Forming an interesting in-class Statistics activity.

This Wednesday I've scheduled an in-class activity that relates some of the ideas we've discussed in my Statistics I class. So far they include surveys, descriptive statistics (mean, median, quartiles, and standard deviation), sampling, experiment design, and observational studies. Writing this on Tuesday morning, I have nothing written down but I have a couple ideas of what I want this activity to include:

• Perform an experiment and/or an observational study. I covered what these are in a lecture, but I really want them to attempt one (or both) on their own so they get a concrete idea of what they are. Doing both would be best since they would then be able to compare the methods of both.
• Take appropriate samples based on the research question. Ideally this would be done with students in the class so they can see how to take appropriate samples.
• Compute and use descriptive statistics to compare and contrast different samples. Quantitative reasoning and analysis are a core focus in statistics. 
• Talk about statistics. Whether it be in groups or in a presentation, forming ideas and communicating them to others is another core skill in statistics.

I also know what I don't want:

• A rote activity that requires no input from students. I want them to struggle with questions about sampling, which statistic to compute, what to do next. Through this struggle I want them to appreciate principles like variability, controlling for certain variables, and how to construct arguments for one action or another.

For an experiment I have a few ideas:

• Performing an intervention dealing with basic math skills. The idea would be to assess if a certain intervention (a logic puzzle, or a game) has any effect on student performance of a basic math test. We would discuss how to create a control, how to create the samples, how to administer the tests, etc. While the direct applicability of this example may be a stretch, it would be a constructed activity that students could perform.
• Have the students develop their own question and proceed from there. This has some issues, primarily because of the unending and uncertain nature of such a question. Without some guidance the possibilities are a bit too large, which generally leaves students not choosing anything at all.
• Provide a context (marketing manager, nurse's aide, etc.), and have students develop an interesting question that can be answered in-class. While being more specific we may not be able to answer such questions in-class. For example, if we wanted to do A/B testing of a website, we would have to construct this example fairly quickly, and perform the experiment on people who weren't part of the development of the website.

For an observational study:

• Given a data set have students develop an appropriate research question, and try to answer it using the data. This could be generalized further by having multiple data sets and have students pick one, or assign different groups different data sets.
• Taking the survey data collected at the beginning of the course, ask an appropriate research question that can be answered either by the previously collected data, or by another survey in class. The survey data is anonymous so it would be difficult to answer some questions.

You can probably guess which way I'm leaning, but I'll post the completed activities tomorrow. Feel free to post in the comments if you have any ideas or suggestions. 

Worksheet Wednesday - Calculus Quizzes Part I

This week I'll be sharing a few quizzes from my 5-credit Calculus I course that meets three days a week. I have students complete a daily quiz that contains fairly simple questions about the material covered in the previous lecture. My intent is to make sure students put in time daily to practice the material and struggle with concepts in the course. So far it has worked fairly well, students seem more capable this term, they understand the expectations I have of them, and their performance on tests is exceptional. I will be doing some rudimentary comparisons of this class with my previous Calc I, so stay tuned for that.

Below are quizzes for a few of the first sections from Stewart's Calculus: Concepts and Contexts 4th edition. We don't cover epsilon delta arguments so a few of the limit questions may seem a bit rudimentary, and in reality they are. I feel uncomfortable asking students to do a lot with limits since we don't cover what is 'really' going on. Then again can we teach it at this level?

Quiz 1 - Review - In this quiz I have them do a few things they should know how to do that are pre-skills for the course; finding the slope of a line, finding the equation of a line that goes through two points, and finding a parallel line.

Quiz 5 - Continuity -

Quiz 6 - Miscellaneous -

Quiz 8 - Definition of a Derivative -

If you have any feedback, or you use them in your class modified or not, please post a comment below! You can also email me at robert dot weston dot 82 at gmail dot com. Thank you for reading!

