What not to do in algebra class...read if you want to laugh!

I think that I decided to write this blog post because I feel that the last 2 days I am a shining example of what NOT to do when you are teaching a problem-based curriculum. I think that everyone should have the ability to laugh at yourself so here I go...

So...I was in the middle of a discussion with my algebra class about the unit problem for IMP Fireworks which involves a celebration where a rocket is shot off of a building and the fireworks need to be set on a timer that allows the fireworks to "go off" when the rocket is at it's highest height. The problem is GREAT because some smart student (who obviously was an ace at physics) has already determined the equation for the height of the rocket with respect to time. My students have to find the answers to the following questions:

• How long will the rocket be in the air?
• When will it reach it's highest point?
• What is the height of the rocket when it is at its highest point?
• ...there are a few more but these are the ones we were mainly discussing...

So...I am supposed to be letting them determine how they might use the equation to answer these questions and I was getting this from my students:

• How are we supposed to know, Mrs. Owens?
• But...we don't know how high the rocket goes...
• So...why don't they just shoot the fireworks off the top of the building instead?
• I plugged in 3 seconds and got _____ for my answer. So, that's the highest point.
• ***I can't even remember what all this one student kept asking me...he was so bothered by the situation and the way they were doing it. He was also bothered by the fact that "quad" means 4 but for an equation to be quadratic the variable has a power of 2. I had another student to help me out and say that the power of 2 means that it is squared and a square has 4 sides.

So...my response was finally something like this:

O. My. Goodness!!!!!!  Just do the math!! You have the equation so the hard part is already done for you. JUST SUBSTITUTE TIMES IN FOR T AND MAKE A TABLE!! Quit getting so bogged down in the context that you forget how to be a math student!!!! You can use the table to estimate the highest height and the time that the rocket hits the ground!! Just be quiet and DO IT!! Use your algebra skills!!

I am laughing as I write this. I had one student to say, "No Mrs. Owens. We are learning how to apply this math in the real world!"  It is funny to reflect back on this because I was so frustrated trying to move them forward. We had worked on introducing the unit for way longer than we were supposed to and I was just trying to motivate the need to find the exact values...which we will learn how to do as we study the unit. I did have a good conversation with them about how the ACT and other standardized tests have portions (especially science reasoning on the ACT) where you are sometimes given a formula and even told what each variable stands for and all you have to do is plug in the values and simplify!! OR that they are given charts and graphs where the information is right in front of them and they have to just interpret what it means. You can sometimes get too bogged down into trying to "figure something out" when you can just read the chart/graph and find the answer!!

IMP Overland Trail - Getting the Gold

I skipped Getting the Gold when I taught Overland Trail last year. I am so sad that I did! This is such a cool activity because it has such real-world applications. In this activity the students are asked to compare the profits of 2 different ways to gather gold. While discussing this activity it is fun to bring up discussions about making business decisions.

The discussion points that are a part of this activity include:

1. Business start up costs
2. Profit
3. Breaking even
4. BONUS...This is the first activity where the starting point is negative so it is fun to watch the students say..."the starting point is negative this time!" 

**I still catch myself wanting to give the student too many hints. I am sometimes excited about how the activities bring in different aspects and I just want to point them out! HAHA!

This activity and Water Conservation are fun "wrap-up" activities that have the students once more create graphs and answer questions. I love to ask questions like:

How do you know how much profit they had on Day 7 according to the graph?

Now, how can you answer that same question using the rule we created?

Could you use a table to answer that same question? 

 

Courageous math teachers

I had the opportunity to spend 2 days last week with some incredible math educators in Alabama. The pilot that was started at Etowah High School last year has expanded through a partnership between AMSTI and It's About Time. There are math teachers throughout the state that are now piloting the full IMP Meaningful Math curriculum. While sitting there listening and learning with these teachers I was in awe. These math teachers are committed to trying something new in order to improve student achievement.

Susan Jeffers says to "feel the fear and do it anyway" and many of these teachers are doing just that. I heard many teachers grappling with the unfamiliar territory of things like assessment and assigning homework within such a different teaching format. It is exciting to be associated with teachers who are willing to get out of their comfort zone in order to try a curriculum that is time tested and has brought results over and over again.

A few years back I came to the realization that I wasn't reaching as many students as I used to and I started seeking a different way to teach math. God directed me down a path that involved coworkers, workshops, Tweets (the MTBoS especially!), and a "chance" meeting of the president of It's About Time in an elevator! Remember that the sky is purple in my world (haha!) but I really believe all these things have lined up to improve math education in the state of Alabama. I am amazed that the small pilot at one school has grown into a state-wide pilot involving many. I am thankful that AMSTI and It's About Time are providing this opportunity for the schools in our state!

