In the past few years, I’ve seen Pick’s Theorem alluded to in various places. This theorem gives a surprisingly simple way to calculate the area of a polygon drawn on a lattice (most people imagine a Geoboard) based on the number of points on the polygon’s boundary and the number of points on the polygon’s interior. I won’t post the formula here in case you’d like to discover it for yourself, but here is one of many websites illustrating the theorem.

For example, the polygons below each have 5 boundary points and 3 interior points. Despite their different shapes, Pick’s Theorem predicts that each will have an area of 4.5 units.

I wanted to explore Pick’s Theorem with our Math Circle, a group of about 8-14 middle schoolers (mostly 6th graders). After examining lots of other math-circle Pick’s Theorem explorations, I handed the students the following much simpler version:

The handout ends right about where you would want to start making conjectures, with students being asked to find polygons with a given number of boundary points, number of interior points, and area. The last question is deliberately impossible.

The ideal math circle as described by its founders, the Kaplans, is a relaxed session in which students are presented with an interesting mathematical situation, then ask questions and make conjectures and discoveries together. I find that my group is a little young for that kind of Socratic seminar style: they really want to be **doing** things. So my aim last time was to given them lots to do, but eventually put them in a position where they would begin to generate some questions and ideas of their own.

Last time, most students got to various places on the last page. For a few, their interest waned before the end of the session. Next time will be tougher to manage! I’d rather not make a handout for the actual discovery of Pick’s theorem because I’d like the investigation to begin with student questions/observations. This means more “talking together” time, which will tax their attention spans.

My plan is to ask students to continue working on the last page, and then whenever enough people grumble about the problems being hard or impossible pool all of our data and see if we can find a pattern that might help us determine which polygons are possible. Depending on what they come up with, I might ask everyone to figure out (B,I,A) for rectangles with dimensions of their choosing, and try to generalize. Or, I might ask them to generate a bunch of polygons all with the same number of boundary points, but to vary the number of interior points, and see what happens.

I’ll bet we can come up with the formula, and at that point some students might lose interest (and I can give them a 1to9puzzle), those who are still with me can think about why the theorem might be plausible. We can come up with a good explanation for rectangles. I’m not sure if we can get beyond that, but the students may surprise me!

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Here’s my conservative solution for this quarter. I’m making a bulleted list of each student’s strengths and “areas to work on”, with a sentence or two of explanation after each one. Often, the explanation will be customized to each student, but I’m also just cutting and pasting explanations in some cases, if I really think the same explanation applies.

Here’s a sample describing a fake student. I made this by combining bullet points I’d made for various real students and changing gender pronouns if necessary.

Strengths:

*Using algebra to her advantage*. XXX is very comfortable solving equations. She will write an equation to describe a mathematical situation if she thinks it will help her to solve a problem. At the same time, she does not over-rely on algebra. She uses common sense to think about what would be a reasonable answer, and she also draws pictures to see what she can learn from them.*Abstract thinking*. XXX seems very comfortable operating in the realm of mathematics. She understands the significance of the concepts she’s learned in class. She can make connections between concepts in order to come up with new ideas.*Enthusiasm*. XXX becomes excited when figuring out new things.

To work on:

*Risk-taking.*XXX participates often enough in class, but sometimes seems unsure of herself. I’d love to hear some of her less-well-developed ideas in class discussion, even if they turn out to be wrong.*Executing strategies that are complicated or take several steps.*XXX has sometimes had trouble getting her mind around the most complicated problems in the course. She may come up with a strategy but not think through all of the parts in sufficient detail.*Clarity of writing*. XXX does put her work down on paper, but it is not organized in a way that is easy for someone else to understand. Working on this should be a priority for XXX: on her lines test, nearly every problem needed more/better explanation.

In the spirit of my newfound format, here are some advantages and disadvantages of doing this.

