Success

My last post was about an idea to use old scantrons as a visual aid to build knowledge of the binomial probability formula before the learners actually were introduced to the formula.

Short post: it worked, I think.

Long post: I passed out the scantrons, which immediately brought forth a groan. We just had the final exam last week, and I was already giving them a quiz! Once we got over that part of it, I asked a question.

I was giving them a 1 question quiz. What is the probability they would get it right? All they had is a scantron, no other papers or anything else, so they asked if they were just supposed to guess at the answer. My response was “yes” and quickly they had in hand that the probability was 1/4. It always surprises me how long it takes to get to that point with some learners though.  It is not as quick as I would think. But we all got there within a minute or two, which is faster than normal.

Next, I told them they were taking a 5 question quiz. And I asked the question, “what is the probability you will get a passing grade on the quiz?”

Now the frustration started. They wanted initially to just say (1/4)(1/4)(1/4) = 1/64.  Not so. I killed that pretty quick. But before I did, I wrote it in exponent notation, so they would be comfortable with the idea of exponents having a meaning in the problem. It worked.

The class started guessing lots of things then. (1/4)^3(3/5). That one was creative, accounting for the 3 out of 5 questions. I did not tell them what was right, but we had in the end 5 options the class thought were possible. One of the options was (1/4)^3(3/4). Pretty close to the basic part of the formula, just missing the fact they missed 2 questions, not 1.

So I asked them which of the options written down actually referred to something bubbled on their scantron. After all, 1/4 means something physical related to their scantron. They quickly ID’d the right equation as meaning something consistent,  and very quickly said they needed an exponent on the 3/4.

The last step was asking them how many different ways to get 3 right out of 5. I did the standard counting and after drawing 3 different options someone said, “there must be an easier way.” The class suggested permutations and combinations, and we quickly settled on the combination of 5 C3.

Done. In 15 minutes, we constructed the Binomial Probability formula using nothing but a stack of old scantrons. I then wrote the formula down for them, and they explained what the pieces were for. They made the connection much quicker than before, and I was really pleased with how fast they could plug numbers into the formula.

I remember last year just screaming inside because they could not get the difference between the probability of the problem as a whole, and the p and q values in the formula. They understand that now.

Success, probably. I like it.  Below is a jpg of the notes I made while doing the exercise.

 click to enlarge.

Really, I found a use for the boxes of old scantrons I have in storage! I didn’t think of it myself, though. It came from here.

Provide each student with a scantron sheet and ask them to guess which would be the correct answer to the first question, if you were unable to see the question.. Talk about the percentage of the students in the class which would have guessed correctly. Extend this to two questions and so on. You can then talk about the math behind probability.

In AP Statistics, as well as the Algebra 3 class I teach, we do binomial probability. It is very abstract, and difficult material. For the Alg 3 course, we incorporate Pascal’s Triangle, as well as the combinatorials, and it becomes a very interesting lesson. But there are still always learners who can never figure out what I mean when I ask:

Now suppose you are taking a 12-question multiple choice test with four possible answers to each question. You need to pass the test in order to get ‘un-grounded’ for that crap you pulled last weekend.

1. If you are totally guessing on every question, what is the probability that you will get a passing grade of at least 60% on the test?
2. Suppose all those minutes of studying pay off and you are able to eliminate one answer for each question. Now what is the probability that you will pass the test and get to go out on Friday?

Yes, this is a real question I ask in Alg 3. The fact that you need 8 out of 12 questions right really stumps the learner, because they try to use that instead of the (.25)(.75) probability required.

But if they actually had a scantron in front of them, would they do better? I don’t know, having taught the material in Alg3 several weeks ago (before Christmas break). But my AP Stats class will be doing the binomial distribution this week. I am going to try it as an introduction. We can then extend the formula to the normal model afterwards.

I will report back what I find out. [which means that one of my goals this year is more active blogging and sharing. I have said that before, but never give up!]

I was not familiar with the acronym T.P.O.V. either until a couple of weeks ago when I was introduced to it at the NCTM’s Reasoning and Sense Making Institute. It was the presentation by Timothy Kanold that introduced the phrase, “Teachable Point of View”, and he brings it up in his book in Chapter 1: The Discipline of Vision and Values.

What brings me to write about this chapter specifically is I have an interview next week for department leader, which is the de facto leader of our PLC. I am going to be (because success is the only option) our department leader, and if that is the case, I better be prepared for the interview and the challenges that come later.

First thing, what is my vision for our math department? What are the values that support that vision? What is my TPOV?

First, let me start with the values of my school. We have 6. Responsibility, Respect, Honesty, Courage, Loyalty and Success. These are the 6 values of the school as a whole. Anything I do as department leader must fit into these values somewhere.

My Vision for the North Valleys High School Math Department

By 2014, the NVHS Math Department will have the highest scores on the District end of course exams, and we will continue to meet or exceed all district mandated requirements for the NV Proficiency exams.

How we will do it:

1. We will implement Reasoning and Sense Making type problems at all levels beginning 2011-12.
2. We will increase the number of RSM type problems over the next 3 years.
3. We will implement the CCSS with fidelity across all levels, starting in 2011-12 (ahead of the district implementation).

