Math Classroom Conversations - IMP Making Friends with Standard Deviation

Math classroom conversations

#mathtalk

The things I have been hearing my students say during this activity...

"Why is the standard deviation the same?!?!"

"Why is the mean so high this time?"

"What?!?!"

"The mean for Set C is...."

"I disagree."

"I agree."

"Is this what you got?"

"Andrew shut up!" - I had to include this one just to "keep it real."

"It says explain why your pattern..."

"No pattern occurred." to which I did a loud "AHEM" and they said "Are we supposed to get a pattern?" 

So...my students had this group investigative task and I helped them to get started. BUT...in this case I made them read the directions out loud and then I asked another student to repeat the directions in their own words AND we did an example with a data set on the board. (This sentence might make more sense if you read this post entitled "Read and Follow Directions!") Then I told them that I was going to sit down and if their group had a question the ENTIRE GROUP had to come to my desk. I usually roam around the room but I believe that I sometimes have students ask me questions that they should really be asking their group members (just because I am close).  I did have a couple of groups come to me to settle an argument...isn't it awesome that they were arguing over math concepts!

MTBoS Blogging Initiative - Better Questions

Boy...when I look at the prompt for this blog post my mind goes in so many different directions. I wrote this post in which I discussed trying to use "quality questioning" in my classroom. I was not really talking about questions to put on an assignment, quiz, or test. I was thinking about questions used to do the following:

• guide students to think deeper during discussions
• guide a struggling student toward understanding a concept 
• scaffold and access prior knowledge
• defend and/or explain answers or reasoning

The prompt seemed to be more along the lines of writing questions for assignments or quizzes. I happened to attend a PD today on Webb's Depth of Knowledge (DOK) in which we discussed asking more level 2 and 3 questions when we assess our students. Of the 4 levels of questioning we were told that the ACT and the ASPIRE assessments have very few questions that are level 1 (basic recall or computation) and the majority of questions are levels 2-3 (harder stuff...HAHA!). Anyway, the majority of textbooks are filled with level 1 questions but not many that are levels 2 and 3. 

For those of you who want to know here is a VERY brief description of Webb's DOK:

Level 1 - Recall and Reproduction - "Right there" questions where you can look it up in a book or follow steps from an example
Level 2 - Skills and Concepts - "think and search" questions where you have to put information together or categorize - these may be open to using different approaches and explanations are often required
Level 3 - Strategic Thinking and Reasoning - More than one way to approach and more than one possible answer - non-routine problems are often used here - often asked to state and support with evidence
Level 4 - Extended Thinking - Extended thinking that takes more time - possible products would be films, plays, research reports (with multiple sources), video games, documentaries, newspaper articles, etc...

(source: A Guide for Using Webb's Depth of Knowledge with Common Core Standards by Karin Hess, E.D - copyright 2013 Common Core Institute)

We were given some strategies on things to do in order to use our textbooks and resources that we have and take the questioning up a notch. One example of moving into the Level 2 questioning is to ask for "non-examples." We often ask our students to give an example of a _________ but they can demonstrate an even greater understanding of concepts if they can also give a non-example.

I had the pleasure of using two non-routine tasks with my algebra classes this week. Both of the tasks seemed impossible at first but when we continued to work toward the solutions we found that there were ways to arrive at a solution.

The first task was the Shuttling Around Problem of the Week #13 in my IMP Meaningful Math Algebra book. It is actually a puzzle where you really have to get out manipulatives to work through it. On the day we introduced the problem only one student found a solution. It took him a while but he finally videoed it so that he could email it to me as part of his POW write-up. The funny thing is that even though I stood there and watched in order to verify that he had a valid solution...I could not do it myself. So, today I allowed another class some time to work on the task and I was going to sit down and figure it out myself so they would see that it was possible...but I couldn't. I called down to the classroom where the guy who found the solution was and he came to my class and showed us the solution again. After watching him do it I had several students go back and work to figure it out themselves...he and I went around the room trying to help and I FINALLY got to where I could do it. The task asks them to investigate other problems too so we weren't taking away all of their fun. It was a great way to end the week! The coolest part of this is that the student who really excelled is not an A/B student. He is rarely ever one who aces a quiz or test. He has an incredible work ethic and tries to do every thing that I ask of him. It was so rewarding for him to have an opportunity to shine!!

