Core Standards for Mathematics Feedback

What follows below is the feedback I provided on the proposed Core Standards for Mathematics.  These represent my own opinions on the direction mathematics reform should take.  As far as I know, the changes I propose have not been sufficiently supported by research.  However, I hope I may provide a fresh perspective on the direction the mathematics curriculum should take to address some of the existing problems.

I'd like to start with some background about myself, to provide a context for this critique. I was an accelerated math student in high school and majored in Mathematics in college. These views are my subjective interpretations of the mathematics curriculum as I experienced it, and the areas of mathematics in college that I felt unprepared for. My objection to the proposed Core Standards is that I do not view them as reforming these areas where I felt unprepared. Instead, all the Core Standards seem to do is set in stone the same failing curriculum I experienced as a student.

Overall I feel that the Standards for Practice expressed on pages 4-5 are solid goals for mathematics education to strive for. However, my major critique of the Core Standards is that I do not think they are sufficient to meet these goals. Two of the Standards for Practice strike me as being unsupported by the proposed curriculum: (3) Construct viable arguments and critique the reasoning of others and (5) Use appropriate tools strategically. The former requires a solid foundation in logic, set theory and critical thinking, while the latter requires an introduction to computation science. The standards that follow do little to reinforce these skills.

In order for students to construct and critique arguments, the students must first know the basic structure of an logical argument. How can students be expected to give valid arguments when the definitions of validity, soundness, completeness, and consistency are omitted from the mathematics standards? The only objective that seemed to actually address this was in the High School Algebra Standards: “Understand that to solve an equation algebraically, one makes logical deductions from the equality asserted by the equation”. I respect that this objective is included, but don't think that the curriculum leading up to it adequately supports it. When I was first exposed to college level mathematics, the notation system of formal logic was used extensively by my professors. My education leading up to that point had not covered this notation, and it felt like I was learning a second language in addition to the mathematical concepts. The notions of formal logic are not complicated, but earlier exposure to these ideas would have made me more prepared for college mathematics.

In my opinion, the K-8 standards are too focused on number and computation. The objectives covered in the K-8 curriculum reflect a notion of numbers consistent with the view of mathematics in the early 1800s. In 1889, when Peano's Axioms were introduced, the mathematical notion of a “natural number” changed. The “natural numbers” have been redefined in terms of set theory since the 1900s. Students need to have a concept of numbers that are consistent with the modern set theoretic constructions. The pen-and-paper computations covered in elementary school are valuable, but have in many instances been replaced by technology. It's not enough for students to know how to perform addition, but the modern student needs to also know that the operation of addition is analogous with joining sets. In mathematics, number theory is built upon the foundation of set theory. Focusing on numbers before sets is illogical considering the hierarchy of mathematical knowledge.

Set theory is also based on logic, which is also missing in action. Several of the objectives mention making logical conclusions about problems, but where's the logic? The mathematical definitions of “and”, “or”, “xor”, “not”, “nor”, “nand”, “implies”, “for every”, “there exists” and “proves” are absent from the standards. The relationship between the basic operations of logic and those of arithmetic needs to be thoroughly established. This is not only important from a mathematical standpoint, but it is essential to learning how computational technology works. The operations of arithmetic can be build from the basic building blocks of logic, and that is how computers manage to produce the calculations that they do. All students will work with technology in some manner or another, and developing an understanding of how that technology works will make them more effective users of technology.

The focus of the K-8 curriculum is to develop students' understanding formal axiomatic systems. Mathematics should be presented in the form a game. The rules of the game determine patterns that are produced in the process. Too much of a focus on the outcomes underemphasizes the importance of the rules to begin with. Algebra, in particular, requires students to rewrite expressions using the properties of numbers. The failure of the curriculum is that students have no prior experience with substitution systems. The algebra student is essentially thrown into the game without knowing the rules. It's no wonder that algebra presents such a challenge for mathematics eduction.

