IMP Alice - Stranger Pieces of Cake and Confusion Reigns

Today I first had my students do Stranger Pieces of Cake. I started off by reading over the into and giving them 5 minutes to think about number 1. When they were asked to put their findings on the ipads and project them (using airplay) I realized that they were in need of help. I did have one girl say that 2 to the 3/5 power would be like Alice's height increasing by 60 percent. I was excited that she used that description but I needed to lead them to a point where we could explain the problem using ideas we had already learned in Alice...so that we could develop the rule for fractional exponents.

So...I then asked them if they could figure out a problem where they either raised a power to a power or multiplied powers with the same base and the answer was 2^(3/5). Luckily, at least one group in each class eventually came up with 2^(1/5) * 2^(1/5) * 2^(1/5). We related this back to the activity where we developed the rule for exponents that are unit fractions which eventually got us to the idea that the 5th root of 2^3. Anyway...I had to do ALOT of leading through this activity but I really believe that the students have a better chance of remembering the rule for fractional exponents after they have had their "hands on" trying to figure it out themselves.

Confusion Reigns is a good activity that makes students just rethink through a few of the rules they have learned. I LOVE the first problem because so many students didn't pay any attention to the fact that they were adding 2 powers with the same base instead of multiplying. I hope this helps them to pay closer attention to the operation signs in the future.

A "case" for a problem-based curriculum with group work

As we have gone through Alice we have seen the need to just pull alot of practice problems for each of the rules. I still have students that approach these problems in a variety of ways. At this point there are some that have memorized the rules. I still have some who use expansion . I have explained to them that expansion is not always feasible but hopefully being able to expand problems correctly will help them to remember a rule.

Yesterday I spent the majority of class answering questions and going over examples from a couple of worksheets that I had left for them to work on when I had a sub. We have developed all the rules they needed throughout our Alice lessons but these problems had more "moving parts" for them to work through. After changing my teaching methods this year yesterday was such a "drag." I asked the students if they liked working through the packet instead of the Alice activities and some of them said yes. Today (while we were working through an Alice activity) I asked them whether they would prefer me just tell them the rule and give them practice problems or allow them to explore their way through a problem and help me to develop the rule. Once again I had some that preferred both ways. HOWEVER, when I asked them which way do you think might help them to remember the material 2 weeks from now not one student chose the "worksheet method." I think we will always have students that battle with you over having to think for themselves. There are many times that my students don't actually get ALL THE WAY to the rule or formula or method that they need to do the algebra. However, once they have had some time to "productively struggle" with a concept our discussion is so much more MEANINGFUL than when I just told the students the rule and had them practice problems using it.

One of my first major AHA! moments came when I attended an ACT Quality Core workshop in which Roy Dean was one of the presenters. I actually had to look back for some email communication between me and Roy in order to get his name. This was the first time that I remember sitting through a workshop (I hadn't attended AMSTI at this point) where MATH teachers modeled how to use strategic teaching strategies. He was very patient with me because I was often picking his brain about how he did things in his classroom instead of doing the actual activity assigned. When I found the emails I just wanted to share them because I learned so much from his answer. Below is the email I sent to him:

Roy, 

Thanks for the information and your willingness to share.  So...when I
came back to school and started telling other teachers about the
movement to teach using the methods we discussed I often get the same
question.  Teachers wonder if the group work translates into higher
test scores since the students take the tests (standardized - like
EOC, ACT, etc...) by themselves.  Concerns are also expressed about
having students in our classrooms to learn primarily through the
methods we discussed when most college classrooms are going to be
lecture based.  I didn't want to ask these questions at the workshop
because I feel like they seem argumentative.  However, I wondered if
you have any direction or advice on how I might answer these
questions.  I certainly share their concerns and understand the
questions. 

I have pulled out the "On Course for Success" book and am trying to
skim it to get answers...but I wondered if you could give me some
direction. 

Thanks!
Teri Owens
Etowah High School
Attalla, Alabama

And here is his reply...

Hi Teri, 

Sorry it's taken a couple days to get back to you. The severe weather yesterday had me "enjoying" the Bham airport for most of yesterday.
In answer to your question, I know the "on Course for Success" book has some data that you can use.  I would also think if you Googled reform math scores, group work and test scores, and such if you would find some data on your questions. 

As far as my personal experience, at my school our scores rose 6 points (a statistically significant rise for Colorado testing) the first year and 2 to 3 points over the next 8 when we switched out curriculum to a group/problem solving approach to mathematics. 