The other task was A Mini-POW About Mini-Camel again from our text. One of the great things about our text is the "key questions" in the teacher resources which helps you have ways to guide the students. In this one all I had to say was, "Who says you have to go straight there?" and I had students to begin to find possible solutions. I even had multiple students to go to the board to try to prove to everyone else that their answer was correct. (Here is a link to my Instagram where I posted a video of them.)

I had one student ask me why we had to do these types of problems and I told him that it is important for him to realize that just because something seems impossible at first glance it does not mean that a solution can not be found. I even told them that I may be helping to save their future marriages (haha!) because they may think one day that the only solution is to give up but remember that one time in algebra class they kept on trying and working at a task that seemed impossible only to find that there was a solution!! I know that is goofy but I got some giggles and I do hope that these problem solving skills stick with them after they leave my class.

I love that I have these tasks included in our textbooks! I wrote my last MTBoS blog about how my textbooks are my favorite tool that I use in my classroom here.

Hey...I still love math:)

I know this is a strange comment for a math teacher to make. For years I have just taught math. I chose to become a math teacher because I love math but to be honest all the years of teaching math to people who do not really want to learn it had just "sucked the life" right out of my math enjoyment! I was covering the standards but rarely ever used any activities that peaked the curiosity of my students.

Thanks to our "Meaningful Math adventure" (see this blog post for an explanation) I am having fun teaching math again. I see my students having more fun learning math. Last week I had a student to tell me that she had fun in class that day. I also had some students tell me that we do the most work of any of their classes but my class is also the most fun. I read Teach Like a Pirate this Summer and I have come to realize that it is okay to have a goal of having fun with your students...as a matter of fact we need to throw in some activities that the students will consider a "fun break" from "regular math."

Today I also found myself sitting at my desk working on the High-Low Differences activity in Overland Trail's supplemental activities. I was "noticing and wondering" myself! I was thinking that I really need to find some extra time to investigate why this works like it does. Then I was so ambitious that I answered one of the questions (in a survey I had to take) to indicate that I considered myself to be a mathematician! (HAHA!) I have found value in addressing problems from a student's perspective. Our new textbooks from It's About Time give me many opportunities to have fun working on math and then turn it around to my students as an opportunity to problem solve and enjoy themselves while they do it. I asked them a few times last week if they wanted me to "introduce" them to the activities or let them just try to figure it out on their own. I was amazed at the number of students who wanted to try it without any assistance.

Now, don't get me wrong. I still have students who sit there like "knots on logs" and wait for the problems to be presented to the class so that they can write down the answers - they just hope that when I roll the dice to call on someone that their number is not called. And when I do call on them they tell me they didn't do that problem...and then I talk them through it until I pull the answers out of them...in some cases it would be easier to pull their teeth without anesthesia. I also still have students who gripe and whine and ask for help before I even get the page number out of my mouth. However, it is so cool to catch that student who says he hates our textbooks truly engaged and enjoying himself during an activity (because he figured it out by himself!). This certain student that I have in mind was "called on the carpet" when I told him that I noticed he had fun working on the activity for the day - which just so happened to be "Getting the Gold" that I blogged about here.

I would like to end this blog with a funny picture of what some of my students did last year after I had gotten onto them for sitting there like "knots on logs" instead of doing their work.

Comedians!! 

Sonya New's Reflection after Year 1 teaching IMP Meaningful Math Algebra

Once again I thought that anyone who is reading any of my blog posts concerning the IMP curriculum might like to "hear" from someone else. We are so blessed to have 3 teachers at Etowah High who implemented this algebra curriculum at the same time. Having the opportunity to collaborate throughout the year was incredible! 

The Mom of  2 little ones (like both under age 1 year little) takes a little more time getting her thoughts together. I once again thought you might enjoy hearing from the 3rd teacher (I posted Gary Webb's reflection in a separate post and my reflections in this post) who taught the IMP Meaningful Math Algebra curriculum for the first time this year. Here are Sonya New's thoughts:

