tag:blogger.com,1999:blog-7691251033406320222.post3097119447720551850..comments2024-03-26T04:19:38.862-07:00Comments on kitchen table math, the sequel: A view from computer science college professorsCatherine Johnsonhttp://www.blogger.com/profile/03347093496361370174noreply@blogger.comBlogger32125tag:blogger.com,1999:blog-7691251033406320222.post-36352168402473728932011-02-10T15:39:26.045-08:002011-02-10T15:39:26.045-08:00I found this intro statistics textbook:
http://ww...I found this intro statistics textbook:<br /><br />http://www.amazon.com/Introduction-Bayesian-Statistics-William-Bolstad/dp/0471270202/ref=sr_1_1?ie=UTF8&qid=1297380961&sr=8-1<br />It seems to be pedagogically sound with lots of exercises.<br /><br />ari-freeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-84799485728092691132011-02-09T06:06:59.697-08:002011-02-09T06:06:59.697-08:00My quick check of the Khan Academy "introduct...My quick check of the <a href="http://www.khanacademy.org/#" rel="nofollow">Khan Academy</a> "introduction to statistics" course seemed to indicate that only first-year algebra was required as a pre-requisite. Here are the first few topics:<br /><br />-- : The Average <br />-- : Sample vs. Population Mean <br />-- : Variance of a Population <br />-- : Sample Variance <br />-- : Standard Deviation <br />-- : Alternate Variance Formulas<br /><br />(I've been meaning to take Khan's course because I am woefully ignorant on this topic.)Gracehttps://www.blogger.com/profile/16355093065582134401noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-70779244596420867602011-02-08T23:09:51.261-08:002011-02-08T23:09:51.261-08:00Anonymous at 2:31 had a pretty good list of statis...Anonymous at 2:31 had a pretty good list of statistics with just algebra 1 topics.<br /><br />You can find lots of examples. Just about every "biostatistics" class in the country assumes only algebra 1, because that is all most biology students master (no matter whether their major requires a calculus class or not).<br /><br />The statistics department here distinguishes between "calculus-based probability classes" as prereqs for the upper-division and graduate statistics classes and the "biostatistics" classes (and other stats classes for mostly innumerate majors).<br /><br />I'm arguing that everyone should get the fairly simple view of statistics that biology and psychology majors get.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-76037697686829592472011-02-08T19:33:36.916-08:002011-02-08T19:33:36.916-08:00I agree that people at the college level have no i...I agree that people at the college level have no idea what they're asking for (or getting) when they ask for "more conceptual understanding." They don't seem to understand that students who can't do their times tables also struggle with more conceptual word problems that require multiplication, because they have NO CLUE if their answer is totally off base. <br /><br />They also don't seem to register that these problems have accelerated SINCE we've tried teaching conceptual understanding without a rote/algorithmic basis. <br /><br />I've picked up my knitting again and I keep wondering what a <i>conceptual understanding first </i> knitting program would look like. I think you'd show them lots and lots of pictures of knitting, let them look at sweaters and scarves and hats and talk alot about "what they notice" and what they like and don't like. <br /><br />Then, nearly as an afterthought, the needles and yarn come out, and there's a quick "here's how to do a knit stitch, here's how to do a purl stitch. BTW, there are youtube videos in case you forget."<br /><br />Then tell them to draw a sweater (or even a scarf or hat) and make it. <br /><br />It won't work. Learning the basic stitches comes first and it's only after hours of practice that you begin to see what mistakes look like. Then after hours more of it, you begin to realize (or can ask, repeatedly) how to go back and fix problems. <br /><br />Then after working up from very simple squares to slightly more complex items, you can start looking at patterns and following them. <br /><br />THEN after doing that over and over, you start to see the bigger picture, you realize how you could make a sweater out of what you know and have done. You can NOW develop and produce a concept. <br /><br />BUT it took a loooooong time of basic practice and then continued practice with gradually increasing levels of difficulty. And there's really no other way to get there.Jennoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-11006427685899230322011-02-08T14:31:50.898-08:002011-02-08T14:31:50.898-08:00Here are some things that I'd include in a sta...Here are some things that I'd include in a statistics course for the general population of students with an algebra 1 background:<br /><br />* Making sense of mean, median, and mode. Knowing when each is the best choice. Realizing that the mean net worth of the people in Bill Gates' high school graduating class is a meaningless number (and less extreme examples).<br /><br />* Basics of distributions, standard deviation. If one student has a 75 on every test and another has a 50 on half the tests and a 100 on half the tests, they both have a 75 average. But would you say that their performance was the same? Practical things like this. Nothing about normalizing areas under curves!!!<br /><br />* Both the number of drowning deaths and the amount of ice cream sold increased in July. Therefore, ice cream causes drowning. Understanding how not to reason with statistics.<br /><br />* Making sense of percentiles.<br /><br />* Making and interpreting histograms and similar graphs. (I do love me some stem and leaf.)<br /><br />* Very, very, very basic probability. Why you should not play the lottery.