Tag Archives: statistics

Apples and Oranges

We compared apples and oranges in statistics today.

The teacher’s resource manual for the textbook comments that standard deviation lets us compare apples and oranges, because we can use it as a “ruler” to figure out how a particular data value compares with the mean. I took that idea and ran with it – to the grocery store, during my 4th period prep. I got 9 apples and a bag of clementines (there were 20 of them in the bag).

We put each piece of fruit on a paper plate with a letter on it (we had to use three Greek letters). We borrowed a balance from a science teacher. Before weighing the fruit, the class made conjectures about which apples would be biggest and smallest, and which oranges would be biggest and smallest. (They did know that our method of measurement would be using the balance.) I told them that in making those decisions, they were comparing apples with apples and oranges with oranges, but what if they wanted to determine which was relatively bigger, the biggest apple or the biggest orange? That, I said, was what they were going to figure out how to do today.

After they made their conjectures, they wrote the name of each piece of fruit on the board. (They decided the fruits needed names, so instead of boring old A, B, and C, we had Amy, Billy, and Caroline as our first three apples. I’ve studied Greek, so I was able to come up with some Greek names for those extra letters.) Then they recorded the mass of each piece of fruit.

The next part was easy: For each type of fruit, which piece was the biggest, and which the smallest? It turned out that the biggest apple was Francis (212.1 g), which had been their guess, but that was the only one they guessed correctly. Caroline was the smallest apple (144.4 g), Wilhelm the biggest orange (84.6 g), and Violet the smallest orange (52.1 g).

Then I told them to figure out the mean and standard deviation for each type of fruit. Doing this REALLY helped some of the students to better understand what standard deviation is (a measure of spread); I was able to point out that the standard deviation of apple masses (19.72) was more than twice that of orange masses (8.26), and the students were able to look at the actual fruits and see that yeah, there’s a lot more variation in size in the apples than in the oranges. They had been pretty confused by it before, so I was really glad to have that visual for them.

Once we had that concept a little more firmly understood, I said that we should look at Francis (biggest apple) and Wilhelm (biggest orange). I asked how we could figure out how many standard deviations above the mean Francis is, and they knew right away how we could do it. We did the same for Wilhelm, and we discovered that Francis was 0.981 standard deviations above, but Wilhelm was 1.746 standard deviations above. So even though Wilhelm is SMALLER than Francis when we look at the measurements themselves, Wilhelm is bigger as a big orange than Francis is as a big apple. We did the same thing with the smallest ones, and I pointed out that since we were subtracting the mean from smaller data values, we were ending up with negative answers…which just show us that we’re that many standard deviations below the mean. They got it.

And then I told them that these “how many standard deviations away” things they were coming up with are called z-scores. They were excited that “it has a cool name.” :) And I’m excited that I think they will actually remember it because of how we got there – they figured it out rather than having me throw a formula at them.

And then? We had a healthy snack to conclude the class. :) (My homeroom may or may not have juggled the leftover clementines right after the statistics class left.)

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Cool Stuff!

E was confused about marginal distribution on her quiz. She came in for a reassessment a couple of days ago…and was still confused, so her score didn’t change. I explained the concept to her again, and she seemed to get it.

We have homeroom at the end of the day. As soon as homeroom was over today, E came running into my classroom.

“Mrs. Dean!” she said. “I made a contingency table in homeroom!”

I said, “Great! Is it on the board in Mr. C’s room, then?”

“Yes – come see!” I was supposed to be going to a meeting, but I figured it could wait a minute or two, so I walked down to Mr. C’s room with her. As we walked, she continued: “Mr. C took a poll – somebody had this toy thing, and we were trying to decide whether it’s an evil fairy or an alien. So I said that we could break down the results by girls and boys. And I got the marginal distribution part and everything!”

We got to the classroom where, sure enough, she’d drawn this:

She pointed out that while there were some votes for alien, ALL of the girls voted for evil fairy. She also pointed out the marginal distribution that she’d written at the bottom. I asked her what percent of the people who voted for evil fairy were boys, and although she couldn’t calculate the percentage in her head, she knew that it was 2/12.

So I changed her score. Because she knows it, and I know she knows it. This wasn’t a scheduled reassessment that I generated for her; she saw an opportunity to use what she’d learned, and then she drew my attention to it because she knew it was a demonstration of her understanding. And that? Is awesome.

Edited because apparently writing a post quickly makes me leave verbs out of my sentences…sorry ’bout that.

Learning Targets and Brainstorming

Status on Learning Targets…

  • Calculus: done
  • Statistics: not all the way through the text, but I’ve gotten through everything we’ve done in the class I’m taking, and I can add to it later
  • Trigonometry & Analytic Geometry: done, unless I get through things and have time for an extra chapter, at which point I’ll add in those LTs
  • Geometry: just opened the book (I’ll spare you a link to an empty document ;))

If anyone feels like giving me feedback, please do so. This has been a bigger task than I anticipated!

Now for some brainstorming…I have an idea for how I want to structure my units. For each unit, I want to put one or more real problems (of the WCYDWT variety) up on the bulletin board. The idea is that I will choose these Big Problems (don’t I come up with creative names?) as things that can be found using what they will learn in the unit. We’ll keep looking at them as we go through the unit, evaluating how our new knowledge can help us reach new understandings about the Big Problems. (I totally need a better name than Big Problems.)

