Here is the first of five posts wrapping up what I learned during the second and final year of the Oklahoma Geometry and Algebra Project (OGAP).
First off, I want to thank the people who made this amazing experience possible. The workshop was coordinated by Dr. Jennifer Bryan from Oklahoma Christian University and Dr. Martha Parrott from Northeastern State University - Broken Arrow. The workshop was facilitated by Linda Hall and Martha Wissler. Linda and Martha are both retired math teachers from Edmond Public Schools, and they are phenomenal! I have learned SO much from them.
The first activity we were given was to sort 12 statements. We weren't given any more guidance than that. Each group was given the same 12 statements. Here's how my group sorted them:
Statements about what teachers should do in the classroom. Strategies that were at a lower-level on Bloom's Taxonomy were placed below strategies that were at a higher-level.
Statements about what teachers should be doing in the classroom. Same ordering of thinking skills.
And, statements about what math instruction in general should be like with the same structuring as above.
I sadly didn't take a picture of the other groups' groupings, but they were vastly different from ours. It was interesting to see how differently the same set of statements could be grouped.
It turns out that these statements were taken from Principles to Actions, a recent publication of NCTM. In Principles to Actions, these statements were sorted into two columns with the headers "Productive Beliefs" and "Unproductive Beliefs." As part of the workshop, we received a copy of Principles to Actions: Ensuring Mathematical Success for All. As one of our homework assignments, we had to read several sections of it. I've really enjoyed what I've read so far, and I'm looking forward to delving into it further. I may never get my Summer Reading List done now since I keep adding new books to read!
|Principles to Actions|
One of the reasons I'm looking forward to reading it is that the chair of the writing team was Steve Leinwand, the author of Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement which I read and LOVED last month! You can see my review of that book here.
This wasn't the only resource we received. Going to one of these workshops is kinda like Christmas! We each picked a subject to specialize in this year. Last year, I chose Algebra 1. This year, I decided to switch my focus to Algebra 2. Basically, we have to collect data from one of our classes. We give that class a pre-test and post-test as well as having them make concept maps at the beginning and end of the school year. The class we focus on is the class we will collect our data from.
We received a new textbook from Big Ideas Math for the subject we chose to specialize in. I haven't had a ton of time to peruse the text yet, but the first thing I noticed was how much thinner the textbook was from a normal textbook. This line of books was written specifically for Common Core. I'm hoping I will be able to pull out some great problems to use with my students from this text!
|Big Ideas Math - Algebra 2 Textbook|
Probably the best resource we got during the week was a 3-inch binder packed with activities!
Several of the activities we did during the week were from The Charles A. Dana Center. Last year, we received a cd of Algebra Formative Assessments Through The Common Core. This year, we received Geometry Formative Assessments Through The Common Core. Each cd has pdfs of 3-4 different books on it!
|Formative Assessment Resources from The Dana Center|
We joked that the next book we received would be considered contraband in our schools now. Even though Oklahoma has pulled out of Common Core, I still found my reading in The Common Core Mathematics Standards: Transforming Practice Through Team Leadership to be relevant to my teaching practice. Whether we are teaching CCSS or not, shifts are happening. And, this book talks about what needs to happen to make these shifts possible.
|The Common Core Mathematics Standards|
One resource we didn't use a lot of this week was our copy of the Common Core State Standards for Mathematics. Actually, I used it three times during the week. But, all three of those times involved me turning to the reference sheet to look up a formula! Next year, I am definitely going to make a reference sheet for my students to keep in their interactive notebooks!
Now that you're jealous over all the awesome goodies I got, I'm going to share the resources we were given on Day 1. Enjoy!
The Camel Problem
The first problem we worked on in our groups was the camel crossing the desert problem. I had just seen this problem the previous week in the copy of Discovering Advanced Algebra that I ordered off of Amazon. I remember reading the problem and thinking it sounded interesting, but I hadn't spent any time trying to solve it.
|Clipart from ClipArt ETC - one of the best sources for free clipart for the classroom!|
The version we attempted reads as follows:
A camel is sitting by a stack of 3000 bananas at the edge of a 1000 mile-wide desert. He is going to travel across the desert, carrying as many bananas as he can to the other side. He can carry up to 1000 bananas at any given time, but he heats one banana at every mile. What is the maximum number of bananas the camel can transport across the desert? how does he do it? Be prepared to present your solution to the class. (Hint: the camel doesn't have to go all the way across the desert in one trip.)
