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Month: August 2016

Resampling – Two Population Proportions

Resampling – Two Population Proportions

On day 7 of instruction in my intro statistics course we spent the class working through a project comparing two population proportions using resampling.

Click here to download a pdf copy of the project: Two Proportion Resampling

Hands-On First

Mike Sullivan recommends a hands-on simulation before turning to the computers, so I began by using poker chips for this scenario: A random sample of 15 students at my college had 9 female students in it (60% female), while a random sample of 8 students at a nearby college had 4 female students in it (50% female). Is this evidence that my college has a higher percentage of females?

I had 13 red poker chips for the 13 females and 10 white poker chips for the 10 males. You could use 13 red playing cards and 10 black playing cards if you prefer. The idea behind resampling is that if the two schools have the same percentage of female students we can randomly assign each of these 15 of these students to my college to determine how often we observe a sample difference of at least 10%. We picked 15 chips from the bag and kept track of how many times the sample contained at least 9 females. (It happened 4 of the 5 times we did it by hand.) Students, based on the one proportion simulations, immediately knew that this was probably not an unusual result if the two population proportions were equal.

Turn to the Computer

I then opened up a StatCrunch urn applet to repeat what we had done with the poker chips. I could quickly generate 1000 repetitions of the experiment, and we observed that my college’s sample had at least 9 females approximately half the time.

Students then used StatCrunch’s Resampling Applet for Two Proportions and investigated two cases where the sample percentages remained the same while the sample size increased. We followed up with a two-tail investigation rather than the right-tail approach we took concerning female students.

Incorporate Student Data

We finished the project using two data sets that students had gathered earlier in the week. They had sampled 50 students, recorded their gender, and asked two questions:

  • Do you plan to vote in the upcoming election?
  • A question of the student’s choice that they felt would produce different results for men and women.

Students then compared the proportion of women who answered yes to the proportion of men who did. Very few of the students were able to conclude that the two population proportions were different, and it gave us a great opportunity to discuss the impact of small samples when dealing with qualitative variables.


It took all of the 50 minutes to get through the project. In the future I might cut out one of the two resampling investigations comparing the percentage of females at the two colleges, and perhaps have the students only ask the one question of the 50 students. I could also spin off their second question into a take home project, with them bringing those results back the next day.

I’d love some feedback on this project, or this approach in general. Let me know what you think. This week we will be taking a look at bootstrapping to estimate a population mean or median and using simulation to determine whether a sample drawn from a population produces an unusual mean.

Learning Catalytics for Turning In Class Assignments

Learning Catalytics for Turning In Class Assignments

Last night my intro statistics students worked through an Interactive Reading Assignment for measures of central tendency. In class today they will be working with four data sets and computing various measures of central tendency. To collect their work I will run through a Learning Catalytics module asking them for certain specific answers.

My plan is for students to work individually while consulting with each other. If they have different results it will be a great opportunity to check each others data entry and computational processes.

Learning Catalytics ability to collect numerical responses is definitely an advantage over the clickers of old that only allowed multiple choice answers. It also does a great job of evaluating numerical answers – it treats all of these as equivalent: 0.6, 0.60, .6, 3/5, 9/15, …

I sometimes use Learning Catalytics in this fashion to collect some written homework. I assign a handful of problems the night before, then collect certain answers at the very beginning of class. I also mix in some interpretation questions or conceptual questions that were not asked in the homework. Sometimes I follow up with a new similar problem to make sure they can do it in class. It’s very efficient and flexible for collecting some “by hand” problems.

Simulation Activity for a Population Proportion

Simulation Activity for a Population Proportion

As part of my effort to bring inferential statistics to the beginning of the semester I had students work through their first inferential project. This occurred on the sixth day of instruction.