Advantages:

- Abandoning the paragraph structure means I don’t have to spend as much time thinking about how I’m going to start, how I will transition between ideas, and especially, how I’m going to introduce some negative comment so as to soften its blow. Everyone will expect to have a few good things listed as well as a few negative things.
- The bullet points will be easier for kids to process (I think). Often they read over their comments too quickly and miss things.
- This ensures that nearly all of each comment will be substantive. This is as opposed to filler like “XXX has been doing a great job in this class; it’s been my pleasure to work with him.”
- In some ways, this format has a structure similar to a rubric, where I am assessing each student along a variety of dimensions. However, not every student gets the same bullet points, and so I can focus on the dimensions that are most relevant to a particular student.

Disadvantages:

- While this format makes it easier for me to get started in my writing, it’s not infinitely flexible. It doesn’t allow for a natural way to talk about issues that are not strengths or weaknesses: for instance, improvement over time. This is only one very particular way to look at students, and so I wouldn’t want to use this format more than once or twice a year.
- There are some students for whom I’m not confident that I know their strengths and weaknesses so specifically! But it’s nice that doing this makes me think about those things.
- One might think that doing this would take less time than writing a paragraph, but I’ve found that the time spent is about equal. This definitely isn’t a way to punt.

What are some of the ways you give summative feedback to your students?

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Click the link to read the article: The Mathematics of Fountain Design

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Last week in 9^{th} grade class we were discussing the following problem, which had been assigned for homework.

Write an equation for a rule a?b, so that the answer is odd only when both a and b are even.

Since this problem admits of multiple answers, I thought we’d hear what several different students came up with and test them. Student #1 offered a rule that was of the form (a+even)(b+even). Everyone agreed it fit the bill. Student #2’s example was less obvious; we’d have to play around with it to see if it worked. So far so good.

Student #3 tried plugging in two even numbers and got an odd answer. Other students were trying similar approaches. “But wait,” I said, in words that I now regret, “remember that this rule needs to produce odd answers *only when* a and b are even. We need to test other kinds of numbers for a and b.”

I waited. The next time a student raised a hand, it was to ask if they could give their own rule. Of course they were disappointed when I replied, “We have to finish checking Student #2’s rule first.” No one else had anything to say, so to help things along I suggested that we try the formula with a and b both odd. One student objected: “the problem is about when a and b are even – we don’t have to do that.” I explained as best as I could why we did, in fact, have to do that.

And then, the hideous cycle repeated itself I don’t know how many times.. I tried to give prompts to students to discover a counterexample that I knew was waiting, and they were genuinely stumped. The more focused students were quiet because they didn’t understand the logic of the problem well enough to understand what a counterexample would look like, and the less focused students continued to raise their hands, enthusiastically wanting to offer their own rules. I cringe to tell you that we spent about 15-20 minutes on this problem before I finally showed them a counterexample, and we moved on. Perhaps worst of all, those students who so desperately wanted to demonstrate their own rules (most of which probably wouldn’t have worked, given their incomplete understanding of the problem) did not even get to do that.

In retrospect, I wish I hadn’t assigned this problem at all. Or I suppose I could have assigned a version of the problem that was equivalent to the students’ interpretation of the problem. Knowing now that the notion of “only when” was not intuitive to these students, even after explanation, I *might* have designed a worksheet with much easier questions designed to illustrate the principle. But I don’t think I would have done that, because ultimately, with another year or two of maturity, these students will come to understand phrases like “only when” mostly on their own. There’s no reason to force the issue now.

And yet I *did* force the issue. Once it became clear to me how difficult they were finding the concept, and how not a big deal that was, I still didn’t pull the plug on the discussion. Why? Because I couldn’t find a graceful way to do it. I could have said, “Yep, you’re right – the rule works,” but I couldn’t stand the thought that we were, officially, as a class, going to make a major error in logic. I could have said, “You know what – this problem is harder than I thought it was. Let’s abandon it.” But that wouldn’t have made any sense to the students, most of whom thought they had solved the problem.

Do you ever find yourself in situations like this? What are your “abort mission immediately” strategies? And what would you have done in my place?