My group got right to work and figured out a way to transport 500 bananas across the desert on our first try. Linda looked at our solution and told us that we could do better than that. We tried, but ever subsequent strategy ended up transporting less than 500 bananas! One group was able to arrive at the solution during the 45 or so minutes we were given to work on the problem. The rest of us had to take the problem home for homework. At home, I tried to solve the problem by employing simpler cases. But, I never did make the breakthrough I needed.
The next day, they provided us the solution. I kinda wish I had left the room during the explanation so I could have figured it out for myself. Oh well... There are plenty of other math problems out there for me to solve!
Fawn Nguyen gives her students a version of this problem that involves transporting 45 watermelons across a 15 kilometer desert. I think this problem would be much more approachable for my students. Plus, you can easily represent 45 watermelons with manipulatives. I wouldn't want to try to represent 3000 bananas! I definitely want to work this problem into my curriculum next year!
Linear and Exponential Model Scavenger Hunt
Next, we moved to an around the room scavenger hunt style activity from Howard County Public Schools. If you're still in a Common Core inclined state, the alignment is F.LE.A.1. 10 graphs were hung around the classroom.
|Picture of Graphs for Scavenger Hunt Activity|
As you can tell, some of the graphs are linear, and some are exponential. We were given pages with 10 equations, lettered A-J, 10 tables, lettered A-J, and 10 verbal models, lettered A-J. The facilitator assigned each group the letter of the graph they would start on.
Now, I've done scavenger hunt style activities in my classroom, but I've never done one quite like this before. A timer was set for 2 minutes, and we had to stay at the graph we had been assigned for those 2 minutes. On our answer sheet, we had to record the letter of the table, equation, and verbal model that matched up with the graph we were standing in front of. When the timer went off, we had to move as a group whether we were ready or not. As we became more accustomed to the activity, the time we were given at each station was decreased to 1 minute.
Some of the groups HATED that everything we were matching was labeled with the same sets of letters. So often, someone in my group would say something like, "This has to match H!" Another person would look at them like they were crazy. "It can't be H!" It would then turn out that one person was talking about the table and the other was talking about the equation! We had to attend to precision with our speaking during this activity. Since I have such a small classroom, the student groups would be very close together if I was doing this activity with them. By reusing the same letters, students can't just listen to the group ahead of them and record their answers. "Okay guys. Our next answer is going to be B." Was it B on the equations, tables, or verbal models???
I've never timed my groups before during this type of activity, but I think I'm going to try that this year. In the past, I've let groups move at their own pace, and that means some groups whiz through the activity and other groups do nothing all hour. Hopefully this will help both of those problems!
You can download the files for this scavenger hunt activity here. You will need to scroll down to where it says "HCPSS UDL Lesson: Comparing Linear and Exponential Functions."
Strategies To Reach Out To Struggling Students
* Build relationships with students so there is trust!
* Focus on the 3 R's
* Get the math out of the textbook, and put it into the lives of students!
* Turn mistakes into learning opportunities.
* Use individual white boards to increase students' willingness to attempt a problem. I've found that my students are more willing to try a problem if they can easily erase it if they mess up.
* Say to students: "Explain to me how you got here."
* Go back to see what your students are actually struggling with. If they are having trouble simplifying radicals, go back to multiplication. Can they fill out a multiplication chart? Start here.
* Color code the steps of your math problem. Allows students to ask safe questions. "How did you get the blue part?" A student who struggles with math vocabulary would be more willing to ask this question as opposed to trying to ask a question using vocab he doesn't know and sound foolish.
* Class time needs to consist of comparing and discussing MULTIPLE solution methods.
* Ask / Expect students to EYB - Explain Your Brain.
The next activity we worked through was a rotating station activity. There were three stations set up around the room. At each station, we were presented with two different ways of working a problem. And, we were asked specific questions about the different solution methods.
