FFAF: Viral Math Teacher Fails

I was never a big math guy, but in today’s episode, we’re going to talk about not my mathematical failures, but the teachers or the question designers. Sometimes they’re the ones who get it wrong. Welcome to the Council of Trent. Mondays and Wednesdays, we talk apologetics and theology, but on free for all Friday, we talk about really whatever I want to talk about. And today I want to talk about some viral math questions that I’ve seen online that people do double takes. Maybe they’re poorly worded. Maybe they’re just frankly, they’re incorrect. And so that’s what I want to talk about. I was never a big math guy. I did the best that I could. I tried calc AB senior year because I wanted to just go to calculus camp with everybody because that was the cool thing to do. I made it about five weeks into the semester and I was like, I thought the limit was infinite or I thought isn’t the limit infinite for everything?

(00:51)
No, apparently it’s not. Had to drop out of that. Probably set myself up for failure in pre-calc by playing snakes on my TI-83 more than paying attention in class. So a rough go for sure. So what I want to talk about is there’s a few problems that I’ve seen online that get shared around and it’s just funny. Usually they’re poorly worded or they’re just kind of strange math stuff. So here’s the first one. It’s probably like from elementary school, they’re dealing with fractions. And it says, question number eight, reasonableness. Marty ate four six four over six. Marty ate four six of his pizza. And Luis ate five six of his pizza. Marty ate more pizza than Luis. How is that possible? Now think about that one for yourself. Marty eats four six of a pizza, two thirds, but we got to keep it together. We don’t want to keep the same common denominator.

(01:48)
And Luis ate five six, but Marty ate more if it was four six. How is that possible? Well, this is an elementary kid. You can see he’s barely written it through and it makes sense. He writes, Marty’s pizza was bigger than Luis’s pizza. That makes sense to me. And what does the teacher write underneath it? Big green X and then writes, “That is not possible because five six is greater than four six. So Louise ate more.” I’m sure it’s the way that the question is worded to make you think there is a possibility or to quote dumb and dumber. So you’re saying there’s a chance. Marty eats four six. Louise ates five six. Marty ate more pizza than Luis. Fact. How is that? And I’m adding fact there because there’s a period, a declarative sentence. Marty ate more pizza than Luis. How is that possible?

(02:39)
It’s not possible. So that goes down to the problem of the question leaves open a false possibility for the student to entertain. But I just love it. It’s not possible. You said it was possible. No, I didn’t. I just asked, how is it possible? How is that possible? It’s not. Here’s the next one. Sometimes these math questions, the reason you get them wrong and you can’t get them right, because they’re not trying to teach you the right answer per se. They want to teach you the method of getting to an answer. And this question’s weird because it wants you to explore the wrong ways you could get to a question. So here’s the question. What is four plus eight? What is four plus eight? And there’s four options here. 14, 13, 12, 11, A, B, C, D. You might think, okay, well that’s an easy one.

(03:28)
C, 12, and you want to write that down, right? And then you get that correct. But then here’s the next question. Hassan chose D as the correct answer. How did Hassan get his answer? What? What is four plus eight? D is 11. Hey, guess what? You know it was 12, but Hassan, he picked D. He picked 11. So here’s the real question, Bucko. How did he get the wrong answer? I don’t know. How did he get his answer? Maybe he was just guessing. Maybe Hasan is sniffing too much Elmer’s glue in the back of class and he start paying attention more. Although I know what they’re getting at here and I was singing it through and this is probably what they mean. It’s probably teaching, this is now addition, basic elementary concepts. How do kids add stuff? They’ll add things together on their fingers, right?

(04:20)
And so what would be three plus two? So one, two, three fingers and add two more. That gets you five. The problem is, okay, four plus eight. So if I have four in my head and then I start doing my fingers, I’m going to add eight more fingers, five, six, seven, eight, nine, 10, 11, 12. The problem would be if you’re adding on your fingers, you think, “Okay, I got four. I got to add eight and I’ll count them up on my fingers.” If you do the first finger or thumb, I guess it would be and you say four, five, six, seven, eight, nine, 10, 11, your eighth finger goes up, it’s 11 that you are counting and adding eight, but you should not have included four in the sum that’s being counted. Now, I don’t expect an elementary student to articulate it in that way, but I think that’s what they’re referring to, that when you’re adding, you have to keep the original, don’t include it in what’s being added, I guess.

(05:13)
Or just don’t count on your fingers. Memorize that stuff. I don’t know. I’d probably be a terrible math teacher. Sometimes also you will see corrections on math homework. Even if you get the correct answer, the teacher will be mad because they wanted you to get the answer their way. I remember getting in trouble for this all the time, that I would do a math problem in my head and teacher would get mad that I didn’t show my work and so would grade me off for that. That was back. I mean, I was not really great at pre-calc and calculus, but I was good. And I would say I’m still fairly good at arithmetic. Just being able to remember strings of numbers and summing them, subtracting the products, divisions, things like that. I think I’m at least still decent at arithmetic. I do stuff and then not show my work and then get in trouble, do an alternate way to get things.

