in a variety of contexts (both familiar and unfamiliar). Validity is given through a deductive argument that is independent of particular instances.

• Letter evaluated
• number assigned to letter at the start (b + 3 = 5)
• Letter not used
• letter ignored (b + 3 = 3)
• Letter as an object
• letter regarded as shorthand for an object or as an object itself (5a +
2a)
• Letter as a specific unknown
• the letter is regarded as a specific unknown (x + 3 = 9, x = 6)
• Letter as generalized number
• letter represents several values (x - 8 < 4)
• Letter as variable
• represents range of unspecified values with relationships between values