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Integrated Algebra – easy and hard – and bias

January 30, 2009 pm31 6:07 pm

The questions on the January 2009 Integrated Algebra exam were all over the place. What did you find easy? What did you find hard?

Let’s get some explanations and answers for the ones you have most questions about.

I’m going to start with the bias question. I know what the State intended, but I think they blew it. Read, and see if you agree:

#23 A survey is being conducted to determine which types of television programs people watch. Which survey and location combination would likely contain the most bias?

(1) surveying 10 people who work in a sporting goods store
(2) surveying the first 25 people who enter a grocery store
(3) randomly surveying 50 people during the day in a mall
(4) randomly surveying 75 people during the day in a clothing store

I don’t know that there is enough information, but I lean towards (2) or (4) having the greatest bias. I know that’s not what they wanted. Do you see how they blew it?

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20 Comments leave one →
  1. January 30, 2009 pm31 7:33 pm 7:33 pm

    Totally inept question. And this is a required-for-graduation test? Ugh!

  2. Rachel permalink
    January 30, 2009 pm31 8:39 pm 8:39 pm

    (3) would seem to me to have the least bias, but assessing the level of bias (as opposed to random error) in the other 3 would seem to me to take a degree in statistics: Are people who shop in a store more similar to each other than people who work in a store? Grocery stores probably have a more diverse clientele than clothing stores, but grocery shoppers don’t distribute themselves randomly during the day, and I’d be pretty sure that the viewing habits of morning shoppers were different from the viewing habits of after-work shoppers.

  3. Rachel permalink
    January 30, 2009 pm31 8:41 pm 8:41 pm

    I don’t think its a problem that high school students are asked to think about questions like this before they graduate. An essay question on “how might each of these be biased” would be pretty reasonable. But pretending that complex questions have simple “rule of thumb” solutions is a bad thing.

  4. TwoPi permalink
    January 30, 2009 pm31 8:50 pm 8:50 pm

    I suspect we’re supposed to lean toward (1) because the sample size is smallest, there is no attempt at randomization (unlike 3 & 4 & arguably 2), and the people are from a homogeneous group (with the cultural assumption that folks who work in a sporting goods store are going to tend to watch sports on TV more than the general public).

    All of the locations raise questions, except that a vast majority of the population at some point shops for food, or clothing, or goes to a mall. If one could get random samples of shoppers in those locations, you might not get a good sample of the general public (with regard to disposable income, say), but it might be an acceptable representation of entertainment tastes.

    But yeah, if they wanted to ask: “is it better to have a sample of size 10 or a sample of size 75?”, or “in what ways are sporting goods store employees likely to differ from the general public?”, this seems like a rather opaque way to get at those ideas.

  5. January 31, 2009 am31 12:51 am 12:51 am

    I agree with Twopi’s assessment of the question. I truthfully have no idea why question like this would be on a beginning algebra exam.

    I teach this topic in one of my college classes.

    The state has got to get out of the test making business. They go from bad to worse in seconds.

  6. NYC Educator permalink
    January 31, 2009 am31 4:51 am 4:51 am

    The answer is number one for which combination would likely contain the MOST bias.

    The sample size is the smallest.
    The population is engaged in the same occupation.
    The population is likely to watch television as they work in a sporting goods store which may indicate they would watch televised sporting events.

  7. January 31, 2009 am31 7:24 am 7:24 am

    I too agree with Twopi and pissedoffteacher – I think 1 would have the most bias. Or be the least unbiased.

    But I spent a semester shadowing an AP Stat teacher. I have no idea how this is an Algebra I topic.

  8. January 31, 2009 am31 8:40 am 8:40 am

    Problem is, Model’s down the road, the cashier’s, manager, stock clerks, they have nothing in common. They don’t live in the same neighborhood, have the same taste, and, I’m going to guess, watch the same shows. But if I stand outside the Pioneer, the people coming in at the same time are the same age, roughly, the same race, live in the neighborhood…

    Part of the assumption here is that middle class suburbs are the whole world.

    But the big problem, it’s the wrong course.

    Now, those of you who’ve made an argument that #1 is best, you have a point. But the opposing point is strong.

    As written, this is a lousy multiple choice item. I like Rachel’s suggestion for an essay question. But not for the math exam that determines whether or not kids graduate.

