Internship for Mathematics Education Leadership

Pamela R. Hudson Bailey

George Mason University

Fall 2009 – Spring 2010

EDUC 994:

Creation of MATH 606

AFDA: Mathematical Modeling with Probability and Statistics

The course is part of a Mathematics and Science Project grant through the University of Virginia of which our school system participated. The grant included two courses that were centered on the new mathematics course Algebra, Functions, and Data Analysis (AFDA). A focus on the study of functions was the subject for the fall term and the spring term was on probability and statistics. Myself and a gentleman from Richmond were selected to write the spring curriculum. Concepts to be covered during the spring session were probability, combinations, permutations, experiments, surveys, bias, box-n-whiskers, normal curve, z-scores, and probability using the normal curve. The course is a mathematics course but there is also to be a focus on how to facilitate the instruction of the content.

We put together the matrix first using email to run things by each other and also met with Vicki Inge, the lead person on the grant for UVA, to streamline our thoughts. At a meeting in Charlottesville we met with the entire team and shared the plans and gathered comments and suggestions. The team consisted of individuals that were heading up the creation of units or teaching in one of the five sites.

One of the major concerns for the spring course was that originally there were only two Saturday’s for face-to-face instruction and the remaining sessions were online. The online sessions would use Elluminate Live! that is on the Blackboard site and the Digimemo so participants may communicate more fluently with each other during these sessions. At the first meeting I brought this to everyone’s attention and proposed that we change it to four Saturday sessions which would allow us to present the four major concepts in a face-to-face setting. This would also give us time to work on calculator issues where some teachers might not be comfortable. Mrs. Inge was concerned about the criteria of the grant but after some investigation it was deemed that the change was possible and very much needed.

The final product was a matrix for quick overview, an expanded document with more detail, and a pre/post-assessment to show student growth. Our goal was to have it submitted to Mrs. Inge prior to the holidays as it needed to also go through an approval process to insure that it would meet the institution’s requirements for that level of course. We did not run up against any concerns and the sites began their first sessions in January 2010.

All participants received a Digimemo and a Calculator Based Ranger (CBR) to be used to gather data during the sessions. If they participated in both sessions they would get to keep both pieces of equipment. Participants also received the text and two books from the National Council of Teachers of Mathematics.

Attachments:

Matrix Overview

Expanded Course Document

Pre/post-Assessment

INTERNSHIP JOURNAL

January 9, 2010 - Saturday Session #1

Today was the first day of class. It was all day with some returning students and three new students. We did activities to get to know each other while getting acquainted with the course expectations. The class is on statistics that include probability, types and ways of gathering and presenting data, and the normal curve. We begin with probability. I have found out that many are anxious about the course so teaching for mastery is a comfort but also scary. The teachers are not used to this type of teaching along with part of the course being online. Since the class was all day long we varied the instruction from whole group to small groups as well as varied those that were in the small groups. This also helped for them to get to know each other.

The new people did not seem concerned about using the digimemo or Blackboard. We have some teachers that have not done any statistics since they were in college, one teacher works with special education so does not have a math degree, and one has a middle school degree. It was good to see them working very good in their groups, not only sharing the work load but also being sure that everyone in the group understood the concept.

I feel good about the class and the communication we have. Our next session is online but I volunteered to go over to a high school to help some use the digimemo. I have also opened up the board office for the online sessions for this course for those with dial up or just feel more comfortable being where someone can help them.

January 12, 2010 - Online Session #2

It was a great evening. Three hours online and it was over before I knew it. I planned group activities interspersed between whole group sessions. We shared information using our digimemos and even shared calculators screens using Elluminate Live! Once we got out of groups to talk about their findings and I had them get back into the same groups and they were able to do so without assistance. Many have picked up on how to share their viewing screens with the groups. Our discussion for the session was again on probability and associating it to a linear and area model. Even online I could tell there were some Ah Ha moments by some of the comments. We were working on one of the problems and a teacher was showing how she came up with her answer when at the end another teacher shared how to do the problem using the area model. The first teacher’s approach was purely symbolic. I could have jumped for joy as many high school teachers are very procedural and symbolic. When many said that it made more sense to envision the problem pictorially I steered into a discussion about methodological approaches with students in our classrooms.

