- "If" Statement. In a basic algebra class, we have the rule "If the base is negative, then even powers are positive, but odd powers are negative". Immediately after that, I'll always have some students incorrectly evaluate something like: −5² = 25 (or worse, 2³ = −8) . Note that the base of the exponent is not negative, but some students overlook the check required for the "if" qualifier.
- "Or" Statement. In an elementary statistics class, we have the rule "To estimate a population mean, we must have either a normal population or a large sample size." Then when I ask the class "Do we need a normal population?", the entire class will always incorrectly respond with "Yes!" the first time. Of course that's not true; they're overlooking that only one case of the "or" needs to be satisfied -- most commonly by a large sample size. It takes several sessions of quizzing them on that before they are sensitive to the question being asked.
- "And" Statement. In practically any class, we might have the policy, "To pass this class you need at least a 60% weighted average, and a 60% score on the final exam." This constantly causes confusion and aggravation. Testy "So, the final exam doesn't count?", or "So, only the final exam counts?" are questions that I routinely have to address. Obviously, students are unclear on the fact that each of two requirements must be satisfied for an "and" statement like that.
2014-05-05
Basic Logic Errors
I constantly wish that students were taught rudimentary logic at an early age (links: one, two, three).
Just musing about that today, here are three common stumbling blocks I
see in different classes due to not being able to read logical
statements properly:
2014-04-15
Multiple Choice Chances
Let's say you have a final-exam assessment that is a multiple-choice test, with 25 questions, each of which has 4 options, and requires a 60% score (15 questions correct) to pass. As one example, consider the uniform CUNY Elementary Algebra Final Exam (link).
How robust is this as an assessment of mastery in the discipline? As a simple model, let's say that any student definitely knows how to answer N types of questions, but is randomly guessing (uniform distribution over 4 options) for the other questions. Obviously this abstracts out the possibility that some students know parts of certain questions and can eliminate certain choices or guess based on the overall "shape" of the question, but it's a reasonable first-degree model. Then the chance to pass the exam for different levels of knowledge is as follows:
Obviously, if a student really "knows" how to answer 15 or more questions, then they will certainly pass this test (omitted from the table). But even if they only know half of the material in the course, then they will probably pass the test (12 questions known: 67% likely to pass). Of students who only ever know about one-third of the basic algebra content, but retake the class 3 times, about half can be expected to pass based on the strength of random guessing on the rest of the test (9 questions known: 19% likely to pass; over 3 attempts chance to pass is 1-(1-0.19)^3 = 1-0.53 = 0.47).
How robust is this as an assessment of mastery in the discipline? As a simple model, let's say that any student definitely knows how to answer N types of questions, but is randomly guessing (uniform distribution over 4 options) for the other questions. Obviously this abstracts out the possibility that some students know parts of certain questions and can eliminate certain choices or guess based on the overall "shape" of the question, but it's a reasonable first-degree model. Then the chance to pass the exam for different levels of knowledge is as follows:
Obviously, if a student really "knows" how to answer 15 or more questions, then they will certainly pass this test (omitted from the table). But even if they only know half of the material in the course, then they will probably pass the test (12 questions known: 67% likely to pass). Of students who only ever know about one-third of the basic algebra content, but retake the class 3 times, about half can be expected to pass based on the strength of random guessing on the rest of the test (9 questions known: 19% likely to pass; over 3 attempts chance to pass is 1-(1-0.19)^3 = 1-0.53 = 0.47).
2014-04-03
Meta-Research Innovation Centre
Interesting article about Dr. John Ionnidis at Stanford founding the "Meta-Research Innovation Centre" to monitor and combat weak and flawed statistical methods in science research papers, especially medicine. Good luck to him!
2014-03-27
Gears of War
When I was a kid, one of my favorite pastimes was the Avalon Hill wargame Bismarck about fighting ships in World War II (see it reviewed on my gaming site here and here). In junior high school, at some point my English teacher asked me what I wanted to do as a career, and was completely apalled when I said "I want to join the Navy and control the main guns on a battleship". (I think I'd share her dismay if someone told me something like that today.)
