Aug 012011

I can’t figure out how to do a complete and thorough archive without running software on a computer. From my phone, however, I was able to activate my account and tell it to archive the hashtag #nctmrsm11. (click the link for some interesting visuals from the archive)

Then, when I got home, I searched for the hashtag on twitter, and scanned the two lists to see if there were any differences, copied and pasted the missing ones into the list. Have I probably missed some? Yes. It is hard to get an archive out of twitter. They don’t make it easy, by any means.

Anyway, this is probably the best I can do at the moment. I still have at least 2 articles to write from the time in Orlando.

Earliest to latest order: Monday, 1 Aug on top, 28 July at bottom, after the break.

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Aug 012011

Viva Hathaway was the third session on Saturday, and I looked forward to her presentation with great gusto. I attended a short session she gave at the NCTM Conference in Salt Lake City in 2009, and walked away from her session with a brain exploding with ideas.

In fact, on her site, she still has a link to all the resources from that event. Just go down to “NCTM09” and use the most obvious password you can think of that rhymes with “nctm09”. In fact, use it twice as username too!

First off, let me tell you that if you have not had the opportunity to spend an hour or more in a room with Mrs. Hathaway, you are missing out. She has more energy, enthusiasm, and engagement potential that myself times 10. Seriously, she is a human dynamo that never stops, and as the self proclaimed “Stats Goddess” in her room, I have no doubt that she is amazing.

Second off, I would like to share just a few basic things she does that I know last time I missed because I was so new, but this time I caught and need to incorporate into my classroom style.

  • She deals with a lot of food. Learners like to eat food. If learners eat food before the lesson is over, the lesson IS over.
  • In order to fight this, she declaims, loudly, that the Stats Goddess has NOT blessed this food yet, and therefor it is TOXIC until blessed. If you eat it, you will become very sick.
  • She is NOT kidding. One of the teachers at some M&M’s before she said it was okay, and she was right there, immediately saying, “Are you okay? I will call the nurse and an ambulance right now, okay. You will be getting very sick soon. Oh, my. Look at you. You are not feeling are, are you?” etc.
  • It worked. Not a single other soul in the room touched an M&M until she said so. Once she was done with the particular lesson, she said, “And now, the Stats Goddess blesses the M&M’s so they are now safe to eat.”
  • All the lessons Mrs. Hathaway does she spirals. EVERY TIME, EVERY LESSON! This is so important. The M&M lesson she says she uses with probability, and then again, the same data is used with Chi Square testing. The Teddy Grahams lesson is used to see if there is a measure bias, and the same data is used to discuss inference testing.
  • She does this, so that instead of saying “Two sample t testing” she can say, “Remember when we did the mood survey?” That is a very powerful idea, that connects something real world to the abstract. The learners understand and connect with her and the material. Total win.
  • She NEVER deletes data. If she enters data on day 1, she can load it at the end of the year to look at again. This is something I need to learn.

We spent about 1/2 the time in the session doing 4 different activities. I will break all 4 down, but we did not cover all of them well. The reason for that I will go into later.

Personal survey of mood: everyone in the room received one of two identical looking cards, and we were told very strictly to not show or discuss the cards, just do what it says.

  • Group 1: For the next 2 minutes write down everything that has been good, exciting or made you happy over the last two weeks. Include ALL the good and fun things no matter how small. 
  • Group 2: For the next 2 minutes write down everything that has frustrated you or upset you over the past two weeks. Include ALL the bad and frustrating things no matter how small.
  • Next, she asked us to think about the list, and then rank ourselves from 1 to 10. We then graphed the results from Group 1 and Group 2. Guess what, we had a bimodal graph. Shape Center and Spread were discussed. Later in the year, this same data can be loaded at the end of the year for a two sample t test. It is always significant.

Wet measure vs. Dry Measure measuring cups using Teddy Grahams. Did you know there are two different kinds of measuring cups? I did, but I didn’t realize why.

  • Have Teddy Grahams available. Have a cup, approximately 8 oz, and a large clear cup, that is marked with 8 oz mark (using water to measure 1 earlier). Mrs. Hathaway actually said she has enough actual measuring cups to do this in her room. She has collected them over the years.
  • The learners measure out teddy grahams in the dry measure to the very top. The learners measure teddy grahams to the top of the wet measure mark. Both are counted. Repeat.
  • Is there a difference? Of course. Here we have an example of a measurement bias, as well as some inference testing later in the year.

Confidence intervals with M&M’s: Goal is to have 100 samples taken from a known population and compare the confidence interval with the truth.

