## Archive for the ‘Uncategorized’ Category

### Talbert’s _Flipped Learning_

June 29, 2017

I just finished Robert Talbert’s Flipped Learning. Here is a brief review.

I will preface the review with a couple of comments. I have “known” Talbert online for years, although I have never met him in person. I was also mentioned in the acknowledgments, although I did not play much of a role in writing the book (you will see below that I have a lot of work to do with flipped learning). Finally, I did not receive any payment of any sort for anything related to this book, and Robert does not know that I am writing this review (he does not even know that I read it).

Talbert gives step-by-step instructions on several things that can improve your classroom (designing the course, creating Guided Practice assignments, etc). This really acts as a how-to guide, in many ways. He also spells out what the point is: you do flipped learning to take advantage of the active learning in the in-class.

The last section of his book is helpful to anyone using active techniques. For instance, he talks through what to do when students express dissatisfaction because the professor “isn’t teaching” (or “I have to teach myself everything”). This is worth reading even if you never plan on doing flipped learning.

One thing that is worth noting is how useful I found it that the book was written by a mathematician. He frequently used examples relating to mathematics (three of his six case studies were on mathematics classes), and this helped me digest the material. In contrast, I have been reading a lot about Team-Based Learning this summer, and there have been zero examples of a mathematics classroom (although I found stuff on statistics and math for engineers), and the lack of relevant examples has slowed me down a bit in imagining how my courses might look like if I implemented Team-Based Learning. Of course, some may view the focus on mathematics as a drawback, but I (and likely those reading this post) found it helpful.

It was also enjoyable to read a book on teaching written by a mathematician because Talbert thinks about education in the same way one thinks about mathematics. For instance, he gives two approximations for the definition of flipped learning before settling on the one he uses. Also, he abstracts his ideas on flipped learning as much as possible. I paraphrased his definition above by referring to “in-class” and “out-of-class” time, but he abstracts this to “group space” and “individual space,” respectively, so that he can accommodate blended and online courses.

In summary, I feel like I am a bit of a veteran with the flipped classroom, but I am changing my planning for next year because of this book. It was quite helpful. I will end with my two favorite quotes from the book.

Q: I am having a hard time finding appropriate action verbs to use for my learning objectives…Is there a place I can go for hints?
A: Yes, and it’s called “the internet.”

(Talbert goes on to elaborate his answer above).

In a flipped learning environment, we instructors have to make educated guesses on the “center of mass” of the students’ ZPDs based on their execution of basic learning objectives and design the group space activities accordingly. Getting this guesswork right is part science and part art (possible part magic).

### Marzano’s _Classroom Assessment & Grading That Work

June 9, 2017

I thought that I would do a couple of book reports this summer. I have been hearing about Marzano for years, and I thought that I should finally read some of what he says about Standards-Based Grading. The book I read is Classroom Assessment & Grading that Work.

I read the book about a month ago, so I do not remember everything. However, below are the ideas that stuck with me.

First, you should use “topics” for your class, and there should be about 15–20 of them. These are akin to standards in SBG. Whenever you test a standard, you should give the student a question in three parts. The first part should be basic details and/or facts that you would expect every student to know, the second part measures whether students understand what was covered in class, and the third part asks the students to go beyond what was done in class. I am teaching real analysis in the fall, so I am going to give an example for real analysis on the topic of compactness:

1. Is the interval $[2,3.5)$ compact?
2. Show that if $S \subseteq \mathbb{R}$ is a compact set, then the supremum of $S$ exists and is in $S$.
3. Give an example of a metric space $M$ with a set $S \subseteq M$ such that $S$ is closed and bounded but not compact.

I don’t love my example, but I hope it gives you an idea. You then grade the students answer according to the following rubric:

• A student receives a score of 4.0 if she is able to answer all three questions (“I can make connections that weren’t explicitly taught.”).
• A student receives a score of 3.0 if she can answer the first two questions (but not the third) without mistakes (“I can do everything that is taught without mistakes.”).
• A student receives a score of 2.0 if she can answer the first question (but not the second or third) without mistakes (“I can do the basics without mistakes.”).
• A student receives a score of 1.0 if she can answer some portion of the questions with help.
• A student receives a score of 0.0 if she cannot do any of the questions, even with help.

Half scores of 0.5, 1.5, 2.5, and 3.5 can be defined in a reasonable manner (Marzano does this in the book). Marzano claims that this scoring system leads to a roughly normal distribution.

