Return of the Triplets: Math and Maths - Q&A

Question

Dear Mitch,

I came across one of your website's "questions and answers" that pertained to a particular person who is pregnant with triplets and asked you about the odds of certain gender combinations. The answer you gave was incorrect.
Your answer to the odds of having 2 boys and 1 girl in fraternal triplets is wrong. You said 1 in 4. I say 3 in 8.

Assuming each child has a 50% chance of being either a boy or a girl, here are the possibilities listed in order of child A,B,C. (Don't worry about birth order- it doesn't matter and only confuses people)

BBB ( A: B ; B: B ; C:B )
BBG ( A: B ; B: B ; C:G )
BGB
GBB
BGG
GBG
GGB
GGG

8 possibilities
3 of which are 2 Boys and 1 Girl.

Thank you,

Erik Wendell

Answer

Dear Mr. Wendell,

Thank you for your interest and your input!

This brings to light a perfect example of a topic that comes up a lot in educational discussions: The difference between something that is the best answer "mathematically" and what may be the best answer "educationally".

The answer you came up with was, naturally, the first answer we came up with (I usually do them myself first, then have an intern or student do them, and then we compare our answers); but this one is akin to the discussion of basic shapes (regular polygons) that you might see on a quality educational program, such as Sesamee Street, etc., where they describe a square and a rectangle as separate shapes. Before I went into education, this used to bother me, wondering why they couldn't find a way to describe squares as "special rectangles" i.e a rectangle with a certain property -- four congruent sides. But then I realized I'm not the intended audience, and it's better to give one kind of understanding at one age, and a different kind at a different age. Just as the 'Right/Wrong' distinction that young people seem to crave -- almost need -- becomes blurred in college ethics courses. Studies show that there is only a certain amount of disequilibrium a given age/stage-of-development can handle before abandoning the hope of ever obtaining any grasp at all on the important concepts. And then, all too frequently, comes the second stage of frustration, which is when they decide that the subject (here, math) is 'not for them'.

The audience in this 'triplet' case (i.e. the ones asking the question here) were quite young, and so we chose to go back and interpret their question from their eyes, which to us meant: How many different ways would you find triplets? So they'd see how many different ways they could have 'been'. A similar question arises when young people ask how many different blood types there are, which is usually asked after they are told theirs -- but -- to complicate things by trying to explain that some blood types are rare, making the odds of getting one out of the four or five or however many there are, much more complicated, and likely to be too complicated for a lower level audience. Such an answer won't help them in any way.

Believe me, when making the difficult decision to switching my focus from specializing in math to specializing in math education, a lot of conflicted situations arose, and for a while I was sure that no one was being served by information that was simplified to the point of becoming (to an adult) incorrect.

I sometimes wonder if this conflict is at the route of why so many gifted mathematicians turn out to be ineffective math teachers, and, conversely, why so many teachers who would not consider themselves particularly mathematical, turn out to be excellent teachers of math. One of the key ingredients seems to be matching the teacher to the right developmental level of student.

The same is true, incidentally, in the areas of grammar and, believe it or not, history, when trying to make sure all students 'get' everything correctly certain types of material turn out to be too confusing for many students to ever recall everything correctly.

Like teaching and learning math, it's a process -- and sometimes a long process. Interestingly, though, as I am sure you can imagine, it's the mathematicians of the world who have trouble with this philosophical approach and, believe me, before a few years of in-class teaching, I also could not agree with the wisdom of 'teaching' something that is, mathematically, 'wrong'. BUT TRY TEACHING PRESCHOOLERS, FOR EXAMPLE, THE SQUARE/RECTANGLE DISTINCTION, AND THOUGH YOU AND I MIGHT HAVE GOTTEN IT, YOU WOULD QUICKLY SEE THAT IT'S A QUESTIONABLE PRACTICE.

BUT I WANT TO BE CLEAR: WHEN I READ YOUR EMAIL, I SMILED, BECAUSE I HAD EXPECTED MORE EMAILS EXPRESSING WHAT YOU EXPRESSED. However, since there were very few, either we hit the audience the way they needed, or -- I shudder to think -- a lot of the people who read what we post don't pause to think each step through to full understanding themselves.

ONCE AGAIN, to be clear, your approach and answer are correct, and I appreciated reading it!

- Mitch

AND NOW, our READER'S RESPONSE to MY RESPONSE:

Dear Mitch,

Thank you for the prompt response. I came across your page and couldn't help but correct the "mistake." I totally understand the difference you explained. One common one I remember very clearly was the example of electrons circling protons/neutrons like the planets around the sun. (which couldn't be more untrue, but does, very much, help illustrate a a basic concept.

I'll probably read more of your site, but will be sure not to send anymore corrections if I find them.

Erik Wendell

AND, FINALLY, MY RESPONSE TO MR. WENDELL'S RESPONSE:

Mr. Wendell,

Oh, please don't misunderstand --

I truly appreciate any potential corrections, because:

1) there's always a chance (a REAL chance) that I/we missed something and are completely wrong in every way; in fact, usually, when 2 people who work side-by-side make mistakes, they often make very similar mistakes, so the fact that a student and I agree is hardly proof of much (especially when you take into account the likelihood that students tend to be eager to get into their teacher's good graces, either for future letters of recommendation, or simply because it's a human thing.)

And

2) ANY input at all is appreciated, because it makes me take another look at what I'm doing, the goals, etc. In this case, it was such a close call that I still don't feel great about the answer we posted and was serious when I suggested that if you're comfortable with it, I would like, in the near future, to post this back-and-forth. I could either do so with your name, without your name, or change the discussion into such a generic thing that you wouldn't have to worry about anyone recognizing you. Also, if I were to do it with your name, I would run it by you first, to make sure you're comfortable with it -- if nothing else, I think you'd find the intro, etc. flattering.

As it happens, somewhere in the archive, I think around last year's Hanukkah candle question, a fifth/sixth grade class did indeed catch a series of substantial errors in my calculations, which I had stupidly done in my head while walking my million pound dog, and if you search for it, you can not only see my public thank you, public apology, public weak excuse, and the fact that I followed up by sending the class a gift!

I am certainly NOT above welcoming input, and, at my age (?) am always shocked that I don't come across more such letters.

As for my grammar, I'm often afraid to look back over past q&a's, because when I do I get queasy with embarrassment. This particular website is, after all, supposed to be not only fun and intriguing, but educational as well!

Happy Holidays,

Mitch