Posted on
by

(Note: I also created an interactive version of this post in the Desmos Activity Builder. Try it here.)

Hey y’all!

I want to tell you a story today about racism and mathematics.

Well, to be more specific, I’m going to make the argument that trends in mathematical notation can have culturally biased consequences that conferred systematic advantages to white people. I’m going to make this argument through a sequence of math problems, so I hope you’ll follow along and attempt them.

I don’t know if I’ll be able to “prove” this argument to you in the course of this activity, but hope this activity at least instills a sense of “doubt” in the idea that math is objectively neutral.

Without further ado, let’s get started.

Place a mark on the number line where you think “1 Thousand” should go.

Got it?

Okay, I’m going to make a guess as to where you placed it.

Ready?

Did you place it at point P below?

Sorry, I hate to be the one to break it to you, but according to the ‘math powers that be’, this is technically incorrect. However, I want you to hold onto this idea because I’d argue that you’re not as wrong about this as they say you are.

Here’s where you “should” have placed it.

When mathematicians present you with a graph, we implicitly assume that the graph is on a “linear scale”. That means that each unit is equally spaced along the line.

One billion divided by one thousand is one million, which means that “1 thousand” should be placed “1 millionth” of the way between 0 and 1 billion.

At the scale of this graph, this number is so close to the 0 that they’re visually indistinct.

So what about that other point?

That’s actually placed at the value 1/3*10^9 or “one third of a billion”.

However, I don’t want you to think you’re wrong if you placed it here. In fact, this is the correct placement of 1000 on another type of scale.

Check out the following scale instead:

So what would you call a scale like this?

Take a second and write it down.

Go ahead. I’ll wait.

According to the “mathematical community”, this is called a “logarithmic scale”.

Under a logarithmic scale, each successive “unit” gets multiplied by a common factor, such as 10, rather than adding. This results in a scale like in the diagram, only the “0” is actually 10^0 = 1.

There’s even research suggesting that human may actually be predisposed to think of numbers this way, and that the linear scale is a learned behavior. We see it in Indigenous cultures, in children, and even other species. Thinking “linearly” is a social norm that is distributed through cultural indoctrination. It’s like mathematical colonialism.

Here’s my problem with that diagram. If I hadn’t been taught to call this a “logarithmic scale”, I would have called it a “power scale” or an “exponential scale”. Doesn’t that name intuitively make more sense when you see this sequence of labels?

Before we go any further, let’s talk about mathematical notation.

What do you think makes for “good mathematical notation”?

What sort of patterns do you see in the ways that mathematicians name things the way they do?

Here’s what I perceive as the primary patterns in mathematical notation.

Mathematicians sometimes name things “descriptively”: the name tells you what the thing is. For example, the “triangle sum theorem” tells you implicitly that you’re going to add up three angles.

Mathematicians sometimes name things “attributively”: the name tells you who came up with the idea. For example, “Boolean Algebra” is named after “George Boole”.

Mathematicians sometimes name things “analogously”: using symbols with visual similarities to convey structural similarities. For example, “∧ is to ∨ as ∩ is to ∪”.

Mathematicians sometimes name things completely “arbitrarily” for historical reasons. Don’t even get me started on the “Pythagorean Theorem”…

Where would you classify “logarithm” in this scheme?

Personally, I think it depends on who you ask.

If you know a little Greek (which is a cultural bias in itself), you might argue that this label is “descriptive”. The word “logarithm” basically translates to “ratio-number”. The numbers in this sequence are arguably in “ratios” of 10, but does that actually convey enough information for you to know what logarithms really are?

I’d argue that this label is in fact “arbitrary” and actually refers to “how logarithms were used” rather than “what logarithms are”.

So what is a logarithm?

Generally speaking, we define a “logarithm” as the “inverse power function” or “inverse exponential function”.

Just as subtraction undoes addition, and division undoes multiplication, a logarithm undoes a power.

For example, 10^5 = 10000 so log10(10000) = 5.

Though a little clumsy, maybe this will make way more sense if we just used the following notation:

log10(10000) = power?10(10000) = 5

The natural inverse of “raising something to a power” would be “lowering it” it right? The mathematical statement, log10(10000) says this:

“What power of 10 will give you the number 10000?”