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According to Jean Piaget (1896-1980), in the same way our body evolves until it reaches a relatively stable state with the maturity of our organs, so our mental life can be thought of as an evolution toward a form of equilibrium characterized by the adult spirit. Thus, since birth, human beings (the subject) tend to reach and maintain a state of equilibrium with their environment, that is, with the world (the object). When this equilibrium is broken (by a change of position, a question, or an emotional experience, for example), an action or response (a movement, a thought, or a feeling) is needed to re-establish the broken equilibrium. These interactions between subject and object continue throughout the rest of life. According to Piaget, there are two ways human beings respond in order to re-establish a broken equilibrium: assimilation (the use of perceptual or mental schemata already existing in the individual to interpret new experiences) and accommodation (the addition of new schemata to those already existing to be able to interpret new experiences). In terms of the individual’s mental development, Piaget constructed a model which identifies four stages (see previous blog On Algebra and Logic): a) *Sensorimotor stage (birth – 2 year-old)*. b) *Preoperational stage (ages 2-7)*. c) *Concrete operations (ages 7-11)*. d) *Formal operations (beginning at ages 11-15)*. For Piaget, the actual mental growth occurring in the individual can be described as the gradual construction of a map of reality. This construction steadily shows connections between any two points—by means of short and direct routes first—which become greater in space and time and take progressively more complex paths. Thus, the first interactions—perceptual assimilations and accommodations—are direct and simple interactions. In the sensorimotor stage (the first in Piaget’s model) interactions include reversals and detours. In the preoperational stage (the second in Piaget’s model), the child acquires representative thought and thus is able to invoke absent objects. This possibility of mental representation makes the child capable of reaching pieces of reality that are not visible, since they lie either in the past or partially in the future. But these representations are more or less static figures, pictures of a mobile reality, which represent only some states or paths out of a great number of possible states or paths. In this stage, the ongoing construction of the map of reality still leaves a lot of blank spaces and not enough conditions to go from one point to another. In the concrete operation stage (the third in Piaget’s model), these static images become mobile, and are able to embrace the dynamic of reality. Also the child’s responses (to regain a state of equilibrium with the environment) follow greater distances in space and time and take more complex routes. The map of reality is in this stage becomes progressively more complete. However, this map is a representation of reality only. In the formal operation stage (the fourth in Piaget’s model) more than just reality is included: The world of the possible becomes reachable. The individual’s mind is freed from concrete reality, and mathematical creation can flourish powerful and unbounded.

