We collected measures!!!

Second time in a row, the class was in a lab. This time, Jackie appeared more sure of herself. She had had time to pick a problem and prepare a Fathom based worksheet to solve the problem. It was one of their homework problems. The activity that the class was going to model was a simple discrete variable based simulation of creating a sampling distribution. This is problem 7.43 from YMS third edition. It is in the section pertaining to the Law of Large Numbers.

Students were modeling the process of solving this problem on a calculator. In all such problems, the first step is to "assign digits". What that means is given the table of probabilities, you have to create a list in your calculator corresponding to a random variable X (in this case discrete). It is a hard process for them to think through because they haven't practiced enough. Fathom is one more way to model this process.

Class strength was at 17 in place of 25, the rest were on a junior field trip to the South for following a Civil Rights trail. Yet again, a class that was far less disruptive than before. Amazingly the screens mask ratehr than provide the distraction.

Spots that glow:

• "Oh! I get it..." - a boy looking at the way they created the collection - by "assigning digits".
• Get good at Fathom - it is a tool for homework - see how much more we could do - said Jackie
• Pointed out where the law comes into effect, by asking everyone to take one sample of 5. Then asking them to compare how many times they got the number "5" with it's occurence in the distribution of the population. After that, she colected number fo "5"s from the entire class(which was a sample of size 85) and pointed out how much closer it was to the theoritical probability of getting a "5".
  
• In our desire to promote exploration based learning, I realized that our Fathom activities rarely model the typical(and endless number of) AP Stats homework problems.
  

3 collections and a graph!

Today there was a scheduled lab session in which they were to work with Fathom. Jackie said later that she wished she had had time to work with this before hand and create a student work sheet. She mentioned to me that the reason she decided to go ahead and do this session was because she knew I was coming - so there I have influenced the running of the class and in particular with regards to the technology use! Anyways...

The activity that the class was going to model was a simple discrete variable based simulation of sample means. This is activity 7b from YMS third edition. The gist of the activity is captured below (will try to add it from the book later)

1. Let X be a random variable whose values are drawn from {1,1,2,3,5,8}
2. Take a sample of two values from this set. (Treat the two 1s as separate values).
3. Compute the mean for this sample.
   
4. Repeat this for all possible samples of size 2.

This session took place in the lab. Each student had an individual iMac to work on. The have assigned computers but they can access all their documents from any computer as they have networked logins. The lab is set up as 6 rows of 6 computers each. Each row is divided into two parts by a vertical aisle with four computers on one side and two on the other. There is a screen in the front that is visible from all computers.

Jackie didn't have a student handout for them to work from. She had only that very morning decided that she would use Fathom after she confirmed my attendance. She reported her worry that if she had tried this activity solo then in case she ran into a snag she would get nowhere. Whereas with a physical simulation, she may only do a few runs but she had the confidence that she would get the point across. This appears to be a common worry with teachers who didn't grow up with technology as an integral part of teaching. As it turned out, the only real Fathom help I gave her was to point out that you could escape out of animation when collecting a 1000 measures. At other times, when she looked a bit puzzled, I just waited for a moment and she figured it out (An example being - which menu to choose from and which collection to have selected when wanting to collect measures)

(Development Note : We know this is a problem - trying to figure out which collection is which and when to work with which inspector)

On the whole the students seemed to not have a hard time with this activity and with Fathom. Jackie modeled the process of "putting together a Fathom document". The began with a collection, she called randomVariable. She called the attribute random_var. And created 6 cases using the case table {1,1,2,3,5,8}. She didn't ask the students for input when creating the document. She asked them later "Why do you think I put in those numbers?"

At the end, she asked them to look at their neighbour's computer and see if their graphs looked similar. This led to a bunch of looking around and comparing but not too much engagement or talk around why they may have slight differences.

About three of them were working with a sample of size ten (the default) when the activity asked for a sample size of two. One of them did sampling with replacement - and realized it only because his graph looked so different from others and asked for help figuring out why that was the case.

