The 2014 Summer PMI Workshops have finished!
Week 1, which ran July 14-18 was organized into a course called Mathematics as a Second Language. We had a total of 20 participants from the following districts:
- Bald Eagle Area SD
- Bellefonte Area SD
- Curwensville Area SD
- Harrisburg SD
- Reading SD
- State College Area SD
- Steelton-Highspire SD
- West Branch SD
- Williamsport Area SD
The participants ranged from a kindergarten teacher through some fifth grade instructors, as well as two participants with middle school and high school experience.
Week 1 was spent exploring arithmetic at a deeper level: picking apart models of understanding, the reasoning behind the procedures, and connections between the operations. We also spent time exploring education ideas such as the iceberg model and evaluating levels of cognitive demand. You can see the daily blog recaps for specifics of each day’s events.
Week 2 July 21-25 was organized into a course called Functions and Algebra. Sixteen of the twenty participants from Week 1 continued on into this week. Here we continued ideas of the previous week to consider proportional reasoning and linear relationships. Teachers of the later grades gained new intuitions while the earlier grades gained a longer view of the topics they teach. These topics also served to strengthen the procedural fluency gained in the first week. The morning discussions of pedagogy topics continued, as we consider issues such as equity in the classroom and the use of open-ended problems.
The workshops ran daily 9am to 4pm, in a style that could be described as “immersion.” Little time was spent in a traditional lecture format. The class time is primarily spent in collaborative problem-solving and discussion, akin to the style sometimes called “inquiry based learning.”
The next post will be on quantitative and qualitative analyses of the outcomes for the summer workshops.
The end of an amazing 10 days! So amazing that it has taken me two days to write this recap!
We started the day with the post-test, which hopefully went better for everyone than the pre-test. We moved into a sample of what a “data and statistics” course might be like as we discussed perspectives on the mean, median, and mode. Then we took some time to talk about professionalism and what it means to be a professional in teaching. Each of us set a goal for ourselves as to how to better our teaching or our school in the coming year.
After lunch we embarked on a sort of “capstone” for our discussions of linear functions with “Bungee Barbie & Kamikaze Ken.” We collected data, modeled that data with a line, and then made some predictions as to how many rubber bands will be needed to give Barbie as much of a thrill as possible (Mathematical Practice #4, not to mention #1, #2, #5, and #6) . The slope and y-intercept had immediate importance (at least, they were important to the doll at the end of the bungee).
We ended the day with each participant sharing one mathematical idea they solidified and one teaching idea that really resonated with them. After handing out the certificates, we all went home exhausted but, hopefully, enlightened.
- Bungee Barbie and Kamikaze Ken (two pages)
- Push yourself to look “under the hood” and find the mathematical underpinnings of why a process works.
- Read through all of Principles to Action.
- Achieve your teaching goals.
The results of activity where we made textbook tasks more open-ended are available here.
It’s hard to believe it’s almost over!
The morning was spent discussing the reading from Principles to Action on eliciting student thinking. A sheet with 8 erroneous examples of student were distributed, the participants split into pairs and took turns being the student and teacher and practicing questioning techniques. This lead into a discussion of funneling versus focusing question-style, and then the exercise was repeated with additional retrospection.
We then moved into a discussion of the homework from Tuesday night on the car-and-van problem and the average speed problem. Then we began the Pool Membership problem, and sought out connections to the previous discussion of lines. We found the cost per visit to correspond with the slope while the cost of membership corresponds to the y-intercept.
After lunch (and a digression on the Doomsday rule), we discussed the magic trick from the previous blog recap post and connected that to multi-step processes and inverse processes. Then we moved into the Hart’s Furniture Problem (and Hart’s Furniture Revisited). Our previous discussions of processes and functions again came into play, as well as graphs of lines to illustrate when one deal was better than another.
- Principles to Action, section “Professionalism” (p. 99-108)
- Get some ice cream.
- Get some sleep.
Today we finally got to Functions!
In the morning we discussed two items. First, how do the ideas about processes and proportions discussed track back to the elementary levels? Second each participant started with a problem from their curriculum and tried to make it open-ended. These will be collected and distributed to the rest of the group.
We then moved into the formal idea of a function, connecting it back to the processes we discussed on the first day. We introduced a lot of new terminology: function, domain, range, well-defined, one-to-one, and onto. We talked about everyday correspondences and whether they might be functions, and we discussed what it means for a function to be the inverse of another.
- Unit 3: Functions
- Module 4: Functions (actually, these are labeled as pages 3.2 – 3.7 for the Unit 3 packet)
- (We will not be discussing Unit 2, but those handouts are available if you want them for completeness’ sake. Likewise for Modules 2 and 3.)
- Principles to Action, Section entitled “Elicit and use evidence of student thinking” (pages 53-57)
- Take a look at the previous blog post’s article “Why do Americans stink at math?”
- Consider the following “magic trick.” (1) Start with your age. (2) Multiply your age by 10. (3) Subtract your age from the previous result. (4) Add up the digits of the previous result. (5) Repeat step 4 with the new result until you get down to a single digit. (6) Subtract 1. (7) Convert your result to a letter, where A=1, B=2, C=3, etc. (8) Think of a US State starting with that letter. Why should I know you are thinking of this state?
