I like the second one best, because it seems to ask for at least a little bit of actual thinking. The one is being fussy about domains of definition in the first problem and not the second (unless they want one to answer #2 with none of the above, which would be unfair). Neither gets at the deeper thinking skills - that's hard to do in a short test.

Well I haven't touched calc or pre-calc in years, but I remember the biggest hurdle for me was being able to deal with a bunch of really convoluted algebraic manipulations.

I just took a trip down memory lane and looked up the current version of my freshman calc 1 class from college... and the differences 12 years make are startling:

"Class business and mini-lectures will occupy about 20-25 minutes of each class period. During the remaining time, you will work individually and in teams on various learning activities. The instructor and two assistants will serve as coaches during the learning activities."

My classes were a bunch of lectures.

They still post a whole punch of previous tests though (I might recommend looking more at actual calc assessments as much as pre-assesments to get a feel for what skills are needed) , and they have a Basic Skills Test Review at the bottom of this page:

The second one is the best. The other two are pretty blah. Both of these suffer the same problem: they focus attention on the wrong topics. Simplifying convoluted exponents, logarithms, and trig identities are some of the LEAST practical math skills I can think of. Calculus deals with negative exponents at times, but never something as crazy as they have here. Forget about log properties. Knowing trig identities might help with it comes to the derivative/integral of another trig function I guess. But trig identities are such a terrible topic with minimal purpose.

If I was gauging readiness for calculus, I would focus on equation solving (linear, quadratic, radical, exponential), simplification of unlike denominators, multiplication of high order terms, summations, a bit on negative exponent equivalency and graph behavior. Calculus is primarily a course in graph behavior.

Seeing Calculus problems through to the end often requires lots of specific knowledge, serious focus, and facility with symbolic manipulation. I don't necesssarily disagree with Jonathan's characterization of the above skills (manipualting convulated exponential / log / trig expression)particular skills as impractical, but maybe the ability to handle them indicates an ability to handle problems at the complexity level of Calculus.

Thanks for sharing this, Michael. When I teach Calculus I just assume that no one is prepared, and proceed from there. Maybe I'll incorporate something like this next time.

For me, key concepts students need to understand going into calculus are:

Comfortable manipulating rational expressions Comfortable with function notation, including composition Graphing sins, cosines, simple quadratics and exponential functions

Long-time AP Calc and Precalc teacher (retiring in 4 months)

This isn't meant to be a complete list, but I've been occasionally writing down some things this year that I want my students to know before they take Calculus. Some of these are going to be hard to test on a short pre-test, especially if it’s multiple-choice only.

* Being comfortable dealing with letters where they usually see numbers. For example, solving literal equations (solve c(x+3)=2d for x), or find the slope of the line through the points (2,r) and (r, 3r+1) in terms of r.

* In particular, solving literal equations with two x’s: c(x+3)=7x+d.

* Knowing that 1 over a small positive is a big positive, etc.

* Comfortable with function notation as applied to formulas, graphs and tables. The basic question, what is f(3). But also more abstract questions such as, for what value(s) of c is f(c)=3.

* Comfortable with the relationship between a formula and its graph. If f(3)=7 then the point (3,7) is part of the graph, and conversely. (They do NOT all know this, even after graphing for years since 8th grade.) I want them to get used to using this in a variety of ways, not just formula --> table --> graph. For example, given f(x)=100-x^2 and given a sketch of an inscribed rectangle from x=3 to x=5, find the area of the rectangle -- I find many students are stumped.

* Determining when rational function graphs have vertical asymptotes, horizontal asymptotes, holes -- based on understanding WHY.

* Being able to graph simple piecewise functions (continuous or not) and understanding why. They do not automatically know that an open circle or a closed circle on a graphis saying. So for example, given a graph with a jump discontinuity, find or estimate f(3), f(2.999), f(3.001).