Simulation for a Population Proportion

On the previous day we were going over different ways to display qualitative data (frequency tables, pie charts, bar charts). We created a pie chart for a sample of 60 students and it showed that two-thirds of those students were female. I asked students if the pie chart was strong evidence that more than half of all students at our college were female. They did not yet have the tools to make that decision, but they all felt it was strong enough to make that conclusion. I pointed out that the same pie chart would be generated from a sample of 3 students with 2 females and they understood that was not sufficient evidence. I told them we would investigate these ideas the following day.

Using a fair coin, P(H) = 0.5, we can simulate a sample of 60 students by flipping a coin 60 times and letting heads represent a female student. Using StatCrunch to simulate this experiment 1000 times, I only observed a sample containing 40 or more females only 3 times. This tells us that getting a sample containing at least 40 female students is very unlikely if the true percentage of female students at the school is actually 50%. That suggests that the true proportion of females at the college is higher than 0.5.


Here is a link to a pdf of the project: Project 1 Qualitative Simulation

Part 1: I worked through the first investigation with the class. They were told that there was a claim that 40% of all college students own an iPhone. A random sample of 1000 students contained 360 that owned an iPhone. We began by building a StatCrunch coin flip simulation applet with P(Heads) = 0.40 (probability that a student owns an iPhone if the claim is true) that would flip 1000 coins (the sample size) and determine whether the number of heads (iPhone owners) was at most 360 (number of successes in the sample). Students then ran the simulation for 1000 trials.

Students observed that it was unusual to obtain a sample that contained at most 360 successes through simulation, and we discussed the fact that this is strong evidence that the population proportion is less than 0.40. I also had students find critical values that separated off the lower and upper 2.5% of randomly generated samples in the simulation. I explained how this approach can be used to weigh whether the population proportion is different than 40% when we don’t have an idea whether the population proportion is higher or lower than the claimed proportion. The critical values are also a great lead in to the concept of a confidence interval.

Part 2: This was similar to Part 1. I had students complete this on their own, conferring with partners as they went. This is another case where the sample would be considered an unusual observation, indicating that the population proportion is above 60%.

Part 3: During the week leading up to this I had students visually survey their classes to find the total number of females and students in their classes. I had students begin with a claim that 50% of the students at our school are female – which is not true. Most students noted that there sample was an unusual outcome in the simulation and concluded that the true population proportion is above 50%.

I then gave students the actual percentage (55%) for this semester, and they rebuilt the simulation applet with P(Heads) = 0.55. Many, but not all, of the students found that their sample was no longer an unusual outcome in the simulation and we discussed what that meant (we cannot conclude that the percentage of females at our school is different than 55%).

This gave us one more chance to discuss the concept of bias in relation to sampling.

Overall, a very effective day when it comes to helping students develop the big picture of inferential statistics. Feel free to use, or alter, my project. If you give it a try I’d love to hear how it went.


Game On in Algebra

Game On in Algebra

This semester I am continuing to use a grading policy in my elementary algebra class that incorporates elements of game design. I begin by telling my students about my discussions with a well-known game designer (who happens to be my son Dylan – check him out on Twitter) that school should be fun, and he challenged me to come up with a grading system that incorporated some of the elements of game design. It took a long time to come up with a system that we both felt would work.

Homework and Leveling Up

Online homework and quizzes do not count directly to a student’s grade. I want students to understand that they must be able to demonstrate their understanding on exams. Homework helps students to learn, and I wanted to reward students who did well on the homework.

Students can level up by scoring at least 90% on each homework assignment during a testing unit (there are 2 assignments per section – a basic HW section that covers all of the topics and a personalized assignment that contains only problems that the student has struggled with) and 100% on each sections 5-question “reflect quiz”. Students are allowed to repeat the quiz as many times as they would like with only the highest score counting.


I give 6 exams, and they are graded as pass/fail. A student who passes earns 1 point and a student who fails receives 0 points. If a student has leveled up that unit they can earn 2 points for a score in the 70s and 3 points for a score of at least 80%.

That means that students can earn up to 18 points from exams.

Other Points

I do have one exam (Rational Expressions and Equations) that is a special double-points test where students who level up can earn up to 6 points. I do not tell my students about this until the end of the semester, and the fun associated with this unexpected reward helps to increase focus on the toughest exam of the semester. The extra 3 points get us up to a possible total of 21 points.

