# Modeling Fun with Paper Fish

Kate Owens, 02/2016

Back in early February, as part of my ongoing work with the Math & Science Partnership, I led a Saturday professional development workshop for STEM teachers on “Proportion, Decimals, and Percents (oh my!).” There were two major projects we worked on that day. First, I split the teachers into teams of two or three and they read over some “Always, Sometimes, Never” statements. Fawn Nguyen’s blog post has some great ideas to get you started on those. Second, we simulated determining a wildlife population. Since this is something I hadn’t seen blogged about before, I thought I’d tell you about how it worked.

I found the idea for this in a book called “Mathematical Modeling for the Secondary School Curriculum.” It’s based on an article called “Estimating the Size of Wildlife Populations” that appeared in the NCTM’s Mathematics Teacher back in 1981*. Here’s how it works. Suppose you have some closed ecosystem that has a population of animals — maybe a large lake containing a population of a certain species of fish. What if we want to know how many fish are in our lake? Can you think of ways we might approximate the number of fish?

Here are some ideas that might spring to mind:

• If we knew something about the social personality of the fish — for instance, maybe they are really independent and territorial and don’t like hanging out together — then we might know that they prefer to have at least X cubic meters of space to themselves. If we knew how big the lake was, then this could give us a rough count on how many fish there are. Problem: Knowing how big a lake is, in terms of volume, can be tricky. The bottom of the lake might not be flat. The amount of water varies based on temperature and rainfall. And what if we don’t know if our fish are social swimmers or solo swimmers?
• We could rope off (fence off? net off?) a portion of the lake and count how many fish are in our section. If we knew we’d roped off exactly 10% of the lake, maybe we could use this information to estimate the total number of fish. Unfortunately, this is also difficult. First, we don’t know the fish are uniformly distributed around the lake. Maybe we roped off a portion of the lake that’s very rich in food source so we have many more fish than we should. Second, it’s tough to know if we’ve gotten exactly 10% of the lake or not. (How do you measure the volume of a lake, anyway? I’m sure there’s some way to do this, but I have no idea how.)

You may have thought of some other ways, too. Leave them in the comments. Here’s the way proposed in the NCTM article. It’s known as a capture-recapture estimate. Let F represent the number of fish in our lake. First, we capture a large number N of fish and tag them in a way that isn’t harmful; then we toss them back. We wait a while. Once the fish have had a chance to do their fishy things, we go back to the lake. We then capture x fish — some are tagged(T for Tagged), some are not. Assuming the fish are randomly dispersed throughout the lake, we might conclude that the number tagged in our sample is proportional to the number of tagged in the entire lake: N/F = T/x.

For a quick example, suppose we capture and tag 1200 fish. When we return to the lake, we re-capture 200 fish and we find that 30 of them are tagged. Assuming that the number tagged (30) in our sample (200) is roughly proportional to the number tagged in the lake (1200), we conclude that 30/200=1200/F so F=8,000.

What could go wrong? Well, maybe our sample isn’t very indicative of the population. We throw back all of the fish and then take another sample of roughly the same size. If we take several different samples, we can use the additional information from further samples to get a better estimate of the fish population. (I’m not going to go into all of the statistics at work here.)

Modeling the Fish Population

I gave each group of teachers a box. A shoe box would work. Inside each box were about 200 squares of paper. I didn’t count the squares as I put them in, and I didn’t want any two boxes to have precisely the same number. Having ~200 isn’t necessary — you just want enough people can’t do a fast eyeball estimate, but not too many because eventually you’ll want to count them.

One teacher “went fishing” and “tagged” a handful of fish (a dozen or so) by marking those squares with a signature, symbol, smiley face, whatever. The fish were returned to the lake before they suffocated. The box was shaken up. Another teacher then took a sample of size larger than the tagged number — something along the lines of 20-25 fish, give or take. The number of tagged fish in each sample was counted. Fish were returned to the pond, the box was shaken, and like it says on your shampoo bottle, “Lather, rinse, repeat.” Assuming the captured sample was the same size, after taking 10 samples, we averaged the number of tagged fish. Using proportions, we found an estimate for the total number of fish in the pond. Lastly, each team counted the actual number of fish in their pond to see how close they were. Most groups were pretty close. As an extension, we discussed how we might modify the method if more than one species of fish were in the pond.