One of the most withstanding mathematics works has been the *Elements* of Euclid (about 325 BC- 265 BC). This work has been become a model for the organization of knowledge. As a matter of fact, Newton (1643-1727) used the structure of Euclid’s Elements to write his new physics *Principia*. In addition to the structure of this work of Euclid’s, the achievement of the *Elements* had remained unchallenged for more than twenty centuries, due to the feeling of absolute certainty that it had brought about. The structure of this work is based on the formulation of definitions, axioms or common notions (these are general statements, not specific to geometry and whose truth is obvious or self-evident), postulates (these are the basic suppositions of geometry), and theorems or propositions (which are the consequences deduced logically from the definitions, axioms and postulates). Euclid proposed five postulates, each of which he believed to be independent—not a logical consequence of the other postulates of the set. The following statement is equivalent to the fifth of these postulates: Through a given point not on a given straight line can be drawn only one straight line parallel to the given line. For centuries, mathematicians had difficulty in regarding the parallel postulate as independent of Euclid’s other postulates and made repeated and failed attempts to show that it is a consequence of the other ones. These attempts date back to Claudius Ptolemy (AD 90—AD 168) followed by other mathematicians after him, and more recently Girolamo Sacheri (1667-1733), Johann Lambert (1728—1777), and Adrien Legendre (1752—1833) made new attempts. These last three mathematicians tried to prove the dependence of the parallel postulate by reductio ad absurdum. In fact, they assumed that “through a given point not on a given straight line, not one, but at least two lines parallel to the given line can be drawn,” and expected to find a contradiction from this assumption. However, for about 150 years Saccheri, Lambert, and Legendre failed to find such a contradiction, and the problem of the dependency of Euclid’s fifth postulate remained unsolved. The first to suspect the independence of the parallel postulate were Carl F. Gauss (1777—1855) from Germany, Janós Bolyai (1802—1860) from Hungary, and Nicolai I. Lobachevsky (1793—1856) from Russia. However, it was Lobachevsky the first mathematician to publicly express—in an article titled *On the Principles of Geometry*—that Euclid’s parallel postulate could not be proved on the basis of the other postulates; therefore, an opposite statement to this postulate, that is, “through a given point not on a given straight line, not one, but at least two lines parallel to the given line can be drawn,” could be accepted as a valid and independent statement instead of Euclid’s fifth postulate. As contradictory to common sense as it can appear, it is as consistent—not generating contradiction when put together with the other four postulates—as that of Euclid’s fifth postulate. This was the origin of what is known as non-Euclidean geometry. Lobachevsky’s geometry revolutionized not only geometry, but the entire fields of mathematics and philosophy. It took him a lot of courage to share this idea with the world. As a matter of fact, Karl F. Gauss—by many considered the greatest mathematician of all time— remarked that he had thought of this possibility before, but didn’t make any public statement about his idea opposing that of Euclid’s parallel postulate, because he was too concerned about mathematicians’ (and non-mathematicians’) reactions. Lobachevskian geometry showed that Euclidean geometry was not the absolute truth it had been considered to be for more than twenty centuries. Even more, it freed geometrical reasoning from spatial intuition. The concept of space also had to be revised as a consequence of this new geometry. Although Lobachevsky became the rector of a university in the city where he grew up—Kazan University—almost until the end of his life, he faced economic difficulties. Even more, during his lifetime his work was not given the consideration that his revolutionary ideas deserved. Only after his death was he given general acknowledgement for such a creative work. Indeed, his perceptions changed our comprehension of the universe.

The concept of infinity has been a struggle for philosophers, mathematicians, and even theologians since the days of the ancient Greeks. Infinity has been always a mysterious or confusing idea, even feared by the greatest minds in history. As a matter of fact, Aristotle (384 BC—322 BC) thought that being infinite was a poor quality, rather than perfection. Also, Galileo Galilei (1564-1642) affirmed that “by its very nature, infinity is incomprehensible to us,” and Gauss was opposed to the use of an infinite quantity as a real object, which, in his opinion, should never be allowed in mathematics. For Gauss, the infinite was only a way of speaking. Galileo did observe that there are as many squares, as there are natural numbers , but he didn’t go further with this observation. In 1872, Richard Dedekind (1831-1916) defined a set as infinite if it could be put into one-to-one correspondence with one of its subsets. This was Cantor’s starting point. According to Dedekind’s definition, both the set of natural integers and the subset are infinite, but Cantor called them “countably infinite.” For him, countably infinite sets are those that can be put into one-to-one correspondence with the set of natural numbers. Cantor went even further and showed that there are infinite sets that cannot be put into one-to-one correspondence with the natural numbers. In fact, he proved in 1973 that the set of real numbers is one of them, and called these sets “uncountably infinite.” Three years later, Cantor proved what neither Euclid nor anybody else had ever imagined to be possible: that the points of a surface, for example a square that includes its boundary, could be put into one-to-one correspondence with the points of a straight-line segment that includes its end points. Thus, one of Euclid’s common notions, that the whole is greater than the part—which stood without question for more than 2,000 years—had been disarmed! But this was only the beginning of Cantor’s unconventional way of thinking. In order to distinguish the countably infinite sets—the sets of integers, rational numbers, and roots of polynomial equations with rational coefficients, to take some examples—from the uncountaby infinite ones—as the real numbers, or planes, or spaces of *n* dimensions, for example—he established that all countably infinite sets have the same power (or cardinality). He denoted this power by using the first letter in the Hebrew alphabet, the aleph, and the subscript zero, , calling it **aleph-null**. Then conjecturing that there were no sets with intermediate levels of infinity between the countably infinite sets—like the set of rational numbers , for example—and the set of real numbers , Cantor assigned to the latter the next power order or level of cardinality. This conjecture of Cantor’s is known as the continuum hypothesis. Although he struggled the rest of his life to prove this conjecture, his daring way of thinking about a topic that had been so elusive as that of infinity, gave rise to a new (aleph-based) number system: The transfinite numbers, where and are the first two transfinite numbers. Cantor also proved that the set of subsets of a given set is always of a higher power than that of the given set—the set of subsets of a given set cannot be put into one-to-one correspondence with the original set. Therefore, the power of the set of subsets of the real numbers is the third transfinite number, the power of the set of subsets of the previous set is the fourth transfinite number, and so forth indefinitely. His work was so original that Cantor himself acknowledged that his ideas about this new system of (nonfinite) numbers were in opposition to widely spread beliefs about infinity in mathematics. He was not only not understood by many of his contemporaries, but also underestimated. As a matter of fact, Cantor could never join the faculty of what he considered a high-caliber school. He remained at the University of Halle, a rather small school, for his entire career. However, as the great German mathematician David Hilbert (1862—1943) said, “Cantor’s work was the most astonishing product of mathematical thought, one of the most beautiful realizations of human activity in the domain of the purely intelligible.”

