AP Calculus AB • Quarter 1 Benchmark • Michael Fenton

Introduction

Up next: first quarter benchmark assessment from Michael Fenton’s AP Calculus AB class. As you read through the assessment and consider adding your comments to the discussion, keep these questions in mind:

  • What do you like (and why)?
  • What would you do differently (and why)?
  • What questions do you have for Michael?

Submission Type

“Share an assessment you don’t hate…”

“Share an assessment you do hate…”

Okay, I’m really in between on this one. The reason I started the Better Assessments project is that I’m disappointed in the quality of nearly all of my assessments. I’ve already benefitted so much from those who’ve shared their assessments. (Thanks Stephanie, Jim, Jennifer, Luke, and Sam!) This one is certainly better than a lot of my assessments, but I know it could be so much better.

Assessment

The assessment is here.

The solutions I provided to students after the assessment are here.

Author’s Commentary

I gave this benchmark assessment to my AP Calculus AB class nine weeks into the 2012-2013 school year. We use Paul Foerster’s Calculus book (here’s a look at my course outline). The benchmark addresses Chapters 1-3

  1. Course Overview
  2. Properties of Limits
  3. Derivatives, Antiderivatives, and Indefinite Integrals

When I first started teaching from Foerster’s books (I also use his Precalculus text) I was quickly drawn to his fourfold approach: Verbal, Numerical, Graphical, and Algebraic. (As a student, I didn’t have a sense of this multiple representations framework.) I tried to include questions that would require students to demonstrate understanding in each of those four categories.

In reading through the questions again, I see myself leading students and offering hints where they shouldn’t really need them. I may have been trying to guard against poor results based on the weakness of that class, or the gaps I perceived in my teaching.

The assessment was given in one 50-minute class period. It was quite a few months ago, but if my memory serves me well, then my stronger students needed only about half the period, while several of my struggling students took the entire allotted time.

Thanks in advance for your comments!

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3 thoughts on “AP Calculus AB • Quarter 1 Benchmark • Michael Fenton

  1. Michael

    First off, thanks for sharing! Here are some thoughts I have

    First, I love how ‘wordy’ this test is. Rather than allowing the kids to merely display mechanical skills, you are asking them to put their understanding into their own language. I think Foerster would be proud if he saw this. Side question – does your school use Paul’s Precalculus book? If not, how tough is the transition for the students?
    I really like questions 6 and 7.
    I like that questions 19 and 20 give the students x values but not the y values.
    In the techniques of differentiation sequence I notice that there are no product or quotient rule questions. Does this text not introduce those techniques early on?
    Finally, I have a question related to one that I asked Sam. I am interested in your layout for this test. Were the students expected to contain their work on the page as it is laid out? If so, I am thinking of a number of my students who would have difficulty finishing their work in this space. I also think that the description questions would benefit from more space. I am also curious about point values here. I assume that questions are weighted equally since there is no point value indicated for any of the individual questions.

  2. Michael,

    It is certainly a comprehensive assessment. Your questions are clear and reasonable – I especially appreciate the series of questions related to the same function, f(x)=x^2-4x, which allow the student to demonstrate multiple understandings without “starting over” each time.

    OK, here come my suggestions…

    Though your introductory comments do a nice job framing each major topic, I do not feel that they are necessary, or even appropriate, for a benchmark assessment. If students aren’t aware, for example, that “The limit is an important concept in Calculus because it helps us define the derivative.,” then you have another problem. I might use those statements as prompts instead, “The limit is an important concept in Calculus because ________.”

    I agree with you – lose the parenthetical hints.

    This may be a Foerster thing (though I haven’t ever seen one of his books published after 1995 or so – maybe it’s changed), but your questions are all in the same direction. That is, they all start with the symbolic. I think it would be more powerful, more interesting, and more time-efficient to start some with graphs, tables, or verbal contexts. Students should demonstrate understanding of multiple representations from multiple directions.

    Which brings me to my personal priority: applications. Is there a way you could assess one of your outcomes in an applied situation?

    Last question, a mathematical one. In, “State the algebraic definition of the derivative of a function.,” are you asking for the limit of the difference quotient? If so, are limits algebraic? I’m not sure, one way or the other.

    I appreciate your sharing! I hope my comments have been constructive; that is how they were intended.

    – Jennifer

  3. @mrdardy: Thanks for your thoughts and questions. We use Foerster for Precalculus as well. The transition into Precalculus is a painful one for most of my students. The transition into Calculus is less so. I could alleviate some of this start-of-the-year pain/frustration by making my Algebra 2 classes (honors and regular) more demanding/challenging. Glad to hear you like #6 and #7. In the past I’ve given the graphs and asked the limits. I think this gets at the issue better since students are more involved in “creating” rather than simply observing.

    After reading your comment on #19 and #20, I’m wondering how my students would respond if (for some functions at some values) I gave the y-value and not the x-value.

    We develop the product and quotient rules in Chapter 4.

    Regarding the layout, students are always allowed to show their work (all or some) on separate paper. Several students in each class typically do for some (or even most) of the questions. I’ve found that by formatting my assessments like this (two-column, medium/small font, medium amount of space) that many of my students have become more organized in showing their work. After reading the discussion of assessment layout and spacing (i.e., space provided for answers) I’m now concerned that I might be restricting students’ answers, which would be alright for non-open-ended questions, but a terrible approach for more open-ended ones. I’ll have to think about this some more. Thanks for bringing it up.

    Equal value for each question.

    @jennifer: Thanks for taking the time to comment! I borrowed the “introductory comments” approach from a comprehensive semester assessment I saw from Foerster. A few words of commentary, then a series of questions related to that topic. I think you’re right that my students should know this by now (or we’re in for some trouble). I like your idea of turning it back on the students in some cases with some fill in the blanks.

    If I have time this week I’ll revise the assessment and post it to see what you and other commenters think. It would be a healthy reflective task for me to put some of these suggestions into action sooner rather than later. I’ll ditch the parenthetical comments when I do make the revisions, whether now or later. 🙂

    The “all one direction” thing is not a Foerster thing, nor is it really a “Fenton thing” either. I didn’t notice it until you brought it up, but now I see it clear as day: No questions begin graphically or numerically! This may have been due to limited planning time. It’s far easier for me to create a questions that begin algebraically and verbally. Even though it’s not my typical approach, I’m glad you brought it up. This comment will serve as extra motivation throughout this school year to consider not only the “destination” representation of assessment questions but also what representation I’m starting from.

    I assess applications on each chapter and topic assessment (I transitioned to SBG in Calculus at the start of the spring semester). I’m not entirely sure why I didn’t include any application questions on the benchmark. I think in a rewrite I’ll try to write a brief series of questions (three or four), related to a single context, that address all of the major concepts and skills we’ve addressed in the first quarter. Essentially a performance assessment within the benchmark (probably at the end). I’ll have to consider the overall length of the assessment in that case, or maybe kick the application portion to a new page (actually, I should do that anyway) and give it to students in the first 15 minutes of class the next day. Good food for thought.

    With the “algebraic definition of the derivative of a function” I am looking for the limit of the difference quotient. I think limits can be represented algebraically, verbally, numerically, and graphically. But in the case of this definition, I’ll go out on a limb and say it’s an algebraic representation. I’m open to wiser mathematical minds correcting me, though.

    Your comments were absolutely constructive! Much appreciated!

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