Teaching Algebra with the It's About Time curriculum is a much needed complete departure from the norm.  I was always the Algebra teacher that would look at the word problems in the textbook and think wow what a great question and would assign it just to have students not attempt it because "it was too hard" or "I didn't understand what it was asking me to do", so as the year progressed I would resign that kids just couldn't do those problems and basically stick to practice of the most basic problems.  Even after "going over" the "hard" problems my students didn't seem to get it.
When we received our new textbooks my students opened them to discover mostly words, very few numbers, and virtually no "traditional" practice problems.  Students are taught Algebra through situations.  Many students have found Algebra to be a very attainable subject that once thought it was "so hard".  As a teacher and lover of math I have also discovered that Algebra doesn't have to be so structured, formulated, and procedural.  The concepts of Algebra are often "common sense" and when approached from that direction make sense to many students.  By the end of the year my students were no longer afraid of the "hard word problems." They were not intimidated to try them anymore.  They would try to make sense of a problem and work their way around to a solution. Still not all would get the correct solution but at least we had something to work with ;-).
There were times during the year I would question the curriculum.  Are my students really getting it?  What about this formula or this method?  When is this concept covered?  I have learned to relax and trust the progression of the curriculum.  Things are not taught in a traditional progression, but the topics do get covered.  I am still working on my balance between completely trusting and supplementing more practice but I am coming around.  I anticipate each year to get easier for me to understand the beauty of the curriculum and to do a better job of facilitating.  I know this one thing for sure...it may have been my first year to use the curriculum and there were definitely flaws in my implementation but I don't want to teach Algebra using anything else!!!

Gary Webb's reflection of 1st year teaching Meaningful Math Algebra

There are 2 other math teachers who have gone through IMP Meaningful Math Algebra with for the first time this year. I have mentioned them both from time to time in my blogs. One of them is Gary Webb. We were asked to write a testimonial about our first year's experience. I thought another teacher's perspective might be interesting. Here are his thoughts:

I really enjoyed teaching from the Algebra I Meaningful Math book this past school year. It was quite different than traditional math text books. The books have few examples, fewer problems, require deeper thinking, and don't have answers in the back either.

One takeaway I have is that your best students will do whatever you ask them to do. Some of the students are not going to do anything no matter what. These are the ones who complained most about the book. However, these are the same students who might do 5 traditional math problems in 30 minutes and complain about having homework.  The many students who are in the middle were the ones that I was able to reach. Students were more engaged because they were able to use their creativity in the math classroom, were in groups much more often, and were encouraged to discover mathematical concepts on their own.  I loved watching my students think and not be a robot and follow set procedures.

Gary Don Webb
Etowah High School

STUDENT End-of-Year Reflections on IMP Meaningful Math Algebra

Wow! What a year it has been! It has been a while since I have written a blog post due to the craziness of the end of the school year. I have so much that I would like to share.

Here are the student responses I got from the end-of-year review questions in the Fireworks portfolio. We did not have time to do the complete portfolio so I just had them answer the questions on pg. 421 in the book. I feel like what they have to say is more important than anything I could add. I only had one student to just absolutely say that he wishes he was taught out of the "old" type of textbook. Of course the responses I am sharing below are the "fun" ones for me to read as a teacher. There were some students that talked about how they didn't like that the book was so "wordy" but those same students later admitted to growing more confident and learning how to work in groups. I also had a few students to tell me that they still preferred to work alone but the overwhelming majority had positive things to say about group work. I told my students to be honest with their responses and give good explanations to support their comments. I told them I really wanted to know what they thought.

The first question set included the following:

How was this experience different from your previous work in mathematics? Did you learn the mathematics differently? How was the math itself different? 

Here are some of the responses: (I really wanted to fix all the grammatical errors...but I didn't because I didn't want to put my "spin" on what they said.)

• The books we used this year was all word problems and that will help me during high school and college. 
• Working in a group helped me understand better cause some of them understand better than I did and they helped me understand it better. 
• It was more fun with the activities and been taught different.
• ...the mathematics itself was longer and a bit harder also
• My past experiences I didn't understand anything but now everything seems a lot easier. (this is a repeat algebra student)
• This book is also different because it never (is) just straight on work it always has a fun story. Also it helps you a lot more than any other math book.
• We actually learned about real work stuff. We did a POW about having a house, paying bills, etc...

The second question set included the following questions:

How have you changed personally as a result of your experience? Has your confidence in your own ability grown? How has your experience of working with others changed?

