What makes for a good task? Rachel Lotan, a teacher educator at Stanford, coined the term *groupworthy task* to describe what we strive for in task design. In a book review (link: http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/240/164 ) that I wrote of Mathematics for Equity, I describe groupworthy tasks as follows:

Groupworthy tasks facilitate students’ interdependence by foregroundingmultiple abilities and multiple representations, requiring students to worktogether in solving complex mathematical problems. These tasks involvesufficient interdependence and challenge; even those students who areperceived as “advanced learners” often experience difficulty completingthe tasks on their own.

Let’s take a closer look at interdependence, multiple abilities, and multiple representations:

**Interdependence****:** If we want students to work together, we need to create a task that actually requires working together to be able to solve it. Moreover, we need to *convince* students that they need to work together. If the task is not sufficiently complex and mathematically rich (See: What is a mathematically rich task) then there will be no need to work together. The typical end of chapter exercises in most textbooks are not mathematically rich; merely teaching a skill and having students practice it (so-called “drill and practice” or more derisively, sometimes called “drill and kill”) is not sufficient to satisfy this criterion.

**Multiple abilities and multiple representations:** A groupworthy mathematical task also requires students to use a lot of different academic abilities (verbal, written, spatial, visual) along with intra and interpersonal skills. Going hand in hand with this, a good task also requires the use of multiple representations—the so-called Rule of Four suggests that we need to use graphical, numeric, linguistic, and symbolic ways of representing mathematics.

Let us take a look at a task that I have used with both high school and college algebra students. This task is adapted from a text called Discovering Algebra.

You have a sheet of paper and are folding it in half, and then inhalf again, and so on. You need to find out how many layers thereare total for a given number of folds. For example, with two foldsthere are 4 layers.In other words, you are searching for a formula that represents therelationship between the number of folds and the number of layers.

This task allows for multiple representations; I generally provide physical paper for students to fold and some find the physical folding helps to make things more concrete. The task asks for a formula, but many students record their observations/answers in a table as an intermediate step. There are also multiple abilities needed to solve the task; students need to be able to count, to notice patterns (such as the doubling relationship), to understand operations conceptually (such as how repeated multiplication becomes exponentiation), and to communicate their ideas with each other. The task requires interdependence; there is not a simple procedure for finding this answer and students have to be able to explore and test different ideas. Moreover, the task is mathematically rich; the concepts of multiplication, doubling, exponentiation, geometric series, exponential functions, and recursive functions are all things that have come up when my students have worked on this problem.

In the next part of this series of posts, I will discuss how to adapt problems that you might already have access to in order to make them more groupworthy.

]]>Some of the posts are done live so attendees can jump in on events while they happen. Other posts are done after the event which allows the authors to give more of an overview of everything that occurred.

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Anytime we have a problem, what do we do? We use an app! Here to save you at JMM is the 2017 JMM Mobile App! You can search for events, personalize your schedule (which can add things to your calendar!), and even give updates/announcements about the conference. The app also has a networking feature that can help you connect to others attending JMM.

Okay, so maybe it’s not a “hidden gem” since it’s the first links on the JMM website and there’s a description in the registration packet but I’m still excited to check it out and take some of the bookkeeping out of my brain and onto my phone. I didn’t get a chance to utilize the app last year so if you give it a shot or have any tips or tricks, let us know!

]]>One outlet for such support that I recently had the opportunity to help implement here at the UW–Madison (sorry for the humblebrag), is a statement of community commitment to, and value of, inclusivity. Based on this experience, I would like to talk about what such a statement is, why I think they are meaningful, and to (not so) secretly encourage others to do similar things within their own departments.

To start, such a statement simply outlines what inclusivity and diversity mean to you/your department, how this group values these concepts, and how they will fight for and support them. For example, here at the UW–Madison, the department adopted the following statement,

“As a diverse group, the Mathematics Department strives to foster an open and supportive community in which to conduct research, to teach, and to learn. In accordance with these beliefs and § 36.12 of the Wisconsin Statutes, the Mathematics Department affirms that all community members are to be treated with dignity and respect and that discrimination and harassment will not be tolerated. We further commit ourselves to making the department a supportive, inclusive, and safe environment for all students, faculty, staff, and visitors, regardless of race, religion, national origin, sexual orientation, gender identity, disability, age, parental status, or any other aspect of identity.