Marzano then suggests that each topic is graded in one of two ways: either you find a function of the form a*x^b “of best fit” for each topic to predict where they will be at the end of the semester (using software). I will not be using this method. He also recommends using the “Method of Mounting Evidence,” which basically means that you keep track of all of the student’s scores within a topic (e.g. 1.5, 2.0, 1.5, 2.5, 2.5, 3.0, 2.5). Once you are convinced that a student’s “true score” is at a certain level, you mark it down and then look for evidence that they surpass it in future assessments. For instance, in the example list of numbers above, the second 2.5 is in italics, which might indicate that our hypothetical teacher thinks that our hypothetical student has convince him that she is definitely at 2.5 level for this topic. On assessments following that corresponding to the italicize score, the teacher will be mainly looking to see if the student jumped to a 3.0, 3.5, or 4.0 as her true score. And is she gets, say, a 1.5 on a future assessment? The teacher just returns the assessment and asks her to correct the missed “easy” work, with the assumption being that the student just had a bad day rather than no longer knows the material.

You can assess students as many times as you like, although Marzano recommends assessing students you are unsure of more. This seems entirely reasonable.

It seems possible that a student could get the hardest question correct but not the easiest question. Marzano mentions this possibility, but basically says that he assumes that a student who can answer the hardest question should be able to answer the easiest. So, ideally, the assessment writer would write questions in such a way that this is true.

At the end of the semester, the student’s score for each topic is just wherever they ended up with from the Method of Mounting Evidence. Marzano then talks about ways of averaging together the topic scores, although this is not particularly of interest to me. His other method for determining a final grade is something akin to what many of us to already, which is creating rules like, “A student gets a B for the semester if no topic score is below 2.0 and the majority or 2.5 or above.”

The two ideas that I am thinking a lot about are:

1. Topics should be assessed at different levels, as with my real analysis example. I have been heading this way for a while now, and maybe this is the year to try it.
2. You can give grades based on whether a student can solve it with help. I think that this is brilliant. However, I still need to figure out how to assess this in a reasonable way with 75 students. But I like it.

### The New Proposed Curriculum Fails

May 10, 2017

I told you a couple of weeks ago about how I was nearing the end of a 4-year process on building a new curriculum. We had the vote last week, and we lost: the faculty decided to reject the proposed curriculum. We lost by five votes (if we had only changed three people’s minds! Actually, I am not sure if I would have wanted the curriculum to pass by one vote—I don’t want 50% of the people unhappy).

This is disappointing, but the people have spoken. There may be a tiny bit of hope for the curriculum, though: I talked to several people (at least three, which would be enough for it to pass) who wanted more details about a separate, but related, distribution requirement that will be decided in the fall. So it is possible that the Faculty Senate will decide to resurrect the curriculum after the distribution requirement is settled, but there is no guarantee of that.

That is the bad news. The good news is that a lot of my time has just been freed up over the next couple of years.

### New General Education Curriculum

April 24, 2017

I haven’t been writing much up until now because I have been in the middle of creating a new general education curriculum at our school. This is year-four of the process, so we have put quite a bit of work into this. We are in the middle of a five-day discussion period, followed by a five-day voting period on whether to adopt the proposed curriculum. Our current general education curriculum is a pure distribution requirement, where students take

• a two-semester writing seminar in their first year
• one Mathematics course
• one Natural Science course
• one Social Science course
• one Fine Arts course
• two Humanities courses
• two Theology courses (I am at a Catholic school)
• one Ethics course
• one Gender course
• one Intercultural course
• one Experiential Learning course
• a capstone course within the major

The Gender, Intercultural, and Experiential Learning courses can be double-counted with other courses; that is, a course could count as both Humanities and Gender.

There are many reasons why our Senate decided to explore changing the curriculum. Historically, very few people were happy with the way our current curriculum was decided upon in 2005ish (I was not at my current school then). Additionally, many feel that these courses occur in isolation, and the students are not given any opportunity to see how they are related.

Here is a link to the website of our proposed curriculum. This was a huge amount of work, and roughly 1/6 of our faculty ended up working on it in some capacity. It is safe to say that this curriculum is no individual’s ideal curriculum, but rather a result of considering 300 faculty members’ needs and desires for their students.

We are happy with this curriculum, although I have no idea how the vote will go. Many people have come out both in favor of it and many people have come out against it. If this does not pass, we (my school, not me in particular) will have to spend a couple of years revising our curriculum to get it up to snuff (many of the current outcomes are unable to be assessed, for instance).

### Galois Theory

March 22, 2017

I am lucky enough to be teaching a course on Galois Theory that only has five students in it. The set-up of the course is this: we are working out of Pinter (which I really like due to its ease of reading, great problems, and price of \$12; I don’t like that he defines subgroups/subrings/etc in a weird way and is sloppy with defining variables), and students present problems from the textbook to each other.

This will be obvious to people who do IBL, but: holy cow do you get a sense of what students understand and what they don’t. A large part of this is the small class size, but this IBL-like format helps. What students understand (or not) is often not what I would expect.

