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What Set Theory is covered in the Integrated Algebra Curriculum?

April 17, 2009 pm30 10:36 pm

It’s really a sort of trick question. New York State’s Integrated Algebra isn’t so much a curriculum as a set of “standards” and “performance indicators.” And the Standards and PIs aren’t really more than guideposts to what will be on The Test. And it is The Test (NYS Integrated Algebra Regents) that mot districts will look at in determining what to teach.

But before we move on, we need to look at two additional documents:

Suggested List of Mathematical Language for Integrated Algebra (HTML version, or page with links to HTML, doc, and PDF versions)

(for teachers) Commencement Level Glossary of Mathematical Terms (available as a document, on the same page)

Hidden (hidden? not really) in the algebra section of the List of Mathematical Language are the following:

  • complement of a subset
    intersection of sets
    interval notation
    roster form
    set
    set-builder notation
    solution set
    subset
    union of sets
    universal set

Here is what appears in the Glossary:

  • complement of a set  (A)  The elements of a universe not contained in a given set; the subset that must be added to any given subset to yield the original set. The complement of set A is indicated by A' or A^C . [there is a colored Venn Diagram]
  • intersection of sets  (A)  (G)  The intersection of two or more sets is the set of all elements that are common to all of the given sets.
    Example:  If A = {1,2,3,6} and B = {0,2,5,6,7}, then the intersection of A and B, denoted by  A ∩ B, is  {2,6}
  • roster form  (A) A notation for listing all the elements in a set using set brackets and a comma between each element.
    Example:  The set of prime numbers less than 10, expressed in roster form is {2, 3, 5, 7}.
  • set  (A) (G) A well-defined collection of items.
  • set-builder notation  (A)  A notation used to describe the elements of a set.
    Example:  The set of all positive real numbers in set builder notation is   This is read as “the set of all values of x such that x is a real number and x is greater than 0.”
  • solution set  (A)  (A2T)  Any and all value(s) of the variable(s) that satisfy an equation, inequality, system of equations, or system of inequalities.
  • subset  (A) (A2T)  A set consisting of elements from a given set; it may be the empty set.
    Example: if B = {1,2,3,4,5,6,7} and A = {1,2,5}, then A is a subset of B.
  • union of sets  (A) (G)   The union of two or more sets is the set of all elements contained in at least one of the sets.
    Example:      if Set A = {2,4,6,8,10} and Set B = {1,2,3,4,5,6}, then the union of sets A and B,  written as  A \cup B is {1,2,3,4,5,6,8,10}.
  • universe  (A)  The set of all possible specified elements from which subsets are formed.  Also know as the universal set.
  • Venn diagram  (A)  A drawing showing relationships among sets.
    Example:  The Venn diagram below shows 14 students.  Five students play basketball, seven run track, two play basketball and run track, three play only basketball, five only run track.  Four students do not play basketball or run track. [there is a Venn Diagram drawn. An unlabeled, outer rectangle is the universe. Numbers are written in the four regions]

So, add to this the PIs:

Patterns, Relations, and Functions
A.A.29      Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form
A.A.30     Find the complement of a subset of a given set, within a given universe
A.A.31     Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets)

And the questions they’ve used:

  • 06/08 #18 (multiple choice, 2 points) Consider the set of integers greater than -2 and less than 6. A subset of this set is the positive factors of 5. What is the complement of this subset?
  • 06/08 #33 (free response, 2 points) Maureen tracks the range of outdoor temperatures over three days. She records the following information: [the exam has three number lines, for Day 1, Day 2, and Day 3 respectively. The ranges indicated are -20 ≤ t ≤ 40, 0 ≤ t ≤ 50, -23 ≤ t ≤ 45]. Express the intersection of the three sets as an inequality in terms of temperature, t.
  • 08/08 #25 (multiple choice, 2 points, I don’t think this should count, either) Which ordered pair is in the solution set of the following system of inequalities y < \frac{1}{2}x + 4 and y \geq -x + 1
  • 08/08 #33 (free response, 2 points) Twelve players make up a high school basketball team. The team jerseys are numbered 1 through 12. The players wearing the jerseys numbered 3, 6, 7, 8, and 11 are the only players who start a game. Using set notation, list the complement of this subset.
  • 01/09 #17 (multiple choice, 2 points, I have included the first response) The set {1, 2, 3, 4} is equivalent to  (1)  { x | 1 < x < 4 , where x is a whole number}

And that’s all we have to go on.

So, what do we have?

A loose definition of set. A loose definition of subset. Curly braces all over the place (the idea of solution set permeates the curriculum, without any recognition that it is a kind of set). A mention of “empty set.”  Symbols also include the vertical bar “such that.” Included are intersection and union, (strangely without binary descriptions), and their respective symbols. Venn Diagrams are included. “Complement” is included, with two possible symbologies.

Missing is the word “element” and the cute, associated symbol: ∊   Also missing is the concept of proper subset. Symbology for the empty set. Symbol for subset. Missing is the word “disjoint” and any hint about the concept. No reference is made to cardinality. And obviously, no associated symbols.

[edit: also missing are the symbols for not an element of, ∉ , and not a subset of, ⊄.  Vlorbik reminded me.]

In addition, notice that the Regents questions have not used any of the associated symbology except curly braces and the vertical line. Empty set has not shown up. Subset has not shown up in a way that requires students to understand what the word means. The symbols for intersection and union have not shown up.

Sounds like we’ve defined a minimum: Teach what a set is, how to write it (including both roster and set builder notation), what union and intersection mean, and what “complement” means. And we’re done?

Unfortunately, in many places, the answer will be yes, we are done. But I will be recommending in upcoming posts that we teach more than this, that we create a small unit, and that we embed hooks to it in other parts of the course.

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5 Comments leave one →
  1. April 18, 2009 am30 12:25 am 12:25 am

    i’m pretty well convinced that the main purpose
    of set-builder notation at the introductory level
    is intimidation.

    i know for a fact that many of my colleagues
    don’t understand it.

    the example at 1/09 #17 is a mess.
    it’ll still be an abomination if you fix it.
    i blogged about this
    (kind of thing) back in the day.

  2. April 19, 2009 pm30 11:47 pm 11:47 pm

    I hate, hate, hate teaching this to freshmen. I don’t even know where the disconnect is. But they don’t get it. I’m reduced to pounding definitions into and out of them for a week, and then I wash my hands of it. Any insight you have will be greatly appreciated.

  3. queens, ny permalink
    April 20, 2009 am30 2:15 am 2:15 am

    i’m a freshman!

    and i don’t get subsets ):

  4. April 20, 2009 am30 8:32 am 8:32 am

    Thanks Vlorbik. I’m counting on you to pick out things I get wrong.

    Kate, it’ll take me most of a week to tease out a full unit with connections.

    And, queens! Lots of people don’t get subsets. Did you learn anything about them in class? Later this week, I’ll attack them backwards: I’ll help you decide what is NOT a subset. More fun that way.

  5. mzadvntr permalink
    June 3, 2010 pm30 4:54 pm 4:54 pm

    Dear God…

    My son is taking this course in the fall and I am immediately suspicious of anything in NY that is labeled “integrated”. I’ve read the powerpoint on the Curriculum, and this course seems like a nightmare for the students. It’s as if they are being asked to translate a foreign language using a dictionary that’s written in another foreign language. Isn’t there a more clear way of teaching them to “speak” in set builder, or is it really too early to introduce that concept. Or are we back to Math A where they figure it’s okay if you don’t “get it” the first time, they’ll loop back in 2 years and you’ll “get it” then??
    Argghh.

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