To earn the Certificate of University Teaching from Duquesne University, I participated in teaching workshops, mentored new TAs, and designed and delivered lessons under formal observation. I was the first student in the Computational Mathematics program to earn this certificate.

My official PRAXIS scores are on file with the Pennsylvania Department of Education. I will provide copies upon request.

Mathematics Content Knowledge (0061): 155

C-PPST Reading (5710): 184

C-PPST Writing (5720): 179

C-PPST Mathematics (5730): 188

Dr. John Kern, Chair and Associate Professor of Statistics at Duquesne

Dr. Zimmerman, Professor of Mathematics at Robert Morris

Mr. Robert Weet, Supervisor of Student Teachers at Robert Morris

]]>To realize these goals, both students and teachers must cooperate and take responsibility for their roles. Both the student and the teacher are necessary components of the classroom. If one is missing, there is no classroom and there will be no learning. The responsibilities of the teacher and the student are different but equally important.

The role of the student is to acquire the knowledge and skills described by the standards of the courses they take. Students must construct their own knowledge by creating meaning for themselves. They must be prepared and available to engage with the material and with others in order to build on their prior knowledge and experiences.

The role of the teacher is to design and facilitate an active, enriching, and safe learning environment that provides opportunities for students to construct their own knowledge and skills. To create this environment, I first need to know several things about my students: their level of math skills and prior knowledge, their preferred learning style(s) and special needs, and their personal goals and interests. I use their level of math skills and prior knowledge to inform where I begin instruction. I use their preferred learning styles, special needs, goals, and interests to differentiate my instruction in ways that are most helpful to all students. I design lessons to accommodate learning disabilities, to modify the use of materials, and to appropriately set up the physical classroom environment for the safety of the students. I also design projects and activities that allow students to choose the topic or format. Some components of such an environment with examples of how I have implemented or plan to implement them follow:

Active Learning Opportunities | Multiple Representations | Multiple Intelligence | Building on Prior Knowledge

Modeling | Collaborative Learning Opportunities

Expectations and Feedback | Motivating Students and Building Rapport | Safe Learning

Whenever possible, I design hands-on activities with a real world focus to engage students with new procedures or abstract concepts. An example of a real-world activity that I used to engage students in the data analysis process and to develop their statistical thinking comes from my *Business Statistics* classes at Duquesne. Students gathered data by weighing packages of M&Ms and counted the number of each color. Students entered the data into an online survey and analyzed the results in a spreadsheet. The applications of this student-collected data included discussions and activities about cleaning data, calculating numerical descriptive statistics, creating graphs, calculating probabilities, and determining independence.

Activity Post: M&Ms Introduce Students to Statistics.

Another example of an activity that I used to illustrate sampling distributions is when I asked students in my *Fundamentals of Statistics* class to draw several random samples from a population of test scores and to calculate the average score for each sample. All students plotted their results on the board to create a histogram of the averages. This exercise illustrated the sampling distributions in a concrete way, which improved student understanding over simply asking them to read the theorem.

Activity Post: Create a Sampling Distribution.

I facilitate learning through a variety of methods so that all students have an opportunity to use their strengths to learn. Students have different learning styles and benefit from lessons that engage them in multiple ways. Mathematics lends itself well to a variety of representations such as verbal descriptions, diagrams, tables, graphs, and formulas. Graphic organizers provide a means for students to represent and organize information in various forms, which helps them process and remember new information. During my student teaching, one student who was an English language learner struggled with the different types of lines and their respective equations. To help him process and remember the similarities and differences of parallel, perpendicular, and coincident lines, I created a graphic organizer where students could model these three concepts verbally, visually, and with formulas. Not everyone took advantage of this, however, the ELL student did fill it out and proceeded to get every related question on the test correct.

Activity Post: Types of Lines Graphic Organizer.

During my student teaching, I created a project where students designed a building or outdoor space with measurements written in the form of a binomial. Students wrote about their design, calculated the area using the FOIL method, and completed their project on a poster board with pictures or drawings. This project allowed students to demonstrate their knowledge and skills through writing, drawing/art, and computation.

Activity Post: Get Creative with Polynomials.

Prior knowledge includes knowledge and skills from prerequisite courses as well as new knowledge recently acquired in the current course. To assist students in recalling concepts or procedures learned in previous courses, I either write relevant formulas or definitions on the board for students to reference or I engage students in a short anticipatory set consisting of problems that use the required prior knowledge.