  9. January 31, 2009 pm31 7:36 pm 7:36 pm

    Once again, the real world is far too messy to be contained in a simplistic little problem. As I said on a different comment, this question was probably written by someone who didn’t teach and who didn’t completely understand the subject. In his mind, avoiding bias was only a function of the sample size, but he forgot that this rule applies to those “chosen at random.”

    I agree with everyone so far that there is simply too little detail to ascertain the randomness of the sample.

    The employees at a particular store are probably quite varied and most random, unless they are all minimum-wage cashiers. Which is it?

    The first 25 people who enter a grocery store would be self-selected as well – none are at work at that time of day, same type of shopper with same type of viewing habits – they might even have scheduled grocery shopping around their other interests – and may be all getting this chore done so as to go and watch the same show. Still, 25 grocery shoppers are a better sample than 10 employees if you assume that the 10 are all cashiers.

    The 50 in a mall during the day have the same problem – if you have time to go to the mall during the day, you aren’t working during the day. Any mall worker could tell you that – the mornings are the older folks doing their exercise and the idle. (I am basing this on the advertisements for such at the local mall. I can’t say for sure as I can’t recall as I’ve ever been in the mall during the day on a normal weekday.)

    The 75 people during the day in a clothing store are probably the most homogeneous – daytime free time and all that, but the clothing store would select for only a very small demographic. In my mind, the advantage of the greater number is dwarfed by the uniformity of the clientèle at the Old Navy Store in the mall during the day.

    I vote (4) as the most biased sample, based on the assumptions I made. I’m not sure how you can tell me I’m wrong unless I’ve told you those assumptions.

    Once again, real world knowledge, intelligence and critical thinking are a hindrance to the finding “correct” answer, exactly opposite to our goal.

    Oh well.

  10. mathman42 permalink
    February 1, 2009 am28 7:50 am 7:50 am

    Really making too much of the particular question. Random surveys show less bias, larger sample size less bias, more diverse population less bias, so # 1 is the answer. The real problem is too many somewhat unrelated topics to teach, and what do you do in a one term repeater class ? I tried the Kaplan workbook but the reading level and lack of motivation did not allow me to get much done. A handfull passed with 31 – 34 scores. Any suggestions as most will have IA redux. ??

    • February 1, 2009 pm28 9:30 pm 9:30 pm

      Yeah… no. They shouldn’t have vague or ambiguous questions. The most homogenous population was going to be the customers at the supermarket or clothing store. I know the State wanted #1 to be the answer, but they wrote a bad question (again. They are showing no improvement. If they were a kid, I’d start thinking about an evaluation). Had they simply used the customers at the sporting goods store, there would be no conversation, right?

      As far as the regents, there are some strategies that are helpful for weaker students, but they depend, in part, on where you teach, what your principal is willing to do/sacrifice, etc.

      What year are these students? Will you get support if you choose to scrap the curriculum and get them to pass at all costs? How many math credits do they already have?

      (if the answers are 1. at least sophomores, 2. yes the principal just wants them to pass, and 3. they have 2-4 math credits already, then I have a set of suggestions)

      (if not, I have a set of less effective suggestions, but still a bit useful)

  11. mathman42 permalink
    February 1, 2009 pm28 11:40 pm 11:40 pm

    Unfortunately 1 and 2 are affirmative, they have between zero and 2 math credits for the most part, and should have taken IA for 3 or 4 terms before they even attempted the Regents in the first place, in my opinion. My guess is the minimum will inch up to 34 over the next few administrations. One suggestion was to pretest with old regents to see areas of difficulty but I’m skeptical. I think, basically pick what you can ignore based on difficulty level or low chance of inclusion and go from there. The question is, what ?

  12. February 2, 2009 am28 12:02 am 12:02 am

    Pull the exams. Pull last June’s, too, if you can. Figure out what some of them are already doing well. Line it up with about 50 points. And clip off the rest of the course.

    Be explicit about it. Tell them exactly what they are learning, what they are not learning, and why. Go for some real buy in.

    Teach them to skip. Teach them to find THEIR questions and ignore the others. Teach three relatively simple topics. Then give them a 10 question multiple choice where they get, say 2 points for each correct, and minus 5 for each wrong. And then trade papers to grade. Repeat. Often.