(06:03)
So here’s the question. Carol read 28 pages of a book on Monday and 103 pages on Tuesday. Is 75 pages a reasonable answer for how many more pages Carol read on Monday read on Tuesday than on Monday? So read 28 pages Monday, 103 pages on Tuesday. How many more pages did Carol read? It’s an interesting thing, not what’s the correct answer. What is a reasonable answer to the question? So explain. And the student wrote, “Yes, 75 is a reasonable answer because 103 minus 28 equals 75. And does a minus one, you need to estimate. You should have put 100 minus 30 equals 70 and that would be a reasonable answer.” Well, no, it’s one reasonable answer. It’s one reasonable answer, but I would say it’s actually not the most reasonable answer because they’re trying to teach these kids, “Hey, sometimes you could just have to estimate things, you won’t get the exact answer.” So what’s so funny is she puts estimate and comes up with 70 pages as if that would actually be unreasonable to get the exact amount of 75.

(07:08)
But you can get 75 by just rounding and doing an estimation. So if you rounded to the nearest zero or five, you would get Carol read 30 pages on Monday and 105 pages on Tuesday. And so if you don’t want to do complicated subtraction nearest zero or five, you would get 105 minus 30 equals 75. That should be the estimate that you use, not 100 minus 30, which is crazy. That’s an estimate that still gives you the incorrect answer when there’s an easy estimate you could use that serendipitously gives you the correct answer. All right, here’s the last one that’s funny. Here’s the problem. Can you solve it? It took Marie 10 minutes to saw a board into two pieces. If she works just as fast, how long will it take her to saw another board into three pieces? I’ll read it again for you.

(08:04)
It took Marie 10 minutes to saw a board into two pieces. All right. Marie saws a board into two pieces, takes her 10 minutes to do that. If she works just as fast, how long will it take her to saw another board into three pieces? Now what’s interesting here is that if you thought about the numbers more than the task, you probably got the incorrect answer the teacher got. If you thought about the task, you get the student. The student put 20 minutes. Now, if you’re trying to math your way through this, you would think really fast. Okay, so 10 minutes for two pieces, that means it was five minutes per piece. So for three pieces, that would take 15 minutes, but that’s not correct when it comes to the task of sawing a board because the number of pieces is not identical to the number of cuts.

(08:50)
So if the question says it took Marie 10 minutes to saw a board into two pieces, what that means is it took Marie 10 minutes to make one cut in a wooden board. So if she works just as fast, how long will it take for her to saw another board into three pieces? To saw a board into three pieces, you have to make two cuts. So if it took 10 minutes for one cut, then that means it’s going to take 20 minutes for two cuts if you’re working at the same rate. That is what the student understands, the task involved, and the teacher understands the numbers, but picked the wrong task for the question because when you saw a piece of wood, it results in two pieces. Now, if you could rework the question, say, “It took Marie 10 minutes to cut through a wooden board.

(09:39)
If she works just as fast, how long will it take for her to cut through another board?” Of course, that’s not that complicated, right? But I just love that, that it’s interesting with a lot of these questions. If you have the numbers and didn’t know what it was referring to, you get the wrong answer. You need to know other contextual elements to get the right answer. In fact, I was doing analogies with my wife the other night. So we were looking at SAT analogies, word games, things like that. And apparently, I guess the SATs removed the analogy section because they claimed it was discriminatory. I think one of the analogies had the word rigata in it. And a regatta, of course, is like a yacht or a sailing boat race, or I think it could also refer to rowing. It at least refers to that.

(10:25)
I’m not rich enough to exactly know what a rigata is, but I know it has to deal with, say, when rich people go out and row their boats or sail, it’s a regatta competition, right? But if you are from your underprivileged kid in the inner city, you’ve never heard of a regatta, then it’s like, “Oh, it’s discriminatory. You won’t be able to get the right answer.” But then I’ve seen so many other things where because of people with different backgrounds, you can’t mention anything in store. You can’t mention birthday parties. What if a kid never got to have a birthday party? I think dinosaurs weren’t even included, even though you’re supposed to learn about dinosaurs in school. So that is interesting there. So those are the math questions. You know what’s funny, by the way, I remember my mom, I had a math teacher once who was teaching me prime numbers and told me prime numbers, it starts one, two, three, five, and my mom was like, “No, one is not a prime number.” And she emailed the teacher and the teacher was like, “No, you’re wrong.” This is like an elementary school teacher, by the way.

(11:19)
It’s not a Cambridge mathematician, but teachers who feel just so confident, I’m like, “You’re a third grade teacher, you’re not goodwill hunting, okay? So I don’t think you should be that dismissive when a parent comes to you and says you got a math problem wrong,” but they’re very dismissive of my mom. Then my mom went to the principal and went to somebody else and printed out all this stuff from early 1990s or mid 1990s internet saying, “No, I have it right here. One cannot be a prime number.” Because if one were a prime number, you couldn’t do prime factorization where you break numbers down into primes would have endless solutions. It could be like three, two, one or three, two, one, one, one, because one multiplied with itself is the same amount. And I remember going through a bunch of it, but no, one is not a prime number in that regard.

(12:05)
All right. I hope that was interesting and helpful for you all, and I hope you have a very blessed weekend.