I also had a summer school class (not exactly your star students) that I taught with the reform/group/prob solve math for 6 weeks in the summer instead of the usual fraction ws, decimal ws, etc. The students seemed to enjoy the class more.  Of the 24 students I had that actually attended thes ummer classes, 23 improved the test scores the next spring and 8 moved from unsatisfactory to proficient (we have unsatisfactory, partially proficient,proficient, and advanced) and all but one improved their scores a substantial amount. 

As far as students to college, students that returned from college to chat mentioned that the college classes were different (having more lecture) but they weren't hard.  It seemed to me that since they knew the concepts and not algorithms, they could adjust. 

Sorry I don't have any hard data for you.  I at least hope this helps some.Thank you for your hard work during the training. Have a great rest of the year. 

Cheers,Roy

Going from the "Context" to the "Abstract"

I have been mulling over the advice of a veteran IMP teacher (Michael Reitemeyer). He mentioned in an email how he tries to use the context of the lessons to introduce topics but then moves as quickly as possible to the abstract. Sonya and I were discussing today how we would like to develop around 8-10 "homework/classwork" problems that reiterate the concept of the day/lesson. This is not appropriate for every lesson but there are several that it would work well with. For instance, each time we develop/investigate a new exponent law in Alice we could assign 8-10 practice problems. There is a point in Overland Trail where I wish I had given them more problems in which they practiced graphing lines without a context.  The problems they will see on standardized tests won't always have a story to go with them.

Now...having said all of the above...I have been practicing solving equations with my 5th period.  We have been working through a packet together of different types of equations. Today I once again heard some grumblings. I told them that they didn't realize how much they liked the textbook we were using (IMP Meaningful Math Algebra). They were talking about LOVING the textbook. They enjoy the math so much more when there is a story/context to help make sense of it all. I have especially found this to be true with my students who struggle with the math.

IMP Alice Days 9-10 - Continuing the Pattern and A Half Ounce of Cake

The last 2 days we finished some discussion of Having Your Cake and Drinking Too and then moved forward. Continuing the Pattern develops the rule for negative exponents and A Half Ounce of Cake starts the development of fractional powers.

Today I handed back the worksheets I left for them to do on Tuesday.  The worksheets were an exploration on exponents.  I gave them feedback on the ones they missed and told them to make corrections. I started class with a warm up which reviewed the rules which will be quizzed tomorrow.  I had each group project their answers using the Ipads. (Airplay allows up to 10 to be projected simultaneously.) I went over each problem by having students to defend their answers. It was great. I have really been enjoying the Ipads!

Some students used expansion...some used the rules...and some used the Alice context... fun stuff!

IMP Alice Day 8 - Having Your Cake and Drinking Too

When I saw that the teacher's guide allowed 60 minutes for this activity I thought there was no way. I thought we would get through it quicker...haha! We didn't even finish and we had OVER an hour. This activity helps the students to explore and HOPEFULLY discover the rule for dividing powers with the same base. I love it! It is really revealing the lack of number sense that my students have. The questions are not really that difficult...that is why I thought we would breeze right through.

I have allowed myself to "sit" on this activity and not rush it because I really believe it might help to cement the concept. I am going to be absent tomorrow from my classes and I have some good exploration worksheets on exponents to leave for my students. However, it has reminded me how blessed we are to have found a book that combines the exploration with the context. Someone had a wonderful imagination!

**Update 3/7/15** Original post was made 3/16/15
This year when I taught this lesson I took the time to help the students to explore #2. I did not let anyone blurt out an answer when we first started looking at it. I have been using the random integer generator on the TI-83 (which is awesome but I just now figured out how to do it!!) to call on students. In my classes I just happened to call on students who were unsure of what to do so I just asked them to start by making a guess. I told them to tell me a number of ounces of cake and a number of ounces of beverage and then I showed them how to "test" their answer to see if this would give an answer where Alice's height was multiplied by 8. Then I randomly called on more students. The first 3 students guessed more beverage than cake (ugh!) so I asked the next student to make a "conjecture" on whether or not more beverage than cake would EVER allow Alice's height to be multiplied by 8. My 2nd class arrived at the correct number of ounces much faster than my 1st one did but I feel that modeling to the students how to "guess and check" was valuable. I am always telling them not to ever erase their guesses. I want them to learn to look back over the ideas that didn't work in order to help them to identify new theories that might work!! Anyway...the day I taught this lesson this year I felt really good about what we had accomplished:)

Show me how to teach it differently!

