Monday, November 16, 2015

Joyous Excitement

Did you know that this week is the 100th anniversary of Einstein's completion of General Relativity? Specifically it was November 18, 1915 when Einstein drafted a paper that realized the final fix to his theories that would account for the previously unexplainable advance of the perihelion of Mercury. The next week he submitted this paper, "The field equations of gravitation", to the Prussian Academy of Sciences, which included what we now refer to simply as "Einstein's equations".

Einstein later recalled of this seminal moment:
For a few days I was beside myself with joyous excitement.

And further:
 ... in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now.  

(Quotes from "General Relativity" by J.J. O'Connor and E.F. Robertson at the School of Mathematics and Statistics, University of St. Andrews, Scotland).

Monday, November 9, 2015

Measurement Granularity

Answering a question on StackExchange, and I came across some very nice little articles by the Six Sigma system people on Measurement System Analysis:
Establishing the adequacy of your measurement system using a measurement system analysis process is fundamental to measuring your own business process capability and meeting the needs of your customer (specifications). Take, for instance, cycle time measurements: It can be measured in seconds, minutes, hours, days, months, years and so on. There is an appropriate measurement scale for every customer need/specification, and it is the job of the quality professional to select the scale that is most appropriate.

I like this because this issue comes up a lot in issues of the mathematics of game design: What is the most convenient and efficient scale for a particular system of measurement? And what should we be considering when we mindfully choose those units at the outset?

One key example in my D&D gaming, is that at the outset, units of encumbrance (weight carried) were ludicrously set in tenths-of-a-pound, so tracking gear carried by any characters involves adding up units in the hundreds or thousands, frequently requiring a calculator to do so. As a result, D&D encumbrance is infamous for being almost entirely unusable, and frequently discarded during play. My argument is that this is almost entirely due to an incorrect choice in measurement scale for the task -- equivalent to measuring a daily schedule in seconds, when what you really need is hours. I've recommended for a long time using the flavorfully archaic scale of "stone" weight (i.e., 14-pound units; see here), although the advantage could also be achieved by taking 5- or 10-pound units as the base. Likewise, I have a tendency to defend other Imperial units of weight as being useful in this sense (see: Human scale measurements), although I might be biased just a bit for being so steeped in D&D (further example: a league is about how far one walks in an hour, etc.).

The Six Sigma articles further show a situation where the difference in two production processes is discernible at one scale of measurement, but invisible at another incorrectly-chosen scale of measurement. See more below:

Monday, November 2, 2015

On Common Core

As people boil the oil and man the ramparts for this decade's education-reform efforts, I've gotten more questions recently about what I think regarding Common Core. Fortunately, I had a chance to look at it recently as part of CUNY's ongoing attempts to refine our algebra remediation and exam structure.

A few opening comments: One, this is purely in regards to the math side of things, and mostly just focused on the area of 6th-8th grade and high school Algebra I that my colleagues and I are largely involved in remediating (see the standards here: and I would highlight the assertion that "Indeed, some of the highest priority content for college and career readiness comes from Grades 6-8.", Note on courses & transitions). Second, we must distinguish what Common Core specifies and what it does not: it does dictate things to know at the end of each grade level, but not how they are to be taught. In general:
The standards establish what students need to learn, but they do not dictate how teachers should teach. Teachers will devise their own lesson plans and curriculum, and tailor their instruction to the individual needs of the students in their classrooms. (Frequently Asked Questions: What guidance do the Common Core Standards provide to teachers?)
Specifically in regards to math:
The standards themselves do not dictate curriculum, pedagogy, or delivery of content. (Note on courses & transitions)
So this foreshadows a two-part answer:

(1) I think the standards look great.

Everything that I've seen in the standards themselves looks smart, rigorous, challenging, core to the subject, and pretty much indispensable to a traditional college curriculum in calculus, statistics, computer programming, and other STEM pursuits. I encourage you to read them at the link above. It includes pretty much everything in a standard algebra sequence for the last few centuries or so.

I like the balanced requirement to achieve both conceptual understanding and procedural fluency ( As always, my response in a lot of debates is, "you need both". And this reflects the process of presenting higher-level mathematics theorems: a careful proof, and then applications. The former guarantees correctness and understanding; the latter uses the theorem as a powerful shortcut to get work done more efficiently.

Quick example that I came across last night: "By the end of Grade 3, know from memory all products of two one-digit numbers." ( That's not nonsense exercise, that's a necessary tool to later understand long division, factoring, fractions, rational versus irrational numbers, estimations, the Fundamental Theorems of Arithmetic and Algebra, etc. I was happy to spot that as a case example. (And I deeply wish that we could depend on all of our college students having that skill.)

I like what I see for sample tests. Here are some examples from the nation-wide PARCC consortium (by Pearson, of course; I'm looking at the 7th- and 8th-grade and Algebra I tests. They all come in two parts: Part I, short questions, multiple-choice,  with no calculators allowed. Part II, more sophisticated questions, short-answer (not multiple choice), with calculators allowed. I think that's great: you need both.