<br /><br />* Extremely concrete examples of conditional probability, but without using that term and without using Bayes' Theorem. (It can be done just in terms of fractions and percents.) Examples like universal AIDS testing. (Upshot: So few people in the US are HIV+ that universal testing would end up with a lot of false positives -- way more than real positives.)<br /><br />* Good and bad surveys. <br /><br />* If you have a population of 10,000 people and you ask 1000 of them what their shoe size is and take the mean of these 1000 people's shoe size, is this the mean for all 10,000 people? Problems like this, but not going into too much detail about distributional assumptions.<br /><br />* What margin of error means in a poll.<br /><br />* Very basic hypothesis testing, such as the chi-square test with 2x2 tables. It is OK to let the students accept the chi-squared table as a magical black box.<br /><br />* Interpretations of statistics seen in the newspaper, in science publications for the general public, in the pamphlet that comes with your prescription medication, etc.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-41048004444712989352011-02-08T13:58:09.982-08:002011-02-08T13:58:09.982-08:00...we can also prove the other cases (ASA, SSS, AA......we can also prove the other cases (ASA, SSS, AAS) directly using the trig approach as implied in the other links and I believe we can avoid the problems of weak proofs for SSS described here:<br /><br />http://www.cut-the-knot.org/pythagoras/SSS.shtml<br /><br />ari-freeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-25322509280958583462011-02-08T13:30:10.565-08:002011-02-08T13:30:10.565-08:00gasstationwithoutpumps wrote: "Geometry is ac...gasstationwithoutpumps wrote: "Geometry is actually a terrible subject for teaching proofs (direct instruction in logic is probably more productive),"<br /><br /><br />Incidentally, I've always been bothered by the SAS postulate. The proof using superposition always seemed a bit dodgy for something that was so essential for many other theorems. Last night I realized that it could be proved directly using the trigonometric laws of sines and cosines which don't rely on the SAS postulate.<br /><br />http://mathworld.wolfram.com/SASTheorem.html<br /><br />All you have to do is make another triangle A'B'C' where sides a=a', c=c' and angle B = B', then prove that b=b', A=A' and C=C' by substitution.<br /><br /><br />ari-freeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-91915843775285291172011-02-08T12:54:37.007-08:002011-02-08T12:54:37.007-08:00Alliosn wrote "Exactly what topics should be ...Alliosn wrote "Exactly what topics should be taught in a statistics course that has *no prereq other than algebra 1*?"<br /><br />You can be sure it will be the frequentist approach and not Bayesian. <br /><br />ari-freeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-28732273850692231942011-02-08T12:52:55.360-08:002011-02-08T12:52:55.360-08:00Steve's point
"this argument means someth...Steve's point<br />"this argument means something completely different to "math brains". They are usually astounded when they compare notes with educators."<br />cannot be overstated.<br /><br />Tech folks and educators can nod in agreement all day about how students need conceptual understanding and how procedural fluency without it is a waste. But they don't know that they aren't agreeing on anything beneath that. Many *many* educators think that teaching the standard algorithms for addition, subtraction, multiplication, and division to proficiency INHIBITS conceptual understanding. Go ahead, ask them--do the standard algorithms support or detract from conceptual understanding?<br /><br />I think most tech folks would have their mouths hanging open to realize how many educators they thought they agreed with believe the standard algorithms should not be taught.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-6658052733591779582011-02-08T11:49:46.862-08:002011-02-08T11:49:46.862-08:00Well, yes, it does depend. Which is why I want to ...Well, yes, it does depend. Which is why I want to hear what "statistics course" you and GSW/OP think high schoolers should receive. I doubt you meant high schoolers should receive what first graders know. What <br />do you mean precisely?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-71649915666637631792011-02-08T10:33:20.080-08:002011-02-08T10:33:20.080-08:00Exactly what topics should be taught in a statisti...<i>Exactly what topics should be taught in a statistics course that has *no prereq other than algebra 1*?</i><br /><br />I don't know, but in NY statistics instruction starts in pre-k. From the "Statistics and Probability Strand" for first grade:<br /><br /><i>Students will collect, organize, display, and analyze data.<br />Collection of Data<br />Pose questions about themselves and their surroundings<br />Collect and record data related to a question<br /> Organization and 1.S.3 Display data in simple pictographs for quantities up to 20 with units of one<br />Display of Data <br />Display data in bar graphs using concrete objects with intervals of one<br />Use Venn diagrams to sort and describe data<br />Analysis of Data<br />Interpret data in terms of the words: most, least, greater than, less than, or equal to<br />Answer simple questions related to data displayed in pictographs (e.g., category with most, how many more in a category compared to another, how many all together in two categories)<br />Students will make predictions that are based upon data analysis.