On the whole, an exciting and satisfying class. even the usual disruptive suspects had less effect on the class - maybe it was the fact they were in front of a computer and all doing something. The individual computer in fact to some extent lessened the disruptive effect because it was easier to stay on task with this screen blocking the distractions.

The person who had caused quite some disruption in the earlier class this time spent a lot of time fooling around but was also one of the first three to finish. So is he disenegaged because he is bored adn this material is too easy? Hard to say...

At the end of all this - they had fun, were more engaged than in any class before this, but did they learn better? Now that is a question for a later time...
Spots that glow:

• Even for a bright and dedicated teacher like Jackie, it is hard to get over the mental hurdle of doing an entire lesson based on technology.
  
• All it took in this case was the knowledge that a "technology expert" would be present for her to take the brave step of plannign this lesson
• "Why does Fathom create the third box?". "I don't know; for fun?"
• "Why do you have different numbers?" "Oh because we got different samples!"
  
• Fathom was very well suited to modeling this simple activity. There was none of the cognitive baggage associated with putting together a probability simulation involving cards or a "real-life" situation. Perhaps for students, whose big hurdle on a AP like test is reading and parsing the question, there are two simultaneous needs fighting for their attention. One is the need to understand the language - the other is to model the situation.
  

Westvaco- what does that number mean?

Today was the first real classroom observation of Jackie's class. There was very little Fathom in today's class except for a demonstration at the end.

The class began by the students constructing test taking barriers. A little context - the class is held in a room that serves as the school's library. It is a large hall with round tables that each seat four.

The students all face each other and there aren't any spare desks for them to spread out at to take tests. This has been a problem that Jackie has grappled with last semester. Today, the students working in pairs created physical barriers by taping together two large pieces of cardboard and dividing the table into four spaces. The students took a short quiz on computing variances and standard deviation by hand (5 data values between 1 and 2 up to one place of the decimal). 4 students finish early and go into the adjoining computer lab to look up answers to earlier homeworks.

The teacher has all her lessons organized in Goggle Docs and the class has a website which has the homework answers, notes from each lesson, homework assignments and all class policies. She did an overview of whats on the website and for what she expected from them and what tools she was providing for them to make it through the course. The students know what thye need to finish in one class because the schedule is up on the board. She has just taken over active teaching because the first semester was being taught by her student teacher.

This was followed by a hands-on simulation of the Westvaco case from YMS book. They used graphing calculators to simulate a sample of ten workers from 100. Jackie made them go through the 4 Steps that are expected in an AP answer for a simulation question. She models all the calculator behaviour - spelling out the keys and the functions therein. She also explained "whats syntax?"

On the calculator, they use randInt(1,100,10) to take a sample of ten from 100 workers. They have in their head/ on paper assigned 1-24 as "old" people (she tries to steer them to call them people over 55, but the instinctive categorization is old peole ;-)). They count the numbers 24 or under and plot this count on a graph she has set up on a white board in front of the class. Five to six of them did this more than once - they ended up with 39 dots. The graph was nice and triangular and the most common point was 2. This allowed her to illustrate the relationship between the 24/100 proportion of the population and the 2/10 of the sample.

The class concluded with a Fathom demo of the Westvaco activity. Jackie demonstrated how to create a collection of 100 people, how to use the formula table to assign "55+" and "young" values to the attribute of age(she actually had instructions in the document for two other ways to do this) and then how to take a sample of 10 people from that collection. She moved onto defining a measure but didn't spend much time explaining this(we were running out of time). That concluded the lesson.

Spots that glow :
Things she said -

• Put down 52.6 I am picky
• Don't make my job hard
• says "did good"
• says they all have a chance at passing the AP

She spends a significant amount of time managing motivation. expectations and class policies. This in the end seems to her at this point to be the differnce between a pass in AP or not