    For short answer, again, teach them a question. Just one. And give a 3-pointer or 4-pointer or even 2-pointer as a quiz, with just that question, or with that and a nuisance question (to be ignored). And then exchange papers, and share the rubric. Teach them to grade the partial credit. Let them see where the deductions come from. And now, since a classmate graded it, let them challenge the scoring. When you hand stuff back, they accept the lousy grade because they have confidence you did it right. But now the level of interest is high. They know how the grading should work. And the deductions were taken by a classmate.

    Oh yeah, make it real. Red pencils for grading. Even teach them about initials in the margins. Use existing questions (especially at first, as you can use the moronic Albany rubrics), and when you move to your own questions, lay them out just like IA. Times New Roman 11.5 (bold) for directions. TNR 12 for the questions.

    I have a fake IA; e-mail me if you’d like it. There are some poorly worded problems, but it is a Word file and can be fixed up. I have a draft of a second, but I lost the formatting and never saved the original. I can snail mail a physical copy, or e-mail a goofy computer file with gaps.

  13. mathman42 permalink
    February 2, 2009 am28 2:24 am 2:24 am

    So we only teach what, say at least one third of the class can do and ignore the other topics. probably take at least two weeks to figure this out I suppose. Concept is worth thinking about, thanks. How about requiring work to be shown on most short answers ?

    I like the fun part about the grading; no different than what the teachers do.

    Finding 50 points ( raw ? ) they can do will be difficult.

  14. February 2, 2009 am28 2:38 am 2:38 am

    That would be the concept, yeah. The bad thing, you really are teaching the kids less math.

    I would go for 60 – 70% of the course. But at the start, I would go for a handful of questions that have been on each test and are easier. I like some of the plug in multiple choice for that one (check each x and y in both equations to see which pair works). Linear equation in one variable is also good.

    If they know, for sure, 10 multiple choice, they probably pass with a little test-taking training.

    Just, different from what usually happens in low-level classes, teach for mastery. Get it so the kids get that question right 95% of the time before you introduce the next question. And super-spiral. Once they have it right, they have to keep doing the same question, each day, every other day, at least once or twice a week, until they’ve passed the test.

    Think for a moment, if you had to memorize the cities, rivers, mountains, bays, towns, cities, provinces, etc of a country that you’d never studied, with words and names that sounded strange and were hard to remember, and you needed to get a 30 on a test on this stuff in a few months, and it mattered bigtime, wouldn’t you do something similar?

  15. mathman42 permalink
    February 2, 2009 am28 5:31 am 5:31 am

    You’re a great help. Much appreciated. Of course if they knew for sure 10 mult choice, they would have been in my geometry class and struggling like the majority. Another interesting situation that you may be familiar with. Perhaps the best opportunity in their high school career to really use their brains if the class is organized somewhat by ability level and you have sketchpad and other resources and maybe 3 terms. ( not sure about the last part )

    • February 2, 2009 am28 5:51 am 5:51 am

      The getting to geometry based on 31 IA points is why I asked about credits. If they had 3 now it would be better.

      And 10 multiple choice can be hard – but it’s got to be so much easier when you’re not trying to learn everything.

      I would also look at the problems that your kids always get wrong, and ban them from even trying them.

      Geometry? Go ask pissed off teacher what she is doing about geometry…

      • mathman42 permalink
        March 2, 2009 am31 7:04 am 7:04 am

        So far in my three repeater sections, attendance averages about 50 % of roster. Hardly anyone seems to know very much. I am going to do 7 or 8 main topics ( finished modeling expressions, equations and inequalities ), and will be emphasizing use of calculator for answer testing and graphical representation. Plus whatever else I have time for that’s not too difficult. Based on past Math A results I can usually get a 10-15 point increase on 75 % of the students who have 80% or higher attendance. Hopefully 22-25 from the three sections would be a good result for a 65 pass score.

  16. Anonymous permalink
    May 27, 2009 pm31 8:37 pm 8:37 pm

    this is stupid the states wants the students to understand that there is more bias in less people tested and also these people work at a sporting good store. If they shopped at the store there would be less bias because almost everyone has bought a ball or a bat or shoes. Also everyone shops at grocery stores and almost everyone has step foot in a mall and if you dont shop at a clothing store i soppose you are naked right now. This is a simple question. What you are all arguing is that your students or you do not have the life experience to answer this, but all you are really doing is coming up with an excuse

  17. Barbara permalink
    March 14, 2010 pm31 6:01 pm 6:01 pm

    I think that the answer is (1) because according to statistics, bias means that something is NOT random.

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