One student has his television on loud and we did have to ask everyone to minimize outside noise. It worked for everyone but the one student. This student is also very opinionated, very procedural, and does not view change as productive. Everyone seemed to tolerate this teacher during the Saturday course but a few were unhappy during the online. I am able to write notes to students so did request for him to minimize noise. (This student also requested that all class session dates be changed as well as meeting times.) ;)

January 26, 2010 - Online Session #3

Probability is the topic once more with the addition of conditionals. The students just amaze me with the acceptance and abilities to adapt to online. I was anxious about how we would teach statistics online with the vision of the course we are modeling, Algebra, Functions, and Data Analysis. This new course in our state is all about experimentation, laboratory, student-centered classes that are relevant to the students. How was I going to do that online??? This session we used our calculators and shared data and screens with each other. Each online session seems to be better than before. We worked in groups, shared out to the whole class including calculator screens and what was hand written on the Digimemo.

February 6, 2010 - Saturday Session #4

We are such a small group (10 altogether) so we decided to have a working lunch. A real sense of community is developing. There is actually 2 different groups which I tried to mix up a little to get them out of their comfort zone. Topic for the day is relating Venn Diagrams to probability and the multiplication principle. We worked in various groups and a lot of rich discussion came out of the activities. There was a disagreement on one question and the remarks were not just an answer but were combined with reasoning and justification. This is just what we want out students to do. At the end of the day we talked about our actions as participants in the lessons. Some still seemed to believe that they were able to reason and justify because of their current knowledge. Reference was made to some of the comments which were based on what we were experiencing and the illustrations that were being used. Many had also experienced discomfort due to their lack of knowledge and understanding of the concepts. This led to discussions about how our students feel.

The class ended with a discussion of the “Task Creation”. Students were to read an article about increasing students intellectual needs which encouraged teachers to create experiences for students that would acknowledge past concepts but lead students into wanting to know more or to learn more relevant mathematics. We broadened the topic from statistics to whatever the teachers were doing with their students so they could use the material with their classes. I gave an example using rational expressions in which students could take many routes to find the answer and there could also be multiple answers. The assignment is due prior to session 6 so they have time to think and ask questions.

February 9, 2010 - Online Session #5 (Made up on Feb 23)

We were getting in to types of experiments and surveys and when each would be appropriate to use. Bias will be another issue that will be discussed in the next few lessons as well. The lesson began with questions about the task creation assignment – there were none. Next, leading in to experiments we began activities about sampling techniques. This lesson was overwhelming based on using the calculator and a lot of data crunching. A few of the teachers are not as fluent with the calculator as others so they did help each other but being online did pose some problems. I let the teachers pretty much lead the lesson with their comments and group work sharing while I put calculator screens online so those having trouble could follow along easier. Everyone did, by the end of the lesson, understand the various kinds of sampling methods - in groups they were explaining to each other and giving examples of each.

March 2, 2010 - Online Session #6

I began the lesson with having vocabulary words on the screen and asking them in groups to discuss the words. We came back together and I just called on individuals to share out what they knew. This helped me to gauge their knowledge and helped the teachers to recall information that they may have not used in many years. They seemed to like that. We discussed the article but there was little talk – don’t believe most read it and many had not downloaded their reflections of the article. We used “Navigating through Data Analysis” book to encourage discussions on bias, validity of the study, what was actually being studied, and does the conclusion answer the question stated. A lot of discussion. I believe many felt that they could contribute because it was what they assessed from reading the short scenarios of various studies. Big discussion on treatment groups and control groups for an experimental design. Some felt that there could be a treatment group and a control group and other’s felt that there had to be three groups, treatment, control with placebo, and control with nothing. We agreed to research the topic more and discuss at our next online session. This also led to a discussion that every class does not need to end with everyone happy and knowing the material presented. There is nothing wrong with students wanting to know more and going off to try to find answers. Curiosity is good and a want to know more but teachers need to be careful not to discourage students. For the assignment teachers were to find an article in the news and assess it for several things like what was the treatment and control groups, question being asked, results, bias, placebo …