Piaget’s interpretation of his mental development model as a gradual construction of a map of reality, up to the point where the subject’s mind frees from this concrete reality and makes reachable the world of the possible, is wonderfully illustrated by Lobachevsky’s and Cantor’s mathematical achievements. Not only did they liberate themselves from the constraints of a petty reality that became short to the greatness of their thoughts, but also had the courage to share their ideas with the world, despite the hard opposition they had to face for going against “common sense.” The history of these achievements also shows us, mathematics teachers, that among those students who think differently, those who don’t follow conventional ways of thinking and surprise us with estrange ideas—which may even seem like silly ideas—there might be the ones who will transform the world. The requirements of this extraordinary 21^{st} century demand extraordinary thoughts, capable of reaching what could be seen as impossible achievements. The future achievers of this kind of accomplishments, exponents of the new generation of Lobachevskys and Cantors ready to reach those heights, may now be in your classroom. Don’t ignore them.

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Tensegrity polyhedra, made from 3/16″ dowels and standard rubber bands, based on this little article by George Hart. The coolest thing about them is that no two sticks are actually touching each other (which makes me wish I’d used different color rubber bands). Or maybe it’s that they can collapse like this…

…and then snap back into shape. Or maybe that they bounce. (Yep.)

In any case, they were really fun to make–the dodecahedron turns out to be a great spatial reasoning puzzle as you get close to the end–and I think they’ll make good toys or decorations. And there’s tons more inspiration for mathy crafts at Math Monday (as well as at georgehart.com and vihart.com). Maybe I should crowdsource this by offering it as extra credit–that ought to get the classroom looking good in no time

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I wanted to teach 9^{th} grade this year because I realized that I was not doing nearly as much as I could be to teach students how to be learners. Assigning students nonroutine problems has its drawbacks: though we have great class discussions and kids learn to see math for the open book that it is, students also have the perfect excuse to say, “I just didn’t know how to do this homework problem”, and teachers have the perfect excuse to give large hints that don’t empower kids to feel that they could have solved the problems themselves. 9^{th} grade seems like a good opportunity to focus on changing some of my practices. We’re all making a fresh start.

Here are the things I most want to work on with my students this year, things that we need to establish in the earliest days.

**What it looks like to have tried a problem**. The message of our first habits lesson, “Tinker”, is conveniently relevant to this question. It means that you’ve written down something,

**What counts as a good written explanation of a problem.** Students have plenty of opportunity in class to explain their work orally, and they get lots of natural feedback from their peers about how clear they are being. However — and shame on me — I‘m generally not giving kids much feedback on their writing until there is some kind of graded assignment. Meanwhile, all sorts of bad habits are going on in their homework notebooks. I plan to have students swap papers early in the year and give each other feedback. The principle I hope they’ll discover: we don’t need to stick to arbitrary standards like complete sentences or the formatting of the page, but we do need to be able to communicate our ideas in a way that is simple to read.

