Algebra for Precalculus and Calculus Students Mindful Manipulation

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Algebra for Precalculus and Calculus
StudentsMindful Manipulation

• Philip Cheifetz,
• Ellen Schmierer
• Nassau CC

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Recent Events

• 1960-1970s
• New Math
• Viet Nam War Deferments
• Open Enrollment less talented student pool
• 1980s
• Graphing Calculators
• High salaries for computer science and
  engineering grads attract mathematics majors
• Less talented mathematics teachers

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Recent Events

• 1990s
• Symbolic manipulators
• TIMMS Report (1995)

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Really Recent Events

• 2000s
• No Student Left Behind Lowering of Standards
• Rescaling SATs
• 2007 NYS 9th Grade Algebra Exam
• 44 passing
• 2008 NYS 9th Grade Algebra Exam
• 34 passing
• 50 of all elementary middle school teachers
  take ALL the mathematics they will ever take at a
  TYC.

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Who are some of the students we teach? (A true
story)

• Student  "I need to find my math textbook."
• Librarian "Sure, which course are you taking?"
• Student "Premeditated math."
• Librarian " Premeditated math?"
• Student "On no, that's not it. I know, it's
  primordial math!"
• Librarian "Primordial math?  Could you possibly
  mean remedial math?"
• Student "That's it!!! ( smiling brightly).
  Remedial Math 001. That's it!"

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Recognition After a Unit on Number Systems

• There are 10 kinds of mathematicians. Those who
  can think in binary and those who can't...
• NO recognition

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Mistakes My Students Truly Made Last Month in
Calc II
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Mistakes My Students Truly Made Last Month in
Calc II
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Mistakes My Students Truly Made Last Month in
Calc II
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Issues My Students Routinely Have

• Not knowing when to distribute (when to, UGH,
  FOIL.) GET RID OF FOIL!!
• Pattern recognition
• Reading tables of integrals
• Copying what I write rather than what the problem
  solving strategy is
• Refusing to think

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Mistakes My Students Routinely Make

• Everything is commutative
• Everything is distributive
• Things behave the way we want them to as in
  (fg)f g since it works with sums
• Everything is linear f(ab) f(a)f(b)
• Canceling ?
• Fractions ??
• Im ANGRY. Who Is doing the teaching, and what is
  being taught??

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Mistakes with Cancelling My Students Routinely
Make (A Joke)
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Mistakes with Fractions My Students Routinely
Make (A Joke)
The chef instructs his apprentice "You
take two thirds of water, one third of cream,
one third of broth..." The apprentice "But
that makes four thirds already! "Well - just
take a larger pot!"
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Mistakes People Make (NOT A Joke)

• Item costs 200
• There is a 40 discount
• Then.

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Mistakes with Commutativity Young Students Make
(Not a Joke)

• Teacher "Who can tell me what 7 times 6 is?"
• Student "It's 42!"
• Teacher "Very good! - And who can tell me what 6
  times 7 is?"
• Same student "It's 24!"

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UNFORTUNATELY,

• Students truly believe, and indeed expect, that
  if they come to class, they will receive a
  passing grade and, more likely than not, the
  grade will be a C or better.
• Students believe that coming to class is their
  sum responsibility as students.
• The vast majority of students do no homework,
  ever.
• Students have come to expect long review sessions
  before an exam, and that the exam will contain
  only the problems covered in the review, with
  altered constants.

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UNFORTUNATELY,

• Mediocrity is now acceptable and even encouraged.
• Partial credit for mostly incorrect work
  encourages mediocrity.
• Having a cursory knowledge of a topic is
  considered sufficient.
• Multi-step reasoning is an unfair expectation.

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UNFORTUNATELY,

• Students do not have a number sense.
• Therefore, they do not have an algebra sense,
• Nor do they have the sense to ask Does the
  answer seem reasonable?

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The Fall of Rome

• No student left behind has produced
  expectations aimed at the lowest common
  denominator. Skills have become so weak that
  examinations have had to be rescaled.
• Making students and parents feel good is more
  important than setting realistic goals and
  limits.
• The concept of on time is very loosely applied.
  
• Better Each child to his/her maximum potential

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Im Angry!

• If a student does poorly on an exam, it is deemed
  the fault of the exam.
• If a student fails to live up to expectations, it
  is deemed the fault of the teacher.
• A college degree today is significantly different
  from such a degree of 25 years ago.
• Many experienced, well-trained teachers are being
  replaced by faculty who are weaker.