To all members of our community, we, the members of the Department of Mathematics, welcome you.“

Of course what precisely such a statement should say, and how it should say it varies person-to-person and group-to-group. I see power in these statements having unique, personal voices—serving as honest expressions of one’s beliefs and commitments, laid bare for the reader to hear. So if you are thinking about drafting such a statement, consider trying to make it personal. Be honest and speak to those who you hope eventually read it.

As a second example of such a statement, and to highlight a slightly different voice, here is (in some sense) my own statement. (Note my statement was influenced by Federico Ardila-Mantilla’s amazing recent article in the Notices, “Todos Cuentan: Cultivating Diversity in Combinatorics”.)

No student may be denied admission to, participation in or the benefits of, or discriminated against in any service, program, course or facility of the {UW} system or its institutions or centers because of the student’s race, color, creed, religion, sex, national origin, disability, ancestry, age, sexual orientation, pregnancy, marital status or parental status.” ~ § 36.12, Wisconsin Statutes

I recognize the importance of a diverse, inclusive, and supportive community. In accordance with these beliefs, as well as § 36.12 of the Wisconsin Statutes, I am committed fully to the following axioms:

Axiom 1: All members of the department/university community should be treated with dignity and respect.

Axiom 2: I strive to promote a supportive, inclusive, and safe environment for all students, faculty, and staff, regardless of race, religion, national origin, sexual orientation, gender identity, disability, age, parental status, or any other aspect of identity.

Axiom 3: Incidences of hate, bias, discrimination, or violence have no place in a department/university community, and will not be tolerated.

Axiom 4: I will, to the best of my abilities, aid those facing instances of bias, discrimination, hate, or violence; directly and by helping individuals find the appropriate campus and community resources.

These axioms serve as guideposts in teaching, research, outreach, and all aspects of my career/life. Moreover I am dedicated to working/partnering with other campus organizations (Multicultural Student Center, University Health Service, LGBT Campus Center, End Violence on Campus, McBurney Center, Campus Women’s Center, etc.) to promote these axioms.

Finally, to all members of our campus community, I would like to say:

I seeyou, I acceptyou, and I affirmyou.

At this point, some of you reading this might be wondering what the point of these statements is. You, or well the conveniently constructed strawperson I so often use rhetorically at this point in my writing, may be thinking, “Surely people know that our math department is a nice friendly place, which is inclusive of all people.”

Well, a couple of points:

First, be careful in thinking your department and university are friendly places free from bias, hate, discrimination, and violence. The fact of the mater is that most are not. (Some privileges checklists!) So taking a moment to write this, and honestly evaluating your, and your department’s, commitment to inclusion and diversity can be eye opening. You may begin to see incidences of bias, hate, and violence you were previously blind to.

Secondly, even if your department is a magical land free of bias and hatred—it’s not, but regardless—not everyone knows this. Sure you might, but I am certain that there is someone, another grad student, a visitor, and a calculus student, who doesn’t. Someone who is unsure of whether the department will respect their identity, or whether there is someone who they can turn to for support. Letting this person know that you recognize them and will do your best to support them is meaningful. There is power in being an ally and power in recognizing and affirming the identities and rights of others, especially those who often face bias, hate, discrimination, and violence. This can even more true in turbulent times like these where many people feel uncertain, unwelcome, and unsafe.

All this is to say that creating a statement of commitment to inclusivity can be both meaningful and powerful. However, they are not ends in themselves. **They will not end bias, hatred, and discrimination within your department, university, or the mathematical community more generally. So while I encourage everyone to the time to write such a statement, I also think it is important to recognize that such a statement is best seen as a starting point, a guidepost that can direct your actions going forward.**

*Did you say your department is a safe place free from hate?*Well how will you, or others, handle a situation where someone creates a hostile environment in the classroom? Moreover is there training to help instructors identify and handle incidences of bias and hate in their classrooms?*Did you commit yourself to being an inclusive place for people of all gender identities?*Well are the health plans your department offers inclusive of trans-persons? Do you have gender-neutral washrooms in your department? Do instructors ask students what pronouns they use?*Did you commit yourself to support victims of violence and harassment?*Well do students know who they should report incidences of violence and harassment to within the department? Are your department’s policies on these matters clearly articulated and publicized? Do instructors know what campus resources exists to help victims of violence?