There you have it. That is my vision for my math department. We are currently in the middle of the pack on the end of course exams, and we are meeting AYP proficiency goals. We were on a year 4 needs of improvement classification, but through a lot of hard effort, we pulled ourselves out (our consistent red cell was mathematics ELL and Special Ed.)

It will definitely take some Courage and Respect to implement this vision, but implementing will lead us to Success.

One reason I think this is achievable, is that out of the 700 teachers at the RSM Institute this summer, only 5 people from my district attended, and only 2 of them are high school teachers. Both are from my school. We have an advantage in learning, and we can leverage that advantage to overcome the obstacles that stand in our way.

Any comments or feedback on this vision statement? My interview is on Tuesday morning next week, so I need to refine this and sharpen it quickly!

Thank you!

I have done a very poor job of writing about advanced algebra, the course I helped co-author 4 years ago with 5 other teachers in my district. I would like to rectify that this year, and explain more about the course and honestly, get better ideas for the course.

The course is more project based, and it has four distinct, but overlapping sections. Quarter 1 is financial math, quarter 2 is math in art, quarter 3 is math in technology and quarter 4 is math in health / human body. If you go to the site http://mrwaddell.net/AAA you will see the basic structure and some of the lessons / sites I use to teach the course.

Starting off this year, I am going to do something different with the quarter 1. It always felt a little disjointed to me. We use the materials from NEFE to get us started, and then we jump in much much deeper than NEFE goes. We spend a lot of time on spreadsheets (which are nothing more than giant algebra problems using variables) polynomial equations and rational equations (another way to think of Pert and other compounding equations, ie. purchasing a car and annuities) as well as some basic ideas of personal finance.

This year, instead of just teaching each module as a standalone, I am going to tie all of the quarter 1 together with this outline.

At the end of this project, you will need to show the Banker your portfolio that demonstrates you have the financial knowledge and ability to purchase a house in the North Valleys.

Part 1: Setting S.M.A.R.T. Goals

1. Short term, medium term, & long term goals

2. Tracking your goals

3. Adjusting and rewriting your goals

Part 2: Savings & Compound Interest

1. Calculating compounding interest for n

2. Calculating compounding interest for continuously

3. Finding rate or time in both kinds of compounding (working backwards)

4. Knowing and demonstrating differences between APR and interest rate

Part 3: Career & Income

1. Identify 3 careers for yourself

2. Calculate lifetime earnings

3. Calculate $1,000,000 earnings timeline

4. Comparing 2 and 3 for your careers

Part 4: Purchasing a Car

1. Calculating your payment

2. Deciding on years of repayment

3. Comparing years to payment and making a good choice

Part 5: Investing & Credit

1. Risk vs. Reward

2. “Safe” vs. “Intermediate” vs. “Risky” investments

3. Annuities

4. Credit Cards

5. Credit Scores

Part 6: Budgeting your spending and savings

1. Creating a budget (will be working on all quarter)

2. Projecting income

3. Projecting expenses

Part 7: Buying your home

1. Put it all together for the Banker, and using the financial information gathered to justify to the banker that you are a good loan candidate

That’s right. The end goal and purpose of the portfolio will be to purchase a house. It is the biggest investment a person generally makes, but I also know of 4 learners who graduated within the last 3 years who are now homeowners. It is a reality they can achieve now, whereas 3 years ago it was out of reach.

The purpose of this structure is to create a buy-in. Now they see the end goal. This goes hand in hand with backwards design and Understanding by Design principles. I have emailed out a draft to the other Advanced Algebra teachers, and once I have it more fleshed out I will email out another copy. I also hope to get some feedback from them to see what they would add as well.

So, what do you think? Is this a viable way to put together a quarter on personal finance? Let me know in the comments.

In Montana, there is a town called Haugan on I-90 near the Idaho border with an interesting building. As you approach, you see signs like this every couple of miles. [click on the pictures for a higher res image]

Outside, the place looks like any tourist trap.

Do they really have 50,000 silver dollars inside?

And at the back of the bar they have a sign saying what the current count is. I visited twice, 1 year apart.

In July 2010, it looked like this:                    In July 2011, it looked like this:

Any Questions?

[Please leave me some challenging questions in the comments. I am looking to develop this into a classroom material lesson. I also have more pictures of the inside, I just posted a few of them here.]

This will be the last post I make on the specific topic of the 3 days spent in Orlando at the Reasoning and Sense Making Institute. I will expect that I make many more posts on things I do in the classroom that are based on the ideas I have learned, however.

A collection of all the posts, organized by days.

Day 1:

Am I creating more “Clever Hans’” – Notice the plural on the Hans. My answer in general is no, but that may be overly simplistic. It is something that must be fought every day. The rest of the post is about Dan Meyer’s challenge to reform math education is even more important!

Day 2:

Reasoning and Sense Making day 2 – Most of the great resources I found are in this posting, along with some fabulous ideas for Algebra and factoring.

Beth Chance, Henry Kranendonk, and an overwhelming task in stats – Beth Chance and Henry Kranendonk went through how to teach a lesson using Reasoning and Sense Making. Very powerful to actually have someone demonstrate it.