To help students navigate the new content in *Fundamentals of Statistics*, I provided a course map at the beginning of the class as part of the syllabus and course schedule. The map illustrated the “big picture” of doing statistics and showed students how each section fit into that big picture. I referred to this course map and to prior lessons to promote understanding of what we were doing and why we were doing it.

As a math teacher, I teach mathematical concepts and procedures as well as problem solving skills. I engage students in hands-on activities to help them discover the underlying concepts. After this introduction to the big ideas of the lesson, I implement an “I do it, we do it, you do it” scenario. I first demonstrate how to solve a few problems. Students then work together in groups to solve additional problems while I am available to assist them. Finally, students work through homework problems to reinforce the skills they developed in the classroom. To help students develop problem solving skills, the problems students practice in groups and on their own get progressively more difficult, requiring them to recall and use prior knowledge and skills.

Cooperative learning allows students to gather and organize information, problem solve, and process results together. As students complete tasks, each one moves toward reaching academic goals and objectives. A successful cooperative learning environment requires the following elements:

- All students in the group must participate and take responsibility for their own learning
- Students must engage with the other students, i.e., use communication and leadership skills to assist and encourage the others
- Students must be accountable to the others and demonstrate mastery of the information
- Students must participate in a group reflection on each other’s progress and the effectiveness of their contribution (peer evaluation)

Students know what I expect from them through lesson objectives and rubrics. Each one of my lessons begins with an outline of learning objectives, which are written to the student in the format, “After this lesson you will be able to…” For projects and large assignments, students receive a rubric that details the project requirements and how the points will be distributed among them.

Activity Post: Activities with Rubrics.

Students receive feedback from me in many forms. After a lesson, I may ask students to solve a problem relating to the lesson so I can verify they understood the material and address any misconceptions before moving on. I also provide solutions to homework and assessments and encourage students to compare their answers and ask questions during review. On graded assignments or assessments, I provide written feedback in the form of short notes about errors or student growth/improvement.

If a student seems to struggle with the material, I work with the student individually or in a small group during workshops or office hours. One ELL student in my Algebra class was struggling with foundational skills that were required to learn the new material. Even though this student’s English was quite good and he was very close to being released from the special education services he had been receiving, there were gaps in his knowledge from when his English was not as good. I worked with him to help him learn the skills he had not completely grasped in his earlier years.

If a student is unprepared or misses excessive classes, I reach out to him or her and to the parents to discuss the school policy on attendance and the importance of attendance and preparation to performance. One student in my *Fundamentals of Statistics* class was taking the class for the third time. She had a significant amount of math anxiety so my mission was to get her to my office hours as much as possible. She responded whenever I reached out to her. I worked with her on her homework and research project, and she successfully completed the course.

Mathematics intersects many different fields of study and I strive to create problems or design activities and projects that allow students to discover mathematics through a topic that interests them. Real-world activities or word problem scenarios help students to see the relevance in what they are doing and can motivate students to be prepared and available.

The research project I assigned in *Fundamentals of Statistics* allowed students to perform real research to answer a question that interested them. After choosing a topic and developing a research question, each student designed a research study to answer their question. Students engaged with statistics in a real-world context as they conducted research and applied what they were learning in class to summarize and analyze their results.

Activity Post: Activities with Life Application.

I encourage students to ask questions and to be considerate when others ask questions. There is validity to every question. All students benefit from the answers or discussion generated from other students asking questions. The obvious can sometime elude any of us when we are learning a new concept or skill; it is okay to need a reminder or to need something explained in a different way.

Homework is essential for reinforcing concepts and practicing skills. I use homework as a formative tool giving credit for completion not accuracy. Students must attempt every problem in order to get credit. It is easy to watch an instructor do problems and to think you understand. But this is a passive learning scenario that does not give the students the practice they need. Instead I challenge students to evaluate their own work and to ask specific questions that will help them solve the problem. Cooperative workshops where students help each other find their errors and where I am available to assist if they need it is a valuable exercise.

I believe it is necessary to assess student progress in a variety of ways because no two students have the same background knowledge, life experiences, learning styles, or personalities. I use both formative and summative assessments. Formative assessments show me where students are in the learning process so I can help them improve before taking a graded test. Summative assessments show me how well a student learned the material and are incorporated into the final grade.

My formative assessments include open-ended questions, observations, conversations, journaling, quizzes, and performance assessment problems. The information I gather from these sources guide my instruction. If only a few students seem to understand the content, I can re-teach the material using different strategies. If most students know the material, I can work with the few who do not know it in a one-on-one or small group setting to help them master the material. The information I gather will also enable me to provide constructive feedback, which will enable students to monitor their own learning.