Anyway, over at Ars Technica, a wonderful article has been written by Sean Gallagher (former Navy officer and IT editor) on exactly how the fire control systems on those ships did their jobs -- solving 20-variable calculus problems in real-time (accounting for moving, pitching, rolling, recoiling, Cariolis-spinning projectiles on both ends) with shafts and gears, with accuracy that is hard to beat even today with digital computers and GPS-driven rocketry. There are lots of insightful videos about the components and gears used to do input, sums, multiplies and divides, and spinning disks that can do complicated functions like trigonometry and more.
To me, this stuff is completely like crack. Check it out:
(Also: Further commentary and links at recently-established news site SoylentNews.)
Anyway, over at Ars Technica, a wonderful article has been written by Sean Gallagher (former Navy officer and IT editor) on exactly how the fire control systems on those ships did their jobs -- solving 20-variable calculus problems in real-time (accounting for moving, pitching, rolling, recoiling, Cariolis-spinning projectiles on both ends) with shafts and gears, with accuracy that is hard to beat even today with digital computers and GPS-driven rocketry. There are lots of insightful videos about the components and gears used to do input, sums, multiplies and divides, and spinning disks that can do complicated functions like trigonometry and more.
To me, this stuff is completely like crack. Check it out:
(Also: Further commentary and links at recently-established news site SoylentNews.)
2014-03-20
FiveThirtyEight
Nate Silver (statistician who famously predicted all 50 states voting in the last election) recently expanded his FiveThirtyeight blog to a full-blown "data journalism" site. His first post was a manifesto on data, science, statistics, politics, journalism, and honest storytelling in general. I agree with almost all of his observations here. They guy really knows his stuff and has a fiery passion for his particular mission. Great stuff.
2014-03-10
Faulty Factoring
Here's something I think I see a few times in any college algebra class: a really weird way of accomplishing quadratic factoring. (More generally, this might go in a larger file of "things students swear are taught by other instructors which are semi-insane" -- including whacked-out order-of-operations, keep-change-change for negatives, the idea that -42 means (-4)(4), etc.).
Anyway, let's say we want to factor what I call a "hard quadratic", i.e., Ax2+Bx+C, in integers, with A≠1 (hence "hard"). I prefer the method of grouping: i.e., factor AC so it sums to B, use those factors to split the term Bx, and then factor the four terms by grouping. Pretty straightforward.
But here's what a few students will insist on doing every semester: (1) Find factors of AC that sum to B; call these factors u & v (so that step is the same); (2) Write the expression (Ax+u)(Ax+v); (3) Look for a GCF of A in one of those binomials and strike it out.
Here's an example: Factor 5x2+7x−6.
Step (1): Note AC = −30 = (10)(−3), factors which sum to B = 7.
Step (2): Write (5x+10)(5x−3)
Step (3): Divide the first binomial by 5, producing (x+2)(5x−3).
So while this procedure does produce the right answer, what irks me tremendously is that the expression written in step (2) is not actually equal to either the original expression or the answer at the end. (Compounding this issue, students will nonetheless usually write equals signs by rote on either side of it.) Riffs on this procedure would be to write something like this on sequential lines, if you can follow it:
5x2+7x−6 → x2+7x−30 → (5x+10/5)(5x−3/5) → (x+2)(5x−3)
Again, the primary grief I have over this is that none of these expressions are equal to any of the others, and the students using this procedure are always oblivious to that fact. Second issue: They're likely to trip up over a non-elementary problem where the factor A does not appear in either of the binomials, e.g.: 4x2+4x+1 = (2x+1)(2x+1). Third issue: If there's a GCF in the quadratic itself and you overlook that, the standard grouping technique will still work (even if it's not the easiest way to do it), whereas I suspect users of this technique will be prone to incorrectly striking out any GCFs they discover at the end of the process.
Now, technically you could modify this and turn it into a correct procedure this way: Note that for quadratic Ax2+Bx+C, values u & v satisfy uv=AC and u+v=B if and only if Ax2+Bx+C = 1/A(Ax+u)(Ax+v). (Proof: 1/A(Ax+u)(Ax+v) = 1/A(A2x2 + (u+v)Ax + uv) = Ax2 + (u+v)x + uv/A and equate coefficients). So you could find u & v as usual, then write this latter expression, and simplify. The 1/A does always cancel out, but I've never seen a student actually write that factor in the second step.