IMP Pit and the Pendulum Days 1-2 - The Question and Initial Experiments

On the first day of Pit I was actually issuing books. Therefore I used the WONDERFUL advice of Jim Roebuck and we listened to a Youtube video in which the entire story was read. If it were not for the fact that I needed the time to issue books anyway I probably would have gotten too impatient and skipped to the portion that read the excerpt from our algebra books.

After the story I assigned The Question where they draw a sketch of the situation and look for information in the story to answer the big question...does the story's hero really have time to carry out his escape plan? On the first day we didn't completely finish the sketches and didn't even start the discussion so we continued it on the 2nd day. We made a list of what we know from the story (especially the items that are mathematicaly relevant). They include the following:

• The ceiling is 30-40 feet high
• The pendulum swings perpendicular with his body
• The pendulum was 3 inches from his body
• He thinks he has 10-12 sweeps or vibrations (back and forth) before it will touch him
• It will take the rats one minute to eat through the rope (yuck!!)

Then we arrived at a "revised question" - How long does it take the pendulum to make 10 swings? We are also going to use the assumption that the ceiling is 30 ft high. 

Initial Experiments pg. 201

Okay...the fun part was then asking the students what we needed to know to answer this unit question. I had one boy who instantly said that we needed to know how long it took the pendulum to make one sweep. We had a class discussion about all the things that might effect this time and we narrowed it down to 3 things that we could actually test in class: weight of the bob (end of pendulum), length of pendulum (which is the height of the ceiling in the problem), and the angle of release (amplitude??) of the pendulum. I am not used to using the word amplitude for this type of problem. I keep picturing the perpencicular distance between opposite sides of a parallelogram!

Anyway...the students enjoyed building pendulums. I assigned each group one variable to test but I didn't give them any guidelines other than to test each weight (or length or angle of release) 10 times. Therefore they did 30 total trials. Also they only timed one sweep. After the experiments they decided that the angle of release didn't matter but that the other 2 variables did. Then we discussed issues that may have effected our data like not using the same angle or pendulum length when you were trying to test different weights. Or not using the same pendulum length and weight when testing angle of release. Also, some student held the pendulums in their hands and kind of helped the swing by moving their hand up and down. I look forward to doing more investigations and tying in the statistics as we go!

Students working on their sketches

P.S. - Thank you Jim Roebuck for giving us helpful hints on how to best teach this unit!

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

I have been challenged by Brian Lawler to answer the same End-of-Year Reflection questions that my students answered. He reworded them a little and I am going to paste them into this blog and answer them. They are all very thought-provoking!

1. How was this experience "teaching mathematics" different from your previous work teaching mathematics?  How was the math itself different? Did you learn the mathematics differently?

This teaching experience has been different in numerous ways. First, I have never taught a curriculum that had unit problems or "themes." Having a context for almost every algebra topic that I taught this year truly did make the subject more meaningful to my students. Secondly, the tasks are written in a way that students are given the opportunity to discuss and "struggle" with the problems even if they do not initially understand the math behind it. The teacher's guides always provide you with great "leading questions" that help you to guide your students to discovering the math without you just saying, "This is how you do this problem. Write it down." Having the teacher's guides AND having seen this style of teaching at AMSTI training were huge helps for teaching this curriculum the way the authors intended (or at least close to the way it was intended to be taught). We also received training from It's About Time in which we were able to go through many of the activities as "students."

When going through the training as a student (at AMSTI and It's About Time training) I was reminded often to quit thinking like a teacher. I think that one piece of advice was one of the most helpful. At first I would only see the training from a teacher's perspective and I would be worried about what formula I should use to solve the problems. As I taught this curriculum I have realized that the students are asked to use common sense, repeated patterns, and the context in order to solve the problems. The formulas can also be used (and taught, of course!) but when a student is taught to totally rely on formulas and then they get on the ACT (or other standardized test) and forget the formulas they don't have the problem-solving experiences that will help them to persevere and be successful.

This is my 2nd year to teach the entire year with my students sitting in groups of 4. Although I had already taught with students grouped last year, the majority of the year the only function the groups had were that my students could check to see if they had the same answer on a problem and help each other if someone was confused. This year the IMP Meaningful Math Curriculum provided my students with opportunities to utilize group work in a whole new way. The problems were presented in such a way that the students would start discussing their ideas on the best way to solve the problem. Sometimes a few of them would work quietly until they felt like they had an idea to share with the group. Other times they would sit there and talk about it before they tried to put pencil to paper. The exciting thing was that the groups this year were used for actual mathematical discussions about how to solve problems.