I could go on—in fact, that might make an interesting future post—but to keep it short: **creating a inclusive, supportive, and safe community takes dedicated, deliberate, and thoughtful action; writing a statement of commitment is just a first step.** (I know that this is my goal here at UW–Madison.)

Finally to all those reading this—especially those who may be spinning, either a bit or a lot—let me reiterate what I said above:

*I see you, I accept you, and I affirm you.*

My story of surfaces starts in a beautifully weird morning when I got up to realize that life in the usual Euclidean plane had changed dramatically. Vectors had shortened, areas had shrunk, and infinity was just a few feet away!

Here is the aftermath (why the term “after math?!”):

What had happened was this: A vector of unit length in the usual sense was now much shorter, of length where is the distance (in usual sense) of its base (=initial point) from the origin. Yes, here it does matter where the base of a vector is.

The vectors that used to be perpendicular are still perpendicular. By looking at parallelograms, we see that areas have shrunk by a factor of . So, in the above diagram, the square feet of the two shaded areas are in fact almost equal.

Let me do some measurements.

**Length of an infinite straight ray.**

Consider (see picture.) To calculate lengths we must add up lengths of tangent vectors. Tangent vectors are all , unit length in ordinary plane at each point, so their new length must be taken into account. The result is:

Hmm! Seems that the infinity is only feet away!

**The area of the whole plane!** (But wait, isn’t that infinite?)

We add up areas of infinitesimal rectangles . As discussed above, in our new *metric *(Ok, I had to finally use the term!) this rectangle’s area is shrunk by our factor. So, we get

Maybe my whole success in posting these series depends on my ability to convince you now to experiment with this new plane I have created. Be bold about it. Ask questions, do calculations for yourself, get a feel of the structure of this space. Ask what could be asked about this space? What other familiar constructions are possible on it? Could we talk about spaces of functions? What is the measure? How will we integrate a function? What is the shortest path, say between and ? Notice that going on the straight line isn’t the best, because we are better off bending away from the origin so that lengths are shorter. But how much to bend?!

And finally answer this: the plane with this tampered metric is a copy (in almost all aspects) of a familiar 2-d shape, can you guess what shape?!

To be continued with more fun…

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This month’s riddle is actually fairly well-known, but was communicated to me by C. Cook. As always, if you have a nice mathematical riddle that you would like to share, then let us know and we’ll see if we can post it!

You, an intrepid explorer, have found a mystical fortune-telling shrine. This shrine is particularly mystical because it has *three* mystical oracles (all in the same room), making it manifestly superior to other run-of-the-mill mystical shrines, which have only one oracle. Complicating matters further, however, is the fact that not all of the oracles in the shrine tell the truth. **In fact, exactly one oracle always tells the truth, exactly one oracle always lies, and exactly one oracle always answers randomly.**

Your job is to determine, using no more than three true-or-false questions, which oracle is which. Each question must be addressed to a single oracle, and may (if you wish) be about the other oracles in the room. You are required to ask only questions which have a definite true-or-false answer, even if you suspect the oracle to whom you are addressing the question is the random oracle. (Note that a question such as “would Oracle A say ‘true’ to such-and-such a question” would thus be allowed only if you were certain that Oracle A was not the random oracle.)

It is not required that each of the three questions be asked to a different oracle (so you could ask the same oracle all three questions). Successive questions may of course be based on the answers to previous questions.

**Can you come up with a set of three questions that will allow you to determine the identity of the oracles? **Give your answers in the comments below!

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What does Arrow’s Impossibility Theorem actually say? Like Euclid’s parallel postulate, Arrow’s Theorem is frequently transformed into a variety equivalent formulations that are either more intuitive to grasp or more relevant to problem at hand (to read Kenneth Arrow’s original paper, click here). The formulation below is based on a paper by John Geanakopolos, who according to the acknowledgments, consulted Kenneth Arrow prior to publication.