In the High School Standards, my main objection is to the separation between the general curriculum and the STEM curriculum. Most of the objectives labeled as STEM objectives, should be included in the general curriculum. STEM students need a more thorough picture of mathematics than that presented here. As an example, complex numbers are treated as a STEM only topic in the standards. The Fundamental Theorem of Algebra depends on the field of complex numbers and all students should be exposed to this result. Students preparing for STEM need to go beyond complex numbers to constructions such as dual numbers and quaternions which would help prepare students to acquire a more general notion of Clifford algebras in college. These tools may seem obscure to traditional educators, but are essential tools to physicists and engineers. Quaternions have several useful properties that make them ideal for modeling rotations, and dual quaternions can be used to represent rigid transformations.

One of my pet peeves about high school mathematics is that the picture of physics presented there was radically different than the picture presented in college. As an example, consider the following algebra problem: “Joe and Sue live 10 miles apart. Joe heads towards Sue's house at 5mph and Sue heads towards Joe's house at 3mph. If they both leave at the same time, how long until they meet?” Questions like this are usually represented as linear equations, like “3x+5x=10”. However, this gives students the false impression that this is an accurate model of velocity in the real world. The fact of the matter is that this equation is only reasonable for small numbers. A change in the numbers included in this problem could invalidate the model. Consider the modified question, “Joe and Sue live 10 light-years apart. Joe heads towards Sue's house at .8c and Sue heads towards Joe's house at .9c. If they both leave at the same time, how long until they meet?” Under these figures, a linear model of time is no longer accurate. Time slows for Joe and Sue relative to a stationary observer, and the question of “how long until they meet” is more complicated than it appears on the surface. If they each start a stopwatch at the moment of departure, their clocks will have different readings when they meet. Students in a STEM course of study need to understand that velocity in space-time is fixed at the speed of light, and what we perceive as motion is a rotation of this space-time vector.

There are several other topics that I consider important for STEM students that go beyond the standards. The study of triangles in geometry needs to extend to ordered triangles, and the linear algebra needed to manipulate such triangles. The notion of “angle” as presented in the high school curriculum is insufficient. Students need to start thinking about angles as they relate to the dot product (inner product) of vectors. The trigonometric functions of sine and cosine need to be connected to complex exponents by way of Euler's formula: “eix = cos x+i sin x”. The properties of logarithms also need to be explicitly covered in the curriculum. Covering these notions in the high school curriculum would make students better prepared for subsequent studies of calculus and geometry.

Some of these suggestions may seem like obscure areas of mathematics, but my argument is that they shouldn't be. If the purpose of K-8 is to develop an understanding of what formal axiomatic systems are, then the focus of 9-12 should be on discovering the useful properties that result from the standardly accepted axioms. I once conducted a job interview where a mathematics student, like me, was applying for a software engineering position, also like me. My employer had brought me into the interview to determine whether or not the candidate was ready to apply his mathematically expertise to computer programming. During the interview, the candidate mentioned quaternions as one of his areas of interest. In the software developed at this company, orientations and rotations are routinely stored and manipulated as quaternions. When I asked how the candidate would use quaternions to compute a rotation, he was stumped. He also became extremely interested in the interview at that point and was eager to learn more about the technique. My question had revealed that the abstract mathematics he was familiar with had a real purpose behind it – a practical use within the field of computer science. It's this kind of disconnection between abstract and applied applied mathematics that seems to be one of the major problems with mathematics education.

Abstraction plays a large role in mathematics and it's usually the use of mathematics in other disciplines which connects it to students' real world experiences. I was fortunate enough to have learned most of my mathematical knowledge in the context of computer science. Mathematics and computer science share a good deal of common ground. Furthermore, working with computers has become an essential skill for career-readiness in modern times. Learning how technology works adds to ones ability to effectively use that technology. When the Standards for Practice call for students to use computational tools proficiently, the lack of standards addressing how that technology works will hinder the obtainment of that goal. When students use computers or calculators to produce computations, they need to know that they are not working with “real numbers”. Not all real numbers are computable. Until the mathematics curriculum prepares students to tackle such notions, students will not be college or career ready.

