Fun Math for Families

All Out May 1st

I’m planning to keep my kids out of school on May 1st (see: #AllOutMay1 and #SCforEd, and news coverage), but I’ve begun planning for mathematically themed activities for us to do together. Recognizing that I spend hours every day thinking about math, my kids, and math with my kids, I have a lot of resources and ideas about things to do, so I’m hoping to compile a list of resources here.

My goals for this list:

  • Activities must be fun for both adults and kids. Fun is my top priority.
  • My kids are young (age 5 and age 8), so I want things that don’t require too much background.
  • I want the list to be usable for parents of all sorts, not just those with backgrounds in STEM
  • Everything required should be stuff available in your house, so no fancy classroom toys, expensive building blocks, board games, etc.

I’ll toss together a few things I know about off the top of my head and I hope to add to this list as I find more resources! If you know of others that should be added, send me a tweet with a link: @katemath 

Fun & Cheap Activities for Family Mathematics

  1. My favorite PDF ever is by Joel David Hamkins (@JDHamkins)  and it’s on the “Fold & Cut Challenge.” All you need is to print it and find some scissors: PDF File for the entire project
  2. Alex Bellos (@alexbellos) has authored a couple of different pattern-based coloring books (like Patterns of the Universe), and they’re fabulous for both kids and adults. PDF File of some Sample Pages ready to print.
  3. Paula Krieg (@PaulaKrieg) is constantly doing beautiful math, usually involving geometry, origami, and tiling projects. She has so much great stuff I don’t know what to link to! Here’s “Origami Boats and Meandering Number Lines with 4-year-olds” and here’s a ton of materials about hexagons. Check the bottom for a link to a ton of PDFs for hexagon printing.
  4. Mike Lawler (@mikeandallie) is famous for Mike’s Math Page & his enjoyment of math with his kids, with hundreds (thousands?) of videos as evidence. Here’s his post “10 pretty easy to implement math activities for kids
  5. Dave Richeson (@divbyzero) made a cool Rubik’s Cube themed hexaflexagon. It’s a fun paper-folding project and turns out really well. My kids thought I was a magician when I showed them the one I made from Dave’s PDF.
  6. …stay tuned!

Right now I’m proctoring around 80 College of Charleston students working diligently on their Final Exams. They’ve all worked so hard this semester & I apologize for the typos in this post — it’s difficult to blog while cheering students on at the same time!

Memories from MAAthfest 2018

0. At #MAAthfest this past August, Drew Lewis asked, “How can we make sure we are providing adequate opportunities for all our students to demonstrate mastery?

This year’s MAAthfest was held in Denver, Colorado from August 1st through August 4th. I went, I had a great time, and I want to tell you about some of the things I learned. While there, I presented twice: once as an invited panelist for Project NExT and then as a speaker in the Special Session on #MasteryGrading. Info about my talks is available here in my blog post called “MAAthfest 2018“.

Now I hope to give you a quick summary of some of the many great take-aways from the rest of the #MasteryGrading session.

1. Many of us noted that Mastery Grading reduces stress levels, both for the instructors and the students.

2. It’s hard to get by on a partial credit strategy. Mastery Grading holds students accountable for their own learning.

3. Many of us moved toward Mastery Grading after spending a long time really considering questions like “Why do we assign grades?” and “What do we want grades to tell us?”

4. I really like, respect, and enjoy these people.

We didn’t spend the entire time working. We also had shared some great meals:

5. Traditional Grading expects all students to learn material at the same pace, but Mastery Grading allows learners to find their own path.

6. Mastery Grading really changes the way you write questions. If your goal is for students to change how they answer questions, sometimes you have to change what you’re asking them.

7. In Traditional Grading, instructors give students points. In Mastery Grading, students have accountability for gaining and then displaying knowledge.

8. There are many different ways to implement Mastery Grading. The real challenge is  finding the one that works best for you, your courses, and your students.

I’m excited to read an upcoming issue of PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies) devoted to Mastery Grading. Submissions are due October 15th, 2018 and more information is available here.

9. In all learning, there’s struggle. Mastery Grading supports and encourages students through the struggle.

Austin added, “Assessment should guide students toward productive struggle” and I really like this quote. On Deadlines, he also gave us two pieces of important advice:


10. Mastery Grading allows a path for improvement and success for all students, while still keeping clear, high expectations for learning.


Bevin Maultsby (NC State) shared with us course grade distributions for students in a course on Matlab, a computer programming language. Over 60% of her students earned As! So impressive.