Consider a society of N voters and at least 3 candidates. Each voter has transitive preferences, meaning they can mentally rank all the candidates (with ties allowed). A voting system is a system that produces a single ranking on behalf of society (i.e. it determines a winner, but also a second place runner-up, third place, etc.). Consider the following three fairness criteria:

**Universal Domain:**Voters can rank their candidates in any order they like. No matter what combination of ballots it receives, the system will always produce a ranking.**Unanimity:**If everyone prefers A to B, the voting system should rank A above B.**Independence of Irrelevant Alternatives (IIA):**Whether the voting ranks A above B should not depend on how voters rank C (or any other candidate).

**Theorem:** **The only voting system that respects universal domain, unanimity, and IIA is a dictatorship.**

Note: The dictator in this case is not a candidate, but a rather voter who has the power to decide the winner every time, no matter what the rest of society thinks (more precisely, a dictator can always force the voting system to rank A above B).

A two-page proof, accessible to any outsider who is used to proving things, can be found here. To give you a taste, I’ll walk you through the first lemma (or you can skip to the next section see why Arrow’s Theorem might not be so doomsaying after all):

**Lemma:** **If every voter ranks one candidate as their highest or lowest choice, then the voting system must as well.**

*Proof:* Suppose, on the contrary, that even though every voter has put B either first or last, the voting system ranks A>B>C. Thus every individual voter ranks B>(A and C) or (A and C) > B. Suppose every voter decides they like C better than A, but leaves B’s rank unchanged (they’re allowed to rank them however they want under *universal domain*). By the *independence of irrelevant alternatives* condition, the voting system must still rank A>B and B>C. Now, every voter’s mental ranking is now C>A>B or B>C>A. By the *unanimity *criterion, the voting system must rank C > A. The two rankings A>C and C>A are not compatible so our original assumption is impossible. The voting system must rank B either first or last. QED

The proof then uses this lemma to show that there is a unique voter who can cause B to go from being last to first in the rankings and that this voter must in fact be a dictator.

Before we throw up our hands and declare that no voting system can be fair, let’s pause and consider how reasonable these assumptions and “fairness” criteria actually are. Unanimity does not sound like an assumption we ought to relax: directly contradicting the wishes of literally every voter sounds about as bad dictatorship. Likewise, we could limit the voting system’s universal domain, but prohibiting voters from ranking candidates a certain way because it will cause the system to malfunction hardly seems fair either. If we force there to only be two candidates, then all our contradictions go away, but presumably you will need some system of primary elections to narrow your choices down to two, so this only kicks the can back up the road. Even if we don’t need our voting system to produce a ranking but rather just a single winner, unanimity affects affects “winner” just as much as any other choice. But what about IIA?

On its surface, this criterion seems like the sort of fairness we would want in a system, though like Euclid’s parallel postulate, it does seem a bit more complex than the other basic assumptions. Most systems around the world violate IIA. The 2000 Gore versus Bush election, for example, likely constituted an IIA violation: that is, presence of third party candidate Ralph Nader tipped the election from Gore to Bush, even though the Nader voters did not necessarily like Bush more Gore. This does seem unfair, at least to Gore and Bush, though not necessarily for the voters. Arrow’s Impossibility Theorem assumes the people actually vote according to their mental ranking of the candidates: that is, they vote *naively, *not *strategically. *If you assume these Nader voters knew the potential outcome of their actions (and that’s a big “if”), then the system was taking into account their preferences: that is, they cared so much about Nader that they were willing risk Gore losing even if they preferred him over Bush. Note that Arrow’s Impossibility Theorem only takes into account *rankings*, not *ratings**. *Allowing voters to rank each candidate on scale of 1-10 (or grade them A-F), a system known as range voting actually avoids Arrow’s paradox altogether. However, if at least some of our voters are strategic, then this system is highly manipulable (e.g. even if don’t think your second choice candidate is so bad, you give her an F if you think the election is going to be close). In fact, the even more consequential (though less famous) impossibility theorem known as **Gibbard–Satterthwaite Theorem**** **states that every voting system is either:

- A dictatorship,
- Prevents one of the candidates from ever winning, or
- Is vulnerable to strategic voting (i.e. savvy voters can tip the election by misrepresenting their true preferences).

