CMP Digging Deeper Day 3

June 22, 2016


Algebra, a word that strikes fear in many and is the topic of many jokes about how irrelevant to real life high school mathematics is.  I was good at Algebra, or what I though Algebra was.  I could follow a set of "like" problems so well, that I spent the majority of class writing and folding elaborate origami notes after finishing early (which were occasionally intercepted and read aloud to the class for entertainment.)

Algebra: Function based verses Equation based?

In answering this question we began our work today with Thinking with Mathematical Models Problem 1.3.
Students are given a situation of building trusses with rods (see above picture.) This is a bridge with a length of 7 trusses using 27 rods.  Student think about different length trusses in terms of a pictorial, table, graph, and final an equation.  There are many ways to push algebraic thinking by just beginning with different ways of seeing the picture.


 

Truss	Rods
7	27
1	3
2	7
3	11

Looking at the picture, some students see that for each truss added, four rods are needed. Three to create the triangle and one more connecting rod (except the first one which is missing one (-1). Students may use this picture to find the equation R = 4T-1 without even using a table or graph (although they could certainly use them as well.) This change for each truss can be represented in the  picture, table, graph and equation.

The equation R= 4T -1 shows that for every truss added, there are 4 more rods (except the first)

That is not the only way to see the problem. Students can also count the triangles and multiply by 3 to get the rods and then add the connectors (which will always be one less than the trusses because the top is one less).  R = 3T + T -1.  This is equivalent and gives a powerful visual example as to why.  

They could also count the total number of triangles and subtract out the double triangle pieces (since they have 2 in common for each triangle. R= 3(2n-1) -2(n-1)

We looked at the connections between different representations.  We then examined an entire unit of 7th Grade Moving Straight Ahead without having them sequenced for us. We explore linear functions, but the one piece we had the most trouble with was the equations section using pouches and coins. Where does this fit in?  Then we read a article from Texas Mathematics Teacher explaining the 2 different methods often used to teach algebra, equations and functions.

Texas Mathematics Teacher

CMP is a functions based curriculum that connects solving equations with what students know about functions.  In the case of the truss/rod problem questions are asked if there are 27 rods, how many trusses can be build.  Students can use the table (construct values using rate of change and initial value), graph (find 27 rods on the y axis and find where the line is at 27), or equation (substitute 27 in and solve) to answer the equation that is build from this specific instance. This grounds solving equations in the relationships defined by the function in multiple ways with more meaning to students.

To end our evening, we were given several articles to think about including one on questioning and assessment (with an in-depth example exploring the circle area and circumference standard in 7th grade.)

Questioning our Patterns of Questioning

Assessing for Learning

Day 2 CMP Digging Deeper

June 21, 2016

Orchestrating Mathematical Discussions

I have control issues. I like to be in control of making sure students learn. I would love to use my imagined superpower to open their brains and force them think and reason. I find this personality trait is common to many teachers. Several people have asked me how I am dealing with that using inquiry based learning in math. Honestly, orchestrating productive mathematical discussions gives me more control over learning than any traditional way I ever taught. You can't really force students learn (that doesn't stop us from trying though- or wishing for superpowers!) Students need to invest and connect in order to truly learn. During the summary of the problem students are primed and ready to make those connections and finally I can facilitate a semi-controlled learning opportunity by previously anticipating responses, monitoring discussion, selecting student work, sequencing the order, and finally connecting the ideas! Superpower!

Today we revisited our work from the orange juice problem and connected it directly to the excerpt we read on Orchestrating Mathematical Discussions We were given student work from the Connected Math site (from these we were given 7 different fairly randomly chosen pieces of student work (pieces 2,3,7,8, and 18). Then we were asked to look at the work from each of the 7 groups and choose which pieces of student work would be a part of our summary (selecting).  After selecting, we then we worked together to figure out how to best sequence them in a class discussion as well as questions we would use to connect. This was a very valuable experience, especially with the collaboration of our table groups. It was definitely not easy, but low risk (we weren't doing this on the spot) and high benefit (we saw our problem from new perspectives and got to wear our teacher hat without messing up our first period!)

We debriefed our choices with other groups; some chose a different order and also different groups' work to present. It was interesting to see how many different orders would still produce a math "story" and allow for students to recognize connections.

Linear relationships both proportional and non prorportional

As promised we continued to connect proportional reasoning beyond the Comparing and Scaling Unit from 7th Grade.  We explored problem 2.1 in Moving Straight Ahead where two boys race (one with a head start.) This problem is great opportunity for students to explore the difference between a proportional and a linear relationship that is not proportional.  

Inversely proportional relationships

In 8th Grade Thinking with Mathematical Models, students look at inverse variation (y=k/x) using several circumstances (holding the area constant while the side lengths both change- first explored in 6th grade, and holding the distance constant while the rate and time change problem 3.2.)  I have always understood inverse variation based on graphs and equations, but when I looked at the relationship between each variable in a table I really developed a deeper understanding. 