  • First off, a yellow or red M&M is a success. Have the learners scoop out a cupful of M&M’s that is about 100 to 125 candies. Then the learners count how many red & yellows they have and the teacher circulates and gets from every learner the total success and the total number of candies.
  • The teacher now has, with a bit of calculation a census of the population of red & yellows and total candies.
  • Now the learners put their candies back in their cup and shake. The learners then sample with n = 20 enough times to get 100 samples.
  • Graph with post it notes or something else on the board.
  • Calculate CI of the 100 samples. Compare with true proportion. Does the CI include the true mean?

Chi Square with M&M’s we did not have time to go into, but is on her site.

Two last things.

  1. She made a point to show how her lessons fit into the EXCEL framework. Engage, eXplore, Communicate, Empower and Launch. She said quickly that this is a requirement from her principle, but it fits with the idea of getting learners engaged, talking, and launching them to more critical thinking skills.
  2. We got the opportunity to play with the new Casio Prizm calculators. This was fun, and I found out that Casio re-wrote some of the statistics routines in the calculator to make it easier. I was really unhappy with the stats menu on the old Casio calculators. They were hard to use and didn’t do all the AP Stats things I needed, or at least it was so difficult that I couldn’t figure out how to do it. The down side is we spent 1/2 our time on this. Some teachers were struggling and asking questions, other teachers picked them up and were delving into the guts (me and some others), but we really did spend about 30 minutes on this.

I know Mrs. Hathaway wanted to give us the opportunity to spend some time with new tech, but I really wanted to see what she was doing in the classroom more. Much more.

Still, it was a successful session, and I thank Mrs. Hathaway for giving us her time. I will go again the next time she offers as well.

Aug 012011

I have to say I almost left Mr. Godbold’s presentation. It started off on a odd note, where he told us what he wanted to go over, then said we couldn’t do all of it, but we would be lucky if we did half, and then repeated himself for about 10 minutes. He looked and came across as very nervous.

But then he got rolling, and he had some gems that were worth sticking around for.

One of the first things he said when he was interacting with the AP Stat teachers in the room was that he forbids the use of pronouns in his room. A teacher responded to a question he asked with something along the lines of, “It is ….” He asked what “It” was and waited for a response, and then we could move forward.

That moment was actually very powerful for me. Pronouns really are hard to use in conversation without perfect understanding of what is being talked about. In writing it can be just as dangerous. Consider the fact that in this posting, I used two pronouns as shortcuts in the first paragraph alone, when I KNEW I was going to be talking about pronoun usage and how bad it is to use them (and I just used another!)

Our learners in AP Stats use pronouns all the time, and they know exactly what they are meaning when they write / say them. Other people don’t, however. The readers certainly may not. So one little step we, as teachers / facilitators must do is sharpen their usage of language to make them better communicators. Banning pronouns may be unwieldy at first, but useful later!

Second, Mr. Godbold had a fantastic way of demonstrating a very tough idea, what happens when someone says “given that”. I know my learners really struggled with the idea that given that removes things from being considered. Here is the entire process Mr. Godbold used to demonstrate the process.

First thing, collect some data. It really can be anything, but he used “principle residence in FL” and “traveled to conference without flying”, among others. Those two are the important data points we need for this explanation. We had an n = 21 in the room (excluding Mr. Godbold) and 2 pp lived in FL, 7 pp drove.

Second, Mr. Godbold explained that he has a number line on his floor that goes from 0 to 1. Any stat teacher immediately grasps the use of that particular arrangement. Next he had all the people in the room line up on a wall, and the first person was on Zero, the last person was on One.

He asked who were the 2 people from Florida, and had them move to the Zero and just off Zero spots. Now, we can see that the P(FL) = 2/27. All he did was rearrange the people a small bit, but it did make the probability clear.

Next, he asked, “What is the probability of living in Florida GIVEN THAT you drove.” Now he asked us, the teachers, this question, and he said he would not do that with a class that was not completely familiar with the idea. Even with teachers of stats, it caused some confusion. A couple of people sat down immediately. Other people were unsure what to do, but we eventually all agreed that the flyers needed to sit down, and the remaining 7 people needed to spread out!

Now we had 7 people from 0 to 1, instead of 21 pp, and they took up the same amount of space, and now we just had to count to see how many Florida residents were among the 7! (Both were among the 7, so the P = 2/7)

Bingo! The “Given That” removes people from consideration for the second calculation. Mr. Godbold then showed a contingency table on the screen, and it is clear how this exercise applies to contingency tables early in the year, as well as probability later in the year.