March 16, 2010 - Online Session #7

Titles for lists were put online – Types of Samples, Types of Bias, and what is quantitative and qualitative methods. Discussions resulted as we filled in the lists. We picked up on the discussion we left for all investigate during session 6. We agreed that everyone was correct, it depends on how the study was set up. I began the lesson by giving the teachers a scenario about needing a study on homework being assigned in a high school. They got in to groups and planned how they would set up the study, the types of sampling they would do and why, the bias that they needed to be careful of or what might exist with their approach, and how they would report the results. We ended up with only two groups and shared out giving each other constructive criticism. This led to the concepts we needed to cover for this session. The last activity was applying what we had discussed and discovered so far with regard to a study on diets and cholesterol. Writing questions about what you as a reader would want to know, from the data gathered what questions could be answered, and what additional data might be gathered to answer the remaining questions. To help with understanding we simulated those who might have problems with cholesterol by flipping a coin. My overall goal was for the teachers to realize how important the simulation was to their understanding – the topic of our closing discussion of class concepts. We also discussed the mini-project that is due and presented at the next Saturday session. I gave them a few minutes at the end to get in to groups if they wished and to decide what they were going to do. This may be group or individual due to the wide area of where everyone lives and works.

April 17, 2010 - Saturday Session #8

When I began planning for this lesson I at first thought it was going to be boring for the students. We were going over basic statistics that many students learn in middle school – mean, median, mode, range….adding in IQR and outliers. Knowing high school teachers and the tendency to be formula focused I decided to really play up the meaning of mean, median, and mode by working on a conceptual understanding of balance using two different approaches. Boy has it been eye opening. We got so much in to it that we didn’t get done what I had planned. That’s ok…..I want the participants to really understand the meaning of the measures of central tendency so we can build on that for the measures of spread. We compared histograms with box-n-whisker plots. Math talk is getting better. The teacher’s are explaining….if not correct then the group has and is becoming a learning community. We talked about the development of the learning community and about how it has taken time. CBR’s were used again today. This time the teachers got right in to it with only a couple of them needing help due to the application to run the CBR not being on their calculator.

One of the students still is very vocal about traditionalism. Likes to talk about “my students wouldn’t do ….” or “..it would never work with my students”. Today another participant asked why they (the students) wouldn’t participate. A good answer was never given. I can see that if these teachers were also in the same high school and collaborated together that there would be some pressure put on this teacher to change. Is that a good thing?? Creating disequilibrium will promote change according to researchers. I can also see that collaboration will also help some of the others continue growing with the support of each other.

We began the day with students entering class with upbeat music. They liked that! We then used a technique to find a partner and even talked about our actions in group work. The class started with stating the goals of the day and as we accomplished them it was also noted. At the end of the class we reviewed the goals and how they were met, using it as a review and as closure. Looking at Loucks-Horsley, Stiles, Mundry, Love, and Hewson (2010) framework for professional development --- We are working toward the vision set forth for the new course, we did not look at student data, we did set goals for the course and for each session, working on plans, doing, and evaluating. Participants are working on lesson plans for a lesson study, gave me the dates today when the presentations will be made for the groups, and then doing a self-evaluation and a debrief after the presentation. Are we changing beliefs? I think we are progressing.

April 27, 2010 - Online Session #9

The goals for the afternoon online session was on the introduction of the normal curve and z scores. Very concerned about presenting this topic online but it did work out great. We used the digimemo, which allowed us to write the mathematics just like we would on any paper on the board, students used their calculators and I used SmartView. Both allowed us to share information with each other so we had visuals about what we were verbally stating. Tonight we referred a lot to what we did on Saturday…spread and histograms. A great deal of time was spent on entering data….discussion about how we could combat that . We spent time this evening doing a lot of that…discussing how we would work with our students to gain the conceptual understanding. I was happy when the questions began flying among the participants. What would happen if …. and what would we do to show….and what is expected of students in an Algebra I or II class on this same topic. This was not time wasted. I felt that it supported the teachers in their growth.

Lesson study presentations are all done. One group presented their lesson implementing everything I had ever hoped for. They made up a scenario about tiling the floor…the students at the end of the class asked if they could come and help lay the tile. They, the teachers, were very convincing and the students were engaged in learning and exploring. Multiple representations were applied by the students and the math talk was insightful. All of the sessions were good and some were better than others but in all cases the teachers learned by discussing what they could have done and by hearing about how the students perceived the lesson. Several made comments that they would like to do this more often – observe lessons and discuss them afterwards. One group saw that they had manipulatives for the students to use but never modeled their usage or gave them time to do so.

