AP Stats

Discussion in 'Secondary Education' started by JimG, Apr 20, 2018.

  1. JimG

    JimG Companion

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    Apr 20, 2018

    I am confident in MY understanding. It is more the curriculum structure and pedagogy that I am researching.
     
  2. futuremathsprof

    futuremathsprof Connoisseur

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    Apr 21, 2018

    Happy to help a fellow math teacher out! :)
     
  3. a2z

    a2z Maven

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    Apr 21, 2018

    Great to hear. I know a few AP Calc teachers who say they survived stats in college during their major but would never be able to teach it. It wasn't my most favorite math course.

    Good luck with your decision and if you do end up teaching it. As others said, definitely take the AP course to get your footing.
     
  4. TeacherGroupie

    TeacherGroupie Moderator

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    Apr 22, 2018

    In California, teaching computer science may not be that simple. See https://www.ctc.ca.gov/docs/default-source/leaflets/cl603.pdf?sfvrsn=b8e7b402_0. Better yet, talk to the credential analysts at the local county office of education. Topics in computer applications (yes, I know this isn't your issue) come under the heading of business education; topics in computer science per se come under the purview of industrial and technology education; both business education and industrial and technology education are considered part of career and technology education as opposed to belonging to math or one of the other core subjects.
     
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  5. futuremathsprof

    futuremathsprof Connoisseur

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    Apr 22, 2018

    This is great to know! Thanks TeacherGroupie. However, Intro to CompSci (Javascript) and AP Computer Science A are the only computer classes offered at my private school besides keyboarding, digital animation, digital arts media, film production, advanced film, etc, etc. My P has been looking for someone to teach the AP CompSci A and Intro to CompSci classes since the current teacher is looking to retire soon. I’m interested in taking on the position because there is a 10k incentive (;)) to do so as computer science teachers are the most difficult teachers to come by.
     
  6. JimG

    JimG Companion

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    Apr 27, 2018

    How much do you emphasize being able to do by hand vs. being able to use the calculator functions? In my first exposure to stats, we were able to do everything with scientific calculators and some distribution tables. E.g. if X~Bin(n, p), I would physically work out P(X=x) = (n!/(x!(n-x)!)(p^x)((1-p)^(n-x)). Do you still require them to know the formulaic way to do things, or do they only learn the graphing calculator functions?
     
    Last edited: Apr 29, 2018
  7. futuremathsprof

    futuremathsprof Connoisseur

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    I teach them the formulaic way first along with the theory behind it and then after they do enough exercises using the formula by writing it out, I just have them use the calculator functions. However, they are still required to identify the setting, so whether it’s normal, binomial, or geometric, and then set up the problem using normpdf/normcdf, binompdf/binomcdf, or geometpdf/geometcdf. After they compute the probability, they have to provide context for the situation.
     
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  8. JimG

    JimG Companion

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    Apr 28, 2018

    Makes sense. Thanks.
     
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  9. HMM

    HMM Cohort

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    Apr 29, 2018

    you have a small typo

    P(X=x)=(n!/(x!(n-x)!)(p^x)((1-p)^(n-x)).
     
  10. JimG

    JimG Companion

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    Apr 29, 2018

    Fixed.
     
  11. PetrMishikoff

    PetrMishikoff Rookie

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  12. JimG

    JimG Companion

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    Jun 11, 2018

    I am using some time this summer to plan ahead for the course. I looked for a way to send a private message but could not find a way to do so.

    Regarding the layout of the course with the four main themes, I have a concern. Another teacher who has taught AP Stats before had the course laid out with "Planning and Conducting a Study" as the first major unit and "Exploratory Analysis" as the second major unit. Apparently, this was a tip picked up at an AP Summer Institute. I guess the thinking is along the lines of learning how to collect data first before representing and summarizing it. However, the textbook has the units laid out in the order you wrote which also corresponds with what is written on the AP Statistics College Board Website.

    Have you any experience teaching the first two themes out of order? If so, what are your thoughts on which way works better?
     