I think what I "learned differently" was that the students will really and truly try different approaches to solving problems if you give them the freedom. When I used to stand at the board and show them how to do a particular type of problem that is the way they did it. However, I have seen multiple times this year that if I give them a task and then give them the opportunity to figure it out on their own (with the support of their group members) they will solve it with various approaches that make better sense to them. I use to teach them the way that I thought was best. This year has taught me that struggling math students do not interpret and work through a problem in the same way that an algebra teacher does!

2. 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 math-teacher colleagues changed?

I had a day or two that I would kind of go back into my "old teacher" mode and stand at the board doing examples and "giving notes." I would actually stand there and think that I was boring myself to death! HAHA! I have learned a new way to teach that is much more engaging. I do not want to go back to my "old teacher" mode again.

My confidence in my ability to teach math has grown. I have always been a confident math student. I was good at math and so I wanted to be a math teacher. In the past I believed that math was something that some students were gifted at and others were not. The way this curriculum is written gives students more than random "number crunching" in math class. This is a problem-based curriculum in which they are constantly applying the math within a context that gives it a purpose. Using this curriculum literally helped me to reach students that had failed my class in previous years because of lack of interest.

I am blessed to be a part of a terrific team of math teachers at Etowah High School. Sonya New and Gary Webb were also implementing this algebra curriculum. Sonya and I were able to discuss our lessons on a daily basis because we had the same planning period. It was harder to have discussions with Gary but we did have lunch with the entire math department so we were able to talk to him some during lunch. I do not believe we would have been as successful without the opportunity to collaborate and learn from each other.

I also reached out to other IMP teachers via email and Twitter throughout this year. I have found so many helpful teachers who have shared their teaching ideas and resources.

3. What are your mathematics-teaching goals for next year? How have those goals changed over the past year and why?

My main goal is to keep improving. There are many times I felt that I was blindly going through the curriculum this year. I would sometimes hesitate to introduce a particular "math formula" because I didn't want to "steal the thunder" of a future lesson. There are so many concepts that the curriculum kind of allows the student to develop his/her own understanding instead of a teacher just telling them how to perform the problem using a formula or particular process. Another goal I have is I want to do a better job teaching my students how to present their work next year!

My goal of teaching students to present their work is different because the types of tasks that they do in this curriculum are different. For example - If a student is asked to solve a system of equations where they are already given the 2 equations there is not a lot to discuss. They can go to the board and tell the class the method they chose (substitution, elimination or graphing) and then work it out. In the IMP curriculum the students would be given a scenario in which they have to write their own equations and then solve the system. They would have the opportunity to discuss how they assigned their variables, wrote the equations, solved the problems mathematically, and verified that the solution was viable within the context. There is so much more to discuss!

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

Results after 1 year of IMP Meaningful Math Algebra

I just wrote a post in which I shared some student reflections after their first year of IMP Meaningful Math Algebra. I have written several posts reflecting about the differences of the curriculum. I have learned so much about how to facilitate "productive struggle" in the classroom. The new curriculum along with thing I learned from blogs, Twitter posts and professional development (esp AMSTI and It's About Time training) have all combined to help me to make improvements in my instruction and test results. Many teachers ask about how the "new curriculum" is going and it went GREAT. Even though we were told not to expect growth in the first year of teaching the curriculum we analyzed the data and WE SAW GROWTH. Woohoo!

We had 3 algebra teachers at Etowah High School - Sonya New, Gary Webb, and I - who implemented the curriculum 7 weeks into the school year. There are 5 units and we only had time to complete 4 of them. We did not cover the Pit and the Pendulum. In the state of Alabama the last 3 years 8th graders took the ACT Explore. After our students complete algebra they take the ACT Quality Core Algebra End-of-Course test. We looked at this year's 2015 9th graders (who had the IMP curriculum) and compared them to last year's 2014 9th graders (who were taught with a more "traditional" curriculum). It is a little difficult to explain but I will try so that you will see that the results are valid. Instead of just looking to see if the average scores on the EOC tests improved we compared students who came in with the same score on the 8th grade Explore and then compared each group of same-scoring Explore students from the 2 years. We averaged each group's Algebra EOC test results and compared them. Then we just did a +/- on whether or not the scores improved or declined for each category (i.e. students from each year who came to us with a 12 on the Explore). When we factored in all of our students we had a +1.94. All year long we have said that we think the curriculum is beneficial for all students but that it really seems to make a bigger difference for our non-honors students. Therefore we took out the honors students from each of the 2 years and then did the +/- for growth again. We showed total growth of +9.02 which we believe to be an average of about +1.13 per student. The EOC Algebra test scores range from the 130s to the 150s so we feel that the improvement is significant.