I give 4 points to students who pass a final exam review quiz (34 questions) and score at least 90% on the corresponding personalized homework assignment containing problems that they missed on their first quiz attempt. Adding these 4 points brings the possible total to 25 points.

Students can earn up to 5 points by completing the Real World Math project. This is a project in which students choose a real-world math topic they are interested in, plan a strategy for learning about it, and devise a plan to show me how they learned about the topic. (I will blog about this more in the near future.) These 5 points bring the possible total to 30 points.

Students can earn up 5 points by passing an optional cumulative midterm exam. In order to qualify to take this exam a student must level up on one of the first 3 exams and earn 2 or 3 points on that exam. Possible point total after these 5 points is 35 points.

Final Exam

The cumulative final exam, which is often a common exam taken with several other classes and graded by a team of instructors, is worth 100 points. That means that the final exam is worth 100 of the 135 points that are possible during the semester. My students have a clear goal – they must be able to demonstrate understanding of the course material at the end of the semester and everything they do all semester is to put them in position where they can do well on that exam.

Grading Scale

I have set 86 points as the minimum total to earn a C. That is equivalent to 6 points for passing each test without leveling up, 5 points for doing the Real World math project, 5 points that were possible on the optional midterm, and 70 points on the final exam. Since the midterm is not available to students who have not leveled up a student would need to score 75 on the final exam in order to pass the class (80 if they choose not to do the Real World math project). Essentially students have to pass each exam and score 80 on the final in order to pass the class without doing all of the work outside the classroom.

The choice for B and A are somewhat arbitrary – 98 for a B and 110 for an A. I chose these numbers based on previous semesters where a B required 12 more points than a C and an A required 12 more points than a B.


I will continue to post throughout the semester about this system, as well as share progress and results. I have had a great deal of success with this approach and I hope you will consider adapting it to use with your students.

– George

Statistics Classroom Activity for Sampling Techniques

Statistics Classroom Activity for Sampling Techniques

This week I devoted a class period to sampling techniques (random, systematic, cluster, stratified, and convenience). Students worked on an Interactive Reading Assignment before coming to class, and I began class with a quick discussion of the different types and their strengths/weaknesses. After the discussion I followed up with a class activity designed to help students understand how to select a sample using the systematic sampling technique as well as introduce them to some of the sampling features of StatCrunch. Most importantly – I wanted to demonstrate to my students that samples vary, a very important concept for them to understand moving forward in this class.

Click here to open a copy of the activity: Sampling Activity Fall 2016

I provided the students with a list of the genders of 76 students, numbered from 1 through 76. Together we worked through one sample of 10 students with a starting point of p = 1. I then had students repeat this process for different values of p and I wrote their outcomes on the board. We were able to see that the percentage of female students in the sample varied from 40% through 60%.

I then showed students how to create a list of random integers between 1 and 76 using StatCrunch. After we looked at my sample, students paired off to do the same. The results were more varied, but we did have approximately 30 samples instead of 7. This helped to drive home the idea that samples do vary. Although we only get to draw one sample, students need to know how samples vary in general.

For the last part of the activity, students joined my StatCrunch group and opened a file containing the responses of 76 students to a 3-question survey.

  1. Are you a smoker?
  2. Do you own an iPhone?
  3. How much did you spend on books and supplies last semester?

Students had StatCrunch select a sample of 20 students and computer the percentage of students who were smokers, the percentage of students who own an iPhone, and the average (we haven’t learned to call it mean yet) costs for that sample of students. Again, we saw a wide range of results. I then showed them the results for the entire group of 76 students so we could think about how close our samples were to the actual measures of the 76 students.

If you would like to see the StatCrunch data set, use this link to the data set.

My students left the classroom with a solid understanding of sampling techniques and for the benefits of using StatCrunch.