Course Observation - Definitions

Life has been pretty hectic lately, I moved apartments (in the same building), interviewed for a full-time position at the college I am currently teaching at, and have generally been a mess teaching 20 credits, two new classes (to me), developing interesting lectures and activities, and grading (oh the grading). I enjoy it and love the challenge all of this offers me, but makes it hard to post regular updates. Hopefully you've enjoyed (or been amused by) the worksheets, activities, and quizzes I am posting every Wednesday. Feel free to drop me a note or post a comment below if you find anything on this blog particularly engaging or interesting.

This week a seasoned math instructor asked if he could observe my class and I gladly said yes. Sure, it can be stressful having someone in your class that isn't one of your students, but I am always looking to improve my teaching, and if that means a bit of discomfort, so be it. My desire to become a better teacher is greater than my desire to avoid criticism, constructive or otherwise. Lucky for me the feedback was very constructive, and I plan on discussing it in a few posts. Here I will only talk about one part of that feedback, the balance between core skills and more abstract ideas.

They observed a day when we went over graphing functions based on their derivatives. I have had previous conversations with this instructor about this lesson, and it was by chance the they observed this class. The previous conversations were about an esoteric point about the definition of a phrase that impacts how we answer certain questions. I mentioned this in class and how the book and I differ on this one definition, and how it could change our answer.

The observer commented that this fine distinction might not be the best use of time and energy. They were worried that few students would understand what the impact of this difference is, and most students would get nothing from it. Students have a difficult enough time understanding the book's definition, mentioning my definition may confuse them.

I understand this concern, and appreciate the fact that this instructor has spent much more time with this student population. I will likely not mention this difference in definitions in future classes, but I do want to spend time explaining why I included this difference. This lesson is at the end of a section that is very computation focused where they apply different rules. With this small diversion into how the book and I differ, I was hoping to demonstrate that mathematics isn't just a matter of following rules and computing things. This aspect of having and using precise definitions is also important and one they are going to be exposed to in future classes. I still like this idea and would like to incorporate it into another lesson. If you have any ideas about how to do this, feel free to share below.

I'll be discussing the other feedback they provided in future posts. I'm doing so partially to help myself think through the feedback and to reflect on how I can use it to become a better teacher. I also want to be able to talk about these ideas without sounding defensive. I think about my teaching pretty carefully, so sometimes I get wedded to an idea without talking to others about it. When this happens and I get feedback to the contrary I can feel myself respond defensively, and I don't find that response very useful, or frankly, professional. Part of it is this math culture of being right that I've internalized, but it also comes from my own self-doubt, both things I need to shed.

On that happy note, feel free to share your comments or email me directly.

Worksheet Wednesdays - Complex Numbers, Power Series, and Taylor Series, oh my!

For this week I have another twofer, two in-class activities dealing with complex numbers, power series, and Taylor series.

Complex Numbers and Power Series - Students were introduced to some of the properties of complex numbers, and the first page of this activity has them explore operations and graphs on the complex plane. The second page has students identify the radius of convergence of given power series, with a little analysis of the coefficients on the third page. The last two pages have students graph partial sums of power series, with a hint as to what they approximate.

Taylor Series - In this activity students are given two different functions and are asked to analyze their power series about different points, and look at a very special relationship. The first function is a simple rational function, the second being the exponential function. Students are asked to numerically compare Taylor Polynomial Approximations of the rational function centered at different points, and are asked to provide a basic comparison of the approximations of the exponential function as well. The last page walks students through a derivation of Euler's Identity. For the most part students were pretty impressed with the identity, with the exception of the student asking "Is this going to be on the test?" right after the derivation.

Worksheet Wednesdays - Trigonometric Trickiness

For this week I have a fairly basic in-class trigonometry activity, having students develop a deep understanding of these ratios, and to use them in application questions. During lecture I completed a number of 'naked' examples, without an authentic context. Here I have them start solving application questions using trigonometry with some fairly basic questions. I do include one question that increases the difficulty, but with a few hints dropped students seemed to get the basic idea.