Along with state standardized tests, my summative assessments will include unit or chapter tests, semester exams, and other performance-based projects or activities. Final grades will be based on a variety of summative assessments that allow students to demonstrate their mastery of the content and skills. The information I gather from these assessments will help me determine if any given student knows enough to move on to the next course or if the student would benefit from repeating the course.

Because students are active participants in the learning process, I communicate my assessment system with them at the beginning of the year. I also ensure that parents understand that my assessment system is designed to facilitate student learning as well as to be an overall indicator of student achievement.

]]>When I first did this activity, I used only plain M&Ms. If you have enough students (to gather a large enough sample for analysis), I recommend using both plain and peanut M&Ms. I had my students gather data on peanut M&Ms during another class period but the novelty had worn off by then. Gathering data on both types of M&Ms allows you to explore more concepts than with just one type. See Concepts to Explore with the M&M Data below for ideas of how to use this dataset.

After these lessons, students will be able to…

- Gather and record data
- Define basic statistical terminology
- Apply terminology to the context of the activity

- Kitchen scale that measures to at least two decimal places
- Fun size packages of plain and peanut M&Ms (enough for each student to have at least 1 package)
- Copies of the anticipatory set of questions and post-reading guided notes (one set per student)

*Note:* I also have Hershey Kisses on hand for students who cannot eat M&Ms due to allergies. Eating the M&Ms is a perk of the activity.

This data gathering activity engages students in the data analysis process on the first day of class. Students will weigh the package contents and count the number of M&Ms of each color. The worksheet has a table for recording the data. Students can also enter the data into a survey or spreadsheet. I set up a survey in StatCrunch designed to record the weight and color distribution of one package of M&Ms. With this survey, students can submit more than once if they have more than one package of M&Ms. Click here to check out how I set it up and design your own using StatCrunch or another survey platform. StatCrunch creates a table with the variables Type, Weight in Ounces, Brown, Yellow, Red, Orange, Green, and Blue. Each record (row) stores the data of one package of M&Ms.

The pre-reading questions introduces basic statistical terminology from the context of the activity.

The terminology is from the first chapter of *Business Statistics: Communicating with Numbers* by Jaggia/Kelly. The homework I assigned was to read chapter 1 using the LearnSmart technology by the publisher McGraw-Hill.

On day 2, I discussed the terminology using the M&M activity and other examples. I projected the key on the board but encouraged students to fill in their guided notes with whatever they thought would help them remember the definition. Writing the definition word for word is not necessarily effective.

Terminology Guided Notes

Terminology Guided Notes Key

- Cleaning data – It is very likely that someone is going to enter bad data (typos, misunderstanding instructions, or just being funny) and this is a good opportunity to discuss whether records with questionable data should be ignored, fixed, or deleted).
- Numerical Summaries – Collecting weights and color counts allows students to create both categorical and continuous summaries (proportions and means).
- Graphs – I used this dataset to demonstrate creating bar and pie charts and histograms in Excel. Students compared the distribution of weight and colors for plain and peanut M&Ms.
- Probability – I used this dataset to illustrate relative frequency, probability, and independence. What is the proportion of red M&Ms? That
*depends*on what type of M&M you are talking about.

Mathematics is a tool of practical application—the art and science of solving real-world problems. The problems and their solutions come in many forms and include, but are not limited to, proving numerical and geometric relationships, finding unknown values, developing processes, and analyzing and representing data.

To engage in mathematics means to apply inductive and deductive reasoning to a set of assumptions, definitions, and patterns for the purpose of solving a problem. This mathematical reasoning leads to logical processes, implications, and conclusions that can then be applied to another set of assumptions, definitions, and patterns to solve a different problem.

]]>- Description (10%) A typed, double-spaced, one-page description of the question the student wishes to address and his or her plan for gathering data.
- Draft (30%) A typed, double-spaced draft of the project that includes the Introduction, Methodology, and Data Exploration sections outlined below in the Project Outline.
- Final (60%) A typed, double-spaced final report incorporating any suggested changes to the draft and the remaining sections—Results, Conclusion, and Future Study—as outlined below in the Project Outline.