The answer to that question is 5. This is the essence of a logarithm.

So why don’t we just call logarithms “inverse exponents”?

Well, we call them “logarithms” because this was the term popularized by a guy named John Napier in the early 1600s.

Normally I’d be okay with the guy who created something getting to name it. However, I don’t think that honor should necessarily go to Napier. This idea of “inverse powers” had shown up in the early 800s thanks to an Indian mathematician named Virasena, and the way Napier employed logarithms was similar to an ancient Babylonian method devised even earlier than that.

What Napier did, in my opinion, was convince other white folx of the power inherent in logarithms.

Please, allow me to demonstrate with some “simple” arithmetic. Try to do these two problems without a calculator.

Problem A: 25.2 * 32.7

Problem B: 1.401+1.515

Go head, take as much time as you need.

Which problem was harder?

Problem A, right?

Here’s the mathematical brilliance of the logarithm. It turns out that this “inverse power function” can be used to take a very difficult multiplication problem and turn it into a much simpler addition problem. Problems B can be used to provide a very reasonable approximation to Problem A in a fraction of the time if you look at it through the use of logarithms. They didn’t have computers back then, so they used tables of precalculated logarithms to drastically speed up the computation of large products.

Here’s how it works.

Start by looking up the logarithms of the numbers you want to multiply:

log(25.2) ≈ 1.401

log(32.7) ≈ 1.515

Once you’ve taken the logs, add them together.

1.401+1.515 = 2.916

Finally, do a reverse look up to find the number that would produce this logarithm.

log( ? ) = 2.916 = log( 824.1 )

This reverse look-up is really the power function: 10^2.916 ≈ 824.1.

Pretty neat?

The result 824.1 is pretty darn close to the actual value of 824.04. It’s not perfect because we rounded, but it’s reasonable enough for many applications.

This idea of using logarithms to speed up calculation resulted in the invention of the “slide rule”, a device which revolutionized the world of mathematics. Well, more accurately, it revolutionized European mathematics at approximately the same time that the British Empire just happened to start colonizing the globe.

Let’s spell that timeline out a little more explicitly:

~800 CE: Indian mathematician Virasena works on this idea of “inverse powers”.

~1600 CE: John Napier rebrands this idea as a ‘logarithm’.

~1700 CE: Invention of “slide rule” using logarithms to speed up calculations helps turn Europe into an economic powerhouse.

~1800 CE: The British Empire begins colonizing India.

Do you think this is a coincidence?

I don’t.

I think European mathematicians were quite aware of the power this tool provided them. Naming this tool a “logarithm” was a way of intentionally segregating mathematical literature to prevent other cultures from understanding what logarithms really are.

When a tool provides this much computational power, the people using that tool have a strong motivation to keep that power to themselves.

It’s like the recent linguistic shift in the usage of “literally” and “figuratively”. People have used the word “literally” to describe things “figuratively” in such large numbers that they have literally changed the meaning of the word.

Logarithms have been used to describe exponential behaviors for so long that the relationship between “powers” and “inverse powers” has become blurred by mathematical convention.

This has far reaching consequences for mathematics education where we need students to understand the implications of exponential growth. This is even more important considering the recent COVID-19 pandemic.

Consider the following example from the New York Times:

If I accidentally referred to the second graph as an “exponential scale” instead of “logarithmic scale”, would you still know what I was talking about?

What’s the point of calling it that?

I think we need to acknowledge that there exist “self-reinforcing power structures” in mathematics. These mathematical tools provide power to people, so those people fight to keep that power to themselves. This is an act of “segregation”.

After time, these tools become unavoidably common place. Now we’re in a situation where mathematicians argue that these norms should be “assimilated” because their use has become so widespread.

As a result, these cultural biases have resulted in a sort of mathematical colonialism that masquerades as objectivity. I believe that this ethnocentrism systematically disenfranchises BIPOC by hiding the true history of these mathematical ideas. This, in turn, results in systematic biases in test scores between whites and BIPOC, which then reinforce the original stereotypes.

It’s a vicious cycle of racism.

Wouldn’t you agree?

It might be too late to stop the use of the word “logarithm” in mathematics. It’s now something of a necessity to understand a wealth of other mathematical advances that have been built on top of this concept. However, that doesn’t mean we should pretend that this term is completely neutral either.