<br />Predictions from Data<br />Discuss conclusions and make predictions in terms of the words likely and unlikely<br />Construct a question that can be answered by using information from a graph</i><br /><br />I guess it depends upon what you want to call a "statistics course." ;)Gracehttps://www.blogger.com/profile/16355093065582134401noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-14309855926722770882011-02-08T09:56:48.366-08:002011-02-08T09:56:48.366-08:00Way back when I taught a data structures course, B...Way back when I taught a data structures course, B-trees referred to B+ or B* trees, and binary trees were called binary trees. My students had to write B-tree routines. I suppose nowadays students (at most) need to use a B-tree class structure rather than to create one.SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-71973654957937131562011-02-08T09:32:21.251-08:002011-02-08T09:32:21.251-08:00I just realized, in rereading the comments, that S...I just realized, in rereading the comments, that Steve H is probably talking about binary trees, not the B+trees that I was thinking. Binary trees are supposed to be covered in data structures, but since it is the last topic, it is common to run out of time in the semester before getting there. But certainly I agree that binary trees and tree traversals are fundamentalBKMnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-36656093856223903442011-02-08T09:22:14.010-08:002011-02-08T09:22:14.010-08:00"They say that if a student has deep understa..."They say that if a student has deep understanding, he can always have figure out the answer, even if he has to go back and puzzle out out an algorithm."<br /><br />"Math brains" say this only because they haven't had a child take Everyday Math. I've seen the transformation that takes place when that happens. So many arguments sound fine on a superficial level, but fall completely apart when you get to the details. Many educators don't have the knowledge or experience to get to that level. Also, this argument means something completely different to "math brains". They are usually astounded when they compare notes with educators. Higher expectations get translated into lower expectations. Somehow, understanding gets translated into no need for mastery because the kids can eventually figure it out from some sort of conceptual understanding. It just doesn't happen. There is linkage between mastery and understanding. <br /><br /><br />It's hard to progress past these simple pre-conceived ideas, especially when that's all that educators have ever been taught. We can claim "false-dichotomy" until we are blue in the face, but they continue to ignore it over and over and over.SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-42382637446021733432011-02-08T06:58:57.179-08:002011-02-08T06:58:57.179-08:00Exactly what topics should be taught in a statisti...Exactly what topics should be taught in a statistics course that has *no prereq other than algebra 1*?<br /><br />What conceptual understanding of statistics is achievable with no prereq knowledge other than algebra 1?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-59479682652872427552011-02-08T06:55:59.941-08:002011-02-08T06:55:59.941-08:00Math for all citizens should include algebra 1 and...<i>Math for all citizens should include algebra 1 and statistics.</i><br /><br />That sounds right to me.Gracehttps://www.blogger.com/profile/16355093065582134401noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-60416680456020064382011-02-08T06:49:36.499-08:002011-02-08T06:49:36.499-08:00I would guess that your friend would want students...<i>I would guess that your friend would want students to know their math facts, such as times tables?</i><br /><br />For the moment, I don't know what my friend thinks about that. I imagine the answer might be that students should ideally know both the procedures and the concepts, but the concepts are more important if you had to choose just one.<br /><br />Here's what I've heard a few "math brains" say in defense of prioritizing conceptual understanding over procedural fluency. They say that if a student has deep understanding, he can always have figure out the answer, even if he has to go back and puzzle out out an algorithm. I guess that's a pretty standard view among reform math advocates.<br /><br />Having known both mathy and non-mathy students, and having read about the limits of short-term memory, I would respond that it is impractical for most students to forego memorizing procedures. Also, memorization before deep understanding works fine as an effective learning sequence. I’ve experienced this many times.Gracehttps://www.blogger.com/profile/16355093065582134401noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-49425625580264703632011-02-08T06:24:16.343-08:002011-02-08T06:24:16.343-08:00The discrete math issue is somewhat separate from ...The discrete math issue is somewhat separate from the programming issue (computer science isn't just about programmng, after all - that's a big pet peeve for computer scientists!). I agree that the way programming is taught (or not) in K12 is appalling. But that is not the debate that the OP is referring to.<br /><br />Most CS programs will require a data structures course, although the content can be pretty slippery. But it is completely possible to get through a CS major without every programming a B-tree routine. B-trees are often relegated to the database course, which is often an elective (and that is a pet peeve of mine!). I can also tell you that just because a student *takes* a data structures course, it doesn't mean that he or she understood any of it.<br /><br />And I do agree that we get lots of students who are utterly unprepared for the demands of computer science. A lot of HS guidance counselors seem to tell kids (boys especially) that if they like video games, they should major in computer science. We used to flunk 'em out rapidly, but when CS enrollments tanked around 2002, we came under lots of pressure to "retain" any living body.