I remember walking into our Instructional Partner's office a few years ago because of my frustration with my algebra classes. My failure rate was climbing and I was teaching algebra just like I always had - and it had worked well in the past! I remember asking her if she knew of a different way to teach algebra. That year I taught the "repeater" Algebra IA class in the Spring and I chose to teach the majority of the class using the sample units we had received at the ACT Quality Core workshop. That was the first time that I had seen a math teacher model some strategic teaching strategies that I had seen in some ARI workshops. I was often frustrated thinking that the strategies worked for other content areas but the majority of them just didn't fit in the math classroom. That Summer I attended my first AMSTI training and decided (although I wasn't completely sold out on all the activities yet) to make the commitment to put my students in groups and keep them in groups for the entire school year. I hated it at first but once I adjusted I doubt that I ever go back!

That next school year I looked for and used several station activities and review activities. I also came across some inquiry/discovery-based activities that I loved. I was constantly scouring the Internet for resources and making copies for my classes. My student engagement improved in that school year. The following Summer I attended year 2 of AMSTI training and the light bulbs started going off. I realized that the "AMSTI activities" were the type of activities that I had been looking for on the Internet. I had always thought of them as activities to do AFTER I had taught the concepts and I didn't think I had time for that. A phrase that one of the trainers kept saying to me was , "Quit thinking like a math teacher. Your students wouldn't do that!"

Through these experiences I have come to realize that for the majority of my teaching career I have taught all my students as if they were all good math students who were going to pursue educational goals that involved upper level math classes. I do have a few students who fit that description...but the majority of them don't. However, I was teaching them all the way I preferred to be taught...but the majority of my students aren't like me. They don't love math and math does not come easy to them.

Fast forward a few months to my implementation of the IMP Meaningful Math Algebra curriculum...almost everything I teach is tied to a context. I am finding that even my students who are not great at math will interact more with the activities because the context makes the problems more accessible to them. They might not be able to just see a "naked math" problem and figure it out but when they have a storyline surrounding it they have a context that helps them to make sense of the math. So now I not only have a curriculum that is discovery/inquiry based- I have a curriculum that also presents the algebra concepts within a context. It may have taken me a few years to find the resources/ways to teach it differently, but I have definitely found it. Now...I just have to learn how to teach using them. I am working on that!

I appreciate all of the people who have helped me to transform my teaching. I don't want to try to name them for fear of leaving someone out.

IMP Alice Days 6-7 - Piece After Piece, Many Meals for Alice, and In Search of the Law

For Many Meals for Alice and In Search of the Law I really put my Ipads to use. I am learning better ways to implement using the Ipads. I gave each group a few minutes to brainstorm on each problem and then told them that at the end of the allotted time they had to put SOMETHING on the Ipad (using the Educreations app). The Airserver app allows you to project up to 10 Ipads at a time so I have an Ipad in each group (sometimes more). I had told my students that if you have an Ipad you HAVE to be mirroring the screen. I can see all the screens on my computer. This has been a tremendous help for keeping my students from using the Ipads for purposes other than the math! When I turn on my projector where the students can see everyone else's work it is interesting. I am still amazed how my students will approach problems so differently when I don't stand at the board and show them an example of what to do (which causes them to all try to do it my way). We have an easy way to compare and critique the work without it being totally obvious whose group it came from. Then they are able to edit their work when it is needed. Formative assessment on the fly!!

I had a student volunteer to "teach" - she was great!

The context given in All About Alice provides such a neat alternative to simply just giving the students the rules of exponents. Piece after Piece establishes the rule for multiplying powers with the same base. Some of my students wanted to ADD 2^3 and 2^5 so to help them I told them to remember that we are always looking at what we are MULTIPLYING Alice's height by. I am interested to see how other IMP teachers handle this...

Many Meals for Alice establishes the rule for raising a power to a power. The original power is for the number of ounces of cake she eats at each meal - #1 is 3 ounces of base 2 cake so it is actually 2^3. Then the students figure out what is happening to her height after certain numbers of meals. In the end they have hopefully developed the idea that (2^3)^4 means that Alice ate 4 meals where she had 3 ounces of base 2 cake. It takes a few times of looking at it to start making the connection. I LOVE #3.

In Search of the Law actually has the students to explore exponent rules that I have not usually put much thought into. It took me a minute or two of exploring myself - I actually had to tell my students to start working on #1 while I went to read the teacher's guide! When I glanced over the activity the day before I taught it I didn't realize it was going to throw me a curve. Anyway...this activity is actually not difficult AT ALL. It gives the students several scenarios to just work with exponents.