Student responses:

• Working with others - I've started talking about it more than just trying to work on it.
• I'm more conficent in math now than I ever was. (this comment was repeated by several students)
• All year I've had a group to work and to collaborate with so I do believe I have gotten better working with others.
• My confidence in math has grown a lot because at first I never answered out loud, but now I know that I can do it.
• I don't hate math as much. It's not as hard as it was. My experience of working with others has grown alot and I can talk better with other people. (this student is a very quiet and shy young lady) 
• I like working with other people. You get to see what everybody thinks and their ideas. My personality has grown to like math a little bit more. I still kinda don't like it, but I like it more than I did.
• I think I have become more confident. I think I have learned to work with people that I normally don't talk to.
• It's changed (experience working with other) cause if I need help then I can ask my group members.
• It has helped me change by helping me with the word problems to look for clues through the paragraph. I myself had a hard time on word problems till this book helped me out.
• Personally I changed mathematically. My math skills have grown and so has my confidence in my own ability. My experience of working with others has changed like now I can work better with others. I can cooperate with others better. (This student stated in his answers to the first question set that he didn't like the book. I sure did like the results he got though!!)
• Yes my confidence has grown going into ninth grade. I never like working in groups but now I do.
• From a special education student: My confidence grown alot since last semester. It change because I use to just copy people because I didn't know how to do it but now I work together to figure out the answer.
• Working with others gave me more confidence and helped me understand something I didn't know and I could just tell my group and see if they know so we could help each other out.

The last question set had these questions:

What are your mathematics goals for the rest of your high school years? How have those goals changed over the past year and why?

Student responses:

• I also would want to keep learning more math cuase it can actually be fun to do.... But this year in math it has been easier for me and I'm getting higher grades.
• My goals have changed because I feel like I'm trying in my math classes and not just copying.
• I wanna keep improving my math skills for the rest of my high school years and beyond that. My goals have changed over the past year because I learned that I can keep improving my math skills. 

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 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 Overland Trail - solving equations using mystery bags - 5th period

I really enjoyed using Mystery Bags and Scrambled Equations. However, when I gave my students some "naked math" one-step equations they have really struggled. I have one class that is taking all year to work through the 1st half of algebra. They were identified to possibly have a struggle in algebra. I have several incredibly bright students in there. I have often thought - why was this student put in this class...   I had an eye-opening experience today. There were a few of my students who excel when we work in our IMP books who really struggled with today's work. I mean...the ones who are always "piping up" to answer the questions and seem to just "get" the big picture. One student even started the age-old, "When am I ever going to use this? This is stupid!" It took hearing him say that for me to realize that I have not heard that near as often since we started using our IMP Meaningful Math Algebra I books.

Ok...I am rambling a bit in this post so I am going to resort to using bullet points to make sure I get the main ideas that I am trying to convey:

• If I am teaching students who are struggling math students I am going to have to supplement material to help them learn to solve 1 and 2 step equations ...for sure. When integers were thrown in (after mystery bags) it has totally blown their minds. I think I am going to go "old-school" tomorrow and just teach the process in steps.
• After today I have a greater appreciation for how the context and story lines in the books give the algebra more meaning. I think it helps all students but I think it makes an even bigger difference in struggling students who don't usually do well in math.

IMP Alice Day 2 - Graphing Alice - raising to the zero power

I remember Sonya New telling me how much she loved Alice because it gives a context for any number raised to the zero power equaling one. Now I have taught it with the Alice context and it is so exciting! For years I have just told my students to memorize it as a fact. Today I got to use the Alice context... Just in case you might be reading this and you don't know what I'm talking about let me share.

As I'm sure you know Alice (in Alice in Wonderland) shrinks when she drinks the beverage and grows when she eats the cake. The first activity in the unit tells the students to imagine that Alice's height doubles for every ounce of cake she eats and is cut in half for every ounce of beverage she drinks. In graphing Alice they are asked to graph the two situations for 1-6 ounces. For the cake situaion you are graphing y=2^x and the beverage situation is y=.5^x. The outputs (y values) are actually giving you what you will be multiplying Alice's original height by. The  x values represent the number of ounces eaten or drank. The teacher's guide advises you to talk about what happens when she eats zero ounces. This would mean you are raising 2 to the zero power. It was so awesome to have the context to explain that when she eats zero ounces of cake her multiplier (y value) is 1 because multiplying by 1 doesn't change Alice's height!! My students can't appreciate how exciting it is to be able to explain the why. I told them today that I have never been able to explain why anything to the zero power equals one and I just got blank stares. Haha! Some of you may be looking at the computer screen like...