So what I always do if I see this on a test in my college algebra class is to take half credit off for the problem and note that the intermediary expression is "false", i.e., not equal to what comes before or after. This then becomes an opportunity to discuss with the student why that's improperly written math -- went well in my most recent semester, but I can easily see that becoming more combative in a remedial algebra class.
Have you seen this (common) faulty factoring procedure in your classes? What do you as a correction for it, if anything?
Anyway, let's say we want to factor what I call a "hard quadratic", i.e., Ax2+Bx+C, in integers, with A≠1 (hence "hard"). I prefer the method of grouping: i.e., factor AC so it sums to B, use those factors to split the term Bx, and then factor the four terms by grouping. Pretty straightforward.
But here's what a few students will insist on doing every semester: (1) Find factors of AC that sum to B; call these factors u & v (so that step is the same); (2) Write the expression (Ax+u)(Ax+v); (3) Look for a GCF of A in one of those binomials and strike it out.
Here's an example: Factor 5x2+7x−6.
Step (1): Note AC = −30 = (10)(−3), factors which sum to B = 7.
Step (2): Write (5x+10)(5x−3)
Step (3): Divide the first binomial by 5, producing (x+2)(5x−3).
So while this procedure does produce the right answer, what irks me tremendously is that the expression written in step (2) is not actually equal to either the original expression or the answer at the end. (Compounding this issue, students will nonetheless usually write equals signs by rote on either side of it.) Riffs on this procedure would be to write something like this on sequential lines, if you can follow it:
5x2+7x−6 → x2+7x−30 → (5x+10/5)(5x−3/5) → (x+2)(5x−3)
Again, the primary grief I have over this is that none of these expressions are equal to any of the others, and the students using this procedure are always oblivious to that fact. Second issue: They're likely to trip up over a non-elementary problem where the factor A does not appear in either of the binomials, e.g.: 4x2+4x+1 = (2x+1)(2x+1). Third issue: If there's a GCF in the quadratic itself and you overlook that, the standard grouping technique will still work (even if it's not the easiest way to do it), whereas I suspect users of this technique will be prone to incorrectly striking out any GCFs they discover at the end of the process.
Now, technically you could modify this and turn it into a correct procedure this way: Note that for quadratic Ax2+Bx+C, values u & v satisfy uv=AC and u+v=B if and only if Ax2+Bx+C = 1/A(Ax+u)(Ax+v). (Proof: 1/A(Ax+u)(Ax+v) = 1/A(A2x2 + (u+v)Ax + uv) = Ax2 + (u+v)x + uv/A and equate coefficients). So you could find u & v as usual, then write this latter expression, and simplify. The 1/A does always cancel out, but I've never seen a student actually write that factor in the second step.
So what I always do if I see this on a test in my college algebra class is to take half credit off for the problem and note that the intermediary expression is "false", i.e., not equal to what comes before or after. This then becomes an opportunity to discuss with the student why that's improperly written math -- went well in my most recent semester, but I can easily see that becoming more combative in a remedial algebra class.
Have you seen this (common) faulty factoring procedure in your classes? What do you as a correction for it, if anything?
2014-03-05
Presenting at Johns Hopkins
Here's one of these topics that merges my great interests in teaching & gaming, so I have no choice but to cross-post about it here and on my gaming blog.
Last week I had the opportunity to visit Johns Hopkins University, at the invitation of Peter Fröhlich to speak to his Video Game Design Project class in the computer science department there (run jointly with art students from the nearby MICA). A great talk and chance to meet with his students and network a bit with Peter, Jason from MICA, as well as one of my idols from old-school role-playing game publishing.
Bounce on over to my gaming blog for the details!
Last week I had the opportunity to visit Johns Hopkins University, at the invitation of Peter Fröhlich to speak to his Video Game Design Project class in the computer science department there (run jointly with art students from the nearby MICA). A great talk and chance to meet with his students and network a bit with Peter, Jason from MICA, as well as one of my idols from old-school role-playing game publishing.
Bounce on over to my gaming blog for the details!
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