* Algebra skills, including pretty ugly algebra. Hard to assess with multiple choice? I think you want a problem ugly enough that many students won’t get it 100% right, and then assess how badly they messed up. Ideally more than one such problem.

* Solving rational equations: expression/expression = expression, or expression/expression=0.

* Very comfortable translating negative exponents to and from rational expressions, translating fractional exponents to and from radicals. Including 3x^-4 is not 1/(3x^4).

* Knowing when they can and cannot “cancel” in algebraic fractions.

* Finding the equation of a line using point-slope form.

* Knowing what positive and negative slopes look like, knowing what large and small slopes look like.

* Knowing the meaning of slope, both with functions of time and functions of other-than-time. So meaning of slope of a line showing population as a function of time, meaning of slope of a line showing lemonade sales as a function of outdoor temperature.

* Knowing that you need 2 equations to solve for 2 variables; solving systems by substitution.

* Ability to manipulate logs using properties of logs; solve logarithmic and exponential equations; comfortable with working with ln x and e. (But I don’t expect them to know why e matters, just get used to the notation: e is a number not a variable, ln x means log-base-that-weird-number of x.)

* Know sin=y and cos=x on unit circle; know the other 4 trig functions in terms of sin & cos, know special values. Think in radians, not have to translate 5pi/6 into degrees to find the value.

Some small things:

* In solving x^2=4x, you can’t divide both sides by x; how you can solve.

* Being able to solve x^2=9 or x^2=17 and not forget that there are two solutions.

One additional comment. If you haven't seen this, this is great.

Here is Doug Shaw's 10-day Precalc Syllabus, posted on the AP-Calc listserv several years ago. (I share this with my students -- both Precalc and AP -- every year, partly to remind them of Lesson 1 and Lesson 6 but mostly for the humor. And unbeknownst to my students I actually have a 6-foot trout in my classroom closet -- given to me by a former student about 6 years ago -- and once or twice a year I have a student who forgets Lesson 10, and I haul the trout out of the closet and beat the whatever out of the student. It's a VERY soft trout.)

[ap-calculus] 10 day Pre-Calc syllabus Posted: Nov 8, 2002 9:00 AM

Here is my ideal pre-calc syllabus. Please all adopt it.

DAY 1: Teach them that (a+b)/c is (a/c) + (b/c) DAY 2: Teach them that a/(b+c) is NOT (a/b) + (a/c) DAY 3: Teach them that x / ln(x) is NOT "1 / ln" DAY 4: Teach them that you can't solve (sin(kx)) = 1 by saying "x = 1/sin(k)" DAY 5: Remind them that a/(b+c) is NOT (a/b) + (a/c) DAY 6: Show them a movie of a student sitting in a field, writing "(a+b)^2 = a^2 + b ^2" and then getting HIT BY A TRAIN DAY 7: Remind them that a/(b+c) is NOT (a/b) + (a/c) DAY 8: Teach them that if the domain of the a function f is the reals, the graph of y = f(x) is NOT a blank pair of axes, that perhaps they should adjust the "window" DAY 9: Teach them that x/(y+z) is NOT (x/y) + (x/z) DAY 10: Group work: Bring a trout to class. Have them solve sin(kx) = 1. If they get x = 1/sin(k), hit them with the trout. Make it a big trout.

For what it's worth, here's a review packet that I gave to my calculus students on day 1 when I taught it year before last. This class was mostly composed of seniors not planning to take the AP exam.

The packet is broken into sections on functions, algebra, and trigonometry. My goal was to head off these skills as potential "show stoppers", since a lack of comfort with any of these skills and ideas can make any calculus lesson grind to a halt. Automaticity would be great, but a reasonable amount of comfort is passable. Outright terror needs to be dealt with immediately.

All three "readiness tests" assess smatterings: basic understanding of notation, how to manipulate algebraic expressions, and how to solve simple equations, many of which are middle school level skills. None of the questions requires any degree of insight, or a sustained period of concentration to see a complex problem through to its conclusion.