One of the considerations for the Core Standards is that they address 21st century skills. Reading the mathematics standards, I do not think that this consideration has been met. The mathematics included in this curriculum is dated and fails to address the advances made in the past century. The Core Standards for math focus on applying known algorithms to problems with known solutions. A 21st century education needs to focus on creating and analyzing algorithms. Students not only need to know algorithms for solving mathematics problems, but need to be able to think critically about the efficiency of those algorithms. The Core Standards are a step in the wrong direction in this regard. All these standards will accomplish is that mathematics education will be catered to address the specific problems covered by the standards. Teachers will teach to the test and the critical thinking skills hinted at in the Standards for Practice will be lost in the assessment process.

I appreciate that these standards are chosen based on evidence from educational research. However, I think that the research supporting these standards is biased by the currently existing assessments. The evidence showing that American students are behind in math means there is still a gap between the standards in place and the skills students need to be college and career ready. A fixed set of national standards is not a viable solution to the problem. What the education system needs is a solid framework for an experimentation cycle in which standards are continually tested and revised to meet the changing needs of students.

Denialist Misrepresentations of Math and Evolution

This is so me right now.

I generally try to avoid flamebait, but I saw this article linked off of Twitter.  I should have stopped reading after the first section where it's clear that the author is a troll.    Evolution and science denialism aside, the misrepresentation of mathematics in the article is inexcusable.

After attacking Darwin and scientific thought in general, an appeal to emotion, he proceeds into a second hand quote from a philosopher on the subject of "fallacies".  It's kind of ironic that the inclusion of this quote would serve as an appeal to authority.

Next, he goes into intelligent design saying:

we could find incontrovertible evidence that reality, matter, life, has been designed, but that interpretation of the evidence would be discarded because naturalism dictates the exclusion of anything which might lead outside of a naturalistic explanation.

This is absolutely false.  Scientific theories are necessarily falsifiable.  If the evidence implied a "design", that's what the scientific theory would be.  The fact is that the evidence points to the contrary.  Biology shows a picture of  "unintelligent design", consistent with a process of genetic mutations occurring over time.  The naturalistic explanation is the one that the evidence supports.

Then he claims that Gödel's Incompleteness Thereom proves this.

He managed to get Gödel's basic background information right, but incorrectly describes the Incompleteness Theorem.

From the article:

  1. Any system that is consistent is incomplete.
  2. The consistency of axioms (axioms=assumptions that cannot be proven) cannot be proved from within the system.

The real Incompleteness Theorems:

  1. Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.  In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250).
  2. For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Notice how the part about "basic arithmetic" is conveniently left out of the definition?  That's because the author doesn't want you to know that there can exist axiomatic systems which are both complete and consistent.  First-order predicate logic was proven to be both complete and consistent by none other than Gödel himself.  Furthermore, saying that the Incompleteness Theorem "utterly [destroyed] atheist Bertrand Russell’s logical system for arithmetic" doesn't give Russell the credit he deserves.  Gödel's technique was based on the same idea as Russell's Paradox to begin with.  Despite its incompleteness, the development of Russell's work into Zermelo-Fraenkel set theory was an important building block in the foundation of later mathematics.  By referring to him as "atheist Bertrand Russell", it's clear that the author is more concerned about religion than the actual mathematics.

Next we have a very weak analogy.  He describes three items on a table and says:

Now draw a circle around those items.  We will call everything in that circle a system.  Gödel’s first theorem tells us that nothing in the circle can explain itself without referring to something outside the circle.

It's true that Gödel's theorem succeeded in "stepping out of basic arithmetic", but here's where that omitted condition of a "formal system capable of basic arithmetic" comes into play.   Are a half-full cup of coffee, a fishing pole and a jacket capable of arithmetic?  If the answer is no, then Gödel's theorem doesn't apply.  Capable of self reference?  Maybe if the coffee mug says "I'm a half full cup of coffee" on it.