Now...I am definitely not a statistics major so there is room for error in the analysis of this data. However, we are very excited to have seen this growth in our first year! We know that we have so much room to improve - especially since we didn't have time to cover all 5 units.

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 Fireworks - Distributing the Area I and Views of the Distributive Property

I love that Mrs. New has already been through this unit a couple of times. She give me great tips! For "Distributing the Area I" she advised that we make a handout with the 6 area model rectangles so the students could just write inside them. This helped the activity to move along faster than if they had drawn the area models. Some of my students are perfectionists so they would take forever to draw a perfect rectangle and then have little time to work on the actual task in the lesson. My students seemed to really enjoy this activity. After I showed them #1 as an example they pretty much took off with it. I even had one boy who I have to get on to often for sitting and doing nothing to be the first one to get the answers for #6. I was proud of him!

"Views of the Distributive Property" was an activity that made them think about the way that they multiply 2 digit numbers. They are shown through the activity that they have really been using the distributive property and partial products all along. My students whined a little bit about having to do the problem involving the long form but I tried to explain to them that this was just a foundation on which we were going to build as we started multiplying algebraic expressions.

I have taught multiplying and factoring polynomials using the area model for a couple of years. I have called it "the box method" but it is the same thing. Even though I feel like it gave the students a more visual way to work through the problem and it also helped them to organize their work I still had some students who would take forever to get the idea that the product inside the box comes from multiplying the 2 values on the outside of the boxes. I am excited about the way Fireworks has built the idea of using the area model. My students will have the idea of the "lots" changing size from the "A Lot of Changing Sides" activity so hopefully they will feel like the answer is just like adding up all the smaller areas even when we are multiplying polynomials.

Another thing that excited me today is that I got through these activities in the time that the teacher's guide recommends. That is an accomplishment for me!

IMP Fireworks - A Lot of Changing Sides

I really enjoyed doing this activity with my students. It starts with a background story of a housing developer wanting to change the lot sizes for a new housing development. Instead of all the lots being square the city planner wants some of other types of rectangles. The questions tell the students exactly how to change the dimensions of the lots. The first 4 are situations where they increase the size of the lots and the last 2 involve a decrease in the length of at least one side.

Mrs. New had already taught this lesson and showed me the way the teacher's guide recommended the sketch of the lots be drawn. These diagrams will look like "the box method" that they will use to multiply and factor polynomials. The activity asks the students to express the area as the product of the length and the width (which will be binomials) or as a sum of smaller areas. Since the 2 areas are equivalent the students are led to realize that the 2 expressions are equal. I aksed them to look for connections between the sum and the product and a few of them saw it.  I love that the authors have once again provided a context for the formal math to make sense to them!

I led the students through #1 so I could model how to sketch the diagram with the original side length of X. In #5 and #6 I let them come up with their own diagrams to represent the situation. Also, just for the sake of organizing, I labeled the bullets as A, B, and C so that it would be a little easier to organize and discuss.

IMP Fireworks - Using Vertex Form, Crossing the Axis, and Is It a Homer?

In Using Vertex Form, the students have another picture to create with their graphing calculators. Then they are given equations in vertex form and asked to give the vertex. Most of the students could do this without using the graphing calculators but a few still depended on them. I advised them to use the "Trace" feature of the calculator to find the coordinates of the highest point and then compare the coordinates to the equation. This improved their confidence in finding the vertex.

I gave my students a quiz where they had 6 questions in which they matched quadratic equations to their graphs. Then they had a couple where they had the equation and had to list 3 things they know about the equation. The last 2 questions had them describe how to flip a graph so that it was concave down and then sketch a graph with 3 different parabolas and give their equations. I was so excited about the quiz results. I wish I could say that all my students aced it but that is so not true! However, the large majority of my students passed the quiz and many made As and Bs. I have never expected my students to be able to do so much with graphing quadratic equations. The way the activities led the students through the process was so thorough it made the quiz seem easy.

The Crossing the Axis lesson gets students to start thinking about how many x-intercepts the graph of each quadratic equation will have based on the phase shifts. The activity also has students to write the equation given the vertex and another point on the parabola. They have to use the information to solve for the value of a. Numbers 5 and 6 are very important to complete because they give the students the tools they will need to complete the Is It a Homer activity.