In contrast, a solitary multiple part question where the math is subordinate to the complexity of process would tell us much more about a student's potential to handle the kind of problem solving that is required of true college level (i.e., not AP) calculus.

I've been trying to shut up in this excellent conversation so far, because of my relative inexperience in the matter. I'm curious what people think about informal experience with the main ideas of Calculus itself, i.e. experience finding approximations of speed at a point, or experience moving between distance/time and speed/time graphs, or experience estimating volume and area. This experience should help kids better understand the big ideas of Calculus, right? Is there somewhere along this path that kids should be before entering Calculus?

I teach precalc, and all the tests reassure me that I'm on the right track. My kids are usually juniors and seniors. I tell only the A students that they might risk AP Calc AB, but should expect some gaps. Most of them should go into non-AP Calc, where I ideally expect them to get more pre-calc.

"Show them a movie of a student sitting in a field, writing "(a+b)^2 = a^2 + b ^2" and then getting HIT BY A TRAIN"

Laughed and laughed.

I wrote this piece called "Teaching Algebra, or Banging your head with a whiteboard"

I like the second one best, because it seems to ask for at least a little bit of actual thinking. The one is being fussy about domains of definition in the first problem and not the second (unless they want one to answer #2 with none of the above, which would be unfair). Neither gets at the deeper thinking skills - that's hard to do in a short test.

ReplyDeleteWell I haven't touched calc or pre-calc in years, but I remember the biggest hurdle for me was being able to deal with a bunch of really convoluted algebraic manipulations.

ReplyDeleteI just took a trip down memory lane and looked up the current version of my freshman calc 1 class from college... and the differences 12 years make are startling:

"Class business and mini-lectures will occupy about 20-25 minutes of each class period. During the remaining time, you will work individually and in teams on various learning activities. The instructor and two assistants will serve as coaches during the learning activities."

My classes were a bunch of lectures.

They still post a whole punch of previous tests though (I might recommend looking more at actual calc assessments as much as pre-assesments to get a feel for what skills are needed) , and they have a Basic Skills Test Review at the bottom of this page:

https://mthsc.clemson.edu/ug_course_pages/view_course_page.py?course_id=6

The second one is the best. The other two are pretty blah. Both of these suffer the same problem: they focus attention on the wrong topics. Simplifying convoluted exponents, logarithms, and trig identities are some of the LEAST practical math skills I can think of. Calculus deals with negative exponents at times, but never something as crazy as they have here. Forget about log properties. Knowing trig identities might help with it comes to the derivative/integral of another trig function I guess. But trig identities are such a terrible topic with minimal purpose.

ReplyDeleteIf I was gauging readiness for calculus, I would focus on equation solving (linear, quadratic, radical, exponential), simplification of unlike denominators, multiplication of high order terms, summations, a bit on negative exponent equivalency and graph behavior. Calculus is primarily a course in graph behavior.

Seeing Calculus problems through to the end often requires lots of specific knowledge, serious focus, and facility with symbolic manipulation. I don't necesssarily disagree with Jonathan's characterization of the above skills (manipualting convulated exponential / log / trig expression)particular skills as impractical, but maybe the ability to handle them indicates an ability to handle problems at the complexity level of Calculus.

DeleteThanks for sharing this, Michael. When I teach Calculus I just assume that

no oneis prepared, and proceed from there. Maybe I'll incorporate something like this next time.For me, key concepts students need to understand going into calculus are:

ReplyDeleteComfortable manipulating rational expressions

Comfortable with function notation, including composition

Graphing sins, cosines, simple quadratics and exponential functions

Long-time AP Calc and Precalc teacher (retiring in 4 months)

ReplyDeleteThis isn't meant to be a complete list, but I've been occasionally writing down some things this year that I want my students to know before they take Calculus. Some of these are going to be hard to test on a short pre-test, especially if it’s multiple-choice only.