The analogy of a computer is a much better example.  Computer programs are capable of basic arithmetic.  What Gödel's theorem implies for computers is that there exist certain programs which are computationally irreducible.  The only way to determine the output of such a program is to run it.   If we think of Nature like a computer program, the only way to be certain of the future "output" is to let Nature run its course.   This result does not prevent science from making conjectures about the structure of  Nature, but requires that science adopt a Black-box testing procedure which entails experimentation and observation.  There are certainly unanswerable questions in science, such as the precise position and momentum of elementary particles, but evolution isn't one of them.   The evidence for evolution is incontrovertible.

The final second shift the analogy to the universe and the claim is that what's outside the universe is unknowable.  Just because we can't see what's outside the universe, which would be white-box testing, doesn't mean we can make and test hypotheses about it as a "black-box".  The Many-worlds interpretation of quantum theory is one such example which predicts that our universe is but one of many possible universes.  Similarly, M-theory predicts the existence of hidden dimensions beyond space and time.  Just because some questions are unanswerable, doesn't mean all questions are.

The article ends by claiming that evolution and naturalism are "fallaciously circular", but here's the real circular fallacy:

  1. Author misinterprets Gödel's theorem to imply that all axiomatic systems are incomplete or inconsistent.
  2. Author mistakenly assumes that science is an axiomatic system.
  3. Based on this misinterpretation, author concludes that science must be incomplete or inconsistent.
  4. Since author concludes that complete scientific knowledge is incomplete or inconsistent, author ceases to look for empirical evidence of scientific claims.
  5. Since author ceases to look for evidence, author does not find any evidence.
  6. Since author does not find any evidence, author concludes that scientific knowledge is incomplete.
  7. As a consequence, the author's incomplete knowledge becomes a self-fulfilling prophesy.

This whole article is a Proof by Intimidation.   The "average Joe" doesn't know enough about contemporary math and science to go through and verify each detail.  The use of mathematics vocabulary in the article is deliberately being used to distract the reader from the real issue -- the overwhelming evidence for evolution.   The references to Gödel's Incompleteness Theorem are nothing more than a red herring, and the author even misstates the theorem to boot.

Guild Wars Solo Me/D

So I noticed a new comment on my old Guild Wars Me/D video, and it enticed me to play again.  Unfortunately, the build in the video had been completely changed in a recent patch.  Sand Shards no longer deals damage on misses, but instead causes a AoE DoT effect when it ends.  Whereas the old version pumped out a lot of damage if there were multiple foes to miss, the new version has lower damage output and causes the AI to scatter even easier.  Thus, I set out to see if I could make a new solo build with the same profession combination.

The survivability of the old build was still pretty much intact, but I no longer needed the self-blind of Signet of Midnight due to the change in Sand Shards functionality.  I went over to the Zaishen arena and started to play around.  What I found was that I could keep myself alive without blind by using a combination of Mystic Regeneration, Mirage Cloak and Armor of Sanctity.  I changed the elite skill to Signet of Illusions and reallocated my stat points.  This opened up a variety of other options for damage.  After some experimentation, I was able to down IWAY with the following combination:

Signet of Illusions - makes all skills use Illusion attribute (16)

Mystic Regeneration, Mirage Cloak. Armor of Sanctity - keep me alive

Channeling - energy management

Mystic Twister, Dust Cloak - additional damage

Faithful Intervention - just another enchantment

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With the proof of concept working, I decided to give it a whirl in PvE.  I went to my old Sunspear rep farming spot, East out of Camp Hojanu in Barbarous Shore.  Right outside the town are several packs of Heket which are mostly physical damage.  Since the packs were only 4 instead of 8 like IWAY, Channeling wasn't return quite enough energy to cover the enchantment maintenance.  I switched out Faithful Intervention for Auspicious Incantation for more energy.  With the energy problem solved, I didn't really need the blind effect on Dust Cloak anymore and switched it our Heart of Holy Flame to add burning instead.  The only problem with this farming spot are the Blue Tongue Heket monks that spawn randomly in the melee packs.  I had some success using Backfire on them, but they still could take a while to kill.  I decided it would be better to just avoid them altogether.

