I’m happy to say that I’ve participated in the Encompass Summer Institute. After following the rouges gallery of fellows on twitter all summer it was almost like a Hollywood red carpet when they started pouring in.

My role in developing the encompass software is mostly quality assurance; as in, being another pair of eyes to make sure the scripts and codes do what they’re supposed to be doing. Though, my work is not limited to just bug hunting. As an engineer, testing the software and writing bug reports is how I mediate between the teachers and programmers at the math forum. Essentially, I try to translate from programmer to teacher and vice versa.

Either way, it came and went and in the aftermath I’m picking up the pieces and helping to develop software. ESI was quite a boon to us. I’ve been slamming my skull against the concept of “software” and “formative assessment” trying to think of ways to envision pathways to develop new features that teachers at ESI have expressed a need for. Until now, the reverberations of the Institute have been rather fruitful. Damola, Amir, Steve, myself et al have been working hard to create new features and have brought the encompass software miles and miles closer to a final point of development.

On the topic of the event itself, one new tradition that Steve brought to ESI was “Connections”. This was a segment of time devoted to quiet contemplation of life, the universe and everything; though the fellows and staff tended to focus on teaching and the myriad challenges and triumphs therein. It wasn’t entirely silent. One person at a time were allowed to speak their thoughts without expecting a response, praise or judgement. Their words were meant to resonate among us, or as steve put it “ripple”, and inspire new ideas or help us to find new perspectives on things. For some it was very inspiring, and for others cathartic. Some ripples were stories rife with fear, uncertainty or regrets, yet these were just as profound as any other ripple. I’m not one hundred percent sure if this was Steve’s original idea, but even though the ripples were rather deep cuts I think the catharsis helped our fellows focus on the future of the Encompass program. On the penultimate connections session, I had drawn a parallel to what actually occured to a passage in the Tao te Ching.

The thirty spokes unite in the one nave; but it is on the empty

space, that the use of the wheel depends. Clay is

fashioned into vessels; but it is on their empty hollowness, that

their use depends. The door and windows are cut out

to form a home; but it is on the empty space, that its

use depends. Therefore, what has a existence serves for

profitable adaptation, and what has not that for usefulness.

I think that the ripples we created were our best thoughts and our worst memories being emptied from the vessel that is our collective mind. Though concerning Eastern philosophy, perhaps I should stick to Samurai history.

It’s now a few months away from ESI 2014, but the memories and lessons are still rippling with me.

]]>In the latest Trig & Calc PoW students need to find the density of titanium metal. As a Materials Engineering student, I know that this is a difficult question to answer. There are many factors that go into and ways to find the density of a metal, each of these I will get into briefly. Max Ray and I worked together to figure out how to accept answers to this problem regarding the wide varieties in the density of titanium metal, so for now the PoW is as solid as titanium.

Originally, the Ninja Mission PoW relied on the density of pure elemental titanium, though even this is variable. Let’s first consider how a scientist would calculate the density of titanium metal. We know that titanium atoms arrange themselves in a lattice pattern known as “Hexagonal Close Packed” or HCP. This means that the individual atoms can be divided into subdivisions of hexagonal prisms which contain a number of partial atoms. (It’d be a lot easier if titanium was arranged in a cubic lattice, but we can’t have everything.) These geometric subdivisions are known as a unit cell. The atoms are arranged on to various points in unit cell; one in each corner of the face hexagons, one in the center of the hexagon face, and three “floating” within the prism in a triangular formation. But, the whole volume of each atom is not entirely in the unit cell, so some are considered fractional atoms. Luckily, it’s easy to remember how many atoms are in an HCP unit cell. There are six total atoms made up of 12 twelfth atoms in the corners, two half atoms in the faces and three whole atoms in the volume.

I even drew you a picture!

So how do we get from the number of partial atoms in a unit cell to the bulk density of a metal? Well, what do we know?! We know how much an atom of titanium mass based on a weighted average of probabilistic distribution of titanium isotopes that exist on Earth. Essentially, it’s not a direct mass of the atom but, a product of the mass of protons, neutrons and electrons, and the number of each in one atom. The periodic table has this number; 47.867 grams per mole (1 mole = 6.02 x 10²³ atoms). The dimensions of the unit cell are 0.295 x 10^-9 meters for the length of a leg and the height is 0.466 x 10^-9 meters. With this information we can formulate the density of titanium metal.