Learning Catalytics Questions

Learning Catalytics Questions

There is a growing pool of questions available inside Learning Catalytics. Some have been generated by the publisher, others have been generated by the community of instructors using Learning Catalytics. For my first Flipped Classroom I wrote my own questions (sample vs population, descriptive vs inferential, levels of data, …) and it was very easy to do. However, for day 2, I used 7 questions created by the publisher/author and the community of instructors and those questions were outstanding.

The feature that allows you to search for questions works really well. I chose the subject (Statistics > Introductory Statistics) and then refined my search by typing in the author’s name (Sullivan) or by entering tags (explanatory variable, designed experiment, …).

If you are using Learning Catalytics I would encourage you to share your questions by simply checking the box in each question that allows other instructors to copy them. I am sharing all of my questions – look for them under the community option by typing in my last name (Woodbury).

Day 2 of Flipped Classroom/Peer Instruction in Statistics

Day 2 of Flipped Classroom/Peer Instruction in Statistics

OK, this was a good day. No. It was a great day!

I cut down the review of the home content to 10 minutes, and was happy to see so many students participating in the student-driven review by offering their own explanations and definitions. Students know to come to class prepared and looking to participate. I can see where this could turn into a situation where only a handful of students participate though, and will seek to eliminate this pre-review and jump more quickly into the Peer Instruction portion of the class.

I also cut down the number of questions I asked by half. We did not feel so rushed, and it gave more time for deeper discussion. I guess one of my weaknesses is trying to cram too much into my class sessions (and exams too). I will constantly ask myself if I have too many questions, and if can I make due with fewer.

After making our way through a series of experiments where students had to determine the explanatory and response variables I stopped the class to make sure my students understood why I am using the approach. I asked them how confident they were about knowing the difference between an observational study and a designed experiment as well as how confident they were in their ability to identify explanatory and response variables. They were highly confident. Then I asked them if they would have felt as confident learning the material in the opposite order – with me giving the definitions and some examples followed by them cementing their understanding working at home. I saw the lights come on for many students. They understood that their learning was much greater in this system, and I think I have a great deal of buy-in from them now.

Tomorrow we turn our attention to an activity involving sampling techniques instead of a Learning Catalytics assessment. Although we won’t be using our “clickers” I will try to encourage the same level of discussion that I witnessed today.

Statistics – My New Approach

Statistics – My New Approach

This semester I started a new approach in my Statistics classes.

I wanted to

  • focus more on conceptual understanding
  • make class time more engaging
  • introduce inference much earlier in the course – including the use of bootstrapping, simulation, and resampling
  • cover nonparametric options for certain hypothesis tests

The first strategy I decided to employ was the flipped classroom, incorporating peer instruction.

We are using the eText (Interactive Statistics) that I co-authored with Michael Sullivan, and the Interactive Reading Assignments are perfect for flipping the classroom. In these assignments students read a little/watch a little/do a little – students are reading text, watching conceptual videos and example videos, and answering questions that feed directly into their grade book. Students must complete the Interactive Reading Assignment before coming to class the next day. (There is a standard homework assignment that students must complete the following night.)

I am using Learning Catalytics to help with the peer instruction part of the class. Learning Catalytics is like a powerful version of a clicker – students submit answers through their smart phone, tablet, or computer. (The class is inside a computer lab, so most students are using a computer.) Learning Catalytics supports so many types of questions, but at this time I am only using multiple-choice and numerical questions. I post a question and give students time to answer it themselves, then I give them time to discuss their answers with their classmates and potentially change their answers. When most or all of the class has the right answer discussion may not be necessary.

Today I spent 20 minutes going over key concepts from the previous night’s reading assignment. I asked students to define the concepts, to explain the difference between related concepts, and provide examples. I restricted myself to only chiming in when I felt that more detail or insight would help. Most students were highly engaged and many students contributed to the discussion.

I next spent about 10 minutes helping students log into the computers and find the Learning Catalytics session.

