## Help Me Get Away With This

Mean, median, and mode are each important, each easily mistaken for the others, and, depending on context, misleading or completely meaningless.

I came across a scenario yesterday which highlighted their differences and limitations a little too perfectly:

1. Consider every human being in the world.
2. List each person’s total testicles.
3. What is the mean, median, and mode of that list, and what do they mean?

Is this even worth the trouble? If you aren’t trying to shock or pander to or titillate your students, are testicles fair game? ‘Cause the implications and extensions are really awesome here.

Like: the average human being has one testicle.

Or: the median number of testicles is either zero or two depending on gender majority.

And: the median and the mode will be the same number except under a few (also awesome) conditions.

How do I avoid panicking my principal here?

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Mark Barnes

March 30, 2008 - 7:46 am -

Dan, I say getting away with it is simple. When I first started teaching eons ago, a veteran enlightened me with this amazing declaration:

“It’s always better to apologize later than to ask permission, now,” he said.

I’ve done a lot of great things in the classroom, living by this axiom.

Keep us posted on the testicle lesson.

2. #### Joe

March 30, 2008 - 7:48 am -

Huh. How progressive is your school? Your line of inquiry is legitimate from a biological perspective, but good luck dropping that example in class. Personally (and I say this as a science teacher), I’d stay away from it, especially in a math classroom.

What about using ovaries? Less sensational?

3. #### Mr. K.

March 30, 2008 - 8:31 am -

Not every man has two testicles. it’s quite possible the median may be 1. (If you do this, it might be a good extension to figure out how many there would heave to be in order for that to happen, though.)

A quick search of the innertubes did not provide any useful data.

I’d stay away from it, but my kids are 13, and would spend more time giggling than thinking about the math.

4. #### Dina

March 30, 2008 - 9:00 am -

SNORT.

Marzano says loop through your content at least twice, dude. So it’s even worse than you think. By virtue of gender equality alone, you’re going to have add an example using breasts.

Perhaps “eyes” would be equally fun, if not as shocking. Have them *draw* their answers, and then flash a picture of Cyclops up on the ol’ Keynote for comparison.

5. #### Rich

March 30, 2008 - 11:01 am -

Sorry, I have middle schoolers so I pretty much have to leave this one with you. My 6th graders found a cute YouTube video about mean, median, and mode – but then again, your high schoolers would probably turn up their noses at such a cartoon.

However, let me tell you that I’m thrilled when I have students searching for math content on YouTube with absolutely ZERO direction from me!

6. #### Benjamin Baxter

March 30, 2008 - 11:14 am -

I’m with Dina, here, except that part about breasts. You don’t have ’em — as far as I know — so you’re probably better off not talkin’ about ’em.

7. #### Rick

March 30, 2008 - 11:15 am -

Uhhh…as much as I admire your thoughts on here, Dan, I have to question the wisdom of this one. Let us know how it goes, though.

8. #### Joe

March 30, 2008 - 12:30 pm -

I was thinking about the following bilateral human parts that might work: eyes, ears, breasts (yes, men have them too so technically that doesn’t work), arms, legs, thumbs, kidneys, lungs, etc. None of those work though, as both genders have them. You need something gender-specific. What about chromosomes? XX (female), XY (male), XXY (Kleinfelter’s), XO (Turner’s), XYY (Supermale). Would that work? High schoolers would know that stuff by then, plus you could have the bonus conversation of gender/sex determination within our society. Whoa. The med school girlfriend and I are really nerding out here….

9. #### dan

March 30, 2008 - 2:08 pm -

See this is exactly the issue. Outta all the gender-specific bilateral body parts, testicles work like none other.

Guys have breasts. Ovaries require explanation (which goes way outside my content area). Do high school freshmen know chromosomes? I mean like they know testicles?

Maybe I’ve just gotta try this. Next year. At my new school. During my first evaluated lesson.

@Mr. K, the median would also be one if there were exactly the same number of men & women in the world with no genetic abnormalities and no orchidectomies.

10. #### JackieB

March 30, 2008 - 3:17 pm -

Our freshmen take biology. I know they do a unit on genetics and hence are familiar with chromosomes. It might be a nice way to do some cross-curricular integration. One could also tie in probability with the Punnit Squares.

As for the “Maybe I’ve just gotta try this. Next year. At my new school. During my first evaluated lesson.” Yeah… let us know how that goes.