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A non-trivial number of high school mathematics
teachers and/or supervisors

• do not have an undergraduate degree that has the
  word mathematics in it, or,
• do not have an graduate degree that has the word
  mathematics in it, or,
• have neither an undergraduate nor graduate degree
  that has the word mathematics in it.

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What to do?

• We are urged to have higher retention rates!
• I, for one, chose to be part of the cure.
• No student of mine will ever be asked by a
  professor Who did you have in the previous
  course?
• Visit www.matcmp.sunynassau.edu/
• cheifp/Itseemstome1.htm

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Which of the following three questions would you
deem most important for your algebra students?

• What algebraic law would you use to solve the
  equation 3(x 5) 27?
• For what value of x is equal to
  zero?
• Use an algebraic expression to prove or disprove
  that the sum of any three consecutive integers is
  a multiple of three.

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In the following expressions, a and x are
positive. What is the effect of increasing a on
the value of each expression?
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Without solving, decide whether the following
equations have a positive solution, a negative
solution, a zero solution, or no solution.
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A car trip costs 1.50 per fifteen miles for gas
and 30 per mile for other expenses, plus 20 for
car rental. The total cost for the trip is given
by the equation

• Explain what each term represents in terms of the
  trip.
• What units for cost and distance are being used?
• Is the equation linear?

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The number of people who attend a concert is
160 p when the price of a ticket is p.

• What is the practical interpretation of the 160?
• Why is it reasonable that the p term has a
  negative sign?
• The number of people who attend a movie at ticket
  price p is 175 p. If tickets are the same
  price, does the concert or the movie draw a
  larger audience?
• The number of people who attend a dance recital
  at ticket price p is 160 2p. If tickets are
  the same price, does the concert or the dance
  recital draw a larger audience?

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A company uses two different sized trucks to
deliver sand. The first truck can transport x
cubic yards and the second y cubic yards. The
first truck makes S trips to the job site and the
second makes T trips. What do the following
expressions represent in practical terms?
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You plan to drive 300 miles at 55 miles per hour,
stopping for a two-hour rest. You want to know t,
the number of hours the journey will take. Which
of the following equations would you use?
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Precalculus Fostering a deeper understanding

• What is it?
• For whom?
• What topics do we need in such a course?
• Calculus bound?
• Liberal arts?

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Understanding the information a graph imparts
Match each graph with a description of its zeros

• Two real solutions
• One real solution
• c. No real solutions

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Function Notation

• If f (x) 5 then f (2) 10
• If f (x) 5 then 2f (x) 10
• If f (x) 6 then f (2x) 12
• f (a b) f (a) f (b)

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Reading and Interpreting Graphs

• If f (x) h(x) g(x) then the y-intercept of f
  is at
• a. 0 b. 1 c. 2 d. 3 e. none
  of these

g(x)
h(x)
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Reading and Interpreting Graphs

• If f (x) h(x) g(x) then the x-intercept of f
  is at
• a. 0 b. 1 c. 2 d. 3 e. none
  of these

g(x)
h(x)
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Deeper Understanding of Symbols

• In the equation 3x 2y 9, if x increases by 2
  then y
• a. Increases by 2
• b. Decreases by 6
• c. Increases by 3
• d. Increases by 6
• e. Decreases by 3

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Understanding Exponential Growth

• Match the equation to the description
• a. Astronomical growth
• b. A 6 increasing growth rate
• c. A 6 decay rate
• d. Quickly approaches zero

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How to Solve it?Classify each equation as

• Solvable using the exponential function.
• Solvable using the log function
• Not solvable using the log function or the
  exponential function but a solution exists
• No real number is a solution

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Reading Tables

• Which table shows an increasing function?

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Reading Tables

• Which of the following tables shows data from an
  invertible function?

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The Relationship Between Equations and Their
Graphs

• Which of the following are possible equations
  for the graph below?

d
a b c
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In Closing

• Where are we, as a mathematics community, going?
• What are we going to do about it?
• Be an ambassador for common sense
• Talk to anyone you can about the sorry state of
  education in the US and in particular, about
  improving mathematical reasoning

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• Thank you for coming
• Phil philip.cheifetz_at_ncc.edu
• Ellen ellen.schmierer_at_ncc.edu