(I saved the boxes. If I do this experiment again, I need to remember to make sure there are lots of squares of paper. Students were easily “fishing” for 30+ fish at a time, and so sometimes they’d end up capturing all their tagged fish.)

The teachers enjoyed this activity & I hope they’ll try something similar with their own students. We had a lot of great conversations about ecology and how our method could be extended, what flaws it might have, and so on!

*Knill, George. “Estimating the Size of Wildlife Populations.” Mathematics Teacher 74 (October 1981): 548′ 571.

# Fun with Paper Folding

Over the last several years, I’ve been able to work with teachers from local school districts as part of a grant-funded project called “The Math and Science Partnership Program” (MSP). Phase II of this program focuses on “Improving Math & Science Teaching through School Outreach.” We offer free professional development workshops for teachers, held on Saturdays, several times a year. Teachers who are part of our MSP Partner Schools can earn a \$150 stipend from attending each workshop. All workshops are accepted for re-certification credit in the Berkeley & Charleston County School districts. Descriptions of our workshops dating back to 2014 are available online.

Christel and Kate

Last weekend, together with my co-Leader Christel Wohlafka, I held a Workshop called “Mathematical Fun with Paper Folding.” I was inspired to create this workshop as a direct result of Patrick Honner‘s “Scalene Triangle One-Cut Challenge,” which I think I learned about because of a mention of it by Evelyn Lamb. The “scalene triangle” puzzle stuck with me for several hours one day and I was almost unable to function in any capacity until I figured it out.

Christel Wohlafka College of Charleston Department of Mathematics

Our agenda for our “Paper Folding Workshop” is available online. Many of our activities were inspired by great things I’ve learned about on Twitter, and many are available online at their original sources:

1. The “Scalene Triangle” puzzle is part of @MrHonner’s blog series, “Fun with Folding”: http://mrhonner.com/fun-with-folding. The “One Cut Challenge” activities came from his “Fun with One Cut!” Workshop that he gave at the 2013 TIME conference. He blogged about it here: http://mrhonner.com/archives/11863 His templates are available online as a PDF file here: http://mrhonner.com/wp-content/uploads/2014/01/TIME-2000-2013-Templates.pdf
2. “Hole punch symmetry” was produced by Joel Hamkins (@JDHamkins). He wrote about it in a recent blog post: http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cut/ The template itself is available online: https://drive.google.com/file/d/0Bw3BMDqKsMmXRXlXU2xqbXlFYms/view Joel has a whole set of blog posts devoted to “Math for Kids” — http://jdh.hamkins.org/category/math-for-kids/
3. The “Fold & Cut Theorem – Numberphile” YouTube Video we watched is available here: https://www.youtube.com/watch?v=ZREp1mAPKTM The female mathematician featured in the video is Katie Steckles, who finished her Math Ph.D. in 2011 at the University of Manchester. Katie’s webpage: http://www.katiesteckles.co.uk/ or you can find her on Twitter: @stecks
4. Christel’s handout on “Dividing a Square into Thirds” came from an activity on Illustrative Mathematics
5. Christel’s handout on “Paper Folding Proof of the Pythagorean Theorem” came from this “Teachers of India” resource.

Frank Monterisi Jr. folds paper.

I had a lot of fun at this Workshop and I hope we will offer it again next academic year. Between now and then, I need to order more and better-quality hole-punchers. With some of Joel’s “One Punch” activities, the paper ends up folded over itself five or six times, and some of the “well-loved” hole punchers we had with us weren’t up to the task.

# Groceries and Gratitude

Outside of my life as a mathematician, I’m a mom of three kids under age 6. If you’ve ever done some parenting, you know how it is exhausting and joyful and amazing and frustrating and beautiful and impossible–and can be all of these things in a single five-minute window of time. I had a life event recently that impacted everything about my daily life, both in and out of the classroom, in both my roles as “mathematician” and as “mommy”. I want to tell you about it to then share a really uplifting story that will make you feel better about the world.

