Goals and notes for future years

Notes after 2nd semester

Ideas for approach to the content:

  1. Prove quantified statements of both types before ever touching implication

Elicit student thinking by asking them to:

  • take this imprecise sentence and rewrite what the person is trying to say by using language properly
  • ask students to use words like “infer” and “implies” and “means” and “such that” in ways that make sense in ordinary English sentences
  • indicate a certain form for a proof, i.e., a biconditional in which one of the statements is a “for all”, etc. — it's easy to come up with unique ones, so have each student write a "skeleton" and present it quickly to the class
  • indicate every error (logical and notational) in a given proof
  • Give students examples in which definitions, vocabulary, or notation has been misused, and ask them to identify exactly where in the text those things are precisely defined, and to explain why they have been misused. Also ask them to identify whether the misuse results in a statement that is false, or rather in nonsense.
  • To get students to think about type, ask them for things like a set of vectors, a set of numbers, a set of sets of vectors, a set of pairs of sets of numbers, etc.
  • Give students "non-math" statements and ask them to put "implies" or "is equivalent to" in between them, e.g., "Mona is my (maternal) grandmother" and "Mona is my father's mother"

Learning Goals


  1. read and write set-builder notation
    • have students "explain in your own words what it means for x to be in the set S"
    • students recognize "identify an object that is definitely in the set and one definitely not in the set" as a good strategy for getting a concept of what a set is

Habits of mind

  • Techniques of proof
    1. understand the way a proof is like a "thought experiment"
    2. respect that each statement of a proof must follow from the ones before it and that
    3. fluency with fundamental proof techniques
      • can describe the basic online of a proof
        • …by induction
        • …by contradiction
        • …of a universally quantified statement
        • …of an existentially quantified statement
  • Ability to read/write mathematics
    1. emphasis on integrating mathematical notation properly into complete sentences
      • rule number one is make a statement
        • positive indicator: students write mathematics in actual sentences, including things like periods after equations when appropriate
        • negative indicator: student writes down isolated mathematical expressions without any attempt to express a statement of fact
      • rule number two is don't write nonsense
        • negative indicator: student makes reference to a variable before any explicit meaning has been assigned to that variable
    2. grammar of mathematics, e.g., reading and writing "=", "$\Leftrightarrow$", etc. interchangably as "equals" or "which equals"
      • negative indicator: student uses the equals sign in a way that violates its meaning as a verb (e.g., violates transitivity)
    3. emphasis on the importance of being able to read symbols out loud
    4. understand how definitions are written/interpreted
      • the "if" is really an "if and only if"
      • the prepositions that you should use get introduced indirectly also, e.g. "on" or "associated with" — the language you use to employ the defined terms gets set out there too!
  1. respect the primacy of precision in written and verbal expression
    • and appreciate the trade-off between fluency of language and precision of language
    • students read, evaluate and correct example of sentences that represent a failure to respect precision
    • beware of “mathematical weasel words” such as “work”
    • also ambiguous phrasing like “linear combinations of vectors in W are in W”
    • also, beware of pronoun disease!!!
  2. make use of the specialized ways in which human language is employed to express mathematical ideas
    • some idiosyncrasies here are probably specific to English
    • “Let x be” is really saying “Let x represent”
    • such that : these tasks are phrased as "give an example of an x such that…" and therefore it's critical that students understand such that
    • the "if-then" construction is just one of MANY ways to express an implication
  3. conceptualize abstract mathematical objects beyond numbers
    • idea of a "vector" should have been encountered in MA 229 already
    • be able to think of "the set of all…" as its own entity
  4. attend to type : respect the importance of identifying the type of object represented by a symbol or expression before thinking about, talking about, or working with it
    • related to definitions in that a term only has meaning when the “ingredients” from the definition are present
    • this is why I asked questions in MA 229 that start with "write a vector such that" and get answers that are not even vectors… or I ask "give a set of vectors" and get answers that are not sets!
  5. worship the definitions : look to the definitions and understand how to interpret them to reason from them effectively
    • ingredients : understand that definitions and theorems have "ingredients"
    • create examples : be able to use the above skill to effectively create examples satisfying statements (which may or may not be definitions)
  6. read and analyze a mathematical argument effectively
    • understand the idea of inference
      • negative indicator: student uses the word "assume" instead of "infer" or "conclude"
    • understand a proof as a "thought experiment" and other conventions of the way they are written
  7. recognize the role of logic and attend to its proper application
    • recognize the pervasiveness of logic in mathematics at all levels
      • negative indicator: students write a sequence of equations (one above the other) without attending to whether each equation is equivalent to the next one or merely implies it
    • recognize the concept of "implication" when it occurs naturally
    • positive indicator: students can recognize a theorem whose statement is that "X implies Y" even when the theorem is phrased differently, e.g., not as "if X, then Y" but as, e.g., "Y provided X" or "X only if Y" or "X is sufficient for Y"
      • negative indicator: students write things like "since X and Y, …" instead of "since X implies Y"
    • attend to the importance of quantification (and also, specifically, the order of quantification)
      • recognize that in a doubly quantified statement, the entire statement after the first quantifier is “describing” that 1st quantified variable (i.e., there is an implicit “such that”!)
    • recognize different kinds of quantification in context (e.g., that a matrix being invertible is naturally defined as a "there exists" statement"
      • positive indicator: when asked to show that a universal statement is false, a student recognizes that providing a counterexample is sufficient
      • positive indicator: when asked to show that a universal statement is true, a student recognizes that a proof must consider an arbitrary element of the domain of quantification
    • know how to let the form of a statement guide the form a proof
      1. universally quantified statements
      2. statements about consecutive integers

Course Outline

  1. “What is mathematics?”
    • biology is the study of… physics is the study of… chemistry is the study of… mathematics is the study of what's true and what's not
  2. truth and the idea of a "statement"
    • good place to talk about $\Leftrightarrow$ versus $=$
      • in fact, the “what is an equation?” question should be explored here!
      • students should understand why "2+2=5" is an equation and not nonsense
      • meanwhile, something like "The girl was steep." is nonsense because…
        • in a statement everything must be meaningful
    • the idea of "equivalent statements" and the double implication arrow
    • give some of Naomi's good ol' "A implies B", "B implies A", "both" or "neither" questions
  3. variables and the idea of a "predicate"
    • the idea of writing a "truth-valued function" like $P(x)$ to represent a statement
  4. the magical phrase "such that"
    • "give a value for $x$ such that $P(x)$"
  5. forming examples
    • somewhere in the above, the idea of a definition is introduced
    • ask students to write a precise definition of "divisible by 6" or something
  6. quantifiers and quantified statements
    • existential quantifiers first, then universals
    • forming examples of quantified statements
    • universal quantification
      • what it takes to show that a universally quantified statement is false
      • then start in on the idea of proving a universally quantified statement
      • be able to explain the convention of “implicit universal quantification” over all “sensible” values for the variable
      • understanding that order of quantification matters
  7. compound statements
    • statements formed with 'and' and 'or'
    • implication
      1. the many different ways that an implication can be phrased
  8. sets and set theory
    • the concept of the set as its own independent entity
    • set roster notation
    • basic sets like $\mathbb Z$ and $\mathbb R$
    • set-builder notation
    • topics from the algebra of sets like union, intersection, set difference
    • Venn diagrams
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