Essentially, what I’m calculating here is how much mass there is in a 1 cubic centimeter hexagonal prism of titanium based on the weight of the total number of titanium atoms in such a volume. This calculates to 4.53 grams per cubic centimeter. This number represents the density of perfectly crystalline and pure titanium based on the standard distribution of isotopic titanium atoms on earth. What I do not account for are defects, dislocations, grain boundaries and impurities that occur from mechanical and thermodynamic forces. Metals (and most materials) do not crystallize perfectly on a macroscopic scale. What happens when a metal solidifies, lots of small crystals form in multiple crystallographic directions. *Id est* as metals solidify metal atoms arrange themselves into many separate volumes, or grains, with their own orientation. What ends up happening looks a lot like this.

All of those colored shapes are individual grains of crystalline titanium. If you looked at the atomic arrangement of a single grain, all of the atoms would align almost perfectly. Zooming in on a boundary between these grains would look something like this. (The image below is a boundary between a grain of Aluminum and Titanium.)

Though there are two different metals, you can see that the atoms align to form an angle. In the case above the boundary is fairly clean and doesn’t have many gaps, but in most cases, grain boundaries free up a lot of space in a material reducing its density. You can even see that within the two grains there are gaps and inconsistencies in the atomic patterns.

Beyond that, atoms don’t always stack perfectly. There is a sizable laundry list of multidimensional hiccups that can occur when a material solidifies, is heated up or is worked with. These are all classified under the umbrella term “Defects”. These can add or subtract from the density of a material depending on what they are. Single dimensional defects are generally missing, extra or displaced atoms, or other types of atoms crammed into a pure lattice (impurities). Two and three dimensional defects are often created when forces bend, slip and twist atoms out of crystalline alignments.

How does a materials scientist account for defects in terms of density? Short answer: carefully. Considering that in the context of the problem, the wise Sensei grafted titanium spikes to a titanium disks to produce a throwing star, –a process requiring a computer operated plasma cutter– the density of the metal is sure to be changed by defects created via the working and heating of the metal. Theoretically, it’s possible to calculate the density of a metal after all is said and done, though it’d be a massive mathematical undertaking. It’s possible to determine the average number of single dimensional defects using a simple thermodynamic equation based on experimental data. In plain terms, the formation energy of a single dimensional defect wherein an atom is added, removed or displaced from it’s crystalline pattern is the difference between the specific enthalpy of a material minus the product of the temperature and the specific entropy of the defect. Unfortunately, defects with more than two dimensions are only formed by mechanical forces, not thermodynamic, so to measure any changes in density from 2+ dimensional defects, we would need to run a computer simulation of the specific titanium material. All of these phenomenons affect the density of titanium by little more than 0.02 grams per cubic centimeter, but this is enough to affect the final answer by a whole degree.

(Science note: Enthalpy is a measure of energy within a substance based on it’s internal energy and the product of pressure and volume. Entropy is a measure of disorder within a system.)

Why does this matter in terms of our math problem? Well, it doesn’t so much in a practical sense, but mathematically the dimensions of the titanium disks are all functions of it’s density. We know that density is an incredibly variable thing. Though, it’s enough to confuse a student looking up the density of Titanium. It’s enough to know the total weight and volume of the titanium disk, and take the quotient to get the exact density of the particular chunk of metal. Though we don’t initially know the volume, so we have to rely on a carefully calculated density for a material with hundreds of forms and alloys (549 according to matweb). The problem states that the throwing star is made of pure titanium, though pure titanium is impossible to produce on Earth. There are theoretical results for the density of pure Titanium, but we know that it’s a finicky number. I chose annealed grade 1 titanium because it’s the purest Titanium we could possibly produce, and the density is experimental rather than theoretical, as in someone in a laboratory took a known volume of titanium, weighed it and then did the math. There’s lots of reasons why Max and I allowed for a range of answers for the Ninja Mission PoW. It isn’t because the math isn’t solid, but rather the solid isn’t!

]]>- Choose any number with up to four digits and four unique digits. (1234)
- Rearrange that number twice, once from greatest digits to lowest digits and again backwards (4321 & 1234)
- Take the difference (3087, this is also why you need at least two unique digits. using a number like 2222 just hits zero and the algorithm stops.)
- Repeat steps 1-3 until you reach 6174.
- Repeat steps 1-4 once.

Continuing the process for the initial number returns 1234, 3087, 8532, **6174, 6174, 6174, 6174, 6174, 6174, 6174, 6174… ****for most numbers this won’t take more than seven iterations.**

I just really think this is a neat number. It isn’t as practical as π, e or 27… but it certainly instills wonder. This works with other numbers of digits too. A few of them are 495, 6174, 549945 and 631764.