For no other reason that this one exercise I am glad I stayed in Mr. Godbolds presentation. My idea is to use this with the beginning of year questionnaire I do with the class. It gives me a reason to have them pull out the questions later in the year, something I have been struggling with.

Mr. Godbold also discussed independence visually, which is important since it was question number 2 on the 2011 AP Exam.  He also had a great handout for the presentation as well that dealt with probability and independence, and getting the learners to visually understand what independence looks like. It is nice, and I will probably use it in class. I will link to an electronic copy of it, once the NCTM gets it uploaded to the website this week.

Thank you Mr. Godbold!

Resources: Anytime I go to an event like this, I always google the person’s name whose session I am in. When I did that for Mr. Godbold, some great things came up.

He presented at the AAAPSTA, and has some great resources listed there as well, found in the 25 Sep 2010 section.

Aug 012011

Timothy Kanold was the Superintendent of Adlai Stevenson High School District in Illinois. This school is a huge success story, because it went from bottom of the bottom to top of the top. They did this through a strong commitment to PLC’s and communication. Now Mr. Kanold did not spend a second on his Keynote today telling us this. He didn’t have to. Anyone in education knows what Adelai Stevenson did, and it had nothing to do with testing kids to death or any other of the Bill Gates / Michelle Rhee type “reforms.”

Let’s walk through my notes from Mr. Kanold’s presentation today. He was a dynamic and amazing presenter, who knows how to hold the attention of the room.

He began with a story of his 15 year old daughter who has a best friend in a different math class. The friend had a teacher who had rich content and reasoning skills as an expectation, and his daughter did not. This, of course, caused a “minor” conflict and caused him to become involved.

Mr. Kanold was aghast to discover that the school his daughter attended (he didn’t tell us which school that was) did not have certain departmental norms.  Mr. Kanold asked one very simple question, “How is it possible that in 1 department two teachers could have such different expectations that one teacher uses Reasoning and Sense Making and the other teacher does not.” The answer there, as everywhere, is the same, “Because I can.”

And then Mr. Kanold asked the most important question of the morning, “How is that acceptable?”

The answer, of course, must be that it is not acceptable.

Mr. Kanold then challenged everyone in the room to write down 1 element of their TPOV, their Teachable Point of View. A TPOV is something he has taken from Noel Tichy, and is defined as, “A cohesive set of ideas and concepts that a person is able to clearly articulate to others.”  The NCTM has a TPOV, it is found on page 14 of the Reasoning and Sense Making book. But do I have a TPOV? Does my Department? All good questions. (I will write about my TPOV later.)

The next challenge asked by Mr. Kanold was, “How will you close the gap between the vision and the reality of adult action.”

Wow, that is tough. Of course, we have to have some standard to evaluate that gap, which leads us into some methods of distinguishing bad or irrelevant (my words) evidence from actual evidence. There are 5 kinds or levels of certainty in our actions.

  1. Opinion: This is where a teachers says, “This is what I believe.” “I believe this sincerely.” It is an opinion the teacher holds about the ability or capabilities of their learners, either collectively or individually.
  2. Experience: Here the the teacher says, “This is what I have seen based on my personal experience.” My personal favorite example of this is where a teachers says, “Well, based on what I have seen, we will have to agree to disagree.”
  3. Local Evidence: Here the teacher says, “This is what I have seen based on the experiences of my friends and colleagues.”
  4. Preponderance of Evidence: now we are up to evidence based on what we know as a profession.
  5. Mathematical Certainty: finally, at this level, we are basing decisions on evidence that is so certain there is no need for debate.

If we ever come across evidence that is at the level 5, then we need to not even discuss it, we just need to do it immediately! It is so absolutely amazing, and so rare, that not doing it breaks all rules of rationality. Decisions based on Mathematical Certainty are absolutely non-negotiable.

So is there anything that can reach that level of certainty? Mr. Kanald’s answer is Yes.

That thing is the fact that Professional Development that is offered over the course of 6 – 12 months and spans 30 – 100 hours, that deals with non-negotiable behaviors and expectations of success, where the PD is connected to results in the classroom (strong, demonstrable results), and designed to be an ongoing contextual subject matter requirement has an effect size of .73.

Let’s compare that to the well established effect size of poverty in the classroom. The poverty effect size is .57. The effect size of a fully functioning PLC, based on results in the classroom and best practices is .73. PLC’s have a bigger impact on learner success than poverty!

If we don’t act on that, we are crazy. We must act, and act NOW. That is decision making at the level of Mathematical Certainty.

