Presentation on theme: "Algebra for Precalculus and Calculus Students Mindful Manipulation"— Presentation transcript:
1Algebra for Precalculus and Calculus Students Mindful Manipulation Philip Cheifetz,Ellen SchmiererNassau CC
2“Recent” EventssNew MathViet Nam War DefermentsOpen Enrollment: less talented student pool1980sGraphing CalculatorsHigh salaries for computer science and engineering grads attract mathematics majorsLess talented mathematics teachers
4Really Recent Events No Student Left Behind = Lowering of Standards Rescaling SATs2007 NYS 9th Grade Algebra Exam44% = passing2008 NYS 9th Grade Algebra Exam34% = passing50% of all elementary & middle school teachers take ALL the mathematics they will ever take at a TYC.
5Who 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!"
6Recognition 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
7Mistakes My Students Truly Made Last Month in Calc II
8Mistakes My Students Truly Made Last Month in Calc II
9Mistakes My Students Truly Made Last Month in Calc II
10Issues My Students Routinely Have Not knowing when to distribute (when to, UGH, “FOIL”.) GET RID OF FOIL!!Pattern recognitionReading tables of integralsCopying what I write rather than what the problem solving strategy isRefusing to think
11Mistakes My Students Routinely Make Everything is commutativeEverything is distributiveThings behave the way we want them to as in: (fg)’=f ’g’ since it “works” with sumsEverything is linear: f(a+b) = f(a)+f(b)Canceling Fractions I’m ANGRY. Who Is doing the teaching, and what is being taught??
12Mistakes with Cancelling My Students Routinely Make (A Joke)
13Mistakes 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!"
14Mistakes People Make (NOT A Joke) Item costs $200There is a 40% discountThen…….
15Mistakes 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!"
16UNFORTUNATELY,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.
17UNFORTUNATELY, 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.
18UNFORTUNATELY,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?”
19The 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
20I’m 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.
21A 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.
22What to do? ~cheifp/Itseemstome1.htm 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~cheifp/Itseemstome1.htm
23Which 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.
24In the following expressions, a and x are positive In the following expressions, a and x are positive. What is the effect of increasing a on the value of each expression?
25Without solving, decide whether the following equations have a positive solution, a negative solution, a zero solution, or no solution.
26A 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 equationExplain what each term represents in terms of the trip.What units for cost and distance are being used?Is the equation linear?
27The 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?
28A company uses two different sized trucks to deliver sand 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?
29You 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?
30Precalculus: Fostering a deeper understanding What is it?For whom?What topics do we need in such a course?Calculus bound?Liberal arts?
31Understanding the information a graph imparts Match each graph with a description of its zerosTwo real solutionsOne real solutionc. No real solutions
32Function Notation If f (x) = 5 then f (2) = 10 If f (x) = 5 then 2f (x) = 10If f (x) = 6 then f (2x) = 12f (a + b) = f (a) + f (b)
33Reading and Interpreting Graphs If f (x) = h(x) – g(x) then the y-intercept of f is ata. 0 b c d e. none of theseg(x)h(x)
34Reading and Interpreting Graphs If f (x) = h(x) – g(x) then the x-intercept of f is ata. 0 b c d e. none of theseg(x)h(x)
35Deeper Understanding of Symbols In the equation 3x + 2y = 9, if x increases by 2 then ya. Increases by 2b. Decreases by 6c. Increases by 3d. Increases by 6e. Decreases by 3
36Understanding Exponential Growth Match the equation to the descriptiona. Astronomical growthb. A 6% increasing growth ratec. A 6% decay rated. Quickly approaches zero
37How to Solve it? Classify each equation as Solvable using the exponential function.Solvable using the log functionNot solvable using the log function or the exponential function but a solution existsNo real number is a solution
38Reading TablesWhich table shows an increasing function?
39Reading TablesWhich of the following tables shows data from an invertible function?
40The Relationship Between Equations and Their Graphs Which of the following are possible equations for the graph below?da b c
41In Closing: Where are we, as a mathematics community, going? What are we going to do about it?Be an ambassador for common senseTalk to anyone you can about the sorry state of education in the US and in particular, about improving mathematical reasoning
42Phil: email@example.com Ellen: firstname.lastname@example.org Thank you for comingPhil:Ellen: