Prepping for Introduction to Statistics

I'll be teaching MTH 1210: Introduction to Statistics at Metropolitan State University of Denver this summer.

The class is a general studies course that is roughly equivalent to a high school AP Statistics course. The course curriculum is slightly different, not as many topics are covered, and has a slightly different emphasis in focus of the content. The AP Statistics course tends to require a lot more justification of the assumptions and to place less emphasis on calculation formulas.

I have been putting together materials for the course, trying to work in various investigations and inquiry problems that I gathered at the NCTM conference in April. I have 22 two hour periods to cover the material. I will be focusing on building conceptual understanding and problem solving.

I will be highlighting tasks and investigations that I try. My hope is to see how well some of the ideas I have work and then transfer them to my AP Statistics and my Inferential Probability and Statistics classes.

The class starts on June 3 and will open with looking at experimental design and sample surveys.

Keys to Successful Teaching

I was cleaning off shelves and draws, trying to purge old and unnecessary papers. I ran across a note card on which I had written "Keys to successful teaching." The card was mixed in with papers and notes from a mathematics conference I attended three or four years ago. I remember attending several sessions focused on what it takes to be successful in teaching. I don't claim any of these as my own original thought and, unfortunately, I didn't make note of who had said any of these items. They may be quoted in whole or they may be a paraphrase of what was originally said.

THE KEYS TO SUCCESSFUL TEACHING

• Provide students a rewarding experience
• Give six separate exposures to each concept
• Get students actively involved
• Review constantly, especially key concepts
• Teach in increments to build complex concepts over time
• Give many examples and use simple explanations
• Test frequently and cumulatively

I won't comment on all of these. There is already a large amount of research on getting students actively involved. The inquiry/problem-based approach is an attempt to do just that. Anyone who has read through my posts know that I believe in active student involvement.

I will comment on a few of these statements that relate to brain-based research. 

I don't recall reading on six separate exposures being the optimal number of exposures to a concept. But, research does show that repeated exposures with a spaced-timing sequence is optimal. The idea is that a couple of initial exposures should be followed by another exposure in 3-4 days, followed by another exposure in 7 days, followed by 10-14 days, and then in another 30 days. From this perspective, six exposures would help to cement a concept in long-term memory. Exposure could be continued beyond this as needed.

This repeated exposure then ties directly to reviewing constantly. In essence, spacing out the exposure to key concepts amounts to a constant review.

There has been much research recently on the frequency of testing. The research suggests that more frequent assessments may boost student performance. In one study, students who had video lectures broken up by short assessments took more complete notes and performed better on cumulative assessments than students who viewed lectures without interruption. This leads to the possibility of breaking up lectures with quick quizzes and other intermittent assessments.

It's worthwhile to revisit your thinking and focus from years past. It enables you to see what was of concern to you as an educator before. It also allows you to understand how much you have grown and where you still need to grow.

I've come a long way in my teaching abilities but there is still a lot further that I need to go.

IPS - Reflection

Well, the semester is over and it is obvious that I still need to make further revisions to the material. The big issue is that students did not make any connection between the simulations, redistribution sampling, and bootstrap samples to random events. Looking back, I can see that introducing the software too early stripped students from making a connection between their data and what things could look like.

The overall tendency was for students to just find an average compare against a hypothetical and call it good. They did not consider the issue that their sample results will change every time they take a sample. As a result, if their sample was larger than the hypothesized value, they would simply say it was larger.

I am going back through the order of material and how topics are introduced. I think I need to start the class right from the beginning thinking about questions of interest and hypotheses. I also need to then focus on what happens as we repeatedly measure a result; how is the random variations in results accounted for and used to address the hypotheses we made?

I am also planning on revamping the probability section. For this class, students need to understand probability models and expected values as they prepare simulations. They also need to understand what conditional probabilities are and what it means for events to be independent. But the focus needs to be on probability of outcomes given random events.

Students need to be more attuned to random sampling techniques and experimental design. I am thinking of eliminating the hand-washing study since it is an observational study. I think a study requiring sampling will be more helpful to students.

By using more investigations and studies, I can work with students on ways to analyze the data. My hope is this change will make students more adept at understanding how data is collected and what they should do to analyze that data.

I still need to consider how early to introduce the inference piece of the analysis. I think the basic ideas can be introduced early on and that the specific techniques can be developed as the semester continues.

Finally, I need to make revisions to the final exam scenarios. Especially if students are a little fuzzy on the concepts they have no idea of how to proceed.

I was pleased with the trajectory of topics when I taught the class first semester. But the modifications made for this semester actually look like a step backward. I am disappointed in the slippage but will use the results from this semester to revise and hopefully put the course on a better path.

I would be interested to hear from anyone else who is teaching high school statistics classes that are not AP Statistics and not covering classical statistics. Leave a comment and we can get a discussion going.

Discrete Math - Reflection

Another semester of discrete math has ended. The final exam results were good and tracked how I expected things to go. The mean score out of 100 was 78 with a standard deviation of 10.1.

In problems attempted, the C-level questions had consistent numbers of attempts and showed no statistically significant variation (p=0.154). The same was the case for B-level problems (p=0.265). There were not enough attempts made on the A-level problems to conduct a statistical test.

In examining the attempts versus the number of successful completions and partial completions, I identified three questions that should be reworded or replaced.

In talking with students as they left or encountering them later in the hall, the students said they found the class challenging but interesting. They liked the change of pace from regular algebra and that they were allowed to explore and work through problems. This is consistent feedback with the first time I taught this course.

As for changes to next year:

1. I plan on eliminating Bayes Theorem from the probability section and to just spend a little time connecting counting outcomes and sample spaces to probability.
2. I plan to move the graph theory unit to follow the counting unit, enabling more connections to counting and developing proof.
3. Moving the graph theory unit will allow me to include an additional two weeks (approximately) to other graph theory ideas, such as Hamiltonian circuits, graph coloring, and spanning trees. I want to look through these topics to see which provide interesting math that is also satisfactory. If students investigate a problem to find there is no solution or resolution, I want to be sure that a at least it is an interesting investigation or a major topic of mathematical research.
4. I plan on removing the algorithm for solving linear congruences. As long as students understand that the slope for the solution set is the LCM of the divisors, that this value is an upper-bound for the initial value, and they have a strategy or process for finding the initial value it should suffice.
5. I will look at the number theory pieces to see if I can tighten things up just a bit more. 

My plan is to teach the course as is one more time and then look at building in new sections. I have a one semester class, but my goal is to develop a full-year of material. The counting unit and material I have for number theory and cryptography could easily encompass one semester. I've been revising these two units so that I can get a balance with the third unit in graph theory.

Pieces that I will be looking to add-in are set theory and Boolean algebra, logic and proofs, and excursions. The excursion section touches on game theory/decision theory, division and apportionment, and matrix representation and use. I envision that these could fill a second semester of work.

Ideally, I will have core material for each topic and then an expanded set of material. This way, if someone wants to delve deeper into, say, number theory, they can do this and then elect not to cover another topic, such as division and apportionment.

One of the issues I have with current high school textbooks that cover discrete math is that the coverage of division and apportionment tends to be extensive but the math itself is unsatisfying in that there are no optimal ways to apportion votes or divide fairly. As such, the math does not lead to insights or provide a base upon which to build. I'd like to try to avoid those pitfalls if I can.

I would love to hear what others are doing at the high school level with any of these topics. Leave a comment and we can get a discussion going.

Discrete Math - Final Exam

Today was the final exam for the class. The structure was similar to the mid-term exam. The exam was divided into four sections: Graph Theory, Cryptography, Congruences, and Modular Arithmetic. Each section contained four problems: 2 C-level problems, 1 B-level problem, and 1 A-level problem. The C-level problems are problems that I expect anyone in class to be able to understand and answer. The B-level problems are intended to show students have a solid grasp of the material. An A-level problem shows the student knows the material well enough to apply their knowledge in new ways or to extend their thinking beyond what was done in class.

All students need to complete at least 6 questions and they must also complete at least one question from each section. This prevents students from dodging an area of weakness. A minimum of 9 questions must be completed in order to receive an A on the exam: 6 C-level questions, 2 B-level questions, and 1 A-level question.

There was a total of 90 minutes allotted for the exam. Students began finishing the exam after about 50 minutes. About three students took 85-90 minutes to complete the exam.

In general, students seemed okay with the coverage and difficulty of the exam. As one student said, "It didn't make me cry." That's a good thing.

IPS - Final Exam

Today was the final exam for the course. Students have 90 minutes to complete the final.

The structure of the final is a series of three scenarios. The scenarios combine elements of data collection and analysis. There is one scenario that is an experiment, one that is an observational study, and one that is a sample survey. These are matched against analysis techniques: simulation, resampling redistribution, and bootstrapping.

The student picks one of the three scenarios to complete. To discourage students from picking multiple scenarios and hoping for the best, I tell students that if they respond for more than one scenario, I will score the worst response.

Each scenario requires the student to state a null and alternative hypothesis for the question of interest, to describe how they would structure their data collection, to generate a random data set given the analysis technique, to create 5-number summaries and graphs, to analyze data from an inferential perspective, to draw an appropriate conclusion, and to summarize what the big ideas of inferential statistical analysis.

The exam took most students between 60 and 80 minutes to complete. Two students in 40 minutes and one took the full 90 minutes.

Students felt the exam accurately reflected what they learned. As in past years students felt the exam was hard but fair.

IPS - Day 62

Today was devoted to reviewing topics students wanted to focus on prior to the final exam tomorrow. I asked students to take a few minutes to think about questions they had for different topics.

The consensus of the class was to go over expected value, bootstrapping, conditional probability, null and alternative hypotheses, and the 5-number summary and IQR.

The structure of the final exam is scenario-based. Basically, students are to pick one of three scenarios and analyze the situation. This includes developing hypotheses; explaining an appropriate data collection technique; demonstrating they know how to generate random data sets using simulations, bootstrapping, or resampling redistribution; analyzing a data set; and drawing a conclusion.

Conditional probability relates to the idea that we calculate probabilities under the assumption that the null hypothesis is true; therefore the probabilities we calculate are conditional probabilities. As for expected value, this could be a technique used in the analysis but there are other ways to analyze data as well.

The class just wanted confirmation of their thinking about null and alternative hypotheses. Basically that the null hypothesis is what we are hoping to prove incorrect and the alternative is what we want to show is true. We talked through a few examples and the class seemed comfortable with this topic.

We walked through a brief example for the 5-number summary. I asked students how much change they had in their pockets and created a small data set of nine items. We went through the components of a 5-number summary and how to calculate each piece. We also calculated the IQR. From this we constructed a box plot and reviewed how to determine if any data were outliers.

Next, I used this data set to demonstrate bootstrapping, constructing two different bootstrap samples manually. I was able to use one of the samples to construct a box plot that contained an outlier.

The exam is open note, so I gave students the last few minutes of class to organize their notes and get ready for the exam. Unfortunately, not enough students took advantage of this time. We'll see how well they perform on the exam tomorrow.

Visit the class summary for a student's perspective and to view the lesson slides.

Discrete Math - Day 62

Today was a review day. The focus was on topics students wanted to focus on for the upcoming final.

The first question from the class was concerning the topics that will be covered on the test. I told the class that they should expect the following three broad areas to be on the exam:

1. Graph theory
2. Cryptography
3. Modular arithmetic and congruences

Students asked to go through examples for cryptography first. I said that simple ciphering may be included and more difficult problems would delve into the Diffie-Hellman exchange. We walked through a Diffie-Hellman exchange and I provided examples of what might be asked on the exam.

Next we dove into modular arithmetic and congruences. I told the class I would expect students to be able to determine if two integers were congruent mod a given value. I also thought students should be able to write congruence statements for a word problem. More difficult problems would involve solving a system of linear congruences or proving relationships. For modular arithmetic, I might ask them to construct an addition and multiplication table, perform arithmetic for values, or correct a table of addition and multiplication that contained mistakes.

For graph theory, students are expected to be able to draw simple graphs for given criteria (number of edges, vertices, and degrees of vertices), determine if an Euler path or Euler circuit exists for a graph, and prove certain simple properties.

Students felt comfortable with the topic coverage. I allow students to use notes and told the class they should spend some time before the exam going through their notes and making sure they are in good order.

Visit the class summary for a student's perspective and to view the lesson slide.

IPS - Day 61

Today we began the review process for the final exam. To start things off, I showed them Steven Levitt's Ted talk on car seats. This talk walks through an observational study and an experiment that was conducted on the efficacy of car seats versus seat belts for children aged 2-6 years old.

The video provides a nice tie-in to what we have been doing in class. It shows students how researchers actually work. In this case, I was able to ask students what type of studies were used and discuss some of the graphs used to display data.

With that lead-in, I showed students the three major topics that we covered during the second-half of the semester: collecting data, analyzing data, and random events and probability. I asked each student to write down the three big ideas, concepts, formulas, or vocabulary that they associated with each of these topics. I gave students time to get their thoughts down and then share with their groups.

As a class, we went through each topic, listing out what students had written. The lists were extensive and demonstrated clearly how much material we had covered.

I asked students to consider the lists that we created and to determine which items they would want to focus on tomorrow, which is our last day of review.

Visit the class summary for a student's perspective and to view the lesson slides.

Discrete Math - Day 61

Today was the start of the review process for the final exam. To start things off, I showed a portion of an episode from the BBC's The Story of Maths series. This particular segment focused on the mathematics that was developed in China. Of course, it ended with a discussion of the Chinese remainder theorem and its use in modern cryptography.

I like to show videos like this because it shows where the math we learn in school comes from. It also shows how studying and exploring a topic that seems to have no relevance in our daily lives today may, in fact, turn out to be an important linchpin to future generations. The Chinese remainder theorem was first explored over 2,000 years ago but is critical to our digital age.

Next, I asked each student to write down what they felt were the five big ideas for the second half of the semester. I gave the class a few minutes to get their thoughts down and then we shared out as a class. I wrote down the topics as they were brought up and tried to write topics that were related near each other.

This generated some discussion about meanings of some terms and connections. It also spurred students to think about additional topics that had not been listed yet. As we looked over the list, students remarked at how much was covered. I agreed that we covered a lot of material in one semester.

Next, I asked each student to write a question that could appear on the final exam. I did this so that we could have review problems to use tomorrow and to try and incorporate some of the questions into the final exam.

With about five minutes remaining, I asked the class about the last graph that was given to them and whether or not an Euler path or circuit existed. Students readily responded that there was an Euler path but not an Euler circuit since the graph contained two odd vertices, one of degree 5 and one of degree 7.

With that, I asked students to briefly go through their notes and prioritize what they would like to focus on next class as far as topics and problems to review in depth.

Visit the class summary for a student's perspective and to view the lesson slide.

IPS - Day 60

Today we wrapped up the hand-washing project. Students finished their posters at the start of class. As they were completing their posters I walked around and looked at the posters that were complete. Almost every one made no attempt to address randomness in their analysis or conclusion.

This was disappointing and reinforced the need to do things differently next semester. I addressed these issues on a group by group basis and talked with the class about the idea that every time they drew a sample they would see different results. They need to address these random fluctuations in results to be more certain of their conclusion. Next year I must do a better job of getting this idea across. I think more investigations with repeated sampling and trials along with discussing why the results differ each time may help.

As with the last poster, I had students look at what had been produced in class and at two posters created by previous classes. I then had the groups score their posters as before.

There was still about 10 minutes left in class, so I used this to help students understand how they could make use of the techniques we had learned. In this case, there is a bit of controversy about cell phone usage by students. In particular is a concern by some faculty here about students using twitter.

Personally, I have a twitter account and would like to make more use of it in the classroom. My issue has always been that less than half the students have or are willing to create a twitter account. In addition, I typically get resistance from parents about having their children using twitter.

I raised this topic with the class and about half were aware of the twitter controversy. I posed a question of interest: Is twitter usage by students and issue?

I asked students what data we could collect to address this issue. This produced a lively discussion and produced a far-ranging list of potential data:

• length of time a student is on twitter
• tweet topics
• frequency of tweets
• time of tweets
• number of tweets
• number of students with a twitter account
• number of students active on twitter
• hash tags used
• number and type of tweets tagged as favorite
• number and type of tweets that are re-tweeted

I then asked students to consider how this data could be collected. The majority of this data is available through retrospective observational studies. Some must be collected via a sample survey.

The class wondered if we were going to conduct this study. I told them it would be an interesting one to look at but it would take two or three weeks to do it properly. I just became aware of this controversy on Wednesday. Had I known about it a week or two ago I would have definitely had the class tackle this study.

The point was grasped by the students and it was a good way to make connections and close out the final week of class.

Visit the class summary for a student's perspective and to view the lesson slide.

Discrete Math - Day 60

Today we took practiced finding Euler paths and circuits.

I started class by reviewing the Euler theorems we looked at last class. I did this mostly for the benefit of those students who weren't here last time.

I then showed them Fleury's Theorem. This theorem basically says that if you have a connected graph that has either no odd vertices or two odd vertices, an Euler path can be found.

I showed a graph and labeled the degree of each vertex. The graph had four odd vertices, so Fleury's Theorem did not apply. I connected two of the odd vertices with an edge. Now the graph had the required two odd vertices. I followed the algorithm for Fleury's Theorem to demonstrate how it worked.

This led use to the next two theorems about graphs.

Second Euler Path Theorem

If a graph is connected and has exactly 2 odd vertices, then it has an Euler path.

Second Euler Circuit Theorem

If a graph is connected and has no odd vertices, then it has an Euler circuit (which is also an Euler path).


I pulled up the original graph I used for demonstration and said that the way I had connected the two odd vertices was akin to bulldozing through someone's backyard or house. The Eulerization of a graph results from adding multiple edges so the resulting graph now has either zero or two odd vertices. I asked students how an Eulerization of the graph could be accomplished. After a little thought one student offered a way.

With that, I handed out a sheet with three different graphs. They were to determine if an Euler path or circuit for each graph. If it did, they were to draw it. If it didn't, they were to explain why not and then create an Eulerization of the graph so that the revised graph would have an Euler path or circuit.

This exercise went well. The students worked on the problem, discussing terminology and results. Everyone seemed to grasp the basic concepts of what they should be doing. Everyone was able to identify degrees of vertices (although some students found they had to be more careful counting).

After briefly sharing out and discussing the results for these graphs, I passed out a more complex graph. Students were asked to Eulerize the graph if an Euler path or circuit did not exist. There were about five minutes left in class but students jumped right into the task. As students determined degrees of the vertices, they found there were two odd vertices. I would here comments like, "There are two odd ones, this should work, let's do this." It was encouraging to hear them so engaged and so confident in their knowledge.

The bell rang and everyone looked a bit surprised. They were so focused on the sheet they forgot how little time left in class.

We'll finish looking at their results next class. After, we'll start our review process for the final exam, which is taking place in less than one week.

Visit the class summary for a student's perspective and to view the lesson slides.

Discrete Math - Day 59

Today turned out to be an interesting yet productive day.

For whatever reason, there were quite a few students missing at the start of class today. It's AP testing time, so I wasn't sure if half the class was missing due to AP tests or if the students just didn't want to get out of bed on a rainy morning during the last week of school.

I was intending to go through some additional theorems regarding Euler paths and circuits but there were so many students out I knew I had to change direction. I decided to tie graphs back to physical reality. Relating back to how we started with the Konigsberg bridge problem, I asked the class what a graph of our school would look like? We briefly discussed what should represent edges and vertices and I let them work on the problem for a while.

As students worked on the graph I walked around to see what they were doing. Several students wanted to make a physical representation of the school layout with vertices representing the intersections of walls and the like. I referenced back to the Konigsberg bridge problem and the fact the land masses and island represented the vertices and the bridges that connected these land masses represented the edges. With these brief discussions I was able to get these students on a better path to representing the school layout.

As we did this work, more students trickled into class until I had almost a full compliment of students. These students' groups helped to get the new additions up to speed. I allowed students to go through the building to be sure they were getting a good representation of the layout. A few groups availed themselves of this opportunity.

We took a look at the graphs that were created. Most of the groups had essentially the same graph (see below).

The graphs correctly identified the five major landing areas and the fact that two of the school's wings contained four stairways each and were connected by a catwalk on the upper floor and a walkway on the lower floor. The question I posed to the class was whether or not an Euler path existed in this graph?

Students made several attempts but failed to find a path. They asked if they could go out and try walking paths. I let those that wanted go make the attempt. About five or six students remained in class. These students were typically the ones that were not as engaged. I told this group that an Euler path did exist and they needed to find one. They all were busy working on the problem when within two or three minutes one student called out, "I go it!" Everyone else looked up in amazement as this student rarely looked like he even paid attention in class. I had him go to the board and he quickly drew his Euler path. Everyone else was in awe and said how amazing it was. He even joked about being the kid who never pays attention but was able to solve the problem. It was an exciting win for him and I appreciated the positive, supportive reaction from those present.

A few minutes later one of the groups who left came rushing in saying they found the solution and, not seeing any other groups back, that they must be the first. The students who stayed pointed out that the problem had already been solved. As this discussion took place a couple of other groups returned, one who was successful in determining an Euler path and one that was not.

We shared out the solutions and discussed why the attempts to start at one of the even  vertices did not work out whereas starting out at one of the odd vertices did work out. Again, one of the students who tends to struggle in class voiced the opinion that when you start at an odd vertex and leave, you in essence make the vertex even. This even vertex can now be paired up to go back to the vertex and leave again. While not a formal proof, it clearly showed the student was thinking about the problem from a mathematical perspective and using the properties of the graph to reason about the outcome.

We discussed this further. With an odd vertex, at some point you either have to leave or return to it. If it isn't the starting or ending vertex this means that you would have to double-back on an edge, resulting in a path that is not an Euler path.

About this time, a group who decided to make a more detailed graph of the school returned. They attempted to include classrooms, alcoves, and other areas located throughout the school. It was an impressive attempt that made a much more detailed representation of the school, breaking wing floors into subsections. The image below shows their graph.

This graph shows the three wings of the school but provides more detail for each level and for classrooms. Students asked questions to clarify what they were viewing but we all felt they had created a good graphical representation of the school.

I next showed a short video about creating a graph of a house. This reinforced some of the ideas that students had been wrestling with regarding paths and circuits.

With that, I went through some of the material I was intending to cover today. Specifically, we looked at the Odd and Even Vertex Theorems.

Odd Vertex Theorem

Suppose A is an odd vertex of a graph.

1. An Euler path that starts at A cannot end at A.
2. An Euler path that does not start at A must end at A.
3. Every Euler path either starts or ends at A.

Even Vertex Theorem

Suppose B is an even vertex of a graph.

Every Euler path that starts at B must also end at B . This ?means the path is an Euler circuit.

We discussed these theorems in light of what students had been investigating and see with their graphs today. Students seemed to understand why an Euler path must start and end on odd vertices. The video along with our work also helped to make sense of the Even Vertex Theorem. The basic reasoning here was that as I leave an even degree vertex, there are now an odd number of edges left that go to that vertex. Getting rid of even pairs by traveling in and out, I can reduce this number to a single edge that has not been traveled. If the edge is traversed before the path is complete, then the only way to leave the vertex is to double-back, resulting in a path that is not an Euler path.

I asked what this meant for other graphs. I drew an example, where the graph had three vertices of degree 3, one vertex of degree 1 and one vertex of degree 2. See the image below.

I asked students what they could tell me about whether or not this graph had any Euler paths or circuits. Students realized an Euler circuit could not exist because the circuit would have to start at the sole even vertex. But this would mean that the vertex of degree 1 could not be included.

The question of an Euler path was more difficult for them determine. Several students made unsuccessful attempts. Their discussions quickly turned to considering that maybe it was not possible to draw an Euler path. I asked students which of the odd vertices would they start at and which would be used as the starting point. How could you decide?

Again, students quickly realized the degree 1 vertex would either have to be the starting vertex or the ending vertex. I pushed them to consider if they started at the degree 1 vertex, how could they determine which of the remaining odd vertices to use as the ending vertex? I then asked them to consider what happens with the remaining two odd vertices?. 

Pick one of the two remaining odd vertices. You traverse one edge to reach the vertex. To leave you must traverse another edge. This means to go to that vertex and leave it again you must use two edges. However, there are an odd number of edges, so at some point there will only be on edge remaining. This edge must be traveled, but it is not the last edge in the graph, since we are not ending our path at that vertex. Therefore there is not an Euler path in this graph.

After this discussion, I displayed the first Euler Path and Circuit Theorems.

First Euler Path Theorem 

If a graph has an Euler path, then

1. it must be connected and
2. it must have either 0 or 2 odd vertices.

First Euler Circuit Theorem 

If a graph has an Euler circuit, then

1. it must be connected and
2. it must have no odd vertices.

The previous discussions and investigations proved these theorems in a somewhat informal manner, but the reasoning was sound enough to provide a convincing argument.

We'll explore a few more theorems about circuits and paths to wrap up the semester.

Visit the class summary for a student's perspective and to view the lesson slides.

IPS - Day 59

Today was a productive day as students worked on their analysis and posters to present their hand-washing study results. Many groups wanted to go back into the computer lab to create new graphs. by the end of the day just about every group had completed their posters. Next class we'll look at some of the results and discuss consistencies and discrepancies between the different study results.

I did spend some time at the beginning of the class discussing issues in statistical analysis. Students need to realize that there is not a set procedure or process for analyzing data. Students must consider the nature of their data (quantitative or categorical) and the question of interest they are attempting to address. The challenge is to think through these aspects and decide how to bring the data to bear to address the question of interest.

In looking at graphs being created last class, students were creating whatever graph they could grab without considering what data they had to work with and what question they were trying to address. As a result, many quantitative graphs (box plots and histograms) were being created for categorical data. I told the class that the most difficult aspect of statistics is filtering through what data you have and what question you are trying to answer in order to determine what you will do for your analysis.

I felt like students were really starting to understand statistical analysis and the idea of looking at their specific sample results compared to randomness.

Visit the class summary for a student's perspective and to view the lesson slides.

IPS - Day 58

Today was spent in the computer lab with students analyzing the hand-washing data they collected. It is still obvious that many students are not clear about how to approach their analysis. As students asked what graphs they should make, I would reply with my own question, "What is the question of interest that you are analyzing?" We then would discuss what makes the most sense to use to help answer their question of interest.

The other thing I noticed is that some students did not formulate a null and alternative hypothesis that related back to their question of interest. For example, a group's question of interest might be, "Do a majority of people wash their hands?" Their null hypothesis then might be, "Women wash their hands more frequently than men." While these both relate to hand-washing, the question of interest and the null hypothesis are focused on two very different aspects of the issue.

Other groups created histograms, box plots, and scatter plots. I would ask how these graphs help to understand what is happening with regard to their question of interest. Does the graph tell anything about how more or less likely hand-washing is to occur?

All was not bad as some groups clearly understood they wanted to understand what could occur randomly and what their actual results showed. I was able to work with groups on redistribution sampling and bootstrapping, based on whether they were interested in overall hand-washing rates or comparing two groups on their hand-washing rates.

Some groups still need to finish their analysis while others will start to write-up their results.

Visit the class summary for a student's perspective and to view the lesson slide.

Discrete Math - Day 58

Today we continued working with graphs and their properties. I am pleased that the last week before finals and the seniors are still actively engaged with the material.

I briefly went over the two properties we looked at yesterday and talked about how the reasoning that a graph could not contain an odd number of odd vertices was an example of a proof by contradiction. I related this back to proving there was an infinite number of primes. In both cases, an assumption was made that led to a contradiction. This leads us to conclude that the assumption is false and therefore the converse of the assumption must be true.

We then looked at a series of problems. The first question was quickly dispatched. Students argued using the results from last class that a graph with five vertices of degree 1 or five vertices of degree 3 cannot exist because the sum of the degrees would be odd. They were also able to easily draw a graph with five vertices of degree 2.

The problem after this gave students more trouble. As one student said, "This was fun and now it's not." As students examined the problem they began to believe that it was not possible to draw a graph with eight vertices and 29 edges. Part of this feeling arose from the odd number of edges. They were at a loss at proving the graph could not be drawn.

I drew four vertices on the board and asked the class how many edges could be drawn connecting the different vertices. After some discussion and debate, one student said there could be 12 edges drawn. I started drawing out edges, three from each vertex. I pointed out that at some point I was duplicating an edge that was previously draw. Because I was double-counting the edges, I needed to divide by 2, resulting in a total of 6 edges.

I then asked students how many edges can be drawn when there are eight vertices. Again, there was a lot of discussion and debate. To help refocus their thinking I asked the class how many vertices could a single vertex be connected to? The response was "7". How many times can this happen? The response was "8". How many total edges does this represent? Silence. Then a couple of students responded that there would be 56, since 7 x 8 = 56. I asked if there were edges being double-counted? Students realized it was the same situation as with four vertices and saw that there were 28 edges.

The class still was not making a connection to the possibility of having 29 edges in the graph. I asked what the 28 edges represented? Students said that it was the number of edges in the graph. Was it possible to have any more edges in the graph? Students said no, because every vertex was already connected with every other vertex. It was at this point that students started to realize that this meant the graph could not have 29 edges.

I like this problem because it brings in some of the counting ideas that we used at the beginning of the semester. One student commented about being taught stuff that keeps getting reused and why couldn't it be like other math classes where the material just disappeared once you were done with it. I love that previous concepts can be brought to bear on new material or in new ways.

Students were given the degrees of six vertices and asked whether or not it was possible to draw graphs with those degrees. At this point, most students saw the first was not possible because the degrees summed to an odd number. For the second, they had some issues at first until they realized a simple graph did not have to be connected. Many struggled with the last graph as it required five vertices of degree 3 and one of degree 5. They began to believe it was not possible. I pointed out that the sum of degrees was even, so at least we couldn't rule out its impossibility on that basis. Finally a student said they found a solution, a pentagon with a vertex in the middle. The class response was oohs and aahs and comments of how they were copying that result down in their notes.

We looked at whether a graph with eight vertices could have a vertex of degree 0 and a vertex of degree 7. A few students quickly realized this was not possible as a degree 7 vertex would have to connect to every other vertex, which meant you could not have a vertex of degree 0. This was sound reasoning that was easy to understand. A nice little proof.

The final problem was an extension of the previous problem. Could a graph with eight vertices have each vertex with a unique degree value. Students felt like this could not happen but were having difficulty explaining why. I asked students what all the possible degrees of the vertices could be and the said 0-7. I wrote out on the board 0, 1, 2, 3, 4, 5, 6, 7.

Next, I asked what the last problem we worked on told them about the situation. They said that you could not have both a degree 0 and a degree 7 vertex. When asked which they wanted to eliminate the students suggested the degree 0. We now have possible seven possible degree values to allocate to the vertices: 1, 2, 3, 4, 5, 6, 7. There are eight vertices in the graph. By the pigeon hole principle, I cannot assign the eight vertices to the seven possible vertices without duplicating at least one degree value. This was another nice tie-in to the counting work we did at the start of the semester.

To close class, I asked students to record their thoughts on what was easiest to do when proving a statement or creating a counter-example. Then I asked students to consider what they could do in the future to help themselves when proving a statement or producing a counter-example.

We will continue to explore properties of graphs and look deeper into the ideas behind Euler paths and circuits.

Visit the class summary for a student's perspective and to view the lesson slides.

Discrete Math - Day 57

Today we continued to explore graphs and their properties. I asked the class about the problem of creating a graph with two vertices of degree 2 and three vertices of degree 3. Students didn't have any additional comments. I left this to come back later as we explored specific properties in more depth.

I then reviewed some definitions, specifically paths, circuits, Euler paths, and Euler circuits. It had been a few days since we dealt with these terms and I knew that the class would be shaky on these terms and we were going to make use of them in the first two problems we worked on. Once these terms were clarified, we moved on to two problems.

The first looked at creating a graph with eight vertices and 6 edges. The graph could not contain any circuits. This problem proved a little challenging at first because students thought a circuit meant all of the edges were involved. When they realized a circuit could be just part of the graph they quickly found solutions to this problem. We shared out a variety of solutions that included graphs with one vertex of degree zero; two components, one of which had two vertices of degree 1; and two components, both containing four vertices. It is helpful for students to see the variety of solutions that can exist for a given description.

The next problem proved even more difficult for students to solve. The requirement was a graph with exactly three edges that did not contain an Euler path. The definition of Euler path was revisited and this helped students to better understand paths, circuits, Euler paths, and Euler circuits. They tried tackling this without success for several minutes.

There were many different attempts at a solution. The issue that kept arising is that no Euler path existed if you started at a specific vertex but one would exist if you started from a different vertex. As these issues were pointed out, the frustration level grew. Students started to think it was not possible.

Finally, one student modified a graph so that all three edges emanated from a single vertex. In this situation, the graph had three edges but required a doubling back on one edge in order to traverse every edge and, so, fit the criteria. When the class was shown the solution there were many "ah-ha's" around the room.

From here, we moved to the issue of summing the degrees of all the vertices in a graph. The question was if graphs always had to have a sum of degrees that was even. Students started looking at previous graphs they had drawn and creating new ones in the hope of drawing a graph whose sum of degrees was odd. Finally, a student asked if the sum always had to be even because every time you connected an edge between two vertices, each vertex would add one degree, resulting in a net increase of two degrees. This idea was shared with the rest of the class and everyone agreed that, yes, it made sense that you could never get a sum total that was odd.

The next problem asked if you could have an odd number of vertices of odd degree. Students were a bit baffled by the wording posted, so we discussed what this meant. I need to simplify the wording for this so it is more understandable the first time students read it.

I asked students to consider what was just demonstrated about the sum of degrees in a graph. A few students started to say this wasn't possible. When pressed, they voiced the idea that the sum of degrees would be odd which is not possible. This is precisely the idea for a proof by contradiction. I didn't bring this up at the time but will next class.

I used this result to revisit the graph problem of drawing a graph with two even vertices and three odd vertices. I asked students if this was possible. Students quickly saw that it was not possible because the sum of the degrees would be odd, which is not possible.

As class came to a close, students asked if we were going to continue working on these types of problems. I said we would, which they were glad to hear as they wanted more practice. This was the intent and we will continue to work on similar problems next class.

Visit the class summary for a student's perspective and to view the lesson slides.

IPS - Day 57

Today the focus was on the bootstrapping method of resampling. I wanted students to understand fully how a bootstrap sample was created and used.

I first checked on the progress of data collection for the hand-washing study. Most groups had started to collect data and felt comfortable that they would have data to analyze for tomorrow.

I discussed the idea of bootstrapping one more time and used an applet to generate some data that we could use. The free-throw shooter applet allows you to specify how many free throw shots are taken. I set the number of shots to 10, as I wanted a small sample to illustrate bootstrapping.

The question of interest is whether or not the shooter can average 80% free-throw shooting. For our data, we had 7 makes and 3 misses. I numbered these from 1-10 and then generated two different samples drawn from these 10 shots, using replacement. In both instances, the bootstrap samples showed 80% made.

I then used the CPMP tools to generate 1,000 bootstrap samples. In this case, approximately 40% of the time, the results shows a completion rate of 80% or higher. The data suggests that an average free-throw completion rate of 80% would not be unusual if a shooter made 70% of the shots in 10 attempts.

I then asked the class to pick a data set we used (car data, chocolate data, or hand washing data) and generate three bootstrap samples from the data. There were some questions initially but students started to see the process of how a bootstrap sample is generated.

I asked them to consider the results of the samples they generated and what they mean and how they can be used. There were quite a few discussions about this. Again, students gradually started to see how these represented random outcomes that could be used to determine how likely specific results that we observe occur.

I closed with asking students to record in their notes the process of creating a bootstrap sample and how the samples are used as part of the statistical analysis process.

There was a little bit of time left, so I discussed the structure of the final exam that we will have next week. Students will be provided three scenarios and will be asked to complete an analysis and draw conclusions from the analysis.

Tomorrow the class will meet in the computer lab to crunch data they collected for the hand-washing study.

Visit the class summary for a student's perspective and to view the lesson slides.

IPS - Day 56

Today we continued working on the hand-washing investigation. I had students go out and test their data collection procedures. I gave them approximately 15 minutes to try out their data collection methods. When the class returned, we discussed and issues or problems with the data collection.

It became apparent to the class that they were not going to be able to collect data during class-time. There were too many observers and not enough subjects to observe. I told students they would have to collect data outside of class. I asked that they have their data collected by the end of the day Monday. I will remind the class on Monday they need to have their data collected, as we will begin analyzing the data in the computer lab on Tuesday.

Next, I addressed my concern with their analysis. The class is not absorbing the idea that just because they see a difference in two means that the difference is necessarily meaningful. I am rethinking what I have covered during the semester and will make revisions to better emphasize random outcomes and the examination of how likely an event is in relation to random outcomes.

As it is, I decided that some manual manipulation of data may help the class better understand how resampling for redistribution works and how it is used to compare random outcomes to the specific outcome observed in real life.

I used the hand-washing data that was collected today. There were 10 males observed, 40% of whom did not wash hands. There were 6 females observed, 17% of whom did not wash hands.

I asked the class to consider an appropriate null and alternative hypothesis for this situation. A student provided the following:

     H₀: Males wash their hands at the same rate or higher than females.
     H_a: Females wash their hands at a higher rate than males.

I asked students to use a random number generator to reallocate the 16 observed individuals into two groups, one of size 10 and one of size 6. The random numbers 1-16 were used and each value between 1 and 16 could only be used once, since the same person could not be placed into more than one group. The values 7-10 corresponded specifically to the non-hand-washing males, and the value 16 corresponded to the non-hand-washing female.

Students generated their numbers. Some had to start over when I pointed out that the same random number could not be used twice. Once a group of 10 and a group of 6 were created, I asked that the percent of non-hand-washers be calculated.

The point of statistical analysis becomes is there anything different or special about the group of 10 and 6 individuals that was collected versus the random groups that were created?

We had eight random groupings created with percent of the group not washing hands:

     % Male     % Female
          30             34
          40             17
          10             60
          20             50
          30             34
          30             34
          30             34
          30             34

This is a small sample, but it shows that one of the eight random groupings had results as extreme as the one we observed. We need thousands of these random allocations to get a good fix on what random occurrences would look like. This is what software enables us to do. At least some of the students seemed to grasp this idea.

I closed by telling students that many times data differences are reported without considering if the differences might, in fact, just be random chance. This happens quite frequently. Without considering the random nature of outcomes, decisions could be made that are based on supposed fact but are actually no more informed than flipping a coin or rolling a die.

I will have students manually work through a bootstrapping example next class with the same purpose in mind. As the class completes the hand-washing investigation I'll be able to determine if there has been any movement in their ability to analyze data from a statistical perspective.

Visit the class summary for a student's perspective and to view the lesson slide.

Discrete Math - Day 56

We continued to work with graphs, attempting to draw them for specific criteria.

We focused on three problems today:

1. Draw a graph with three vertices of degree 2 and two vertices of degree 3.
2. Draw a graph with two vertices of degree 2 and three vertices of degree 3.
3. Draw a graph with one vertex of degree 1, two vertices of degree 2, three vertices of degree 3, and 4 vertices of degree 4.

I asked the class to work on these three problems. As I walked around, students were engaged, discussing different possibilities, clarifying definitions, and, in general, having healthy mathematical discourse. I saw that all the groups had completed the first task, many had made an attempt at the second problem, and some were working through but struggling with the third problem.

I decided to halt their work to look at their solutions to the first problem. I thought the break and looking at different solutions would help to maintain their focus and persevere through the other problems. The first solution gave students an opportunity to be creative. Students showed connected graphs with no multiple edges, disconnected graphs, graphs with multiple edges and graphs with loops.

Showing these allowed students to see that they might be able to use different approaches to obtain a solution for the remaining problems. The students continued to work on the last two problems, with most focused on the third problem. There were some tentative solutions and students were building graphs that could lead to a solution but they were not complete.

Since no group had created a graph fitting the description for the last problem, I asked a student who said he was close to share what he produced. As he was at the board labeling the degrees of the vertices and sharing what he produced, he and a couple of other students realized that the graph could be easily separated into two pieces to meet the criteria. As this option was discussed, other students concurred that this would work.

Another student had been building a graph but wasn't sure how to complete it to fulfill the requirements. The graph was put up on the board so we could discuss options as a class. The degrees of the vertices were labeled and it was discovered that three of the vertices had not been assigned the correct degree. Additional vertices and edges were added and several students became very involved in discussing how to continue the graph. Jointly, these students came up with a solution that worked.

Both of these discussions provided me some ideas as to assessment questions that I could use. Typically, I would just ask students to construct a graph with a given criteria. I think providing a graph and asking how the graph can be modified to meet the criteria or how the graph could be extended to meet the given criteria would be interesting.

After these two discussions, I asked the class about the second problem. A couple of students thought they had solutions and these were presented. In both cases, there was a vertex whose degree had been miscounted. Students worked on this problem a couple of more minutes without success.

I stopped students and asked if they considered if the graph was even possible to draw. Time was running out in class. I asked students to consider how graphs are put together and whether they could explain why the graph was not possible to draw or to produce an example of the graph.