Finally, that left 20 minutes for the 15 questions in Learning Catalytics. That was a little rushed in places. Tomorrow I plan on 10-15 minutes of discussion about the previous night’s Interactive Reading Assignment, leaving 35-40 minutes for fewer questions. This should allow for an excellent peer instruction session, followed by a time to review what we have learned.

Students were engaged with the questions and worked well with each other. My favorite moment occurred when only about one-third of the students initially got a question correct. Vigorous debate ensued, and in a short while many of the students had the right answer. I asked a student who switched their answer to explain why, and I noticed that several students really gained understanding from that student.

In future posts I will discuss my plans to include inference early in the course (starting in week 2), and the different techniques we will be learning in one of the 20 classroom projects we will be working on.

– George

I am a math instructor at College of Sequoias in Visalia, CA and the author of an algebra textbook and co-author of Interactive Statistics with Michael Sullivan (both with Pearson).

Day One Activities – Focus on Math Anxiety

Day One Activities – Focus on Math Anxiety

At the developmental level it should be no surprise that many of the students have feelings of anxiety related to math. Here are a few of the things I do on the first day of class to help students deal with these feelings.

“Heads Down, Hands Up”

After I take roll on the first day of class, I ask the students to put their heads on their desk with their eyes closed. I then ask students to raise their hands if they feel that they struggle with math or if math is their worst subject. Most students raise their hands, and the students can tell this by all the rustling associated with that many hands being raised. I then ask them to put their hands down and open their eyes.

We talk about how many students feel this way. It’s odd, but so many students feel that they are the only one who struggles with math and that there must be something wrong with them. I tell them about all the hands that went up, and that it’s pretty common to feel that way. I ask them to consider this as their chance to start over, to wipe the slate clean, to develop a positive attitude. Honestly, it does not matter what series of events led to a student being in a developmental class, it’s what happens from there that matters.

This is the time that I can help students to start to develop a growth-mindset and talk about the importance of thinking, trying new approaches, productive failure, effort and perseverance.

Positive outcomes: Students realize they are not alone and that they can start over now.

“A Picture Is Worth 1000 Words”

One fun activity that I enjoy is asking my students to draw a picture of a mathematician. I see lots of pictures of little bodies and big heads, some glasses, some pocket protectors, and some crazy Einstein hair. (I have an ex-colleague that does this activity, and once he had a couple of students draw wizards – math is so “magical”!) The pictures rarely look like any of the students in the room.

Students feel that anyone who understands math is some sort of super-genius. There is a giant wall in front of them that leaves math inaccessible to them. I explain that any student who is willing to devote the time, effort, and thought to learning mathematics can do it. And I’m here to help them. I tell them that if they want to see what a mathematician looks like then they should check out the mirror when they get home.

Positive outcomes: Students realize that math can be accessible to them.

“Tell Me Your Strengths And Weaknesses”

On Day One I give my students a survey that helps me to understand them a little bit better, as well as showing them that I am interested in my students and their success. Near the end of the survey I ask my students to give me 3 reasons why they will pass this class. Basically I am asking my students to list their strengths because I want them to acknowledge that they have student and/or personality traits that can help them be successful regardless of the arena. Even in math class.

I also prompt my students to finish the following statement “If somehow I do not pass this class, it will most likely be because …” Here I am asking my students to identify what they feel is their greatest weakness as a math student. The thought is that the best way to overcome a weakness is to begin by identifying that weakness. I read over the surveys that night, and on the second day of class I go over coping strategies for overcoming these weaknesses. Students at the developmental level have little experience with developing coping strategies, but once this is modeled for them they are more likely to be able to do this for themselves.

Positive Outcomes: Students realize that they have their own strengths, as well as plans to overcome any perceived shortcomings.


Day one is a great opportunity to break down student misconceptions about math and mathematicians, for students to realize that they are not alone in their struggles, and that there is a path to success if they choose to take it. The activities I have shared are great ways to alleviate some of the anxiety our students feel. Give them a try, and let me know how it goes. If you have any activities of your own, please share them with me by leaving a comment or reaching me through the contact page on my website.