Again, whether strategic voting is a bad thing depends on your point of view. It does seem unfair in that some voters who are less informed might naively vote their true preferences, while others may vote strategically and exert an undue influence. Even if everyone had perfect information, some voters (say, voters who prefer a hopelessly small party) are in a much better position to swing the election to their second choice than anyone else. But perhaps this is a difference we can live with—after all, a voter who is fiercely dedicated to one candidate and will vote for them come hell or high water is far less influential than a swing voter who can be persuaded to change their vote, no matter what the system. Still, it seems prudent to try to limit the advantage that a small group of savvy, strategically-minded voters have over the rest of us, just as it seems prudent to lessen the chances of an irrelevant alternative causing an upset.

This observation then, bring us to our solution, if you can call it that, to the seeming impossibility of a fair system. While in theory no system can meet all the fairness criteria we may desire, our challenge is to design a system that will minimize the likelihood of an unfair outcome. Thus, our “perfect design” problem becomes an optimization problem, and luckily there are many candidates: instant runoff voting, approval voting, and the Condorcet method to name a few (Wikipedia’s coverage of these topics is extensive). The plurality system (the most common one used in the US) is almost certainly *not* the optimal solution since its highly manipulable by strategic voters and violates a number of other fairness criteria, though its simplicity does have great appeal. Instant runoff voting, used in Australia and some local U.S. elections, has a number of weird flukes, such as a vote for your favorite candidate actually hurting their chances, though it’s unclear if this is more of a concocted scenario that is likely never to pop up in the real world. **“Most systems are not going to work badly all of the time,” **Kenneth Arrow stated, in the lead up to the 2008 election.** “All I proved is that all can work badly at times.”** (See here for full discussion this paragraph is based on). Perhaps one of you will find a system that is likely to never behave badly in real world conditions. While that may sound modest, finding such a system could still be a huge contribution of mathematics to the the fairness of democracies around the world.

Works Cited:

Arrow, Kenneth J. “A difficulty in the concept of social welfare.” *The Journal of Political Economy* (1950): 328-346.

Geanakoplos, John. “Three brief proofs of Arrow’s impossibility theorem.” *Economic Theory* 26.1 (2005): 211-215.

McKenna, Phil. “Vote of no confidence.” *New Scientist* 198.2651 (2008): 30-33.

Graphics in public domain, available: https://en.wikipedia.org/wiki/Voting#/media/File:Vote_12345.jpg

Also consulted: https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem

]]>One immediate benefit of considering coordinate-free descriptions of geometric objects is that we may talk about “curves” that are not *a priori* embedded in . In other words, we don’t have to start with a subset of to be able to study 1-dimensional objects. There is already quite a nontrivial question we can ask: what curves can be embedded in a plane? The answer will be provided as a condition on and , and this description has the advantage of having nothing to do with or any other non-intrinsic data. Later we will talk about surfaces (2-dimensional manifolds) that do not live in 3-space, but rather in 4-space. Having an intrinsic way of seeing objects is liberating and opens up new possibilities.

**A 1-dimensional manifold**

It may also happen that the data are equivalent only on subsets of the intervals. For example, it could be the case that the data restricted to the first half of interval is equivalent to the data on the last third of interval . Then in this situation, we may “glue” together these overlapping compatible parts and get a longer curve that extends the one on . There will be no ambiguity in measurements over our new curve due to the equivalence of two sets of data in the intersecting parts. This possibility of patching together pieces while maintaining the structures is a fundamental part of the concept of a manifold.

Our definition will be of a Riemannian 1-manifold (because of the metric structures we have decided to keep). A Riemannian 1-manifold is a Hausdorff topological space such that each point in has an open neighborhood homeomorphic to an interval in , along with the set of data as above. If two open neighborhoods have a nonempty intersection, then we require that on the intersection the two localizations be equivalent in the sense of the previous section. There is also the condition of *second countability*: We like for our manifold to be covered by countably many of such neighborhoods.

**Coming up…**

Later in this series on manifolds, I will turn to surfaces which are 2-dimensional objects and re-interpret much of their calculus in the intrinsic language of differential geometry.

And if you missed them, here are Part One and Part Two of this series.