<br /><br />The NSF is taking a huge interest right now in the state of CS education in K12. Take a look at the CE21 program<br />http://www.nsf.gov/pubs/2010/nsf10619/nsf10619.htmBKMnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-73924993876066108402011-02-08T06:12:05.015-08:002011-02-08T06:12:05.015-08:00Actually, I think the problem these days would be ...Actually, I think the problem these days would be that kids come to college after having done some web page development and scripting and think that's what CS is all about. Wait until they get to the data structures course. Then again, didn't someone once say on KTM that now you can get a CS degree without taking a data structures course or ever having written a set of B-tree routines?SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-6011317538758035432011-02-08T05:59:43.000-08:002011-02-08T05:59:43.000-08:00The discussion isn't well defined. Some of my ...The discussion isn't well defined. Some of my old CS students just didn't want to put in the work. They didn't seem to like programming. I wouldn't care if they didn't have calculus in high school, or ever. I never thought "Boy, if only they had class 'X' in high school, everything would be fine". At most, I thought that the CS requirement of a class in differential equations kept many away from the degree program.<br /><br />The post never states what the problem is. It's kind of like saying "There's something wrong with my program, I think we need different inputs? What do you think?" <br /><br />No, what exactly is the error? I had too many students who wanted to use guess and check to debug their programs. Back in the 1980's, Turbo Pascal on a PC was huge; edit-compile-link-run was so fast that students stopped thinking. They just changed something and tried it again. Maybe if these students had a (real) programming course in high school, they would have figured this out by college or would have gone into another field.<br /><br />The "error" I saw was that the students didn't know what they were getting into in terms of work and level of detail. You can't be satisfied with a program that is 90% correct. My solution is to make sure they have a proper programming course in high school.SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-42050856050392322992011-02-08T05:26:53.409-08:002011-02-08T05:26:53.409-08:00The discrete math vs. calculus debate is really mo...The discrete math vs. calculus debate is really more relevant at the college level. In many CS programs, the calculus requirement crowds out math that is more relevant. It is not unusual to see all discrete math topics shoved into a one semester hodgepodge course. I have also heard complaints that math departments use calculus as the gatekeeper course, so that students have to get through calculus in order to take more relevant math courses.<br /><br />I must disgree, though, that discrete math couldn't be taught in high school. There are many topics from graph theory and even automata theory that are accessible. Perhaps this ties to the discussion going on over at Mark Guzdial's blog, about whether HS teachers should be certified in CS? I could imagine that a teacher with CS certification could run a high school level course on discrete math topics. Maybe a course like that could tie in with the new CS principles course?BKMnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-88367583810044851132011-02-08T05:23:29.810-08:002011-02-08T05:23:29.810-08:00At the HS I have mentioned before, with an excelle...At the HS I have mentioned before, with an excellent math/science program, the pre-calc class was called Elementary Functions/Analytic Geometry at the time my kids took the class; called E-FAG for short. That being politically incorrect, the name has since been changed (can't remember to what) but I wouldn't be surprised if the content is the same.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-12882208808923310052011-02-08T05:22:26.785-08:002011-02-08T05:22:26.785-08:00Geometry may not have a lot of usefulness in later...Geometry may not have a lot of usefulness in later schooling or life, but for many students it's a welcome break from the more abstract algebra/trig.calculus (even statistics) progression. It's the only math class where you can see what's going on. It was a treat for me to have a math class that I could excel at and enjoy, and gave me the energy to persist in the following years.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-83237360202619080992011-02-08T04:30:31.525-08:002011-02-08T04:30:31.525-08:00I think that there are some teachers in high schoo...I think that there are some teachers in high schools who could teach discrete math, but I don't think that there is a lot of point to it, as not many students would get much out of it.<br /><br />Geometry is actually a terrible subject for teaching proofs (direct instruction in logic is probably more productive), and most students don't really learning anything that sticks with them from their geometry classes. It mostly ends up wasting time, except for the few who go on to become mathematicians or math teachers.<br /><br />Math for all citizens should include algebra 1 and statistics. Math for carpenters and machinists should add trigonometry to that. Math for scientists and engineers should add algebra 2 and calculus. Geometry is probably the least useful of the math courses taught in high school, though it can be fun if it is done as problem solving rather than as dry-as-dust 2-column proofs.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-86623750747102750842011-02-08T00:23:41.446-08:002011-02-08T00:23:41.446-08:00Students would be better off if they left Geometry...Students would be better off if they left Geometry and Calculus alone and dropped TV, Facebook and video games from their 'curriculum.'<br /><br />ari-freeAnonymousnoreply@blogger.com