Mr. Kanold really motivated and sharpened my thinking on my own PLC time. He gave me a copy (actually he gave 75 pp) of his book, The Five Disciplines of PLC Leaders, which has free reproducibles here.

He blogs at and his twitter is @tkanold. To say I will be following and reading his material in the future will be to say the sun will rise tomorrow.

Some other books he mentioned during his presentation that are worth following up on / reading:

Embedded Formative Assessment by Dylan Wiliam

The Fifth Discipline by Peter Senge. Mr. Kanold said this book was very instrumental in shaping some of his ideas on PLC’s.

Jul 292011

Today had some more very good things happen at the Institute.

I think the first thing is the definition of an “Institute” versus a “Conference”. That distinction was made yesterday, and then briefly discussed again today in a session.

Now these are not formal definitions, found in a dictionary. These are rough and dirty definitions formed through discussion. I attended a NCTM “Conference” in Salt Lake City a couple of years ago. It was huge. Hundreds of vendors, probably a hundred presentations over the course of 4 days, with a schedule that was packed to the gills. Honestly, that “Conference” was a huge waste of dollars.

Oh, yes, I took home some resources. One or two presenters had resources on CD, and I think I even got one website out of the deal. I sat through 10 presentations and listened.

Yup, it was EXACTLY like the worst math class you ever took. Sit down, listen, take notes.

Compare that with the “Institute”. WE are involved, engaged, asking questions, and sharing with each other, both audience to presenter (lots of question opportunities for audience to ask questions) and among audience to audience. We are given working time to share and collaborate built into the schedule, and the presentations are stranded. The stranding means you don’t have 4 stats presentations at the same time so you have to miss out on something. The Institute is designed to have good workflow, with sessions in the middle and end to have collaborative work.


So, let’s start with some of the great recommendations I have taken from today. First off, some books that came very strongly recommended by William McCallum.

1. Mathematical Discovery by George Poyla and is out of print and very expensive to buy on the used market. And by very expensive, if you click the link you will see Amazon has copies, for $165!!!!! Go check it out of the library and photocopy it. It will only cost you $20 and 30 minutes of time. That is highway robbery.

2. The Stanford Mathematics Problem Book, published by Dover Press. $6.95. This book was recommended repeatedly, and is available on Amazon and other fine bookstores. It is nice because it has problems, hints and solutions, and the problems are good problems for high school reasoning and sense making activities.

3. Unpublished Ph.D. dissertation by Sarah Donaldson, 2011, Teaching through problem solving: Practices of four high school mathematics teachers. Not sure where to get this as a high school teacher. College students and teachers have access to everything, but high school teachers get screwed on these types of important works. In this dissertation 5 things were identified as being best practices:

  1. Teach problem solving strategies
  2. Model problem solving
  3. Limit teacher input (hmm, recall yesterday’s post where Dan Meyer said the teacher should stay out of the way)
  4. Promoting meta-cognition by the learners
  5. Highlight multiple solutions by the learners

Some websites I found out about today that will definitely benefit my teaching. There is no particular order to these sites. I just went down my notebook and put them in the order I wrote them down.

  • – very bare. Like mother hubbard’s cupboard bare. I am hoping for more content here soon.
  • William McCollum, who was one of the major authors of the CCSS, has a blog where he has documented and will continue to document developments and resources for implementing the CCSS.
  • Park City Math Institute – Honestly not sure what is on this site yet. It was thrown out by another teacher I overheard, “Hey that can be found on PCMI!” I wrote down pcmi, and then had to do some searching. Honestly, it looks like a promising resource.
  • – a website to help with the CCSS rollout and help for teachers to make the adjustment. Looks like a well made site. I have not examined content yet.
  • – so far it looks like a page selling the NCTM books. There is some additional content, but it is lean on the content and large on the selling. It does not really fit with the approach and styling of the “Institute”. It seems far more “Conference” focused instead. Of course, it is a work in progress. I hope.
  • – This is the official place for the handouts, not the /hsforum. It was another miscommunication with the NCTM that some of the employees of the NCTM handed out the /hsforum on Friday mornings session. Unfortunately, it was not corrected until Saturday’s closing session.
  • – not much going on yet, but the NCTM needs to do some major push to get people to know it is there, populate it with posts from the presenters, and start using it. The biggest need is the presenters without websites need to post their take on their presentation here.
  • – a very nice collection of applets for statistics that may replace the scattered applets that I have been using and suggesting.
  • – Dan Meyer’s site where he is posting all of the materials from his presentations. This is what I would expect of a professional teaching organization. And it is only a one man show!