Let us know if you have any questions on manifolds in the comments below!

]]>After a particularly grueling example involving a word problem and a piecewise function, I asked my students, “What questions do you have regarding the process that we just went through?” and gave them a couple of minutes to review their notes and formulate questions. Gripped by the mid-semester complacency, my students lethargically looked through their notes and shook their heads indicating to me that there were no questions. Instead of taking their word and moving on, I was inspired to not move on and instead ask a question that would lead to discussion and had the potential to pull them out of the complacency that they were experiencing.

The first test was still recent enough that my students remembered the feeling of not knowing how to complete a problem on it so I drew from that experience. I looked out across the sea of students and told them what their lack of questions meant to me was that they were all going to get a problem like this correct on the next exam.

Suddenly my lifeless class had life, and throughout the room there was laughing and many audible no’s. I latched onto the no’s and asked again where were their questions. Immediately I got feedback from the students about where they were struggling and we had an excellent class discussion regarding the questions they had and how to overcome those struggles. Since then, the number of questions in class has increased and the students overall seem more engaged with the material.

Obviously this is not a fix-all solution and I wouldn’t recommend using it all the time or even frequently because it will just become routine for the students. They will expect it and just nod their heads that they can do it. However after a particularly difficult example it might be something to try if you are struggling with student engagement and complacency like I was. After all it is a new question and they can’t just answer like they usually do, and you can catch them off guard and elicit a whole new level of engagement from students that wasn’t there before.

What strategies have you used in your classroom to get your students to genuinely engage? Share your tips in the comments below!

]]>It seems like all anybody can talk about right now is the election. And while it has definitely given me a lot to think about in terms of political, cultural, and social problems in America, there’s also some interesting and potentially troubling math behind our electoral system that I think deserves attention as well. I want to explore election math just a bit, demonstrating that structural changes can seriously affect what kind of candidate wins, and questioning how elections should really be organized.

The Electoral College (https://www.archives.gov/federal-register/electoral-college/about.html) was originally established as a compromise between a full democracy, which would just count the popular vote, and having members of congress elect the president. In most states, the winner takes all of the electoral votes, although a few (Minnesota, Nebraska) assign votes proportionally. The process, as I’m sure you know, begins with primaries, and concludes with a race between the winner of each primary and any third party candidates.

Two years ago, I attended an excellent talk at the Joint Math Meetings in San Antonio, where Prof. Donald Saari discussed many different forms elections can take. The overall point is that, depending on the structure of an election, many different candidates could win the election with the exact same voter preferences. For example, consider last year’s Republican primaries. For simplicity, let’s just look at Trump, Rubio, and Bush. Suppose out of 100 voters, voter preferences followed this pattern:

40 voters: Trump, Rubio, Bush

25 voters: Rubio, Bush, Trump

35 voters: Bush, Rubio, Trump

To be clear, I’m making up these numbers to illustrate a point that I think is prescient.

Now, in a plurality rules primary, Trump wins with 40% of the vote, despite 60% of people, a clear majority, having him ranked last!

Consider another election system, where people rank the candidates, giving their first choice 2 points, their second 1, and their last 0. Assuming nobody tricks the system (and I’ll acknowledge that this might be a faulty assumption, but let’s go with the thought experiment), Trump gets 40 points, Bush gets 95, and Rubio gets 125. This gives Rubio the win, with a pretty large margin.

Consider yet another setup, where Rubio and Bush are grouped in a party, and Trump is in his own. Suppose Trump’s voters vote in a separate primary just electing Trump, and Bush and Rubio compete with the 60 voters who rank Trump last. Bush will win this primary by 10 votes. Then in a general election between Trump and Bush, Trump gets 40 votes and Bush gets 60, so Bush wins.

These are three setups- plurality, rankings, and primaries. Each one has a different candidate winning, but which one makes the most sense? Which one best captures voter preferences? I’m not arguing that we should immediately change the system, because I genuinely am not sure which I think is most mathematically or politically justified. My point is just that, especially after such a volatile election cycle, perhaps our democratic methods deserve greater scrutiny. Our government is deeply divided into parties that cannot seem to work together, and I think more moderate leadership, or at least leadership committed to working across party lines, would be productive and incredibly positive for America.

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