Because I only have five students, I am doing oral exams. The format is this: I give them four problems the week prior to the exam. Students are allow to work together to figure out the answers. They come to the oral exam having written one of the four problems up nicely in LaTeX. The oral exam starts by the student choosing one of the remaining problems and explaining it. I then (randomly) pick one of the two remaining problems. The session ends with me presenting a problem they haven’t seen before (I try to make these easy enough that the student should know what to do immediately). We will do these exams four times during the semester, and they take 30 minutes per student.

Once again: this testing format makes it crystal clear to me what (and how well) students understand things, and it is clear how I should adjust the in-class work based on the information I get from the exams.

I am wondering if this could scale: could I give a class of 25 students, say, 4 questions, have them write up one, and then have them write the solutions to a subset of the remaining problems? I like that this is a learning opportunity for the students, since they get to learn from each other in an exam situation, but they are still individually accountable. However, I am wondering how much is lost if the exams aren’t oral.

Thoughts?

### Teaching Real Analysis

March 16, 2017

I am teaching real analysis for the second time in the fall, and I am excited about it. I used Stephen Abbott’s Understanding Analysis when I taught it in Fall 2011, and the students and I both loved it.

My one issue with it, though, is that I would rather do more with metric spaces (Abbott works with sequences in the real numbers as a foundation for the course); I found that I would often draw pictures of R^2 on the board to illustrate ideas relating to distance, and I would like to leverage this slightly more.

I am sold on the idea of using Abbott: it works ridiculously well for my flipped classroom, the students love it, and I am already familiar with it (I am hoping to stop completely redesigning every course I teach from scratch). Here are my ideas for incorporating metric spaces more:

1. Just follow Abbott’s book as is, and forget about using metric spaces.
2. Start the semester by looking at Abbott’s brief chapter on metric spaces (in Chapter 8), let students know that we are mainly going to be using it for examples in class, and they are not very responsible for knowing it (perhaps I might give challenge problems where they generalize results in terms of metric spaces, but not every student would need to do that).
3. Supplement Abbott with a cheap textbook (roughly \$10) on analysis like Rosenlicht.
4. Supplement Abbott with something like Kaplansky’s text on metric spaces (\$30).
5. Supplement Abbott with something like Keith Conrad’s notes on metric spaces (free).

Money is a factor, so I don’t want an expensive supplement.

I am mainly looking for comments like “It is a bad idea to try to integrate metric spaces with Abbott” or “It is a good idea, and here is the perfect source.”

### What toys would you buy?

March 6, 2017

Here is my short list (as usual, no one paid me to mention these):

What else do you recommend? You can either list things that you have already purchased or things that you wish you could purchase.

### Examples of Math Circle Activities for Young Kids

March 3, 2017

Joss Ives ask for a list of my Math Circle activities for young children. What Joss asks for, Joss gets, so here is a blog post on the activities. Also, my wife wants a list of the activities we did in January and February, so I get to kill two birds with one stone with this post (Note: my usual communication with my wife is not via this blog).

Procedure: We meet once per month in a room at the public library. I usually run two 30-minute sessions, with three to seven kids attending each session. Their parents are rarely in the room, although it happens on occasion.

I stole most of these ideas, mainly from Math from Three to Seven by Alexander Zvonkin and Math Circles for Elementary Students by Natasha Rozhkovskaya. I also steal ideas from Talking Math with Your Kids by Christopher Danielson. I am trying to cite every place where I stole ideas, but I am sure I am forgetting. Please contact me if I should be giving you credit.

Disclosure: I have met Christopher and I am friends with Natasha, but no one has paid me to link to anything in this blog post (or any others).

Here are the activities.

Day 1
1. I gasked questions like “Are there more geese than birds?” and “Are there more women than moms?” I wanted them to start thinking about set containment.
2. I showed them pictures of, say, a bunch of dolphins (and also cats) that look similar (I drew them). I asked, “Is it true that there exists a dolphin with a ball?” “For all cats, the cat has an umbrella?” The purpose is to get them thinking about quantifiers.
3. I showed them a geoboard. I put a geometric design on the left half, and they had to do the mirror image on the right half.

Day 2
I taught them to use Base Ten blocks with the help of a puppet named Yachel. I told them that Yachel is afraid of the number “ten,” so he does not to see ten of anything. The kids organized the Base Ten blocks into groups of ten so that there would be only one big group, rather than ten small groups (so Yachel wasn’t afraid). They kept doing this until all of the blocks were organized. In the end, I asked them to guess how many blocks there were (something like 1287), and I showed them that they can actually tell exactly how many of them there are by just counting how many big groups of each type they made.

I don’t know how much the kids learned about the Base Ten number system, but Yachel was a huge hit; my kids still treat him like he is one of the family.

Day 3
I read them a book called something like 5 Cats, in which the cats categorize their family members into different groups (3 are male and 2 are female; 1 is black, 2 are white, and 1 is calico; etc). I brought hula hoops, and then I asked questions so that the kids could sort themselves (“Stand in that hula hoop if you are wearing something blue today.”). This was a bit of a flop.

