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Algebra for Precalculus and Calculus Students Mindful ManipulationPowerPoint Presentation

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

“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

ReallyRecent 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.

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!"

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

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

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(a+b) = f(a)+f(b)
- Canceling
- Fractions
- I’m ANGRY. Who Is doing the teaching, and what is being taught??

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!"

Mistakes People Make (NOT A Joke)

- Item costs $200
- There is a 40% discount
- Then…….

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!"

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.

- 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.

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?”

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

- 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.

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.

What to do? and/or supervisors:

- 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

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.

In the following expressions, important for your algebra students?a and x are positive. What is the effect of increasing a on the value of each expression?

Without solving, decide whether the following equations have a positive solution, a negative solution, a zero solution, or no solution.

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 theequation

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

The number of people who attend a concert is per mile for other expenses, plus $20 for car rental. The total cost for the trip is given by the160 – 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?

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?

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?

Precalculus: Fostering a deeper understanding for a two-hour rest. You want to know

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

Understanding the information a graph imparts for a two-hour rest. You want to know

Match each graph with a description of its zeros

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

Function Notation for a two-hour rest. You want to know

- 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)

Reading and Interpreting Graphs for a two-hour rest. You want to know

- 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)

Reading and Interpreting Graphs for a two-hour rest. You want to know

- 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)

Deeper Understanding of Symbols for a two-hour rest. You want to know

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

Understanding Exponential Growth for a two-hour rest. You want to know

Match the equation to the description

a. Astronomical growth

b. A 6% increasing growth rate

c. A 6% decay rate

d. Quickly approaches zero

How to Solve it? for a two-hour rest. You want to know 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

Reading Tables for a two-hour rest. You want to know

- Which table shows an increasing function?

Reading Tables for a two-hour rest. You want to know

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

The Relationship Between Equations and for a two-hour rest. You want to know Their Graphs

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

d

a b c

In Closing: for a two-hour rest. You want to know

- 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

Thank you for coming for a two-hour rest. You want to know

Phil: [email protected]

Ellen: [email protected]

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