What I want my kids to recognize is that math can be useful. I think it will also push them in the right direction for completing each semester’s final assessment, assuming I stick with that idea.

So once I get my Learning Targets finished, I think I will tackle the first couple of units for each class and figure out…

  1. Big Problems
  2. Assessments
  3. Instruction
  4. Homework/practice sets

I think doing that will help me feel ready to start the year, though the more units I can prepare ahead of time, the better. Oh, and I also need to work on my Policies & Procedures sheet (to explain things like my crazy new grading idea).

Listening to the Inner Teacher

I’m probably not the only one with an Inner Teacher – that part of me that wants to go into Teacher Mode even when I’m not in the classroom. Sometimes my Inner Teacher just wants to tell store cashiers to spit out their gum. But sometimes she wants to make sure people learn.

My Inner Teacher won out today in my statistics class. There’s this guy who has been coming in to class with a look of defeat; he asks questions, but he keeps struggling to understand. The past couple of days I’ve heard him asking the instructor about the tutoring resources that are available on campus, asking if there are any videos that she can suggest to help explain the concepts, etc.

I had been struggling with the desire to offer him help versus the desire to avoid doing something weird/rude/othernegativeadjective. As much as he was asking the instructor for help, he wasn’t asking me, so I didn’t know how he would take it if I offered. But today after class, I caved to my Inner Teacher and asked the guy if he wants help from me. He seemed appreciative of the offer, and I’ll be working with him after class on Monday. I hope I can help him understand.

Lessons from Statistics

(Refresher background info: I am taking a statistics class at the local community college, since I will be teaching statistics in the coming school year.)

I’m finding the content in my statistics class to be interesting and logically intuitive, and I’m really looking forward to teaching it. The class itself…well, I kind of feel like a spy in there, gaining insights for teaching as I play the role of student. So here are some things that I have learned (or have had reinforced)…

  1. Don’t take over. Allow students to build the connections for themselves.
  2. If someone asks for clarification on how to do something, make sure you understand which part they’re struggling with. If they can’t articulate it for themselves, listen as they continue to ask questions while you go through the process again.
  3. If someone asks a question about application, extension, etc., do not dismiss it as “not what this class covers.” Encourage thinking! It’s even okay to say, “Let me think about it / look at some resources / etc. and get back to you,” or, “I would love to discuss that with you, but can we talk about it after class so we don’t lose our train of thought here?”
  4. Be willing to admit when you could change a question, example, etc., to make it better for the students. Value feedback.
  5. If you tire of hearing from a particular student (or of answering questions in general), don’t let it show. Not all students are concerned enough about their learning to make sure they keep asking questions anyway. Care about student learning.

There may be more to come…three more weeks of the class.

Reading too much into the question

More thoughts on my statistics class from today. We were going over some review questions, and I said that one of them didn’t have enough information for us to determine an answer. The instructor responded,

You’re reading way too much into the question.

Well, no, I really wasn’t. It was a poor question. I’m really not sure why she didn’t just acknowledge that it’s a poor question – it’s not even a pride thing, because these questions are from the textbook publisher, not from her. And even when I am the one who wrote questions, I’ve told kids who challenged them, “You know what? You’re right. I could be asking this question more precisely.” And I make a note to improve the question for the next time I come to that material.

So the first part of my point here is, it’s okay to recognize that a question/assignment/whatever is not the best and could be improved. I think my students like knowing that I will receive their critiques (delivered respectfully, of course) and consider their feedback. I know that I didn’t like feeling like I wasn’t being listened to, being told to just write down the “right answer” and move on.

But the second thing I wanted to say in this post is a question I thought of as a result of this experience. Is there a point at which it’s okay to tell a student “you’re reading too much into the question”? Obviously we want to stop kids from saying,

“Well, Johnny won’t have ANY apples left after he gives 3 to Suzie, because right at that moment Suzie gets turned into a ZOMBIE, and she doesn’t care about the apples anymore but just wants to eat Johnny’s BRAAAAAAINS, so Johnny figures life is more important than apples and he drops the apples so he can run away. Will he survive Zombie Suzie’s attack? Just wait until I turn in my next homework!”

Because that would just be silly.

But. There’s a lot of good thinking that students can do in between “just give the answer you know I’m looking for” and the zombie scenario. I’ve read several bloggers talking about WCYDWT (What Can You Do With This), which is all about posing new problems, digging deeper into what’s right in front of you instead of just relying on the (possibly/probably poorly written) questions in the textbook.

There probably are good occasions for telling a kid he’s reading too much into a problem. But I think we might jump there a little more quickly than we ought to sometimes.

Going back in time?

I’m taking a class that started today. I’ve never taken statistics before, but I’ll be teaching it (one section that will probably have both AP and General kids) in the fall. So I thought that taking it would probably be a good plan.

I’m still processing the reasons why, but it felt somewhat surreal to be in a 2000-level course again. I’m looking forward to learning statistics, definitely; I have thought for a while that I would like to take a course like this. But actually sitting in the class just feels strange. It’s very different than the classes I’ve been in over the last 3 years as a master’s student, and it actually feels much more like I’m back in high school (except the part where the homework is all online).

I’ll have to continue mulling this over. It feels like there are things I can learn through this about becoming a better teacher, but I’m not completely sure what those things are yet. Maybe it will become clearer to me as the next few weeks go by.