D.R. Kaprekar was rather interesting himself. He was a “recreational mathematician” who had no postgraduate training in mathematics and spent most of his life teaching grade school. Kaprekar developed descriptions for several classes of natural numbers. He considered himself a recreational mathematician. He never set out to prove hard mathematical laws or theories; he just wanted to have fun doing math! It just goes to show that Math isn’t all hard work.

]]>Things are still busy at the Math Forum. I was given the opportunity to grade student responses for an algebra PoW based on the Math Forum’s rubrics for problem solving and communication. I found it difficult to grade similar responses with fresh outlooks, but the rubric allowed me to look for things that I wouldn’t ordinarily look for. If I were to grade the responses without the rubric, I would have scored the answers based on how deep the student seemed to understand the answer and scale that based on how accurate their answer is. Though, with the Math Forum rubric, the idea is to look for insights into the thought processes of the student. In separating problem solving into accuracy, strategy and interpretation, I can evaluate better how a student approached a problem and where they may have gotten stuck even if they didn’t explain beyond their strategy. As well, the communication criteria allows for students with less mathematical methods of thought to shine… so long as they actually explain what they’re doing or not understanding. It’s this insight that allows us at the Forum to do what we do.

This week also held the May T3 at the ExCITe center. The Math Forum crew invited me along to see a few short talks about technology development and outreach in Philadelphia, as well as off-kilter examinations on intellectual and academic life nationwide. What particularly interested and entertained me there was Kevin Egan who compared higher education to the development of Punk music. In the simile, he juxtaposed modern experimental and customized education with The Talking Heads, Gary Numan and DEVO. Though, I have to wonder why he left out Joy Division!

In other news, Tracey and I are working on getting the Ask Dr. Math twitter going! Keep an eye out for tweets from our numerical hero.

Coming up, I have plans for personal interest posts regarding STE**M** with a bold **M**. Come back sometime next week, and you might just learn something cool.

For one of the Trig & Calc POWs I found that my solution differed greatly from the examples in the teacher packet. Though, that’s the entire goal of my work so far. The problem was to find the area of a pizza cut in an unusual way. There were only a few solutions in the Teacher’s packet, a lot of them used Heron’s Formula where I developed a system of integrals. I don’t think I ever learned Heron’s formula in any high school classes, but I know how to analyze a geometric situation using trigonometric and calculus concepts. The goal, on my end at least, was to see how my solution differed from other students’ solutions.

I feel as if, by working this way, I’ve stumbled upon the guiding principle of the Math Forum POWs. Beyond deepening my understanding of spatial and geometric analysis, comparing my solution with that of many others from various educational backgrounds helps me to better understand how to help someone else solve the same problem. And, I mean more than just being able to suggest a repertoire of various laws or theorems. In understanding how a student solves a problem, I can figure out how they were thinking, and what ideas came to them as they were reading the problem.

From an engineer’s perspective, this problem screams “integrals!”. I know in the previous post I talked about “no one size fits all solutions” but I like to have one tool does all. To me, integrals and derivatives are more valuable than a toolbox of formulas and theorems. I know that I can find acceleration through derivatives or areas and displacement through integrals, so a huge amount of formulas and equations are basically obsolete. Though, to a student who hasn’t had calculus (or won’t ever take it) the formulas and theorems are the only tools they have. By way of comparison, my toolbox hasn’t necessarily been upgraded. Metaphorically speaking, the comparison is unto a machine shop full of lathes, drills and lots of other precision tools versus a similar shop with a C&C machine. In principle one isn’t necessarily better than the other. The one is a massive collection of tools that may not see use in every problem, but is equally effective as a machine designed to “do it all.”

I also noticed, in general, that other students’ solutions were less savvy about precision. Not in that they were imprecise in their answers, but that they didn’t consider where over-precision would over-complicate a solution or was simply unnecessary given the conditions of the problem. In most sciences, it’s satisfactory to be within 10% of an accepted value while in pure mathematics the answers are more black & white. In my brief time solving POWs I’ve found myself at odds with the answers after carefully considering how precise my answer can be assuming that any measurements given to me are fallible measurements. It’s a habit I’ll have to let go of for my time here at the Forum.

Another problem I worked on was about finding a point on a line such that two other points had a difference in distance of one. I knew I could have solved it via algebra and an established equality, but I knew a more “advanced” way to solve it. (Yes, I wanted to show off.) I solved it via linear algebra and matrices. The problem suggests that there may be more than one solution, so that tipped me off to use my knowledge of linear systems. As it turns out, there were only two solutions which were resultant from the algebraic manipulation of a square root. Even still, I solved the problem correctly, even though I had to submit some clutter with extraneous solutions.

So In conclusion, Math is fun and I’m having a great time at the Math Forum!

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