What is algebra?
Milgram doesn’t think California should adopt what he sees will be weaker standards. Neither does Ze’ev Wurman, a Palo Alto high-tech executive and former adviser to the U.S. Department of Education.
“Essentially we are giving up on the hope of teaching algebra in the eighth grade,” he said. He charges the proposed standards set the bar too low for college readiness.
A lot of the discussion surrounding the 6-8 portion of the Common Core standards has been on the removal of algebra. But has it been removed, really? What needs to be included to call it “algebra”?
6th grade: Understand that applying the laws of arithmetic to an expression results in an equivalent expression. For example, applying the distributive law to the expression 3 x (2 + x) leads to the equivalent expression 6 + 3x. Applying the distributive law to y + y + y leads to the equivalent expression y x (1 + 1 + 1), i.e., y x 3 and then the commutative law of multiplication leads to the equivalent expression 3y.
Is this algebra? When I was in 6th grade there was no exposure at all to algebraic expressions and the simplification thereof. That was part of “algebra”.
6th grade: Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity, and an equation can express one quantity, thought of as the dependent variable, in terms of other quantities, thought of as the independent variables; represent a relationship between two quantities using equations, graphs, and tables; translate between any two of these representations. For example, describe the terms in a sequence t = 3, 6, 9, 12, … of multiples of 3 by writing the equation t = 3n for n = 1, 2, 3, 4, ….
This could, of course, be considered “pre-algebra”, but then: what is pre-algebra? Of the pre-algebra books I have surveyed, if students knew all the material within they would have most of what gets done the first semester of 9th grade already mastered.
6th grade: Using the idea of maintaining equality between both sides of the equation, solve equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Is solving equations algebra? Or are one-step equations too simple? Perhaps they need two steps to be considered algebra …
7th grade: Solve the same word problem arithmetically and algebraically. For example, “J. has 4 packages of balloons and 5 single balloons. In all, he has 21 balloons. How many balloons are in a package?” Solve this problem arithmetically (using a sequence of operations on the given numbers), and also solve it by using a variable to stand for the number of balloons in a package, constructing an equation such as 4b + 5 = 21 to describe the situation then solving the equation.
… rather like the grade 7 standards, which introduce them explicitly.
7th grade: Generate equivalent expressions from a given expression using the laws of arithmetic and conventions of algebraic notation. Include:
a. Adding and subtracting linear expressions, as in (2x + 3) + x + (2 – x) = 2x + 5.
b. Factoring, as in 4x + 4y = 4(x + y) or 5x + 7x + 10y + 14y = 12x + 24y = 12(x + 2y).
c. Simplifying, as in –2(3x – 5) + 4x = 10 – 2x or x/3 + (x – 2)/4 = 7x/12 – 1/2.
I’m sure they are out there, but I’ve yet to see a pre-algebra book that made it to expressions as complex at the above.
What makes the line complex and fuzzy is the history of math education has seen a constant shuffling of curriculum. What used to be called “algebra” in the 1950s is much different than what it is now. Much that used to get taught at the college level now has been moved to the high school level.
8th grade: Understand that the graph of a linear equation in two variables is a line, the set of pairs of numbers satisfying the equation. If the equation is in the form y = mx + b, the graph can be obtained by shifting the graph of y =mx by b units (upwards if b is positive, downwards if b is negative). The slope of the line is m.
Graph proportional relationships and relationships defined by a linear equation; find the slope and interpret the slope in context.
I’m not sure what’s going on in other states, but our 9th graders spend about a month on the above standards, and I have 12th graders who still don’t get it.
While people are busy comparing with international standards, I should point out not every country teaches the form y=mx+b.
8th grade: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because the quantity 3x + 2y cannot simultaneously be 5 and 6.
This is the point where I do some serious head-scratching: how is solving a system of simultaneous equations algebraically not part of algebra?
8th grade: Understand that a function from one set (called the domain) to another set (called the range) is a rule that assigns to each element of the domain (an input) exactly one element of the range (the corresponding output). The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.
Perhaps this is the source of grouchiness? I’ve also written about function notation being introduced too early, and I agree that 8th grade is too soon.
If one compared the Common Core standards to an “algebra” book there are omissions: in particular quadratics are kept out. I believe that’s where the bone of contention is. Somehow if students aren’t factoring by hand it isn’t yet algebra.
As far as I can tell, by 8th grade the Common Core standards include most of what a student needs to know about lines. In my mind, especially for non-STEM students, this is enough. Of course, feel free to disagree in the comments.