It doesn't matter. I know I am a math nerd but today's lesson made me HAPPY! And I felt the need to write about it even though I am sure it is hard to follow.

IMP All About Alice Day 1 - Alice in Wonderland

We have had multiple delays and missed school days in the past few weeks. Therefore I have decided to skip the last portion of the Cookies unit and start Alice. I'm going to try to make up for it by using systems of equations problems as my warm-ups for the next several days.

Today we read the intro and then watched a film clip from the movie. Thank you to Lori White for providing the film clip this past Summer at AMSTI training.  After giving the students about 10 minutes to work on the 4 questions in the Alice in Wonderland activity I had them to write their answers on 2' by 2' marker boards. It has been a while since we have used them.  I like placing all of their boards at the front of the class and comparing/critiquing the answers with the class.  I rolled my 8-sided dice and called on a group to explain the answer to each question. After the explanation we compared the answers on all the other boards. It was interesting to see the different ways the students worded their answers. For the first part of #1 some students said Alice height doubled twice, some said multiplied by 4, and some said quadrupled. A common mistake made for the 2nd part of #1 was students thought they should multiply by 10 for 5 ounces. Some students did say that Alice's height was doubled 5 times. I did have one student to say her height was multiplied by 32 when she ate 5 ounces. He is such a good math thinker. It was sad to me that none of the groups created a table to answer the questions. I showed them how we could use a table to help us to recognize the patterns which helped us to write the rules.

Reflections after 4 months of IMP Meaningful Math

I am totally sold on the idea that anyone who is purchasing math textbooks should take a hard look at the IMP curriculum - whether your district uses the integrated or the traditional approach. I feel like using this curriculum has made me become a better teacher. I know that I am doing more with my students than I have ever done before.

When I first started working through the book (and I still feel like I am a newbie for sure!) I tried to be so literal. It is recommended for you to go through the curriculum without supplementing (especially the first time through). Because I have not been through the book before I had moments where I was unsure whether or not I should introduce a concept - especially technical vocabulary or formulas - because I didn't want to mess up a future lesson where the students would have the opportunity to approach problems with a more intuitive, context-driven method. For instance, in Overland Trail when students are asked to write the rules for the graphs I had several students (even after being introduced to slope-intercept form) who used a table and worked the pattern back to the point where x=0 in order to find the y-intercept or starting point. Teaching them this curriculum has shown me what it really means to allow students to use different approaches for solving problems. I do not think I understood what that meant prior to teaching this curriculum. I allowed students to "approach" solving equations from different ways - if they wanted to solve an equation by getting the variable on the right side instead of the left I allowed them to do that. HAHA! I have now seen what different approaches look like in a classroom. I truly have some students using formulas, others using tables or graphs, and others writing a paragraph which just explains how they reasoned through the problem. What an education I have had!

Another thing I am realizing is that BALANCE is a key. There are going to be times that we need to stop and give notes in which we make the connections to the "naked math" (my AMSTI buddy Melanie Griffis calls it that!) like they will see on state exams or the ACT. I can remember when Jim Delawder made the comment in our training session that what we will do with the curriculum is so much harder than the way they are tested. Now that I have taught it a little while I totally understand his comment. However, if the students don't make the connections between the IMP-style problems and the standardized test style problems then their math ability will not be reflected in their test scores or future math courses. I also remember Jim telling us in our training that he usually pulls sample questions to practice with the students to show them how questions covering those concepts will appear on standardized tests. We just want to create a system that works for us. Sonya New (my often-mentioned algebra teaching buddy) and I have a goal of identifying places in the curriculum where we take a pause and teach the "naked math" version and practice the standardized-test version.

                    


Lately I have been thinking about how I have always found the need to find materials to supplement the textbook we were using. The difference now is that it is so much easier to find some practice worksheets instead of trying to find or create activities that build concepts around a context. Most of the textbooks I have used in the past were mainly a collection of "naked math" worksheets with a few application problems that were stand alone. In the past I rarely ever assigned those application problems. My students seemed to struggle with the basic problems so I rarely ever went to the "next step." Now that I am teaching with the problem-based curriculum that teaches everything within a context things are so different. The students engage with the problems because of the context.

I am going to publish this post because...I just am. However, I have so many thoughts swirling around in my head that I would like to express but I am at a loss right now. I may add more later:)