You can ask students three different types of questions through an activity like this:
- Which is better?
- Why does it work?
- How do they differ?
The problems we looked at involved a conversation of sorts between Alex and Morgan. Here's an example:
|Contrasting Cases Example|
Is anyone else excited about this? You're going to be especially excited when I tell you that there is an entire Algebra 1 curriculum of these that you can download for FREE!
These are from a project entitled Using Contrasting Examples to Support Procedural Flexibility and Conceptual Understanding in Mathematics by researchers from Harvard, Vanderbilt, and Temple University.
Here's a direct link to where you can download these curriculum materials as PDFs. But, I'd suggest taking a look around the rest of their site, too! This study showed that student achievement improved when students were routinely presented with multiple ways to solve a problem. As teachers, we can have a tendency to show our lowest students only a single way of doing a problem for fear that we will overwhelm them with multiple methods. But, our students need to realize that there are many different and correct ways to solve a problem. If we're not showing them this, we're cheating them!
I'm excited to look through their curriculum more closely and see what I can use this year! If you find any awesome resources, send them my way!
As one of our facilitators said, "When students are analyzing, they are thinking!"
Another quote I jotted down in my notebook: "Make students into thinkers - not a location for dumping information."
Ahhh! The dreaded mixture problem! Bonnie and Carmen are lab partners. They need a certain mixture of X% acid and Y% water. But, they only have access to a mixture of A% acid / B% water and C% Acid / D% Water. Whatever can they do?
The specific problem we worked is under copyright by the Dana Center, so I can't post it here. But, I can tell you about how we approached this problem during the workshop.
We were given a blank chart with the following columns
Amount of Mixture A
Amount of Mixture B
Amount of Acid in New Mixture
Amount of Water in New Mixture
% of New Mixture is Acid
% of New Mixture is Water
Acid/Water Correct Mixture?
Since we needed 5 ounces in our example problem, we first tried out 1 ounce of Mixture A and 1 ounce of Mixture B. We used percents and multiplication to find the new amount of acid and water. Then, we divided by the total number of ounces to calculate the percentages of acid and water in the new mixture. In the last column, we had to decide if we had too much acid or too much water and adjust accordingly.
In our example, we ended up with too much water, so we tried a different break-up of the 5 required ounces. We repeated and repeated and repeated the process. Eventually, we discovered that 4 ounces of Mixture A gave us too much water in our final mixture. 5 ounces of Mixture A gave us too much acid in our final mixture. Therefore, we needed somewhere between 4 and 5 ounces of Mixture A, and we went from there.
It was a ton of work to work the problem out this way, but I think it would really help students to see how the process works. I've never actually done mixture problems with my students before. They're not tested, so I've always conveniently skipped over them. The presenter said that she used to teach her students how to set up the system of equations, but they never really understood where the numbers in the equations were coming from.
You definitely wouldn't work every mixture problem out this way. That would be insane! But, doing the process once should help students visualize the process and better understand why a system of equations can be used.
I can also see myself using this as a quick review of percents/proportions/decimals/etc. Then, revisit the problem when you get to systems.
The last problem of Day 1 was Casey's Quesadillas from The Dana Center's series of Geometry Formative Assessments. Casey has a map of downtown. He draws a circle on the map of where he would like to advertise his food truck. Rewrite the equation of his circle in vertex form. Decide where Casey should place his quesadilla cart. It then proceeds to review the meaning of radius and the distance formula.
I can see myself using this activity in Algebra 2 during our unit on conic sections. It reminds me of Dan Meyer's Taco Cart.
Our homework assignment included completing a series of Short Tasks from MARS (Page 2 of this pdf.) This was easy and straight forward. Then, we were to complete a task from the Balanced Assessment in Mathematics Project called Three Circles.
This one took a bit of thinking! At first, I was sure that they hadn't given us enough information. But, I was soon able to set up a system of equations and solve for the radii. Great problem!
You can download this task from the Balanced Assessment Program here.
Overwhelmed yet? This is just Day 1 of an AMAZING workshop!
Disclosure: This post contains Amazon Affiliate links.