In a proportional relationship, when you multiply x by 2, you also multiply y by 2. However in an inversely proportional relationship you multiply x by 2 and divide y by 2! Of course this makes sense when you look at the equation  xy=24  (length x width = 24) because in order to remain equivalent, if you multiply x by 2, you must divide y by 1/2 to keep the equation balanced.  You are essentially multiplying by 2/2, just multiplying x by the numerator and dividing y by the denominator. 

We didn't teach this investigation this year due to time constraints, but I can't wait to see the connections my students will make. 

Proportional Relationship connection in 8th grade

Then we broke out into groups once again and explored other places that connected to revisiting proportional relationships.  Each group took on an upcoming lesson (jigsaw) to see how our work on proportional reasoning could be connected: Thinking with Mathematical Models 3.3, Looking for Pythagoras Problem 5.2, Growing Growing Growing Problem 1.3, Frogs, Fleas, and Painted Cubes Problem 4.3 and Say it with Symbols 2.1. Below are the results.

By the time I left today, I was even more convinced at how connected this math curriculum really is. This also seems to give me a little more of my desired "control" over student learning!

CMP conference day 1

June 20, 2016 CMP Summer Workshop Digging Deeper

Dot Walk

Our class began today with dot posters! After being given a sheet of sticky dots, we placed dots on chart paper diagrams and graphs to indicate our place. For example, a Venn diagram indicated what grades we taught, a coordinate plane poster showed years taught on the y axis and years teaching CMP on the x axis.  We made a dot plot according to our educational role (teacher, coach, administrator) and we plotted where we were from on a almost recognizable "map" of the United States. 

After all the dots were placed, our instructors led us in a Notice and Wonder where we explored patterns and addressed the yellow dot outside our Venn diagram (administrator) and the blue dot living in the ocean (he was from China.)

I see this as a great activity to use returning from summer break. I will need to think more on the categories I will use (maybe a coordinate plane looking at books read and movies watched?)

Survey of Beliefs and Fishing

We then examined our math education beliefs using a survey developed by NCTM in Principles to Action. While we all knew what our answers should be, we had a lively discussion on specific words in questions 7 (master) and question 2 (exactly.)  Knowing the right answers (principles) is at least a step towards implementing (action) mathematics in classroom that will lead to effective learning.   It's tough to see kids struggle, but if we always catch the fish (math) for them, without us they will starve (or at be unable or willing to even start a unique problem.)

How do we teach them to "fish" in math? By not answering their questions....directly. Answer  their question with another question to see just how much help they need baiting their hook! Maybe even wait until the class summarizes the problem if it is appropriate. CMP has a definite lesson flow. Launch the problem, let the students explore, and summarize as a class at the end. Both explore and summarize are excellent places to ask students questions to move them along and check for correct mathematical reasoning. Make sure their questions get answered, but help them get their on their own or you will be fishing all year for them.

Proportional Reasoning, it all Connects

With class sets of 6th, 7th, and 8th grade books we began to look at the critical concept of proportional reasoning.  This is an important step in mathematical development, moving from an additive relationship (more means you are adding) to multiplicative (comparing ratios for more per item). After a quick write on our understanding of proportional reasoning, we were given a problem to work from Comparing and Scaling in 7th grade where students are mixing juice.

Which one is the the strongest? We were given chart paper to explain our reasoning. Our facilitators  walked around, writing down some of our comments and answering questions with questions to move our thinking along. 

Several of us had done the problem already, and took a bit of a back seat. One important comment came up about students having no idea what concentrate was (especially liquid, because they were used to powder.)  

After all posters were up, we started a class discussion. First we looked at the most common solution (creating a part to whole ratio and then looking at percentages.) Then we examined and questioned several others (them most controversial looking at a digram representing 1 cup of water (unit ratio). We also examined students who just divided the first number by the second to get a decimal (2/3= .6 repeating) and then turned it into a percent and why it is not the same at the percent of part to total (40%). We discussed scaling up and down (unit ratio) as well as comparing to benchmark fractions.

We then brainstormed at all the information students could use to access this problem. Then we examined the CMP3 books to see where we could find those concepts.  Below is a summary.

Summary of concepts and where they are found

Orchestrating Productive Discussions

Our last activity of the day involved one of my favorite, and most challenging books, 5 Practices for Orchestrating Productive Mathematical Discussions.  The basic premise of this book involves pre -planning discussions including planning all the ways students will answer the problem, how the teacher will respond to any misconceptions with questions or counter examples, as well as how to sequence the discussion to make a mathematical story that connects smoothly. Quite a task when you have student variables! It is easy to see how amazing and overwhelming this concept is. In several of my lessons I have attempted to chart anticipated student responses. I will continue my work on this next year, as I find charting it makes my discusssions more relevant, efficient, and, when it goes well, beautiful!  The following is my example from the pizza problem in Comparing and Scaling investigation 2.

Chart of student responses and sequencing

Our day ended with a Ted Talk by Diane Laufenburg on How to Learn from Mistakes.