11. Some adaptations of Mastery Grading work well in project-based courses, courses with proofs, etc.

Chad Wiley (Emporia State University) told us about his use of specifications grading, and I’m hoping to adapt some specs-style setup in my upcoming “Math for Teachers” course that starts in October.

If you’re wondering about the difference between standards-based grading and specifications-based grading, Joshua Bowman (Pepperdine University) really summed it up well:

12. Common benefits of Mastery Grading include sustained student effort, clearer learning objectives, and changes in conversations with students.

We had a great three-presenter talk about #SBG happening at three different institutions:

I really need to look back on this list of “14 Characteristics for Evaluating Grading Systems” by Linda Nilson.

13. The Mastery Grading community has begun gathering powerful data about student learning, and we’re seeing that Mastery Grading allows for students to be successful even with differentiated pacing of their learning.

Drew Lewis (South Alabama) had a really amazing slide called “A tale of two students” and I am committed to generating such graphs for my own students this semester:

Honestly, if I had to pick one slide that has stuck with me daily since MAAthfest, it’s Drew’s graph of the learning trajectories of two different students. If we want all our students to have the opportunity to be successful, we must construct our courses that allows for differentiated learning trajectories. 

14. Most University grade systems are already built with Mastery vocabulary in their grading scales.

15. Occasionally I missed Tweeting great stuff.

There are several other talks I don’t have archived on Twitter. Joshua Bowman gave a great talk about his years of experience using standards-based grading. His work was what originally inspired me to make the Mastery Grading jump in my own courses in 2013-2014.

Steven Clontz gave some really great practical tips (and I was too busy taking notes to tweet them!). Thankfully, he did that part himself:

I wasn’t able to attend David Clark’s (from GVSU) presentation when he won the Alder Award, but here’s what the MAA tweeted:

16. Outside of Mastery Grading, I was inspired and found joy in several other places.

 Her slides are available here, courtesy of the MAA.

  • I went to Emily Riehl’s talk in the “Category Theory for All” session and her talk was amazing.

    I mentioned to a colleague here what I had learned about category theory and it turns out one of our graduate students at the College of Charleston is writing a masters thesis in this area. I was invited to join his thesis committee, so now I’m going to have the opportunity to learn a lot more. Emily’s talk reminded me of some of the things I love about universal algebra.

  • One of my best friends from childhood was able to fly to Denver to spend some time with me, and her company was the best gift. Also, this was my first trip away from my three children — ever! — and by the time she got there, I was hug-starved. So great to have someone to offer a hug (and a hundred laughs) each day.

See you at MAAthfest 2019.

Standards-Based Grading in Fall 2018

An Overview of My Semester

This semester, I’ll be teaching three different courses:

  • Pre-Calculus Mathematics (MATH 111), a course designed to review algebra and trigonometry for students who plan to take our (scientific) calculus sequence;
  • College Algebra (MATH 101), a course designed to cover algebra and function basics for students who will continue on either to MATH 111 (pre-calculus with trigonometry) or MATH 105 (business calculus); and
  • “Applications of Mathematics Across the Curriculum with Technology” (SMFT 516), a graduate-level course designed for in-service science & math teachers who are working toward an interdisciplinary M.Ed.

Due to enrollment challenges, my schedule for courses shifted in late July, so I spent a while during the summer trying to ditch my old plans for the semester and start over. Although I was planning to attend Mathfest in Denver, CO and give a talk in the session on #MasteryGrading based on my years of experience implementing standards-based grading in my courses, I must admit that before my trip I had no plans to use SBG in any of my courses this semester.


So I attended every talk in the #MasteryGrading session at Mathfest. And wow, I got a ton of great ideas from all of the talks [stay tuned for future blog post] and, on a personal level, I really enjoyed our conversations, meals, and hang-outs outside of the session itself. (Thanks, y’all!)

Unfortunately a couple of days into Mathfest I realized I just couldn’t go back to traditional grading, so I threw out all my traditional plans for the semester and committed to myself that I would implement SBG/SBSG/MasteryGrading in 100% of my courses this semester.

Did I mention that I got home from Mathfest only 15 days in advance of my semester start?

The Nuts and Bolts of Fall 2018: SBG PreCalculus and SBG College Algebra

Rachel Weir, of Allegheny College, is maintaining a repository of course documents for secondary Mathematics courses that are using Standards-based grading, Specifications-based grading, or Mastery-Based Grading: Rachel’s SBG Repository

Both my Pre-Calculus and College Algebra courses are using the exact same setup. It’s very similar to Tom Mahoney ‘s (@MathProfTom) approach in his College Algebra courses. Here is the basic setup:

  1. I have written 25 standards for each course.
  2. Every time a student completes a problem on a standard, I will assess the solution using a “SGN Rubric” (see below). This assigns either 0 points, 1 point, or 2 points to each attempt.
  3. A student’s score on a standard is the average of their best two attempts.
  4. A student earns total points out of 50 possible (25 standards*2 points max). Together with work in an online homework system, this converts to a usual letter grade*.

For example, for any given standard, I will track a student’s progress as something like “0,1,2,1,1,1,0,2” and this student will earn a 2. After two perfectly correct solutions, the student isn’t required to answer problems on that topic again.

*My department requires a departmental-wide final exam that is graded using a partial credit, percentage system, and this exam must be worth at least 25% of each student’s course grade. So the actual grade computation is (75% performance on standards)+(25% final exam performance).

Links to Possibly Useful Things

Here are some links that I’ve freely distributed to my students. Perhaps reading them will shine some light on how I explained this system to them. Also, there are more details about the “SGN Rubric” I mentioned above and explanation about online homework & how it fits in.

Things To Do Later

I haven’t mentioned that third class (“Applications of Mathematics Across the Curriculum with Technology”). It runs double speed for half the time, during our Express II semester, and it doesn’t start until October. I want this course to be a project-based course, so I’m going to figure out some way to introduce specifications grading into my design. Robert Talbert (@RobertTalbert) has written extensively about his use of specs-grading and it’s my plan to steal as many ideas from his MTH 350 F18 Syllabus as I can. Our courses are very different, but he has so many clever ideas for his course skeleton. Once I write my syllabus, I’ll tell you about it.

MAAthfest 2018

Right now I’m in Denver, Colorado at the MAA’s annual Mathfest. Mathfest is fantastic, and every time I come I tell myself I should come back every year. Check out the Twitter hashtag #MAAthfest to see some of what’s going on.

I’m here to give to talks — one was yesterday and the other is this afternoon. Both are on the topic of non-traditional grading (or mastery grading), which I’ve written a lot about in the past. Here are copies of my slides:

I think my take-away message of both presentations is the same, and it’s the following:

Kate’s Grading Philosophy: Grades should reflect student knowledge and should have a positive effect on student learning.

Standards-based grading is the way I’ve decided to build this philosophy into my courses. Since I’ve written about my implementation in the past, now I’ll describe something I want to do in the future.

I want to implement some kind of portfolio assignment for students to show off their homework solutions. I imagine letting each student pick her best/favorite solution for each course standard, and gathering them all up together, for an end-of-term “look at all the stuff I’ve learned!” binder. This project would fit into course grades as a “grade modifier” on top of a “base grade”. The base grade would come from performance on standards on normal assessments (exams, quizzes, etc) and would be a typical letter grade (A, B, C, …). The performance on the portfolio would modify a B-grade into B-, B, or B+, depending. My rationale for implementing this project is (a) to have the students work on a single thing throughout the term, with changes for feedback, revision, drafting; and (b) to motivate them to work on homework problems or even more difficult problems that aren’t necessarily accessible in an in-class assessment.

My fear is by doing this, I’m asking dozens of students to hand me dozens of problems to review, right at the time that the clock starts speeding up for me to get my final grades submitted. Also, this is probably going to be at the same time that I have to write and grade final exams, and also tackle all of the re-assessments that students are excited to tackle at the very and absolutely last second possible. Until I find some way to schedule my way out of a complete grading nightmare, my portfolio idea is going to be on hold.

10 Steps to Solve a Problem

This semester, I’m teaching our MATH 229: Vector Calculus with Chemical Applications course for the first time. It’s a 5-credit hour course designed for Chemistry majors who have completed Calculus I. Topics include some pieces of Calculus II, Vector Calculus, Linear Algebra, and Differential Equations. (I’m using a standards-based grading system and I hope to blog about that sometime soon.)