May 4, 2010 - Online Session #10

My confidence was up even though we were working on normal distribution again. I supplemented the lesson with an activity that I had found and used with teachers in my school district successfully. The goal of the session was to determine the characteristics of the normal curve and to get in to probability. I gave the students 4 keystrokes steps to illustrate the concepts. They were to work in groups but the questions they asked were so good that we ended up working together. I wanted the others to have the benefits of hearing and thinking about the questions being asked. One of the weaker students was on cloud 9 because she understood and was able to expound on some of the answers. You could hear it in her voice, the excitement. That is good as she will need to teach this topic to our lowest level of Algebra I students next year.

We spent a lot of time drawing this evening. Every instance we drew the curve and acknowledged + 1, +2, +3 standard deviations from the mean. The mean, median and mode was also put on each curve to help us determine if the curve was normal. At one point we graphed a curve with a specified standard deviation and mean. A participant said that the curve was correct because it had to be skewed. We discussed why and then plotted on the curve the mean, median, and mode. I would not tell them what to look for to determine skewedness and which way. We continued to do this. The participants knew the data was not “normal” and by the end of class had hypothesized why and how they could determine the skewedness. At the end of the three hours I knew I was tired mentally and a couple of the participants commented that their brains were fried. After I got home that evening I got an email from the “traditional” teacher that he loved the hand drawn portion and that the students would understand the calculator after going through that activity.

There is one more class so I encouraged the participants to get in all of their work. Several have become lax and are missing several assignments. I try to be understanding as we are all adults with many other responsibilities but some have more than 5 items due. They have a deadline to get them in to me. One last mini project is due at the last session. Each will need to gather data and do all types of statistical analysis and graphs. I stressed that they needed to look at the rubric provided. One thing that has been frustrating is that teachers complain that students do not do what they expect even when they know the rubric and how much everything is worth. The teachers are no different. I have had to ask teachers to do some over as they did not following instructions let alone the rubric. During the last session the students will quickly describe their lesson study class, each will also present their last mini project and all will take a post test which I do not grade but is for the grant. If time permits there are a couple of activity on the normal curve that I would like to go over that we didn’t have time to complete during the session. Most of the day will be spent on housekeeping items and collecting equipment for those who did not complete both semesters.

Overall Comments

I will be interested in receiving the participant’s evaluations. Consciously each session was based on Loucks-Horsley, et al. (2010) framework for professional development. Since the students were from several different school districts it was difficult to look at “data” but I did have a chance to look at their pre-test and of course their work as the semester progressed. My overall goal was to encourage the participants to begin the journey of change. Changing their beliefs on instruction, student learning, and content connections is difficult. According to Guskey (1986) teacher beliefs change only after they have had the opportunity to see positive results with their students. Teachers were requested to go back to their classrooms and put into action what they were learning and to reflect on their actions as well as their students. The reflective journals are also due at the class but they get credit as long as they do it. For the teachers in our school district I can see the model by Guskey coming to fruition. The teachers were engaged in the professional development, some of the classroom practices changed, they saw positive results and attitudes which led to some changes in the teachers beliefs and attitudes toward teaching the concept. I have also seen that teachers who are weak in the three types of knowledge were more reluctant to change anything or to believe that it was possible. Repeatedly we hear that secondary math teachers have a degree in mathematics but they don’t know the connections between the mathematics. We talked a lot during the semester about being open to learning from our students and to truly listening to our students. By doing so we will learn connections and it will encourage us to look at concepts with a different view.

Lastly, this course had only four face to face sessions with the remainder of time online. As noted in my journal there were many times that I had concerns about how to introduce a topic online. The equipment we used was priceless and we would have been unable to be as effective as we were without it. Teachers adapted to and even exceeded my expectations of their abilities to use and incorporate the technology on the online sessions. It did take a different frame of mind to plan each of the online sessions so that all would be participating and would be able to understand concepts. In the back of my mind I kept thinking of the Rule of 5 and how to do that online. The Smartview program or graphing calculators enabled most of that to be possible as well as using the digimemo. Participants had an informal conversation during one of the face to face sessions about technology which led to discussing the appropriate usage of the equipment. During this semester I believe that the technology was used appropriately and to the benefit of the students. Would I teach this course again. Yes.