Day 3:

PLC’s at work with Timothy Kanold – A very powerful look at what the impact of PLC’s can be, and how to evaluate your own PLC. The best quote I wrote down that I must bring back to my department and ask ourselves daily is:

Do we want to be known as a school where the math department’s decisions are based on best practices and evidence or as a school where the decisions are based on opinion?

Landy Godbold & “Given That” shown visually – This session started off slow, but ended with a bang! Great ideas were taken away from this session and will be used in class.

Viva Hathaway & stats with food! – Viva is an epic teacher. We didn’t spend enough time doing what I would have liked, but her personality and energy is really the motivating factor in the room. I will attend more of her sessions if given the chance.

Random thoughts and musings that don’t fit above.

Tweet Archive of #nctmrsm11 – There were very few teacher tweeting at the event. Internet was an issue, power was an issue, and it made things difficult to do create electronic versions of the talks. But, some powered through! I won’t make any claims to the accuracy of the archive, but there it is.

1. One of the goals of the Institute was for the participants to walk away with an action plan to implement RSM in our classrooms. Notice that nowhere did I speak to my plans! I have one. I wrote down two pages of things I will do, but I didn’t want to spell that out yet. Part of the issue is I was blindsided by some news at the institute about another teacher. I can’t go into details, it isn’t public yet. But it radically altered my plans and what I need to do.

2. I noticed that as the institute went on, I became a better participant. My notes became better and my thinking sharpened. Just look at the postings above. Day one to day 3 shows a marked distinction in the quality of postings, and what I took away changed as well. This is only the second conference / institute I have attended, and I need to attend more. Before I do, though, I need to go back and read these postings to develop that sense earlier rather then later.

3. Finally, all this was money and time wasted if I do not actually put forth the time and energy throughout the school year to actually DO reasoning and sense making in my classroom. It must become a habit for it to be successful (Beth Chance), it must be implemented with fidelity throughout the department (Tim Kanold), it must be done with energy and enthusiasm (Viva Hathaway), and it must be done with tools that the learners will relate to and engage with (Dan Meyer). That is the real takeaway I have from the 3 days I spent in Orlando.

At the NCTM Reasoning & Sense Making Institute, I attended the session given by Beth Chance and Henry Kranendonk, two of the three authors of the Reasoning and Sense Making in Statistics and Probability book.

I have that book on my bookshelf at school, and I have looked at it, but have not tried any of the lessons in it. I read the lessons at the end of last year, and said to myself, “Are they crazy!” “I don’t understand what they are getting at!” and “Maybe next year.”

After their session, where they essentially went through several of the lessons in fairly good detail (the Old Faithful Data and the Will Women Run Faster than Men in the Olympics as well as others) I am much more confident of doing the lessons successfully.

They had a handout of the material, which I scanned and you can download here.

I apologize for the missing data on the women column, but my copy was very faint also. However, if you go to the book link above, the “Read an Excerpt” is the full chapter from the book in PDF form! Yay!

I am not here to sell their book though. I feel the book is worth buying, now that I have spent better time with it, but the front side of their handout scared me silly.

Go ahead, download it now. I will wait and get some coffee while you do.

….

See what I mean? Under “Habits of Mind in Statistical Thinking” they have 24! bullets under 5 sections. Here is my problem with that list. A habit is something you do unconsciously, because you have done it so often and so repeatedly that it becomes second nature; like breathing.

How can a list of 24 bullets become second nature? The first and second time I read this list I just got a sinking feeling in the pit of my stomach. If this was successful Stats teaching, then I have been failing. Clearly something has to change in the way I teach Stats and what I teach in Stats.

And then it dawned on me. What must change is the quantity of Reasoning and Sense Making I do in the course. After all, let’s examine the list, not for the whole statements but just the root verb stems.

describe, analyze / analyzing (2x), looking for, making deductions, choosing, creating, considering, drawing conclusions, comparing (2x), evaluating (2x) questioning, applying, noticing, identifying, understanding (2x), connection, considering, determining, justifying, and looking for

When you get rid of the other language, the details of what the statements are, and focus on those verbs right there, it is easy to see how a habit can be formed. It is not hard through repeated effort to construct those habits in learners.

See, all those verbs have something in common. They all have Reasoning and Sense Making (RSM) as their goal. Anything I do in class that does not have the goal of RSM is officially a waste of time. There it is. There is the big takeaway from the session.

I just figured out what every lesson I teach in AP Stats needs to revolve around. Now I just have to rewrite all of my lessons to focus on that. That will be the subject of many more posts.

——

As an aside, Beth used some dice and applets in the session that were very cool and useful. The Dice showed an 11 in 6 out of 9 throws. I searched for something that would do that, and found them here and here and here again. These are very useful to generate some statistical thinking skills in the learners.

The applets she used can all be found on the RossmanChance.com website. There are additional materials on this site from other presentations they have given. I may end up using her applets almost exclusively, just because they are all on one site. It makes life easy.

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 Archivist.visitmix.com 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.

Read more »

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.

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.