New York State writes their own Common Core tests instead of using PARCC, at least at the high school level ( here I'm looking mostly at Algebra I ( Again, a nice pattern of one part multiple-choice, the other part short-answer. I wish we could do that in our system. Now, the NYS Algebra I test is all-graphing-calculator mandatory, which sets my teeth on edge a bit compared to the PARCC tests. Maybe I could live with that as long as students have confirmed mental mastery at the 7th- and 8th-grade level (not that I can confirm that they do). Even the grading rubric shown here for NYS looks fine to me (approximately half-credit for calculation, and half-credit for conceptual understanding and approach on any problem; that's pretty close to what I've evolved to do in my own classes).

In summary: Pretty great stuff as far as published standards and test questions (at least for 7th-8th grade math and Algebra I).

(2) The implementation is possibly suspect.

Having established rigorous standards and examinations, these don't solve some of the endemic problems in our primary education system. Granted that "Teachers will devise their own lesson plans and curriculum, and tailor their instruction to the individual needs of the students in their classrooms." (above):

Most teachers in grades K-6, and even 7-8 in some places (note that's specifically the key grades highlighted above for "some of the highest priority content for college and career readiness") are not mathematics specialists. In fact, U.S. education school entrants are perennially the very weakest of all incoming college students in proficiency and attitude towards math (also: here). If the teachers at these levels fundamentally don't understand math themselves -- don't understand the later algebra and STEM work that it prepares them for -- then I have a really tough time seeing how they can understand the Common Core requirements, or effectively select and implement appropriate mathematical curriculum for their classrooms. Sometimes I refer to students at this level as having "anti-knowledge" -- and I find that it's much easier to instruct a student who has never heard of algebra ever (which sometimes happens for graduates of certain religious programs) than it is to deconstruct and repair incorrect the conceptual frameworks of students with many years of broken instruction.

Before I go on: The best solution to this would be to massively increase salary and benefits for all public-school teachers, and implement top-notch rigorous requirements for entry to education programs (as done in other top-performing nations). A second-best solution, which is probably more feasible in the near-term, would be to place mathematics-specialist teachers in all grades K-12.

The other key problem I see is: how are the test scores generated? We already know that in many places students take tests, and then the test scores are arbitrarily inflated or scaled by the state institutions, manipulating them to guarantee some particular high percentage is deemed "passing" (regardless of actual proficiency, for political purposes). For example, the conversion chart for NYS Algebra I Common Core raw scores to final scores for this past August is shown below (from NYS regents link above):

Now, this is a test that had a maximum total 86 possible points scored. If we linearly converted this to a percentage, we would just multiply any score by 100/86 = 1.16; it would add 14 points at the top of the scale, about 7 points at the middle, and 0 points at the bottom. But that's not what we see here -- it's a nonlinear scaling from raw to final. The top adds 14 points, but in the middle it adds 30 or more points in the raw range from 13 to 40.

The final range is 0 to 100, allowing you to think it might be a percentage, but it's not. If we consider 60% be minimal normal passing at a test, for this test that would occur at the 52-point raw score mark; but that gets scaled to a 73 final score, which usually means a middle-C grade. Looking at the 5 performance levels (more-or-less equivalent to A- through F- letter grades): A performance level of "3" is achieved with a raw score of just 30, which is only 30/86 = 35% of the available points on the test. A performance level of "2" is achieved with a raw score of only 20, that is, 20/86 = 23%  of the available points on the test. And these low levels (near random-guessing) are considered acceptable for awards of a high school diploma (, p. 19):

In summary: While the publicized standards and exam formats look fine to me, the devil is in the details. On the input end, actual curriculum and instruction are left as undefined behavior in the hands of primary-school teachers who are not specialists, and rarely empowered, and frequently the very weakest of all professionals in math skills and understanding. And on the output end, grading scales can be manipulated arbitrarily to show any desired passing rate, almost entirely disconnected from the actual level of mastery demonstrated in a cohort of students. So I fear that almost any number of students can go through a system like that and not actual meet the published Common Core standards to be ready for work in college or a career.

Monday, October 26, 2015

Double Factorial Table

The double factorial is the product of a number and every second natural number less than itself. That is:

\(n!! = \prod_{k = 0}^{ \lceil n/2 \rceil - 1} (n - 2k) = n(n-2)(n-4)...\)

Presentation of the values for double factorials is usually split up into separate even- and odd- sequences. Instead, I wanted to see the sequence all together, as below:

Values of the double factorial function

Monday, October 19, 2015

Geometry Formulas in Tau

Here's a modified a geometry formula sheet so all the presentations of circular shapes are in terms of tau (not pi); tack it to your wall and see if anybody spots the difference.

(Original sheet here.)

Monday, October 12, 2015

On Zeration

In my post last week on hyperoperations, I didn't talk much about the operation under addition, the zero-th operation in the hierarchy, which many refer to as "zeration". There is a surprising amount of disagreement about exactly how zeration should be defined.