!!! Ouch !!!
Two weeks ago, I was hanging out with my kids and my husband in our garage and in our driveway. The kids were playing on their bikes and we were enjoying a burst of Spring-like weather. Between our kitchen and our garage, there is a one-stair step down.While carrying my 7-month-old into the garage, I stepped out on my left foot and I think I twisted my ankle. My immediate reaction was to throw my weight over to my right side, which I did. And then, as if in slow motion, I started falling to the ground, holding my baby.

We landed. Thankfully, my mommy instinct kicked in, and I enveloped him in my arms as we fell. On the ground, he didn’t seem to notice anything had happened. He didn’t cry, he wasn’t hurt, he was completely fine.

Unfortunately, I was not completely fine. I landed on my kneecap with the full force of my body weight (plus his). In the blink of an eye, I found myself getting orthopedic knee surgery less than 48-hours later. I went from full-time care-giver to full-time care-receiver. It was a hard transition and I’m still working on figuring out this “new normal” around my house. I was devastated to learn I won’t be able to return to campus for several more weeks, but thankfully I will be able to do some work from home, teach an online class, and continue interacting with, supporting, and helping my students whenever possible.

A Happy Story

The grocery store closest to my house is Harris Teeter. They offer an online shopping service called “Express Lane“, where you can order your groceries online & then go through a drive-thru lane at the store for pick-up. Their helpful employees bring your groceries out, load your car, and they have a digital, portable payment system if you want to pay with a credit card. You can pay for the service per-order, or per month, or they offer a 1-year subscription.

After my knee injury, I was trying to figure out how things like my family’s grocery shopping would work. I can’t walk very well, I certainly can’t drive, and I even struggle to watch my three kids unless there’s someone else to help me. (For example, actively potty-training a two-year-old requires a hands-on approach by a very patient and mobile adult.) I decided I’d send an e-mail to Harris Teeter’s Customer Service Team and see if they could help me out.

I’ll admit, I wrote a pretty sappy message. I explained I’m a professor, a mom of three kids, a wife, and a grocery shopper. I told them I love their store (which I do!) and I love shopping there with my kids — They love “driving” the race-car shopping carts and the free cookie they get (but only if they listen to Mom the whole time!). I told Harris Teeter about my knee injury and surgery and I asked if they would consider extending me a free one-month subscription to the “Express Lane” online shopping for my family to use during my immediate recovery. This will allow me to shop online from home, and then send friends & neighbors to pick up my groceries. The worst part, I explained, was “I won’t get to visit all the members of my HT Family during my regular shopping trips.

The next day, my phone rang. It was the manager of my local Harris Teeter. He introduced himself and asked how my knee was doing. Before I could ask how his day was going, he said,

“Yes, this is the manager of your local Harris Teeter, and I am calling from your driveway.”

Completely shocked, I sent my parents out to meet him and invite him inside.

The Harris Teeter manager brought with him an amazing bouquet of flowers, a giant gift basket of fresh organic fruit, and a touching “Get Well Card” that was signed, “We hope you get well soon, Your HT Family“.

They also extended us a free one-year subscription to their Express Lane online grocery ordering program.

I was completely blown away by this. My colleagues, friends, neighbors, and family have been so amazing supportive, compassionate, and loving during my recovery. This chain of events has been incredibly tough for me — whether medically, physically, psychologically, mathematically… just NOT fun. I had no expectation that even my local grocery store manager would go so far out of his way to be supportive and do something just to make life easier and my day brighter. I was really, really touched by the gesture and I am very grateful.

As “corporate” and anonymous as modern life has become, it really inspires me that there are complete strangers who will go well above & beyond for someone they don’t even really know.

Even if you aren’t a Harris Teeter shopper, please consider contacting my local Harris Teeter to say “Thank You” on my behalf. I have told them this several times already, but I don’t think they can hear it too much.

Post Script

The flowers were delivered two weeks ago today and they still look amazing. The fruit was delicious (especially the kiwis!) and is long-gone, but I still wake up each morning to see my bouquet. It’s pretty impressive they look as good as they do given how many days they’ve been hanging out at my house.