Ideas that came up that I will want to follow-up on later during the summer and school year.

  • First off, in the keynote today with Gary Martin and Eric Robinson, Eric pointed out that questions like x2 – 7x + 12 have no reasoning required, just rote recall. But what if we replaced that factoring question with this one: x2 – ?x + 12.  What numbers could go in the question mark? What integers? What fractions? Why? Now the problem becomes interesting and has multiple points of entry and multiple answers. Just a simple tweak, but enormous payoff. (Thank you to Eric Robinson for this idea.)
  • Notice that this question is not contextual like Dan Meyer’s questions. It is not always about contextual problems, but sometimes it is just about INTERESTING problems. After all, they publish sodukus in the newspaper not because just math teachers do them, but they are interesting and challenging. (Thank you to Gary Martin for this idea.)

Okay, that is enough for now.

My only complaint, and for me, it is a big one. Thanks be unto the NCTM for choosing a hotel that charges $14.95 per day for internet access to teachers! Like many of you, I had my pay cut this year, and I paid my own way cross country to attend this conference (so far total cost is estimated to be $1300, not counting all the meals.) This lack of internet is pretty much my major annoyance with this beautiful hotel. The Comfort Inn I stayed at in Lewiston MT offers free internet for $80 per night, but the Renaissance at SeaWorld can’t for $149.00, except in the lobby. Really? I call call (cough) Bullhouy.

Jul 292011

That really is the question posed by Dan Meyer in his opening keynote speech at the NCTM Reasoning and Sense Making Conference Institute. Honestly, I had never heard of Clever Hans, and unless you are a fan of esoteric German animal trivia, you might not have heard of him either. Clever Hans was a horse that had amazing powers of reading the people around him. The horse could do any math the audience could do.

I won’t bore anyone with the story, especially since Lisa Henry did such a great job explaining it on her blog, and Wikipedia and other sites have thoroughly explained it as well. The story is extremely relevant to math education today, and more importantly, important to the practice of math educators today.

As I was listening to Dan, I recalled a comment a learner made to me. I had just asked a learner, “Why” when she gave me an answer to something. The learner tried to explain, and couldn’t. Another learner came to her defense, and together they figured out she was correct, and they explained why. At that point, Learner 1 turned to me with a very angry look on her face (all made up, she is a great actress) and said, “Mr. Waddell. You tricked me. You are only supposed to ask why when we give wrong answers.” The class then went on to tell me that I have a great poker face, and that I am supposed to let them know when they get something right by smiling or something.

I had just accidently stumbled upon the Clever Hans Effect, and didn’t know it. I just wanted to know if the learner understood the problem.

Imagine that. The learners in the class understood the issue better than I, the teacher did. They were instructing me when it was okay to ask them if they were sure, and how to non-verbally communicate with them when they are supposed to stop clomping their hoof guessing the correct answer. Thankfully I ignored them.

I didn’t have a name to describe what occurred that day, but I do now, and Clever Hans will now and forever be in my first day of class discussion.

But of course, the keynote was far more than just a talk about a horse. It was about how to take a boring, dry, and thoroughly massacred problem, and turn it into something that might actually have some interest to learners.

Dan breaks his problem solving process into 4 steps.

1. Visualize

2. Abstract

3. Decompose

4. Verbalize

Of these, I personally find the Verbalizing of the problem the most important. Verbalizing a problem is asking the truly relevant and interesting question in just a few words. Taking a long and boring question from a text book (how about the “which cell phone service should I get that is now de rigour for textbook companies) and turning it on its head.

The new question should be something along the lines of:

1. What cell phone plan would you buy?

2. [insert some image of local cell phone advertisements here] Make the insert local, relevant, and complete.

3. And this is probably the most important part as a teacher. WALK AWAY for a bit and let the learners abstract the question. You see, the textbooks do that for the learners. They break the problem down into nice, clean, manageable chunks that are easy to digest. We need to let the learners look at the actual ads, the actual, messy, ugly details of the ad and decide for themselves what is important.

4. After that, then they will decompose the question and figure out an answer and then justify the answer.

Guess what, some may choose a plan I would not. The real question at that point is, “Why?” Did the learners overlook something, or did they just think the 2 gig plan fit their needs better than the 5 gig plan?

Why do I need to always decide for them?

And that is a great way to run a rich question with Reasoning & Sense Making. I will have more to say on this later. Right now, I am in Orlando, and some fun beckons.

After all, it is not always about the math. (yea, right!)