**What to do if you don’t understand a problem. **It’s Polya’s step 1, but how do you get there? I’m always reflexively rephrasing questions for students, sometimes before they ask. This is bad, bad, bad! Instead, I need to give this type of advice. The habits of mind are useful in getting a handle on problems, even before you go looking for a solution. As you read the problem: draw a picture or make up an example of the type of thing the problem deals with. Add details as you read further. Read the problem phrase by phrase. What is the first phrase you don’t understand? Can you ask a specific question that will help you understand the problem? If all else fails, ask a classmate before you ask me. Maybe you’ll be able to work it out together.

There are reasons why I’m not already focusing heavily on these issues: they take time, and I’m always eager to dive into discussions of problems, often at the expense of stepping back. However, looking at my list, it’s woefully obvious that students who develop these skills will be well-placed to learn mathematics… or really any topic. They’re what we should be teaching. I’m going to try. Hold me accountable, please!

What are the habits you want to instill in your students from the earliest days of class?

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This fragmentation implies a divided and incomplete knowledge of reality. It is understandable that people have had the need to divide the world into small pieces, so that they are able to approach some kind of knowledge of it. However, this increasing division and subdivision of the world, while deepening our knowledge of smaller pieces of the universe, increases the disconnection between the components of that universe. In fact, it is evident that one can be very knowledgeable about a particular field of study, while hardly being able to understand any other field or, more importantly, how our actions within our field of work may affect the rest of the world. Therefore, the problems that we face as a consequence of this specialized disconnection ought to make us take a look at the multiple pieces of a totality that should be restored some way.

This divided vision is even experienced within a particular discipline. As a matter of fact, it happens in the way we believe mathematics should be taught. For many years now, there has been a controversy between those who endorse teaching mathematics through real-world problems and those who favor an emphasis on basic skills. In my experience teaching mathematics for more than twenty years, I have observed the limitations that overemphasizing either skills or problem-solving brings to a true conceptual understanding—understood as the connection between a problem at hand and a more general theory from which this problem is a particular case. On the one hand, when the emphasis is only on skills, a mechanical approach to the solution of problems makes it unnecessary for the students to reach a true understanding of the situation involved. On the other hand, emphasizing only problem–solving without enough attention to the development of skills may deprive the students of the power needed to take mathematics—and ourselves—to upper levels of development. Also here I see the need for a unified approach: Skills, problem-solving, and concepts are all of necessary importance.

The evolution of mathematics shows that it moves backwards in a retrospective reflective abstraction—going deeper into earlier mental processes, looking for the roots of mathematical concepts, as the formulation of new systems of axioms shows—and also forward, formulating more powerful theories, which are a generalization of the structures formed throughout the retrospective reflection mentioned above. More than a static frame to be applied to common situations, algebraic procedures are the formalization of general properties of multiple particular past cases. We cannot deprive the students of the power of mathematics by withholding from them either a true understanding of the problems at hand or the retrospective look of reflective abstraction contained in algebraic procedures. Therefore, our work as mathematics educators must be that of engaging the students in activities that keep them deeply focused on what they are doing, as well as in activities that make them reflect on the current and previous procedures, abstracting general properties from them.

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Colleges in the US are being compelled to rethink what the First Year Experience or Seminar ought to be for students who have difficulty with mathematics, and what ought to be the mathematics education of teachers, K-12, given the minimal success most students are experiencing. It will be argued here that toward ensuring a more successful education for all students learning mathematics, and most especially for those who will become teachers, the inquiry process must be made explicit so that the productive practices of a mathematically-inclined mind are considered as content. That is to say, the classroom conversation needs to include discussion of the actions mathematically able thinkers use to gain insight into a problem; such as: considering a simpler problem, tinkering, taking things apart. This paper will make an argument why this is an essential consideration for promoting a robust society, and include instances of how mathematics may be presented in this framework.

If you would like to access the article online, free of charge, send an email to parkmathblog@parkschool.net. Provided you’re one of the first 50 people to ask, we’ll send you the link.

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