# An Adventure in Standards Based Algebra

This semester I am teaching several sections of “Math 101: College Algebra”. One section uses an “emporium” method, where students work independently in a computer lab. Instructors are available for questions and we also hold mini-lessons as needed, during which small groups of students can work on a particular topic at the same time. The other two sections are “traditional” in format and I’ve designed a standards-based grading system for them.

I began by creating a list of 30 standards for our 16-week semester. These are grouped by textbook section. Each standard has one or more “I can…” statements associated with it. Here’s the complete list. I’m giving three midterm tests this semester and each test will have an assortment of problems. The exam I gave this week covered our first six standards and had fourteen problems. Not all standards had the same number of problems.

I graded each problem using a modified “ERMF Rubric” (see http://www.nctm.org/Publications/mathematics-teacher/2004/Vol97/Issue1/EMRF_-Everyday-Rubric-Grading/). If you aren’t familiar with ERMF, I’d suggest checking out this post by Taylor Belcher, or some examples of the ERMF Rubric used in a beginning physics course. I decided I didn’t like the baggage associated with an “F” so I made mine an “ERMN” rubric:

Basically, I’m implementing a “Pass/Fail” system — although I refer to those as “Proficient” and “Not Proficient.” Scores of “E” and “M” are passing scores, and scores of “R” and “N” are failing scores. If a student earns all “E”s and “M”s on problems from a particular standard, then they get a “Proficient”. If there’s a mixture of some “R”s or “N”s, I looked at those case-by-case to determine if the student had shown enough understanding of the relevant ideas to merit a “Proficient” or not.

Overall, grading the exams took about one minute per exam page. I have about 50 students and this exam contained 6 pages. I don’t think this is too far off what it would have taken, time-wise, to grade using a traditional points- or percentage-based system.

I’m allowing students to come to my office for re-assessments, so any standards that earned a score “Not Proficient” can be improved upon later. In an upcoming post, I’ll write about my “Policy for Re-Assessments” and outline my system. From past experience, one key factor I’ve found is limiting the number of standards that can be re-attempted to no more than one per week.

At the end of the semester, 50% of the course grades will come from how they perform on their midterm tests. I’m converting all these “Proficients” and “Not Proficients” into a numeric score using this formula: “Midterm Exam Grade = 25 + 75*(# Proficient)/(# Total)”. Basically, this is the percentage of standards ranked Proficient, plus a tiny bit. Now I have to run off to class to return exams to students and explain more about how this grading system works — and why I believe it is to their advantage.

# Documents related to SBG

This afternoon I’ll be presenting about standards based grading as part of Teaching, Learning and Technology‘s “Faculty Showcase.” I’ll be giving a similar talk at an upcoming conference. In case you’re interested, here are some documents related to my presentations:

A lot of my FAQ document was borrowed from Joshua Bowman (@Thalesdisciple). This semester, I didn’t actually give my students the FAQ document — It turned out that after three semesters of SBG, my explanation to students about how our grading system works & why I think it’s a good idea has gotten a lot better.

Actually, that point speaks to one of the great things I’ve gotten out of using SBG: Implementing my system forced me to give deep consideration to exactly what mathematical content I want my students to get out of the course. Instead of debating if homework should count 10% or 12% of the overall grade, or what I should do if a student misses a quiz for an undocumented reason, or other administrative policies like those, the SBG system made my entire course planning process focus on the math stuff I want to teach and assess — instead of worrying about policies unrelated to mathematics (compliance with the rules, attendance, percentage breakdowns, etc).

# Two Upcoming Talks on Standards Based Grading

In the next month or so, I’ll be giving two talks on my implementation of standards based grading. (Okay, if you want to be really precise, that should say that I’m giving the same talk twice.) The first will be hosted by our “Teaching, Learning, and Technology” (@TLTCofC) division as part of their events for “Assessment Week”, and it will be on Wednesday, April 1st at 2pm. The second will be at SOCAMATYC  — the South Carolina Mathematical Association of Two-Year Colleges Annual Conference. They haven’t finalized their schedule yet, but the conference runs Friday 4/17 through Saturday 4/18. Thanks go to Frank Monterisi (@frank314) for letting me know about this opportunity.