The textbook for the course was written by Jason Howell and although no longer at CofC, he has kindly let us continue to use the book and distribute the PDF to our students for free. Like most math textbooks, each section has some explanation pages and various Examples. We are working through the Examples together in class, and to prepare, I’ve been going through them ahead of time. Here’s an Example from an upcoming section: I started thinking about this problem on January 1st and today I finally produced a solution that made me happy. During the 9-day stretch, I found lots of non-solutions — either methods that I couldn’t get to work, methods that I could get to work but didn’t like, or methods that I realized could work but weren’t suitable to use in my Vector Calculus course. When we reach this problem in class, we will still be in Chapter 1 of the textbook, and my students will know some stuff about three-dimensional space, vectors, spherical and cylindrical coordinate systems, but not a lot of linear algebra or complex variables. I won’t tell you the solution (consider it your homework!). Instead, let’s just consider ways we might find a “suitable coordinate system for the molecule” since that’s really the part I found tricky.

My algorithm was very inefficient as compared to Feynman’s Problem Solving Algorithm, but here it is:

Ten Step Problem Solving Algorithm:

  1. Put one of the hydrogen atoms at the origin, another one along the positive x-axis, and a third somewhere in Quadrant I. Use rectangular coordinates and the Pythagorean Theorem (a lot). Try to find the centriod of the triangle.
  2. Say “Hmmmmm…” aloud often enough that your husband asks what you’re working on, and then do a fantastic sales pitch about how interesting the problem is so that he starts working on it too.
  3. Put the equilateral triangle built out of the three hydrogen atoms on the xy-plane with the origin at the centroid of the triangle, and one of the hydrogen atoms along the positive x-axis. Use what you know about triangles to figure out the distance from the origin to the atom on the x-axis.
  4. Because of input and advice following Step 2, give up on rectangular coordinates and think about using cylindrical coordinates.
  5. Give up on cylindrical coordinates and go back to thinking about rectangular coordinates.
  6. Put the nitrogen atom at the origin and the hydrogen triangle on a plane parallel to the xy-plane, then try to find the distance between the triangle and the origin.
  7. Stop people in hallway and ask for help and input. Convince former students and former graduate teaching assistants the problem is interesting and see what they say.
  8. My office next-door neighbor convinced me that it’s smartest to put the origin at the center of the triangle, so I stuck with that after hearing her arguments about symmetry.
  9. I pitched the problem to another colleague who immediately drew a picture using complex analysis, DeMoivre’s formula, and roots of unity. I had to toss aside this solution since it didn’t follow from the previous material in my Vector Calculus course.
  10. Settle on a solution: Use ij, and k vectors, some vectors of the form aibj, and some known lengths to figure out appropriate constants a and b.

[Another colleague suggested I just get the solutions to the textbook problems from someone else, but (a) I haven’t found anyone who has them and (b) as a matter of pride stubbornness I’ve been doing them on my own.]

I’m not sure how long I spent on this single problem, but an estimate around 4 hours is probably reasonable. I hesitate to mention this since I’m sure the entire internet will leave me a comment of the form, “How can you be so bad at such immediately obvious and simple math??!!” On the other hand, maybe it’s worth mentioning that even those of us who do this kind of thing for a living sometimes find “easy” problems quite challenging and that not being an extremely speedy problem solver doesn’t preclude you from getting a job solving problems.

Also, here are three specific goals I have for myself this semester:

  1. My instincts about problems in vector calculus are not very strong, almost certainly because I have not solved any vector calculus problems since I was an undergraduate. (That statement is probably factually false, but it is a reasonable approximation of reality.) So maybe I can re-awaken those parts of my brain.
  2. I want to get better at drawing things in 3D. I have sometimes wondered if my lack of passion for multi-variable calculus is because I am not happy with my ability to draw the objects? Maybe this course will force me to do more drawing and I’ll get better at it as we go.
  3. I hope to learn some stuff about chemistry — sure, from the textbook and course material — but, more importantly, from my students. I like hearing them talk with each other before class about all the various chem classes they are taking. I haven’t taken a chemistry class since high school (and that one wasn’t didn’t even have a lab associated with it).

Course Currency Model

This semester (Fall 2017), I’m teaching Math 120: Introductory Calculus for the first time in a while. I’ve been debating introducing a standards-based assessment (SBG) system in the course but decided against it this semester. One of the things I really like about my prior SBG experiences is that SBG allows students flexibility when they need or want it. For example, in the past, I’ve allowed students to re-try quizzes during office hours to demonstrate a higher level of mastery on course topics. I have been trying to find a way to allow more flexibility in my Calculus course in a way that limits the amount of time and work it requires on my end.