* Being comfortable dealing with letters where they usually see numbers. For example, solving literal equations (solve c(x+3)=2d for x), or find the slope of the line through the points (2,r) and (r, 3r+1) in terms of r.

* In particular, solving literal equations with two x’s: c(x+3)=7x+d.

* Knowing that 1 over a small positive is a big positive, etc.

* Comfortable with function notation as applied to formulas, graphs and tables. The basic question, what is f(3). But also more abstract questions such as, for what value(s) of c is f(c)=3.

* Comfortable with the relationship between a formula and its graph. If f(3)=7 then the point (3,7) is part of the graph, and conversely. (They do NOT all know this, even after graphing for years since 8th grade.) I want them to get used to using this in a variety of ways, not just formula --> table --> graph. For example, given f(x)=100-x^2 and given a sketch of an inscribed rectangle from x=3 to x=5, find the area of the rectangle -- I find many students are stumped.

* Determining when rational function graphs have vertical asymptotes, horizontal asymptotes, holes -- based on understanding WHY.

* Being able to graph simple piecewise functions (continuous or not) and understanding why. They do not automatically know that an open circle or a closed circle on a graphis saying. So for example, given a graph with a jump discontinuity, find or estimate f(3), f(2.999), f(3.001).

* Algebra skills, including pretty ugly algebra. Hard to assess with multiple choice? I think you want a problem ugly enough that many students won’t get it 100% right, and then assess how badly they messed up. Ideally more than one such problem.

* Solving rational equations: expression/expression = expression, or expression/expression=0.

* Very comfortable translating negative exponents to and from rational expressions, translating fractional exponents to and from radicals. Including 3x^-4 is not 1/(3x^4).

* Adding, subtracting, multiplying, dividing algebraic fractions.

* Knowing the value of 5/0, 0/5, 0/0.

* Knowing (3/4)x is the same as 3x/4.

* Knowing when they can and cannot “cancel” in algebraic fractions.

* Finding the equation of a line using point-slope form.

* Knowing what positive and negative slopes look like, knowing what large and small slopes look like.

* Knowing the meaning of slope, both with functions of time and functions of other-than-time. So meaning of slope of a line showing population as a function of time, meaning of slope of a line showing lemonade sales as a function of outdoor temperature.

* Knowing that you need 2 equations to solve for 2 variables; solving systems by substitution.

* Ability to manipulate logs using properties of logs; solve logarithmic and exponential equations; comfortable with working with ln x and e. (But I don’t expect them to know why e matters, just get used to the notation: e is a number not a variable, ln x means log-base-that-weird-number of x.)

* Know sin=y and cos=x on unit circle; know the other 4 trig functions in terms of sin & cos, know special values. Think in radians, not have to translate 5pi/6 into degrees to find the value.

Some small things:

* In solving x^2=4x, you can’t divide both sides by x; how you can solve.

* Being able to solve x^2=9 or x^2=17 and not forget that there are two solutions.

* sqrt(x^2+9) is not x+3.

* (a+b)^2 is not a^2 + b^2.

As I say, that’s not a comprehensive list.

Evan Romer

Susquehanna Valley HS

Conklin Ny

One additional comment. If you haven't seen this, this is great.

ReplyDeleteHere is Doug Shaw's 10-day Precalc Syllabus, posted on the AP-Calc listserv several years ago. (I share this with my students -- both Precalc and AP -- every year, partly to remind them of Lesson 1 and Lesson 6 but mostly for the humor. And unbeknownst to my students I actually have a 6-foot trout in my classroom closet -- given to me by a former student about 6 years ago -- and once or twice a year I have a student who forgets Lesson 10, and I haul the trout out of the closet and beat the whatever out of the student. It's a VERY soft trout.)

http://mathforum.org/kb/thread.jspa?forumID=63&threadID=171692&messageID=670899

Doug Shaw

[ap-calculus] 10 day Pre-Calc syllabus

Posted: Nov 8, 2002 9:00 AM

Here is my ideal pre-calc syllabus. Please all adopt it.