As students work on in-class activities I usually go around the room and answer any questions they have. I do so using the Socratic Method, answering their questions with questions. They usually find this annoying, me answering their questions with questions, but over time I have noticed that students become more thoughtful with their questions. They anticipate how I will answer their questions, and so modify their phrasing so they are not just asking for numbers of values, but for methods. Near the end of an activity I will hear many more "How do I..." type questions than "What is the value for...". This demonstrates a greater appreciation of method, something I think is the goal of all mathematics educators.

Right Triangle Trigonometry - Students first confirm a few trigonometric ratios for a randomly drawn right triangle, and then answer a few application questions. Granted, it is only two pages, but if this is done the first day the sine, cosine, and tangent functions are defined, it will take 30 minutes to 1 hour.

If you have any feedback, or if you use these worksheets, let me know! Feel free to post a comment below, or email me directly.

Worksheet Wednesdays - Sequences and Series In-Class Activities

This is the inaugural post of Worksheet Wednesdays where I will post some of the worksheets, in-class activities, quizzes, and other assessment items I have develop in teaching college mathematics courses. Today will be a twofer, two in-class activities for my Calculus III course.

I have students start in-class activities during class, but they are free to complete them afterwards. In order to receive credit for completing them I have them show it to me before the next test. When they do I usually go over it with them, providing for some 1-on-1 feedback and developing a repertoire with them. So far this has worked out well for three of my classes, but as I write this it is only Week 2 so that may change. Anyway, on to the worksheets!

Monotonic and Bounded - This activity walks students through a variety of sequences and introduces the idea of monotonic and bounded sequences. They then classify each of those sequences as monotonic and bounded and then see if these sequences are convergent. Students should identify the pattern that all sequences that are monotonic and bounded are convergent, but it is not necessary for a convergent sequence to be monotonic or bounded.

Integral Test - From lecture it was clear that students' integration skills were a bit rusty, if not a bit poor. Thus I created this in-class activity to get students to practice their indefinite, definite, and improper integration skills, and to use the integral test of convergence of a series. There are a few curve balls, and we had a good discussion of how to deal with them.

If you have any feedback, or if you use these worksheets, let me know! Feel free to post a comment below, or email me directly.

In-Class Activities: A Well I Have Fallen Into

This term I am utilizing in-class activities in all four of my classes in various ways. Previously I've used them only in special cases where the topic is best understood by a hands-on demonstration of the concept. After teaching with a flipped classroom model developed by another faculty member, I am becoming more comfortable with students discovering the material in-class, as opposed to me telling students what the ideas are.

College Mathematics - This is the course I taught last term that utilized a flipped classroom model. Students are to print out handouts before class, take a pre-quiz, come to class, work on the handouts in groups, complete the handouts at home, and take a post-quiz within 24 hours after class. They repeat this for a number of handouts, and also have group tests, tests, and homework. Initially I was worried about, well, everything. Students not showing up, not understanding, waiting for me to tell them what to do, etc. It all happened, but those students who would have done well in a traditional class did well here. Those that would not have done well in a  traditional course... I'm not sure if they did any better using this model. The freedom that students are allowed in this format presents them with a choice, to put the time into understand the material, or not. But this is the choice all students have in any course. At this point I see this method doing no harm, and possibly helping students understand the material a bit better.

Trigonometry/Pre-Calculus - This is a 5 credit hour course that meets twice a week, meaning two and a half hour class meetings. To break up lectures I'm using in-class activities to have students develop concepts, or to gain practice with different skills. The first two had students developing the graphs of trigonometric functions. Students were initially resistant, but during the summation of the activity they were able to answer questions students in previous classes were not.

Calculus I - I'm using the POGIL activities (link to come) another instructor recommended. There are only three in the course, but if the first activity is any indication, I think they'll do well.

Calculus III - Weekly activities meant to simulate a structured recitation. The activities have been successful, but the time it takes to develop them is considerable.

I will be posting these activities in my upcoming Worksheet Wednesdays. Feel free to download them, use them, and send along any feedback.