I’ve decided to create a course currency system that I’m calling Calculus Tokens. I think the idea of “class tokens” came about from the “specifications grading” community and I’m not even sure where I first heard of this idea. Each calculus student in my course will begin the semester with 10 Calculus Tokens. These tokens can be redeemed, as needed, for a variety of things, like getting an extension on an online homework assignment, making up a quiz due to absence, or even re-trying a quiz in my office to improve a student’s score. Additionally, tokens can be earned by completing extra online assignments or by completing problems on my Study Guides before each test. If students have a balance of 8 or more Tokens at the end of the semester, they will earn a small (1% or 2%) grade boost on their score on the final exam.

I’m hoping this system works. My goals are:

  1. Allow students flexibility when they miss assignments due to absence;
  2. Allow students a re-assessment procedure for bringing up quiz grades;
  3. Allow students who need or want an extension on the homework a method of doing so that is transparent and fair to everyone.
  4. Let students have more ownership for the course, in the sense that I can be flexible in the direction that benefits them the most (and it doesn’t have to be the same for every student).

I’m not sure about the details of my implementation. The cost of making up a quiz, whether due to absence or just to re-assess, is 3 Tokens. The cost of getting a homework extension is 2 Tokens. Once students reach 0 Tokens, they can’t redeem any more for additional reasons.

I’ll let you know how it goes!

10 Minutes of Thoughts on My SBG Linear Algebra Class

I’ve been meaning to write a post about my standards-based Linear Algebra course for months, but the hectic schedule of the semester has kept me away from this task until now. Today was my last “content” day of Linear Algebra — we have two more classes remaining, one for a test day and another for a re-assessment day. This seemed like a good time for me to take ten minutes to gather some thoughts about how the semester went.

Standards List for Linear Algebra  (Availablle on Overleaf, which is awesome and I can’t recommend enough)

What Went Well: We ended up having 20 standards this semester. This is a little more than one per week (our semester has 16 instruction weeks). Overall, I think this was a good number of standards to have, and I’m happy with how they turned out. I tried to group them again by “Big Questions” to have a reference frame of what it is we’re trying to do in the course. Oddly, we tackled “Big Question 5” last (on inner product spaces), but I kept it numbered like that because of the textbook we are using. My basic idea was to come up with a Big Question for each chapter. For some stuff, this worked well (e.g., eigen-everything) but for other stuff we didn’t cover a whole lot (e.g., determinants).

I think I’m doing a better job of the sales-pitch aspect of a standards-based course. Many of my students expressed to me at various times that they really appreciated the ability to improve on past performance and that they were under less stress than in a traditional class. In a recent class meeting, a student wasn’t happy with the performance on the last quiz, and exclaimed, “Oh, thank goodness we have an exam on this soon!!!” [I asked the student for permission to share this quote.] I think this is one of the best things about my SBG courses — students really want to take an exam just to show what they know, whether that means showing mastery of current material, or showing mastery of material they struggled with earlier in the course.

My SBG approach definitely has some pros and also some cons, but the way it has shaped my interactions with students has always been a huge positive. Even with the sticky details that need to be cleaned up from this semester, I can’t imagine going back to a traditional grading scheme.

Room for Improvement: This semester was a little odd because we lost several days because of weather. Tropical Storm Hermine hit us, and we lost almost a week because of Hurricane Matthew. The re-shuffling of the academic calendar created a speed-bump that I never really recovered from. I hope next semester our calendar runs much more smoothly.

In particular, I am wondering about how I can improve in three areas. First, I want to expose my students to more applications of the material we are learning. I felt rushed all semester (related to shuffling of course calendar, maybe?) and so I didn’t ever feel like I had time to fit in cool applications, or videos on where people use this stuff “in the real world,” etc. A colleague teaching the same course required students to do group projects on applications of linear algebra & I believe the students presented them to the class at the end of the semester. This seems like a great idea, but I’m always nervous about assigning group projects because I remember how much I hated doing them as a student. It’s something I should consider more.

Second, all of my course standards are weighted equally. This has served me well in Calculus II and in other courses. But in Linear Algebra it became a little tricky, because part of what I was aiming to do was to have my students attempt to write proofs of mathematical statements. (The only mathematical background required for entry into my course is Calculus I, and that is for “mathematical maturity” as opposed to content reasons.) So some of my students were concurrently taking our “Introduction to Proofs” course, but others weren’t taking this course and won’t need it for their major. In general, my idea was to ask them to prove elementary results they had already seen in class. The problem I encountered is that a “write a proof” standard is really tough. How do I let them have multiple attempts? Is it okay if they end up never being able to prove stuff about, say, matrix inverses, but they can prove stuff about, say, subspaces of a vector space?