  13. futuremathsprof

    futuremathsprof Connoisseur

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    Jun 12, 2018

    I personally feel that the ordering in the textbook is optimal. The AP exam actually covers very little on planning and conducting studies. There is a just a small handful of questions in the MC section on that and maybe one part of one problem in the FRQ section, if any. Thus, I don’t spend a lot of time on that section because of this — I have exams all the way back to 1997 and that is the common spread.

    What helped me structure the class was using a spreadsheet that listed the relative frequencies of the types of questions that are asked year to year on the AP test. For example, I noticed that interpreting the slope and y-intercept of a least-squares regression line shows up time and time again, as well as using z-scores to compare data sets, and explaining what a type II error means in context, and so I gave my students plenty of practice on those. I, of course, review the lessons on planning and conducting studies, but it mainly just entails having them identify the different types of sampling methods (i.e. cluster sampling, stratified sampling, multistage sampling, systematic random sampling, and simple random sampling) and knowing when it’s more advantageous to use which depending on the situation. However, I definitely make sure they know how to design an experiment using a completely randomized design (CRD) or block design and how to randomly allocate their experimental units/subjects using a random digits table, the hat method, the randInt( function on their graphing utilities, and other miscellaneous methods.

    Look for problems that show up again and again and that will tell you how long you should focus on certain topics.

    You can switch up the order, but I feel there is no need for that. My students do just fine going along with the textbook and I think your students will do fine, too.

    Lastly, I feel students should be able to model and manipulate data before they collect it. My thinking is this: If students collect the data and can’t interpret it or know what to do with it numerically or graphically and/or cannot identify periodic trends, then what’s the point?
     
    Last edited: Jun 13, 2018
  14. JimG

    JimG Companion

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    Jun 13, 2018

    That makes sense, and I agree on many points. I also like your strategy of the spreadsheet with past exam question topics; I may steal that myself!

    I am still waivering on which order I will present for a couple reasons.

    1) The very first section of Part II in this book focuses on distinguishing between population and sample, and I feel those are very fundamental concepts that are good to wrap one’s mind around early on. I remember in my first exposure to stats as a student, I did not understand the importance of knowing the difference, and that led to struggles down the road.

    2) The first-semester project involves students designing a survey that will intentionally be biased and then interpreting the results. Last year’s teacher told me their plans/drafts were due for approval by the end of October with the actual project due in early December. I am not quite sure how the timing of instruction will work out yet, so I would fear not having proper time to cover Part II before the due date of their plans/drafts. Perhaps as I delve more into planning this summer, I will have a better idea of the timing of everything.

    As usual, I appreciate that you took the time to give such well-thought-out advice. Thanks!
     
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  15. TrademarkTer

    TrademarkTer Habitué

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    Jun 13, 2018

    I have not taught AP Stats before, but the difference between population and sample is usually explored in a typical algebra 2 course so they should have at least some exposure coming in.
     
  16. JimG

    JimG Companion

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    True, ASSUMING the Algebra 2 teacher made time for a statistics unit amid all the material they are already hard pressed to cover AND that the teacher did an adequate job covering it. Personally, I am going to teach as if the students have little to no statistical background to avoid any confusion caused by overly optimistic assumptions.
     
  17. TrademarkTer

    TrademarkTer Habitué

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    True. In our school, every algebra 2 teacher spends around a month on stats as it is covered on PARCC assessments. Every teacher who teaches the same course teaches the same thing each day so we are all on the same page, and I am able to make assumptions about what they had the previous year.

    I guess it would be more difficult if there is more a "free-for-all" mentality in a particular school.
     
  18. JimG

    JimG Companion

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    Our department is similar to yours when it comes to syncing up what is taught in each class regardless of the teacher. Just there is not a big emphasis placed on stats in Algebra 2 in this area.
     
  19. JimG

    JimG Companion

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    Jun 15, 2018 at 9:53 AM

    Alright, after thinking it over quite a bit, I think I have decided to teach Part II first, and the main reason actually deals with teaching students about variance and standard deviation in Part I. Bear with me, and please point out any evident flaws in my logic that you may see.