- Introduction – Describe the project and scope of the problem addressed.
- Methodology – Provide any details of the approach to the problem solution. State methods of data gathering, difficulties, assumptions and/or constraints, and the statistical analyses/tests used.
- Data Exploration – Provide both graphic and numeric summaries of the data and document the data exploration with explanations of any and all calculations used. Provide tabular results in an appendix to the report.
- Results – Provide calculations and results of statistical tests/inferences.
- Conclusion – Provide a clear rationale of the conclusion(s) reached in this project.
- Future Study – In a project of this nature [with time being a factor to properly complete a full investigation of a question/project] provide a description of what should be included in further study of the problem.

Project Instructions

Stats Project Rubric

SAT Scores Income Spring 15

Nurse Anesthetists Salary Spring 15

Lucky 7s Spring 15

NFL Running Back Spring 15

Hockey Period Scores Spring 15

Penguin Wins Spring 15

Each student will obtain and calculate the mean for eight different simple random samples of size 4 from a population of exam grades. The class will construct a histogram of sample means to illustrate a sampling distribution. The teacher will demonstrate a sampling distribution from the same population using sample size n=2. Students will consider the sampling distributions of sample means and the population distribution in light of the purpose of statistics, i.e., to make inferences about the population from sample data.

The Central Limit Theorem will be discussed in the next lesson. It is not practical to use this lesson to illustrate the Central Limit Theorem because students would have to calculate the means of multiple samples with n=25 or more. The more difficult concept here is the sampling distribution itself. This activity lays a strong foundation for students to understand the Central Limit Theorem by illustrating what a sampling distribution is. I then use computer simulations to illustrate the Central Limit Theorem.

Students will be able to…

- Generate a sampling distribution and construct a histogram of sample means.
- Differentiate between population distribution and sampling distribution.
- Compare and contrast the various distributions and draw conclusions about how sample data can be used to make inferences about the population.

Students will be interested in this lesson because it is highly visual and interactive and involves data they see on a regular basis. Students will benefit from this lesson as a necessary step for understanding a crucial theorem in statistics, i.e., The Central Limit Theorem.

- Computer, projector, and screen
- 35 copies of the activity worksheet

(Recall)

- The population distribution describes how individuals vary in the population.
- A normal distribution is a continuous distribution that is bell shaped and symmetric
- Illustrate that samples are taken from a population (usually too large to evaluate as a whole) for the purpose of calculating statistics that can be used to make inferences about the unknown population parameters, e.g. population mean and standard deviation.

- Construct a sampling distribution from a large number of samples.
- Each student obtains eight simple random samples (SRS) and calculates the means (worksheet).
- Students construct a histogram on the board; each student marks his or her eight sample means.

- Define sampling distribution and contrast it with population distribution.
- The sampling distribution describes how sample means vary in repeated sampling.
- A sampling distribution is the distribution of a statistic, such as the mean, drawn from a population and calculated for all possible combinations of samples of a particular size, n.

- Demonstrate using smaller samples, n=2, from the same population, N = 10.
- Compare and contrast the population distribution and two sampling distributions.
- Discuss the implications of the similarities with respect to the purpose of statistics.
- Define the sampling distribution of the sample means:

If the individual observations are normally distributed, N(μ, σ), then the distribution of sample means, with samples of size n, is normally distributed with mean equal to μ and standard deviation, called the standard error of the mean, equal to σ/sqrt(n), N(μ, σ/sqrt(n)).

Student evaluations (exit slip)

- Describe an aspect of the lesson that you found interesting or practical.
- Describe an aspect of the lesson that left you confused or questioning something that I left unanswered.
- Was there any part of the lesson that was especially good?
- Was there any part of the lesson that I could have handled better?

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**Reflection**

Some students were really involved in this activity and obviously had fun finding their facts. They found some really interesting ones such as the estimated number of blades of grass on the Earth, the number of gallons of water on the Earth, the radius of an electron, the number of individual M&Ms made per day, the number of people employed by McDonald’s, etc. My students were very competitive and started competing for the largest and smallest numbers.

Scientific Notation Webquest Instructions and Worksheet

Scientific Notation Webquest Student Samples

The Posttest Write Abouts are part of the curriculum at Fort Couch Middle School. The teachers grade the write abouts using the PSSA Mathematics General Rubric, which is a holistic rubric that measures the level of understanding exhibited by the student in the answer (advanced, excellent, satisfactory, almost satisfactory, and minimal understanding). My cooperating teacher suggested I create and use my own rubric without first seeing the PSSA rubric. I created an analytic rubric so I could give specific feedback on the solution and the explanation as well as award partial credit for answering only one of the questions asked in the problem. I awarded up to 3 points for problem accuracy and up to 2 points for the process description.