Here’s a blurb about my talk:

In this presentation, we will give an overview of standards based grading (SBG) including helpful answers to questions of the form “What?”, “Why?” and “How?”. While an implementation specific to Calculus II will be discussed, the method outlined could be applied to courses in any discipline. If you’ve ever wondered about alternatives to traditional grading and how to avoid hearing the question, “What percent do I need to make on the final exam to get an 82% in the class?” then this is a great place to start.

Once I have put together my slides, I’m hoping to upload them here, along with some updated SBG documentation from my Calculus II course, like my current list of standards and the information provided to students about how the grading system works.

In a way, it feels a little strange to prepare a talk about standards based grading when I feel like the relative newbie to this topic. My entire system came about after many conversations and interactions with fellow educators on Twitter, and I am still indebted to them for all of their helpful support and guidance. In particular, I couldn’t have gotten my course running smoothly without inspiration from Frank Noschese (@fnoschese) and Joshua Bowman (@thalesdisciple). A quick google search just told me that Joshua gave a similar talk about his transition to SBG; I stumbled on his slides here.

# A useful quote

At lunch today I spent a few minutes reading a recent edition of the AWM Newsletter. One article, written by Jackie Dewar, is called “Situated Studies of Teaching and Learning: The New Mainstream.” In it, she gave a great quote that I want to keep handy for later. The quote is from the keynote address at the 2013 ISSOTL Conference, given by Dr. Lee Shulman:

What advice did [Dr. Shulman] offer [Scholarship of Teaching and Learning] investigators? “Do not look for generalizations. Try to figure out what to do tomorrow because it matters.” (emphasis mine)

Whenever I think about my teaching approach and philosophy, I always stumble across the following problem: I have the tendency to think about what I want my classroom to look like, say, five or ten years down the road. I think about what I want the student experience to be and about big, radical changes I’d like to fully implement to get things there. Usually what happens at this point is I am jolted back to reality. I have so many different things pulling me in different directions that, in the end, I never feel like I’ve got momentum in the direction I’d like to go.

This is why I wanted to keep Dr. Shulman’s quote handy. Instead of thinking about big, long-term changes and projects, I really should spend my energy figuring out how to make class better tomorrow, or this week, or this semester. Hopefully small changes over time will have an additive result.

# The range of a function

A recent Tweet led me to discover an old blog post written by Christopher Danielson (@Trianglemancsd) back in 2013. His post was titled, “College algebra teachers! Please try this and report back!” Although I don’t teach College Algebra, my current semester includes two sections of our “PreCalculus” course. I tried a modified version of his activity and I’ll describe my version below.

I gave each student a small (roughly 3″x5″) piece of green paper and a small piece of orange paper. I wrote on the board that green meant YES and orange meant NO. Then I passed out a one-page quiz that had six “Yes/No” style questions. The questions were about the function $f(x)=x^2 - 1$. Each question asked, “Is (number) in the range of the function?” I displayed each question on the projector and then asked the students to vote on the answer by holding their green or orange paper above their head.

The first question was, “Is 8 in the range of the function?” In both sections, all of the students answered this question correctly. I asked for a volunteer to explain their reasoning. In the case the reasoning was too short or wasn’t quite right, I asked for another volunteer to explain the answer in a different way.

Next, I asked, “Is -3 in the range of the function?” In both sections, all of the students answered this question correctly. Most explanations cited the fact the graph of the function would show a parabola with a minimum y-value of -1, so -3 was “too low” to be on the parabola.

With the next three questions, most of the students answered correctly and gave appropriate and correct reasoning. The questions asked if 1/4, -1/2, and π were in the range of the function. The last question, though, had an interesting result.

“Is ∞ in the range of the function?”

• “Infinity isn’t in the range because it’s so big it has all the negative numbers in it too, and we already said -3 wasn’t in the range.”
• “Infinity isn’t in the range because it’s not a concrete idea.”
• “Infinity isn’t in the range because whenever we write a range using interval notation, we always use open brackets or parentheses for infinity. So this means it’s always less than infinity so infinity doesn’t count.”
• “Infinity isn’t in the range because it’s not a real number.” (Another student then asked, “But &pi; isn’t a real number either because it just goes on and on and on, and we already said it was in the range.” I really think this is an interesting connection! At the end of the activity, I spent a few minutes talking about real numbers and rationals, and how there are plenty of real numbers that aren’t rational and don’t even have patterns in their infinite decimal expansions.)