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Having Signet of Illusions leaves plenty of room for variation in the build, as it can use any skills with an effective attribute of 16.  I found that having Armor of Sanctity up was more important than Mirage Cloak in most cases.  The damage from letting Mirage Cloak drop and the synergy with Auspicious Incantation made it worth keeping, but it might be possible to do without it in some locations.  A Dervish primary could probably use a similar setup and might be able to get away without the energy management skills needed on my Mesmer.  Although Fast Casting is kind of nice against the Heket because they like to interrupt.  Not bad for a proof of concept, but more damage and a speed boost would be nice.

Anyways, good luck and happy hunting!

Bleach Bicameralism

This article is just for fun and is not targeted toward an audience unfamiliar with the Bleach series. However, if you're a fan of Tite Kubo's Bleach and have never heard Julian Jaynes and the Bicameral Mind I'm hoping that this will provide an entertaining introduction to this daring psychological hypothesis.

[spoiler alert!]

I originally saw the first episode of Bleach on Cartoon Network and have been delightfully following the series ever since. It's about a orange-haired teenager named Ichigo, who becomes a Shinigami, which roughly translates as a “death god”-- like the Grim Reaper in western tradition-- which sends departed souls to the afterlife in “Soul Society”. Some of these lost souls turn into “Hollow”, evil spirits which accumulate power by consuming other lost souls and occasionally will turn to attack humans. This serves as a never ending source of conflict for the wide cast of Shinigami to fight off evil in extravagant action sequences. Each of the colorful characters is complemented with a unique weapon called a Zanpakutou which would is considered to be a manifestation of the wielder's soul in a sword.

This relationship between the Shinigami and the Zanpakutou has several qualities about it that remind me of Julian Jaynes's Bicameral Mind. The Shinigami are portrayed as conscious actors, in a Jaynesian sense, while the Zanpakutou represent their unconscious instincts to fight and kill. A recurring theme in the series is that Ichigo's instincts tend to take over in times of severe distress, but he gradually improves at harnessing the Zanpakutou consciously to control the amount of devastation unleashed. The universe of Bleach is one of fiction, but much like Jaynes considers language of the Iliad as a metaphor for the mind of the ancient Greeks, might modern fiction also serve as a metaphor for modern social perceptions of consciousness? I'm going to focus primarily on the bicameral nature of the Shinigami-Zanpakutou relationship, but I'd note that Ichigo represents a slightly more complex model that still has the potential to revert to this bicameral state.

The first thing to note is that Jaynes' model of consciousness is not the same as awareness, as it is commonly used in language but rather refers to something a bit more technical. There are four key features of Jaynes Consciousness (J-Con): (1) an analog “I”, (2) a metaphor “me”, (3) inner narrative, and (4) introspective mind-space. These four features enable an individual to “test” potential behaviors in the mind-space before trying them out in the real world. In contrast, an “unconscious” being acts instinctively and is immediately focused on the “here and now”. The reason I think Bleach is a great example of J-Con is because Ichigo's Hollow form personifies the “unconscious” mind and poses a stark contrast to the behavior of Ichigo while he is “conscious”.

Ichigo's consciousness normally resides in his human body, but when he becomes a Shinigami, his consciousness separates from his physical body. His analog “I” and metaphor “me” are manifested in his Shinigami form. Shinigami can influence their environment, including damage and destruction, and can be also be influenced by their environment, including injury and death. Ichigo is often portrayed narrating fights, consciously breaking apart his opponents fighting style. When Ichigo's Hollow takes over, he doesn't bother so much with reading his opponent. He just attacks relentlessly with no concern for how much damage is caused. Ichigo's conscious mind strives to suppress and control this instinct, so that he may uses its power to protect his friends. Ichigo's internal mind-space is depicted visually at various points in the series. Ichigo's world resembles a sideways metropolis. In one of my favorite episodes, Ichigo literally fights against his Hollow self within this inner world.