Is It a Homer is an awesome activity to me as a former softball player and coach. It was also fun to the students. Mr. Webb shared a link with me of a Youtube video of a "dramatic reading" of a poem about the "Mighty Casey." We watched it before we read the activity. They are challenged to figure out if the ball clears the fence and they must prove it mathematically. I had the students sketch the graph with height on the y-axis and distance from home plate on the x-axis (which was advised in the teacher's guide). I gave the students the hint that they will be using the same process they did on 5 and 6 of the previous activity. After giving the students some time to think I went to the board and sketched the graph and labeled the vertex. I asked the students if there was another time when we knew what the height of the ball was. My 2nd block students chose to use (0,0) and but my 4th block students pointed out that the ball was not hit off the ground so they used (0,3). They struggled some with the computation of this problem but a few students in each class got the answer correct based on the height of the ball at contact. I had one student who told me that he estimated that the ball would fly 400 feet because the maximum height was after the ball flew 200 feet. Even though we have not yet talked about the symmetry of the graphs he had recognized it and used it for his reasoning. Unfortunately he didn't tell me his thoughts until AFTER class. I will be sharing it with my classes tomorrow.

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 Day 4 - A Wonderland Lost and Scrambling Equations (Overland Trail)

We started class today with another warmup on solving systems of equations. My students will have their 3rd and final opportunity to retake the quiz in order to improve their grade for this 9 weeks.

Once we went over the warmup I assigned A Wonderland Lost. We read it together and discussed it some first. I love the question the teacher's guide advises you to ask about how long it will take the rain forest to be completely gone.  Several students think it is 10 years. I did an example at the board using a beginning amount of 100 and worked through decreasing by 10 percent per year. I led them to the "shortcut" of taking 90% of the previous (or beginning) amount each time. I am learning to be more comfortable writing exponential equations within the contexts we've used so far. I don't know if my students are going to retain these lessons but I'm quite sure I will. This was a time when the teacher's guide gives you the rule.  I had to stop and think about it to be able to explain it. AND relating back to the walk thru we did for the Alice problem certainly made it easier for me to understand and explain! After today's lesson I took a few minutes to give notes on the differences between exponential, quadratic, and linear functions. I also made sure they heard me use the vocabulary terms of exponential growth and decay.

My 5th period finished the Scrambling Equations activity from Overland Trail. I invested a little more time in this activity this time through and I think it went better.  I found many mistakes in the students' work and tried to clarify some misconceptions. I had my students write their "complicated equations" on notecards. I also provided a problem I made up and wrote on the board for students to use if they didn't feel certain of their problem. After they worked each other's problems I asked them to write down a brief description of the process they used to solve the equation. They used phrases like "canceling out" and "did the opposite." We also emphasized that looking on the side with the variable tells you what to do. After the activity I passed out a worksheet for them to practice 1-step equations. We didn't have much time to work on them today. We will build up in difficulty and have a quiz by the end of the week.

IMP Alice Day 3 - Logic POW and Mystery Bags revisited

Today my Algebra IB students took a short quiz on systems of equations. The grades were not all that great but Ms. Whitt (my incredible coteacher) and I did our best to talk to each of the students about what they missed. The students will be given the opportunity to retake the quiz within the next week.

Ms. Whitt tutoring a student one-on-one on solving systems of equations.

 After the students turned in the quiz they were instructed to start working on the Logic POW. This is a very different POW than any that we have done before. It gives the students several pairs of statements and asks them to figure out if there is a logical conclusion that can be made. Some of the pairs do not have logical conclusions. I think this is a great way to introduce them to the type of thinking and reasoning that is required in geometry!

My 5th period Algebra IA class started Mystery Bags this week. Today we did the More Mystery Bags activity. I appreciated the context the first time I went through this last semester. I went back and read the blog I wrote after that day. The first time through most of my students seemed to have a pretty firm grasp of solving basic equations. However, my 5th period class needed more prompting. When the students had a problem like 5M +3 = 2M + 15 the context gives me such an easy way to explain why you would first want to take 2 Ms (or mystery bags) from each side of the equation (balance). In the mystery bag activity the constants are referred to as the weights that are on the balances. Therefore it once again makes logical sense that the next step is to take 3 ounces of weight from each side. We want to get mystery bags only on one side and weights only on the other side. Almost every student then understands to divide the amount of weight by the number of bags. (They are told that each mystery bag has the same amount of weight in it.)

Tomorrow they will be thrown some problems with some negative numbers. I explained to them today that the concept of getting all the mystery bags (variables) on one side and all the constants (weights) on the other will still work.

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.