    In Part II, topics covered include population vs. sample, bias, and simple random samples. As I have been thinking about how I would explain variance and standard deviation to my students in Part I, in a way that gives them a clear, concrete understanding of what is going on, I feel I need to invoke each of those terms in the explanations, as follows.

    The population variance is the average squared distance from the population mean. The population standard deviation is the square root of the population variance and gives a relatively good estimate of how far any given data point may be from the mean, give or take.

    The sample variance is used to estimate the population variance, and the sample standard deviation is used to estimate the population standard deviation. So why does each of those formulas divide by (n-1) instead of just n? Well, if you take a bunch of simple random samples from a population, and compute each sample's standard deviation using the formula that divides by n, the distribution of those calculated values would always have a center that is smaller than the population standard deviation. The reason for this is that in picking a simple random sample, you are unlikely to pick many extreme values at either end of the distribution, resulting in a smaller spread and an apparently smaller standard deviation/variance. This means that such a formula would have a negative bias in approximating the population standard deviation. To correct this bias, the sample standard deviation divides by (n-1), a slightly smaller denominator, which makes the overall calculated value slightly bigger. If you plot of distribution of all of the sample standard deviations using this correct formula, its center would be exactly at the population standard deviation, making the formula with (n-1) the ideal, unbiased estimator of the population standard deviation.

    To aid in the visualization of what is described above, I have created a demonstration that I plan to present that illustrates this. In it, I listed the population of all possible sums from rolling three fair, six-sided dice (e.g. {3, 4, 4, 4, 5, 5, 5, 5, 5, 5, ...., 17, 17, 17, 18}). Its standard deviation is about 3. To avoid unnecessary confusing details, I would probably just tell them it is a made-up population and not discuss where it comes from. I then took thirty simple random samples from it with ten values in each, incorrectly calculated the sample standard deviation of each using Sigma(xi - xbar) / n, and then correctly calculated the sample standard deviation of each using Sigma(xi - xbar) / (n-1). The five-number summary of the former was 2.0, 2.6, 2.9, 3.1, 3.8. The five-number summary of the latter was 2.1, 2.8, 3.0, 3.2, 4.0. Looking at the box plots, it is clear to see how the former (n formula) is negatively biased while the latter (n-1 formula) is centered exactly on the population standard deviation 3.0. This supports that the formula to calculate the sample standard deviation is an unbiased estimator of the population standard deviation and why it is crucial to use (n-1).

    I know that is a lot. Like I said, based on your experience, feel free to poke any holes in this that you may see. Standard deviation is such a central idea in the course, so I really want to make sure the students have a good intuitive sense of what it is going on. The description above is how I best understand it, but I want to make sure that I don't present things in such a way that it goes over students' heads, so to speak.
     
    Last edited: Jun 15, 2018 at 12:09 PM
  20. futuremathsprof

    futuremathsprof Connoisseur

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    Jun 15, 2018 at 12:11 PM

    I actually think this is a logically sound demonstration if you wish to teach Part II first. However, I would define what an unbiased estimator is and use lay terms when defining statistical terms in the beginning. For example, your formal definition for population variance makes sense to us as we’re mathematicians, but I think it safe to assume that it won’t mean jack for most students, even the top-level performers. That’s another reason why I model data first and have them identify trends beforehand in Part I. Yes, we want to encourage students to use academic language, but this course is very vocab heavy and students can quickly get lost just with the vocabulary words, let alone the formulaic parts.

    I use lay terms initially and various graphical displays to get them to develop an itutitive idea of what a standard deviation is, for example, (i.e. the spread about the mean) before I introduce definitions. This is because they need to see working examples with pictorial representations for the various concepts and not just formulas and definitions. In your case, I would include a side-by-side box plot of the five-number summaries so students can see how one is an unbiased estimator through visual inspection, and not just computationally.
     
    Last edited: Jun 15, 2018 at 12:20 PM
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