The standard Peano axioms defining the natural numbers stipulate a single operation called the "successor". This is commonly written S(n), which indicates the next natural number after n. Later on, addition is defined in terms of repeated successor operations, and so forth.

The traditional definition of zeration, per Goodstein, is: \(H_0(a, b) = b + 1\). Now when I first saw this, I was surprised and taken aback. All the other operations start with \(a\) as a "base", and then effectively apply some simpler operation \(b\) times, so it seems odd to start with the \(b\) and just add one to it. (If anything my expectation would have been to take \(a+1\), but that doesn't satisfy the regular recursive definition of \(H_n\) when you try to construct addition.)

As it turns out, when you get to this basic level, you're doomed to lose many of the regular properties of the operations hierarchy. So there's nothing to do but start arguing about which properties to prioritize as "most fundamental" when constructing the definition.

Here are some points in favor of the standard definition \(b+1\): (1) It does satisfy the recursive formula that repeated applications are equivalent to addition (\(H_1\)). (2) It does looking passingly like counting by 1, i.e., the Peano "successor" operation. (3) It shares the key identity that \(H_n(a, 0) = 1\), for all \(n \ge 3\). (4) Since it is an elementary operation (addition, really), it can be extended from natural numbers to all real and complex numbers in a fashion which is analytic (infinitely differentiable).

But here are some points against the standard definition (1) It is not "really" a binary operator like the rest of the hierarchy, in that it totally ignores the first parameter \(a\). (2) Because of its ignoring \(a\), it's not commutative like the other low-level operations n = 1 or 2 (yet like them it is still associative and distributive, or as I sometimes say, collective of the next higher operation). (3) For the same reason, it has no identity element (no way to recover the value \(a\), unique among the entire hyperoperations hierarchy). (4) It's the only hyperoperation which doesn't need a special base case for when \(b = 0\). (5) I might turn around favorable point #3 above and call it weird and unfavorable, in that it is misaligned in this way with operations n = 1 and 2, and it's the only case of one of the key identities being added at a lower level instead of being lost. See how weird that looks below?

So as a result, a variety of alternative definitions have been put forward. I think my favorite is \(H_0(a, b) = max(a, b) + 1\). Again, this looks a lot like counting; I might possibly explain it to a young student as "count one more than the largest number you've seen before". Points in favor: (1) Repeated applications are again the same as addition. (2) It is truly a binary operation. (3) It is commutative, and thus completes the trifecta of commutativity, association, and distribution/collection being true for all operations \(n < 3\). (4) It does have an identity element, in \(b = 0\). (5) It maintains the pattern of losing more of the high-level identities, and in fact perfects the situation in that none of the five identities hold for this zeration (all "no's" in the modified table above for \(n = 0\)). Points against: (1) It isn't exactly the same as the unary Peano successor function. (2) It's non-differentiable, and therefore cannot be extended to an analytic function over the fields of real or complex numbers.

There are vocal proponents of related possible re-definition: \(H_0(a, b) =  max(a, b) + 1\) if a ≠ b, \(a + 2\) if a = b. Advantage here is that it matches some identities in other operations, like \(H_n(a, a) = H_{n+1}(a, 2)\) and \(H_n(2, 2) = 4\), but I'm less impressed by specific magic numbers like that (as compared to having commutativity and the pattern of actually losing more identities). Disadvantage is obviously that the possibility of adding 2 in the \(a+2\) case gets us even further away from the simple Peano successor function.

And then some people want to establish commutativity so badly that they assert this: \(H_0(a, b) =  ln(e^a + e^b)\). That does get you commutativity, but at that point we're so far away from simple counting in natural numbers that I don't even want to think about it.

Final thought: While most people interpret the standard definition of zeration, \(H_0(a, b) = b + 1\) as "counting 1 more place from b", it makes more sense to my brain to turn that around and say that we are "counting b places from 1". That is, ignoring the \(a\) parameter, start at the number 1 and apply the successor function repeatedly b times: \(S(S(S(...S(1))))\), with the \(S\) function appearing \(b\) times. This feels more like "basic" Peano counting, it maintains the sense of \(b\) being the number of times some simpler operation is applied, and it avoids defining zeration in terms of the higher operation of addition. And then you also need to stipulate a special base case for \(b = 0\), like all the other hyperoperations, namely \(H_0(a, 0) = 1\).

So maybe the standard definition is the best we can do, and the closest expression of what Peano successor'ing in natural numbers (counting) really indicates. Perhaps we can't really have a "true" binary operator at level \(H_0\), at a point when we haven't even discovered what the number "2" is yet.

P.S. Can we consider defining an operation one level even lower, perhaps \(H_{-1}(a, b) = 1\) which ignores both parameters, just returns the natural number 1, and loses every single one of the regular properties of hyperoperations (including recursivity in the next one up)?