One idea I’ve had is to have the students keep a “Proof Portfolio” and grade it as either “complete” or “not” at the end of the semester. I’m sure there’s some specs-based approach I could implement for this, but I haven’t worked out what it would look like yet.

Third, trying to put together all my course materials on the fly is hard. All of the time, I was working on: Plans for class, writing exams, writing quiz questions, writing reassessment questions, putting together online homework, meeting with students for several hours a week outside of class, updating the list of standards regularly… I would admonish my summer-month self that I should do more of this “in my free time” before the term begins so I’m not under such a time crunch during the semester. But I am not great at this because I like building a course as it goes, as I see how the students are responding, as I see how the pace of the course unfolds, etc. Having to get all this done ahead of time would probably help me out a lot, but it’s tough to do. Thankfully some of my stuff from this semester can be re-used when I teach Linear Algebra next semester.

My ten minutes are done so I have to move on to the next task on my queue! I hope to add more later.

Standards-based Linear Algebra

This semester I’m teaching our introductory linear algebra course. As I did for Calculus II, I’ve implemented a standards-based assessment system. I’ve taken our course content and split it into “standards”, or little pieces of mathematics that I want my students to master. These standards are grouped together by what I call “Big Questions”. Here is what we’ve covered so far this semester:

  • Big Question #1: What are the tools for solving systems of linear equations?
    • 1.1: I can solve systems of linear equations using row operations. I can use Gaussian elimination with back-substitution to solve systems of linear equations. I can use Gauss-Jordan elimination to solve systems of linear equations.
    • 1.2: I can characterize the solutions to systems of linear equations using appropriate notation and vocabulary.
    • 1.3: I can use matrix inverses to solve systems of linear equations.
    • 1.4: I can find and use an LU-factorization of a matrix to solve a system of linear equations.
  • Big Question #2: What is the fundamental structure of the algebra of matrices?
    • 2.1: I can perform algebraic operations with matrices, including addition, subtraction, scalar multiplication, and matrix multiplication. I can compute the transpose of matrices.
    • 2.2: I can find the inverse of matrices using Gaussian elimination. I can find the inverse of matrices using a product of elementary matrices.
    • 2.3: I can demonstrate theoretical connections about properties in the algebra of matrices.
  • Big Question #3: How can we characterize invertible matrices?
    • 3.1: I can find determinants using cofactor expansion. I can find determinants using row or column operations.
    • 3.2: I can demonstrate theoretical connections between statements equivalent to “the matrix A is invertible.”
    • 3.3: I can demonstrate theoretical connections between matrix equations, vector equations, and systems of linear equations, and their properties and solutions.
  • Big Question #4: What are vector spaces & how can we describe them?
    • 4.1: I can prove whether an algebraic structure is a vector space (or not) using the vector space axioms. I can prove whether or not a subset W of a vector space V forms a subspace. I can determine and characterize subspaces of $\mathbb{R}^n$.
    • 4.2: I can write a proof showing whether a subset of vectors from a vector space forms a spanning set for the vector space (or not). I can write a proof to show whether a subset of vectors from a vector space is linearly independent (or not). I can determine whether a set of vectors forms a basis for a vector space. I can find the dimension of a vector space.
    • 4.3: I can find a basis for the row space, the column space, or the null space of a matrix. I can determine the rank and nullity of a matrix. Given a consistent system Ax=b, I can describe the general solution in the form x=xp+xh
    • 4.4: I can demonstrate knowledge of the theory of vector spaces by proving elementary results and theorems.

The remaining Big Questions are:

  • Big Question #5: What are inner product spaces and how can we describe them?
  • Big Question #6: What kinds of functions map one vector space into another while preserving vector space operations?
  • Big Question #7: What are eigenvalues and why are they useful?

Our first exam was last week, so today has been Re-Assessment Central in my office. I’ll hand back our exams tomorrow and I’m hoping to talk with my students more about standards-based grading and how they can improve their standing in the course.

Modeling Fun with Paper Fish

Kate Owens, 02/2016

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) PDP-fishby 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 pythag-foldedschool 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”: 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: His templates are available online as a PDF file here:
  2. “Hole punch symmetry” was produced by Joel Hamkins (@JDHamkins). He wrote about it in a recent blog post: The template itself is available online: Joel has a whole set of blog posts devoted to “Math for Kids” —
  3. The “Fold & Cut Theorem – Numberphile” YouTube Video we watched is available here: 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: 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.pythag1



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.