Day 4 (March 2016)
This was a Pi Day celebration. I found a bunch of circular lids of different sizes, and cut up a bunch of pipe-cleaners so that they were the length of the diameters of the lids. Then I hot-glued googly eyes on them, and called them “Diameter Worms.” I made up a story about how Diameter Worms find circles to live in, just like hermit crabs. The Diameter Worms need to have a circle that is exactly the right size for it. Then I asked them to figure out how many Diameter Worms can lie end-to-end around the outside of their circle home. First, they made a prediction, then they actually wrapped the worms around the home. Of course, everyone learned that “a little more than three worms” could fit around, regardless of the size.

I don’t know how much they learned about pi, but it started a Diameter Worm craze in my son that lasted for several months.

Day 5
1. More work on subsets and quantifiers, as we did in Day 1.
2. I did something with 3×3 patterns, but I don’t remember what.
3. I gave students cut-out polygons and scissors, and they had to do certain challenges. For instance, cut quads into 2 triangles, or cut a quadrilateral into 2 quadrilaterals, or cut a quadrilateral into a triangle and pentagon.)

Day 6 (11/2016)
1. We did Danielson’s Which one doesn’t belong?
2. We played Nim on graphs, which is a game developed by Marie Meyer and me for her senior thesis.

Day 7 (12/10/2016)
We played (in an unstructured way) with Base 10 blocks (there were only two kids that week).

Day 8 (January 2017)
1. Danielson’s How many? to get at the idea of units.
2. We talked about “Doot Aliens.” When Doot Alien A touches Doot Alien B’s nose (which makes a “Doot!” sound), both Aliens disappear and some Doot Alien C reappears in its place (A-Doot-B always results in the same C, although B-Doot-A might not result in C). There are Ghost Aliens such that A-Doot-Ghost yields A, as does Ghost-Doot-A. I asked them: “What happens when one Ghost touches another Ghost’s nose?” They didn’t come up with the answer, but a couple of them have asked me about Doot Aliens (without me prompting) since then.
3. I gave them a bunch of statements like, “The sky is blue” and “There are seven people in the room,” and I asked them whether each statement was true or false. Then I asked about “This sentence is false.”
4. More of Danielson’s Which one doesn’t belong?

Day 9 (February 2017)
One-cut hearts to celebrate Valentine’s Day (plus, one-cut stars to celebrate Betsy Ross).

Day 10 (March 2017)
I am working a new Pi Day activity, but I haven’t thought of it yet.

### Use Mathematics for the Social Good?

February 23, 2017

I am going to keep this short today: I am really excited about Moon Duchin’s plan to create an army of expert witness mathematicians for gerrymandering cases. This is going to be a summer class at Tufts, with other courses planned for Wisconsin, North Carolina, Texas, and San Francisco. I am really interested in doing this, but I want to educate myself more on gerrymandering first.

How many other such volunteer groups are there? I can think of:Statistics without Borders. I thought there was a similar one for “Operations Research without Borders,” but I can’t find anything on it.

Can anyone think of other organizations?

### Inquiry-Oriented Instruction

February 15, 2017

I was part of a grant last semester to implement a set of teaching materials that has been refined over the last decade. The materials use a teaching method called inquiry-oriented instruction, which I would say is a subset of inquiry-based learning (IBL). I used these materials in my abstract algebra class, although there are materials for both linear algebra and differential equations, too.

A very brief description is “intuition comes before definitions.” The materials introduce quotient groups by discussing Even and Odd integers, which students could easily see is a group at that point (using rules like “Even + Odd = Odd”). Once they got familiar with the idea that we could have sets of elements make up a group, we slowly backed our way into the definition of coset. It was pretty impressive to see students very naturally come up with definitions—having the right prompts helped a lot.

As part of the grant, I went to training at North Carolina State to use the materials. I also had funds to have student video my class, which will be used to analyze how well instructors who were not involved with the development of these materials can implement them.

We also used the class video as part of a weekly online working group. The purpose of this group was to prepare us, both in terms of pedagogy and course materials (not everyone was an algebraist), to teach the class. We discussed the purposes of the prompts, talked about what was going well and poorly, and watched video of each others’ classes. I found this immensely helpful.

I would use these materials again (in fact, I am planning on using the linear algebra materials next year). My sense is that my students had an abnormally good grasp of the definitions; previous students have struggled to understand what a coset means, for instance. My focus for the next time I use the abstract algebra materials is to work harder on the technical proofs—I think that my students did better on writing proofs than the previous time I taught the course, but not by a lot. Still, I think that the gains in intuition were worth it.

Links to the abstract algebra, linear algebra, and differential equation materials can be found here in the middle of the page.