I thought it was really great to hear everyone’s different ideas of thinking about this question. I was happy that the activity resulted in about a dozen students each talking in class. This was only the second day of our semester, so the students aren’t yet used to each other. Hopefully this activity made them more willing to share ideas and answers during class time.

# Reflection on Standards-Based Calculus

Our semester is wrapping up and we only have one more class meeting day after Thanksgiving. I’ve been teaching two sections of “Calculus II” using my standards-based grading system that I’ve mentioned before. I think I made several improvements this semester and I wanted to share them, along with a couple of things I’m still contemplating. But first, here are things I thought went well:

• I really liked having my standards organized by Big Questions. This is probably something I could have implemented outside of my grading system. Somehow writing and organizing my list of standards gave me the motivation and time and priority to think about the take-aways I wanted my students to get from our course.
• Last Spring, I had approximately 18 standards, meaning about one per week. They were large learning targets. Take, for example, the “Techniques of Integration” standard that encompassed a couple of weeks of class time spent talking about integration by parts, by trigonometric identities, by trigonometric substitution, by partial fractions, and so on. This semester, I wanted more standards that were more specific. I hit my goal of 30 standards for the semester and this number worked well. On the one hand, the standards were specific enough that students could focus on just one idea at a time. On the other hand, there weren’t an unreasonable number for me to assess. Roughly they correlated to one standard per textbook section, spanning about 1.5 classes per idea.
• Originally, I had a “policy of replacement” where a score would be updated each time a problem was attempted. In some cases, this seemed to harsh, since prior good work was “erased” easily. In some cases, this seemed to lenient, because sometimes an easy problem would earn a high score, but replace more thoughtful work on a harder problem. This semester scores were defined as the average of the scores from the last two attempts. This also made picking problems for re-assessments easier on me since I wasn’t as concerned about having them all be exactly the same difficulty. It also means that a score of 4 means a student demonstrated a strong level of mastery on two problems of a particular type, and that seems to work well.
• I limited the number of re-assessments to one re-assessment topic per week. For example, if a student were struggling with Taylor Polynomials, they could come in throughout the week and try re-assessments. In some cases, they would just solve one problem. In other cases, they might solve four or five problems, each time getting a little more of the correct solution. Previously I let them do 2 standards per week but I found two problems with this: First, some students would just always pick their lowest two scores and try them, without really ever focusing on a single idea and working toward mastering it. Second, having multiple re-assessments on multiple topics times multiple students meant my grading workload was higher. So, one per weeks seems like a more manageable number for them to work on and it makes my grading workload lighter. Lastly, since we had 30 standards (but only 16 weeks) this policy pushed them to demonstrate mastery on in-class assignments (quizzes, exams) without just punting them to re-assess in my office later on.

Two things I don’t have data on yet:

1. This semester I tried assigning online homework, with the homework contributing 5% to the overall course grade. I found assigning just textbook problems (and not grading them) did not work well. Perhaps I was not very good at motivating students to solve more problems on their own? I haven’t taken a detailed look at homework scores compared with course standing, so I am not sure if homework correlated with success on in-class assignments or not. I also feel a bit “icky” about assigning and grading homework, given some of the research I’ve seen.
2. The other change I made was I separated “during semester scores” from “final exam scores.” So 70% of course grades will come from a letter grade assigned based on the scores on standards that were accumulated during the semester and 25% of course grades will come from a letter grade assigned based on scores on standards that will be accumulated on the final exam. This breakdown was in response to some conversations with students from last semester who felt that the old policy (“average of semester score and final exam score”) was too strict. We will see how this works out and if there is much movement in pre-final letter grades to post-final letter grades.

I’m teaching calculus II again in Spring 2015 and I plan to continue using this system. I am still entirely undecided about trying it in Pre-Calculus. I have several worries about trying it in that course.