11. #### Dina

March 30, 2008 - 4:45 pm -

Here’s a thought. What does your kids’ health class look like at your grade level? Are they going to come across human anatomy or sexual reproduction this year? (Thinking of ways to CYA with a cross-curricula defense…)

12. #### Dee

March 30, 2008 - 6:44 pm -

(collapsed on the floor, holding my sides, and laughing so hard I’m cryin!)

Can I be there? Seriously this is the best post and comments I have read in a while. If you have a “bookworthy” file – put this in it.

13. #### sandy

March 30, 2008 - 7:08 pm -

How about using cows and their udders, or some such thing from the animal kingdom that is still fun and gimmicky (in a good way) but not so touchy? My sensitive and brilliant son (who also reads this blog, BTW, and is thinking of becoming a math teacher) would have been embarrassed TO DEATH to have had to sit through that discussion in early high school. Aside from any objections the administration might have, why make adolescents psychologically uncomfortable when you don’t have to?

14. #### dan

March 30, 2008 - 9:53 pm -

@Dee: I had a bookworthy file at one point but inside of a few weeks I was copying in basically all my posts, most of my e-mails, tweets, text messages. It got out of control. I’ll let you know when that future chart-topper snags a publisher.

@Sandy: Okay. I definitely don’t want to make adolescents psychologically uncomfortable. There is a very good probability that I would’ve been psychologically uncomfortable back when. But I have to believe that the class whose teacher approaches the material dry, factually, like it’s no big deal, like he isn’t trying to scandalize anybody, might just dodge that heavy-caliber bullet.

Plus, I dunno how comfortable you are with the math, but udders won’t work. Cow testicles would but, I mean, c’mon, right? I’m hurting for alternatives, basically.

15. #### John Larkin

March 31, 2008 - 12:55 am -

Dan, what about a different beast? The moose. Correct me if I am wrong but I think only the male of the species have antlers. Took me a while to nut that one out.
Cheers,
John

16. #### Dina

March 31, 2008 - 6:08 am -

“But I have to believe that the class whose teacher approaches the material dry, factually, like it’s no big deal, like he isn’t trying to scandalize anybody, might just dodge that heavy-caliber bullet.”

That’s what I meant about investigating health-curriculum connections, smartass. (Smartietesticles?)

I get a lot of mileage out of using other people’s handouts. Perhaps you can capitalize on the fact that the word might be one on a list of 25 to which they’ve already had to write out definitions and study for a matching quiz. Nothing like rote teaching to defuse the explosive nature of language…

17. #### Tony

March 31, 2008 - 7:49 am -

I have to agree that the cross curricular argument just might be your best defense. Either way, you have to be really careful, because this one could get your picture in the paper real quick. Testicles definitely have shock value, but there has to be something else out there with instructional value equal to, if not exceeding, testicles. I’m not sure exactly what that is, but is has to be out there somewhere.

18. #### Ben

April 1, 2008 - 6:11 am -

I think I’d go with the testicle example for several reasons:
(1) at my school 9th graders take Health, which includes information on testicles, so it’s probably a curricular connection,
(2) It’s effective. To be honest, I couldn’t’ve adequately described the difference for mean, median, and mode off the top of my head until this post,
(3) testicles are attention grabbing. You’re keeping it scientific (i.e. you’re not using “balls”) and discussing the mathematic principles of testicles is bound to catch their attention.

If I were in your shoes, I’d probably use it. Of all the comments, I think Sandy’s worry about creating psychological stress in young adolescents is the only real worry. You know your students. If you think they’re up to it, go for it. Don’t let the “man” tell you what words can’t be used in class. :-)

19. #### Tony

April 1, 2008 - 7:51 am -

Sorry, Ben, but I have to disagree. Attention grabbing? Sure. Will it leave an impression? You bet. But you have to weigh how much more effective using “testicles” are vs. the potential of getting yourself in a wee bit of trouble. Depending on your administration and your students and how “uppity” their parents can get, it could be a bad decision.

20. #### Ms. Libb

July 15, 2008 - 1:47 pm -

This is way late but…

What about putting up a series of animals and counting their legs (mammals w/4, birds w/2, fish w/0)?

You could arrange it so that the mean would be 1 and the median 3, or something like that, and then only the mode would be useful.

I would LOVE to do testicles w/my 9th graders, but I don’t think I should in my first year at a school :-/ I’ll wait to build up some goodwill first :-)