My 3 R’s and Mindset

My 3 R’s and Mindset

I took last semester off (bank leave), so Monday will be the first day teaching students in about eight months. What did I do during that time? I worked on the three R’s: resting, reading, and redesigning my classes.


I have taught a full load of classes every semester (and nearly every summer) since I started at College of Sequoias back in 1994. I did not think I needed a break because I did not feel the burnout that many teachers discuss. I love my students and I treasure working collaboratively with many colleagues at the college. However, about two months into this break I felt full of energy in a way that I have not felt in a long time.

What to do with all this energy while I am not in class?


I started reading … a lot! I have read 45 books since January 1. I read a few fiction books and baseball related books for fun, but I focused mostly on books dealing with math, creativity, and business leadership. Books in these categories always have lessons that can be incorporated into our classrooms.

Two books that had a big impact on me this year were Carol Dweck’s Mindset (Amazon link) and Jo Boaler’s Mathematical Mindsets (Amazon link). Dweck has a chapter for teachers and coaches that was enlightening, along with chapters on what the mindsets are, ability & accomplishment, business mindset and leadership, and how to change mindsets.

I’m a big believer in the power of classroom atmosphere and the impact it can have on students, especially on their attitude, motivation, and persistence. While developing a growth mindset may not be the magical solution for all students in mathematics, making students aware of some of the key tenets can do no harm. Math ability is not an innate talent, it can be grown through effort. Students with a fixed mindset believe that you are either smart or you are not, and choose not to try rather than giving their best effort and falling short because that would label them poorly. I want my students to understand that we learn even from failures. They can increase their mathematical ability with full effort, persistence, and an open mind.

I plan to be careful with my language in the classroom – praising students effort rather than their intelligence.

We all know that math has been a difficult subject for many students, and they often tell us about how they are not a “math person.” I plan to use what I read to break the connection to past struggles and launch my students on a new path to success. Based on this I added two questions to my Day 1 survey to make students aware that they (and others) can learn material that is difficult and achieve accomplishments that they felt were not attainable.

Give an example, in detail, of an area in which you once had low ability but now perform well.


Tell me about a person that you saw learn how to do something you never thought that person could do.

(I’ll share more about my revamped Day 1 survey in a later post.)

Redesigning My Classes

Besides incorporating what I learned about the growth mindset into my classes, I have made a few other major changes to my classes.

  • Statistics
    I have made the commitment to go “early and often” with inferential statistics, devoting 20 of the approximately 60 class sessions to inferential projects. In week 2 I will spend one day on estimating a population proportion through simulation and sampling, and a second day on estimating the difference between two population proportions using resampling.

    I am also going to be using the flipped classroom often, opening up time for peer instruction and group activities using Learning Catalytics.

  • Elementary Algebra
    In my daily class I will continue to use my game design based approach with a couple of tweaks. Students who level up and pass one of the exams in the first half of the course will be able to take an optional midterm cumulative exam in order to earn more points.
    Students can also earn points by completing a real-world mathematics project in which they will learn about any math related topic of their choice.

    In my class that meets twice a week I will use a traditional approach, but the online homework assignments and quizzes will have increasing weights as the semester progresses. I got this idea from reading James Lang’s Small Teaching (Amazon link). I’ll share what I have learned/incorporated from that book in a future post, but I will tell you know that James Lang is a must-follow on Twitter (@LangOnCourse on Twitter).

    I am planning on incorporating the use of Plickers in my course. Students hold up a card with answers A through D corresponding to the orientation of the card and I scan their answers with my iPhone. Think of clickers without the students having to have any technology.

I’m looking forward to a great semester, and I am looking forward to learning from you as well as sharing what I am doing with you. Please leave comments or questions, and let me know if you’d like to be friends on Goodreads.

– George

I am a math instructor at College of Sequoias in Visalia, CA and the author of an algebra textbook and co-author of Interactive Statistics with Michael Sullivan (both with Pearson).