This question is a good starting point for proof using graphs. We will look at drawing graphs with circuits and Euler paths next class and delve more deeply into trying to prove the existence of certain graphs.

Visit the class summary for a student's perspective and to view the lesson slides.

Discrete Math - Day 55

Students presented their cryptology research papers to the class. The presentations are to highlight the topic they wrote about in their report.

Most of the presentations were well structured and indicated an appropriate level of research into the topic. One group tackled women in cryptology and drew comparisons of the participation and growth in this field, especially during and after World War II with growth in women's rights and equality. Several groups looked at Bletchley Park and the enigma machines used during World War II. A couple of groups focused on the Navajo code talkers used by the United States Marine Corps during World War II. Students made some solid connections to the topics we studied in class and asked additional questions about different topics.

I then used some time to show more of the video The Music of the Primes. Specifically, I started to show segment 3, which deals with Alan Turing, his work at Bletchley Park, and his development of an electronic calculating machine, the fore-runner of the modern computer. Besides connecting directly to many of the presentations made it also made more explicit connections to topics we studied and to our world.

I pointed out to students that one aspect of this story that struck me was that Alan Turing conceived of his idea of a computational machine as a tool in helping to prove or disprove the Riemann Hypothesis, a purely theoretical undertaking with no practical value at the time. As it turned out the computational machine became the foundation for modern computing. When students ask why they are studying a topic and when will it ever be used, I think about a situation like this where it may not be used for decades or centuries, but it can eventually be of great value. This is an aspect of human growth, the pursuit of knowledge and insight that pushes us, collectively, to greater things. The ancient Greeks looked at many purely theoretical topics in mathematics that had no practical purpose, yet two millennium later so of these purely theoretical topics are so pervasive in our daily lives that we do not even take notice. Sometimes we examine problems and dive into topics for the pure adventure of where they might take us, challenging our thinking and pushing our experience, exploring uncharted landscapes to make discoveries not thought possible.

To wrap up class, we plunged back into graph theory. I started by having students share what they had drawn for the problem of six vertices that contained four loops and two multiple edges. There was quite a variety of drawings and students could see and appreciate that there were many different graphs that could represent the same situation.

We then tackled a graph with five vertices all of which had degree four and without any loops or multiple edges. Students struggled with coming up with this graph. Many wanted to draw segments coming out of a vertex but not connecting to another vertex. This was an opportunity to revisit the definition of an edge, which states that it is a segment connecting two vertices. Finally, I drew a pentagram and connected each vertex to every other vertex. I asked the class if this met all the criteria. After a little consideration the class agreed that it did meet the criteria.

I then asked if there was another graph that could also work. They jumped into attempting to create another version. Some people made elongated pentagons but then realized they had just stretched the figure. As class wound down, I drew a second figure, in essence folding one of the pentagon vertices inside the pentagon. The class agreed that this worked, but only after arguing that the figure contained multiple edges. Once they realized the edges were not multiple ones they realized that this figure worked as well.

We'll continue working through some of these. My goal is to build comfort in the class with the definitions and terms used in graph theory.

Visit the class summary for a student's perspective and to view the lesson slide.

IPS - Day 55

Class started today with students evaluating their posters. I first had them look at two posters that were created by previous classes. These posters were scored in the 85% to 92% range. After reviewing the posters, I asked students to score their on posters on a 20 point scale. I told them I wanted an honest evaluation of what they produced and how well they communicated statistical thinking, specifically with providing purpose, hypotheses, processes, analysis, and conclusions.

I will also score the posters and adjust the score the students produced by reducing their score or raising their score. I take the difference in my score and the students' score and base the adjustment on this amount. If they were overly generous in their scoring I will take the difference and subtract it from my score as a penalty for not providing an honest evaluation. If the were too hard on themselves, I will adjust the score upward to a value halfway between their score and my score, in case I am being too generous. This process provides an incentive for students to consider honestly the quality of their work.

Afterward, I introduced the idea of bootstrapping a sample. I differentiate the idea of re-sampling for redistribution and re-sampling for bootstrapping with the idea that a bootstrap sample is developed from a single sample while a redistribution is used when there are two samples. I had seven students come up front as an example. I numbered the students and then randomly selected students seven times, using replacement. We measured the percentage of females in the sample. With this process, we can look at what random samples may look like if the units in the sample being used are somewhat representative of the population.

I then used the CPMP software to demonstrate how the software can generate these samples. I used a simple data set of four values: 10, 10, 10, and 50. After generating 10,000 bootstrap samples, a clear picture of what random samples look like appeared.

The bootstrap method is the last analysis technique that I will introduce for the semester. Students now possess three techniques to consider what randomness may look like and to use for determining probabilities for seeing real samples with these characteristics.

The next project was introduced to students. This project has students consider the issue of whether or not people wash their hands after using the restroom. I asked students to develop a question of interest, create a null and alternative hypothesis for their question, consider what data they would collect, how they would collect the data, and what types of analysis the would use.

The groups had quality discussions about what they would analyze, their hypotheses, and what data they would collect and the procedure they would use for collection. I told the students I would need to sign off on what they were doing before I would release them to test their collection procedure.

As I walked around, I helped groups to focus their question of interest. Some wanted to consider differences between hand-washing rates of males and females, others wanted to look at overall hand-washing rates, others wanted to examine the use of soap when washing. The hypotheses that were created were well thought out and consistent with the questions of interest.

The discussions around data collection were also good. I discussed randomness in their data collection. Most groups had already considered this facet of their collection process. As students considered the physical collection process, I would ask them how they could be sure that their collection technique would not influence the behavior of the subjects they were studying. This led to interesting discussions about how to conduct observations.

Most groups got to a point where they were ready to test their data collection techniques. Unfortunately, at the end of the class period there wasn't much in the way of traffic in the restrooms. We'll continue working on the study next class. Hopefully students will be able to test their data collection procedure and actually collect their data.

Visit the class summary for a student's perspective and to view the lesson slides.