DAY 1: Teach them that (a+b)/c is (a/c) + (b/c)

DAY 2: Teach them that a/(b+c) is NOT (a/b) + (a/c)

DAY 3: Teach them that x / ln(x) is NOT "1 / ln"

DAY 4: Teach them that you can't solve (sin(kx)) = 1 by

saying "x = 1/sin(k)"

DAY 5: Remind them that a/(b+c) is NOT (a/b) + (a/c)

DAY 6: Show them a movie of a student sitting in a field,

writing "(a+b)^2 = a^2 + b ^2" and then getting HIT BY A

TRAIN

DAY 7: Remind them that a/(b+c) is NOT (a/b) + (a/c)

DAY 8: Teach them that if the domain of the a function f is

the reals, the graph of y = f(x) is NOT a blank pair of

axes, that perhaps they should adjust the "window"

DAY 9: Teach them that x/(y+z) is NOT (x/y) + (x/z)

DAY 10: Group work: Bring a trout to class. Have them

solve sin(kx) = 1. If they get x = 1/sin(k), hit them with

the trout. Make it a big trout.

...I am done venting now. Hard day of teaching.

Evan, That list may not be comprehensive, but it's a good one. Thanks.

ReplyDeleteFor what it's worth, here's a review packet that I gave to my calculus students on day 1 when I taught it year before last. This class was mostly composed of seniors not planning to take the AP exam.

ReplyDeleteThe packet is broken into sections on functions, algebra, and trigonometry. My goal was to head off these skills as potential "show stoppers", since a lack of comfort with any of these skills and ideas can make any calculus lesson grind to a halt. Automaticity would be great, but a reasonable amount of comfort is passable. Outright terror needs to be dealt with immediately.

https://drive.google.com/file/d/0B28vOuIUYxAiS1JCWDFrQjBPRWM/edit?usp=sharing

All three "readiness tests" assess smatterings: basic understanding of notation, how to manipulate algebraic expressions, and how to solve simple equations, many of which are middle school level skills. None of the questions requires any degree of insight, or a sustained period of concentration to see a complex problem through to its conclusion.

ReplyDeleteIn contrast, a solitary multiple part question where the math is subordinate to the complexity of process would tell us much more about a student's potential to handle the kind of problem solving that is required of true college level (i.e., not AP) calculus.

https://docs.google.com/document/d/1P55zavU1fzJaKzO0WMtzE9_rXam1LSosQy2oVr-xKpM/edit

I've been trying to shut up in this excellent conversation so far, because of my relative inexperience in the matter. I'm curious what people think about informal experience with the main ideas of Calculus itself, i.e. experience finding approximations of speed at a point, or experience moving between distance/time and speed/time graphs, or experience estimating volume and area. This experience should help kids better understand the big ideas of Calculus, right? Is there somewhere along this path that kids should be before entering Calculus?

ReplyDelete"xo<=x,90" in the first question makes no sense and shold be omitted. Otherwise the questions are OK.

ReplyDeleteI meant to say "0o<=x<90o" makes no sense.

ReplyDeleteI teach precalc, and all the tests reassure me that I'm on the right track. My kids are usually juniors and seniors. I tell only the A students that they might risk AP Calc AB, but should expect some gaps. Most of them should go into non-AP Calc, where I ideally expect them to get more pre-calc.

ReplyDelete"Show them a movie of a student sitting in a field,

writing "(a+b)^2 = a^2 + b ^2" and then getting HIT BY A

TRAIN"

Laughed and laughed.

I wrote this piece called "Teaching Algebra, or Banging your head with a whiteboard"

http://educationrealist.wordpress.com/2012/05/21/teaching-algebra-or-banging-your-head-with-a-whiteboard/

and call that particular sin the equivalent of clubbing cute little baby seals.