- Convert the trick into a chain of 7 functions. Which are one-to-one? Which are onto?
- Does it matter what age you start with? Why or why not?
We opened the day with a discussion of equity and access in the classrooms. Equal treatment may not always be fair treatment, and how that plays out in both our classrooms and districts.
Along these lines a few resources came up:
We then reviewed the previous night’s homework by creating posters detailing our solutions and then sharing the posters. We found this to not work as well for the problems that did not allow for multiple solutions.
The day’s content centered around problems which involved lines. Yesterday’s problems were best described as proportions, as 0 of one quantity (e.g., rice) corresponded to 0 of the other (e.g., water). Today’s lines had a y-intercept which was something other than 0. We also saw how the slope of a line corresponds to a rate of how quickly the line rises as one moves to the right. These were explored via the Currency Conversion with Exchange Fee and Burning Candle problems.
- Reading assignment “Turning Traditional Textbook Problems into Open-Ended Problems”
- Module 1
- Read “Turning Traditional Textbook Problems into Open-Ended Problems”
- Page 1.30, problem 2 (both a and b). Also look at Problem 3 as a challenge.
In the morning we made sure everyone learned names with the Name Game. Then we got out Fran’s motion detector to get some experiential basis for reading lines on a coordinate plane. Following break we did a bit of catch-up by thinking about processes and inverse processes from last week’s Unit 3.
Following lunch we started the Functions and Algebra Unit 1. We first considered feet-to-inches conversions and the inverse operation (1.1 – 1.4). We then skipped over the currency conversion problems and opted instead for the rice-and-water problems. We saw how te inverse relationship between the two processes showed up in the graphs as a reflection about the diagonal. After the break, we tried the 3-act math problem “Nana’s Lemon-water” from Dan Meyer’s blog dy/dan. We then delved deeper into why the reflection about the diagonal should correspond to the inverse relationship, and then pushed forward into pages 1.16-1.18.
- Dan Meyer’s Blog, where we talks about 3-act math and more: dy/dan
- Connecting Arithmetic to Algebra.
- Functions, Algebra, and Geometry Course Overview (pages 0.1-0.2)
- Unit 1: Proportion and Linear Relationships (pages 1.1-1.32)
- Reading: Principles to Action. Section “Access and Equality” (pages 59-69)
- Problems on Processes: MSL Unit 3
- Page 3.21, problems 3 (all) and 4 (all).
- Page 3.22, problems 1a and 1c
- Film Developing Recap at Dynamic Developers 1.19 – 1.20. Problems 1-5 (we will go further on Tuesday).
Whew! What a week!
In a large horseshoe we discussed norms in these workshops and norms in the classroom. We watched two videos about establishing norms for your classroom, then soon after discussed ways to establishing a “productive struggle” norm.
As for mathematical content, it was “Fractions Friday,” and we saw a lot. We began with models of fractions, and how to motivate and underscore equivalent fractions. By lunchtime we used the adjective-noun theme to motivate the need for a common denominator when adding or subtracting fractions. During the working lunch we wrote problems requiring addition, subtraction, multiplication, and division of fractions. We saw that some problems that may appear to be subtraction may actually involve multiplication depending on the “nouns” involved (e..g, “I ate a quarter a pizza, then a third of the remaining portion. How much is left?” actually calls for multiplication because the “third” is of the remaining 3/4 of a pizza). Fractions as lengths or distances (as opposed to part-of-a-whole) often provide a more natural context for addition and subtraction problems.
We closed the day with a discussion of multiplication and division of fractions. We saw two ways to justify the “invert-and-multiply” (or “keep it, switch it, flip it”) rule. The first involved fact-families and some algebra. The second involving finding common denominators and observing the results. We also used the ribbon problem from last night’s homework as a context where dividing by fractions is natural (again, length-based problems make for a more natural context than part-of-a-whole).
- Solutions for Unit 3
- Solutions for Unit 4
- Solutions for Unit 5
- Unit 6
- (There will be no solutions for Unit 6)
- Look at the 17 horses problem. What’s going on?
- Relax and enjoy your weekend.
A farmer died leaving his 17 horses to his 3 sons.
When his sons opened up the Will it read:
My eldest son should get 1/2 (half) of total horses;
My middle son should be given 1/3rd (one-third) of the total horses;
My youngest son should be given 1/9th (one-ninth) of the total horses.
As it is impossible to divide 17 into half or 17 by 3 or 17 by 9, the three sons started to fight with each other.
So, they decided to go to a farmer’s friend who they considered quite smart, to see if he could work it out for them.
The farmer friend read the Will patiently, after giving due thought, he brought one of his own horses over and added it to the 17. That increased the total to 18 horses.
Now, he divided the horses according to their father’s Will.
Half of 18 = 9. So he gave the eldest son 9 horses.
1/3rd of 18 = 6. So he gave the middle son 6 horses.
1/9th of 18 = 2. So he gave the youngest son 2 horses.
Now add up how many horses they have:
TOTAL IS 17.
Now this leaves one horse over, so the farmer friend takes his horse back to his farm.
HOW IS THIS POSSIBLE????