I hope you’ll leave here today with a better understanding of why it’s important to look critically at our mathematical conventions and acknowledge that math is not exempt from cultural biases.

Thanks for reading!

Posted on
by

It had been years since I’d been in a live chat on Twitter (I miss you #mathchat!), but I decided that this year I was going to participate in the #ClearTheAir Spring Semester. Live chats can be intimidating. It literally took me 30 minutes to tweet a response to the first question! I panicked when I saw that I was already 15 minutes late to the second question:

ACTUAL Q: What does it mean to choose Whiteness? In what ways did your family choose it? What do you think was lost? How does Whiteness impact your life? What do you notice about it now that you didn’t before? 2/ #ClearTheAir— ClearTheAirEdu (@ClearTheAirEdu) January 9, 2020

I hastily assembled a response that, looking back, is shallow and deflects from the real question. This tweet had triggered defense mechanisms I’m only recently coming to terms with. I didn’t perceive myself as choosing whiteness, and I’m only now realizing that the fact that I didn’t need to choose whiteness in order to reap its rewards is the very definition of white privilege.

When the chat returned to this topic the following week, a thread caught my attention with the following tweet:

I think I’m trying to figure out what “woman of color” means. @ClearTheAirEdu talked about closure. I want it. some definition when I’m between languages, between latinx countries, between Americanness, lighter on one side of the family and dark on the other, like the moon.— dulce-marie is still angry. (@DulceFlecha) January 16, 2020

Even now, this tweet conjures all sorts of emotions I’m not sure how to process. They seem to stem from a childhood memory, which I believe was from my 5th grade class. We were given an assignment to learn about our heritage and bring in a food that representative of that culture with the class.

I go home and tell my parents about the assignment. My dad identified himself as mostly German with a little bit of Swedish. My mom identified herself as mostly Yugoslavian with a little bit of Native American (Note: these statements are problematic on many levels, but this is about how I understood it at the time). I asked my mom if there was some kind of Yugoslavian food I could bring in. She pauses in thought for a moment, then says that it would be a lot of work and suggests that I bring something German instead. We head to the grocery store for a loaf of pumpernickel bread and some butter. It’s my first taste of “German food”, so I was kind of excited about it.

The day of our multicultural potluck, I present my story to the class and pass out my slices of pumpernickel. As I do so, I hear someone whispering something. I only make out one word, but that one word is enough to make me feel dizzy:

“Nazi“.

Did I really hear that? Did someone really say that? How could someone possibly come to that conclusion based on me being German? I tried as best as a kid could to put these thoughts out of my mind, but no 10 year old is really equipped to handle that level of guilt.

I think this is the origin for many of the defense mechanisms that would eventually develop into my own white fragility. I might even go so far as to say this was my childhood model for race in general. My dad’s side of the family was obviously “white”, but my mom’s side of the family was “less white”. What really drove this idea home was the fact that I started to see the difference within my own family.

My dad had three children from a previous marriage, and there were noticeable differences in skin tone between them and myself. They were “white”; I was “tan”. When they stayed out in the sun too long they turned “red”; when I stayed out in the sun too long I turned “brown”. My full-brother’s skin was like mine, so it was clearly linked through our mother. Even if my mom’s story about our Native American heritage was true, I knew that it was only a tiny amount. I didn’t even know how I’d begin to verify the story, but it explained an observable phenomena in my immediate family so I didn’t have any reason not to believe it.

I knew better than to try and identify myself as “Native American”, but it made me question where to drawn the line between “white” and “not white”. I came to the conclusion that the distinction was bogus and refused to make a choice at all. In reality, distancing myself from whiteness was just a defense mechanism to avoid a collection of painful memories.

When I think about “choosing whiteness”, I think I have to break it down specifically in the context of my family. Looking back, I chose to identify with my mom along multiple vectors. My mom and I were the left-handed liberals while my dad is a right-handed conservative. Since I associated my dad with both “white” and “male”, I thought perhaps it might be helpful for me to look at how I chose to incorporate the latter into my identity. I’m not sure of the exact moment, but know there was a distinct shift in my thinking about gender sometime in my teens.