Now that I've established what J-Con is, the next thing I need to define is the bicameral mind. Jaynes argues that prior to the development of J-Con, human beings behaved according to auditory hallucinations originating from the right hemisphere of the brain which commanded them to act. These hallucinations were often perceived to be the voices of “gods” or “ancestors”, and commanded the individual to act. This mode of thinking is very similar to the behaviors of schizophrenics in modern times. In hypnosis, the analog “I” gives up its power to an outside authority and the body follows this sources command. In the schizophrenia and bicameral mind, this authority is a hallucination.

In Bleach, the Zanpaktou often calls out to its Shinigami master through dreams. In the case of Captain Hitsugaya, he had a recurring dream of an icy dragon calling out to him, but he could not hear its name. When he finally heard its name, that's when he became a Shinigami. The Zanpaktou is often portrayed as its own person, but resides within the soul of its Shinigami. Shinigami become more powerful by communicating with the Zanpaktou. When Shinigami and Zanpaktou fight as one, they come closest to meeting their full potential.

While not part of the manga, episode 255 of the anime involves a fight between Ichigo and Muramasa, a Zanpaktou with powers of hypnosis. Zangetsu, Ichigo's Zanpaktou, speaks to him:

“Ichigo Feel him” “Zangetsu?” “His hypnosis no longer works on me. I shall be your eyes. But for this to work, we must truly communicate with one another as master and Zanpakutou.” “I understand, old man” (Bleach ep255)

The fight with Muramasa starts to turn around. When Ichigo gains the upper hand, he confesses the change to Muramasa:

“I finally understand what he's been trying to tell me.” “What?” “We have to acknowledge each other's existence and accept one another. That's how Zanpakutou and Shinigami are supposed to interact.” (Bleach ep255)

I feel that this communication between Shinigami and Zanpaktou is much like the bicameral state of mind described by Jaynes. When communicating with Zangetsu, Ichigo does not descend entirely into instinctive behavior, as he does in Hollow form, but rather becomes aware of these instincts and uses them to obtain his goals. Much like bicameral humans followed the commands of auditory hallucinations, Ichigo enters a state of mind where Zangetsu dictates his actions. The wall separating Zangetsu from Ichigo's analog “I” dissolves to a point where the two parts of his mind act as one. In fights such as the one with Muramasa, the part that is Zangetsu dictates the behavior while the part that is Ichigo listens and obeys. This bicameral state is where Ichigo's power is greatest.

In the Origin of Consciousness, Jaynes finds support for this theory in the language of the Illiad. In the Illiad, the gods dictated the behavior of the actors. In contrast, the Odyssey presents actors which behave on their own accord. Jaynes argues that this change reflects the development of J-Con taking place in that period. If anything is to be learned from Bleach, it's that modern culture acknowledges both modes of thinking. While humans generally exhibit behavior consistent with J-Con, the bicameral state is still partially accessible to the mind. As human beings, we need to accept that we have certain instincts. Consciousness provides us with the power to observe these instincts, and choose when and how they manifest themselves.

Ichigo's story suggests that although humans are still capable of this bicameral state, there are risks associated with entering it. Ichigo's Hollow self and Zanpaktou are closely related. In relying on his Zanpaktou's powers, Ichigo runs the risk of his instincts taking over. While Ichigo obtains power by descending into a bicameral-like state, he needs to make sure that he doesn't completely relinquish conscious control over his actions. Hollow Ichigo says that he is the “horse” and Ichigo is the “king”, but if Ichigo were to let his guard down he will be quick to “take the crown” (Bleach manga 221). In essence, the Hollow Ichigo represents what would happen if Ichigo descended completely into bicameralism. In bicameral individuals, the hallucination is the “king” and the self is the “horse”.