Growing up, I rejected a lot of things that would be considered traditionally masculine. I spent more time playing with “My Little Pony” than “G.I. Joe”. I prefer musicals to football. However, I didn’t really distance myself from the word “male” the same way I did with the word “white”. I think this was made possible because I had strong feminist role models who taught me to look critically at gender. By “choosing male”, I accepted the fact that I was a male and that being one gave me privileges. Rather than distancing myself from the word “male”, I owned it. I acknowledged that “men” had a history of doing terrible things and that I would take it as a personal responsibility to challenge those statistics and disrupt the patriarchy however I could.

If “choosing white” were to be the analogue of this notion for race, I think the turning point for me reading Chris Emdin’s “For White Folks Who Teach in the Hood… and the Rest of Y’all Too”. By even purchasing this book, I had acknowledged that I was “white folk” and that I needed help understanding how to “teach in the hood”. In fact, this gives me an approximate date for “choosing whiteness”: November 14th, 2016. I think this his the first time I sincerely applied the label “white” to myself and acknowledged the social implications thereof.

Just reading wasn’t enough though, and what really drove this idea home for me was Emdin’s “homework”. He had challenged me to step out of my comfort zone to visit a Black barbershop and a Black church. Something about being the only white person in a “Black space” really puts one’s whiteness into perspective. The barbershop has become a routine occurrence now. I’m still processing the experience of visiting a Black church; I’ve never felt so simultaneously out of place and welcomed at the same time.

Coming to terms with what it meant to be white was only the beginning. Emdin also made me reassess significant aspects of my sense of self-identify by using the word “neoindigenous” to describe students. He had instilled this idea that the classroom didn’t belong to me, but my students. It forced me to take a hard look at what it really meant to be “Indigenous” and come to terms with the fact that I wasn’t.

Being Native American was never a matter of “blood” as I thought when I was young. It’s a matter of tribal membership, which I clearly do not meet the definition of. Even if my family story was true, I had long been socialized into the culture of the “Colonizers”. I have no right to call myself “Native American”. This concept that the relationship between Indigenous and colonialism was completely separate from “race” forced me come face-to-face with a very uncomfortable truths about my own heritage.

Even if my family story was true (and I have little evidence to back it up), it implies that my white ancestors traveled across the country during the western expansion. These white ancestors most likely came into contact with my supposed Native American ancestors somewhere around present day Idaho. Their decedents most likely traveled south through Utah and Arizona to end up in California, while gradually assimilating into white society.

Here’s the part that I struggle with. In order for this Native American ancestry to be true, it also implies that my white ancestors were complicit in the colonization and genocide of the Indigenous people. This is a heavy burden to bear, knowing that some of my ancestors may have committed one of the greatest atrocities in American history against the Indigenous people who may also be my ancestors. I feel like my very DNA is filled with violent contradictions. Maybe that’s what it really means to “choose whiteness”.

During the #ClearTheAir Zoom with Paul Gorski, he made a remark about “embracing whiteness” being akin to “self-hate”. I felt a little uncomfortable about the statement until I started digging into the language a little deeper. First, I wasn’t trying to “embrace” whiteness. I was just trying to “understand” it and “accept” it. Second, it helped me recognize that the feelings I carried towards whiteness weren’t characterized by “hate”. I felt guilt, and I felt shame, but not “hate”.

To “choose whiteness”, for me, was to accept the all the moral contradictions in the last 200 millennia of human history. Not only am I likely to be a decedent of murderers, rapists, colonizers, slavers, and thieves, but I am also likely a decedent of their victims. I will not let the crimes of my ancestors define me, but I will acknowledge that their actions have left me a heavy burden to carry. I will prove myself better than them by fighting to undo the damage they have caused because that’s the kind of human I want to be.

I hope this answer finally does justice to the question that prompted it.

It seems strange to think that so many of my misconceptions about race could be tied back to a single childhood memory, but maybe that’s how this process of socialization works. I think I now have an obligation to find out how much of my family story is really true. I promise to continue digging and see what else I discover about myself.

I’d like to thank the following people who have both challenged me and given me inspiration on this journey (in no particular order):

Posted on
by

I have a confession I need to get off my chest. I think Geometry is racist. Well, not geometry (the discipline of mathematics), but specifically Geometry (the course commonly taught in American High Schools). I realize that in presenting this argument I’m going to have to make some rather sweeping generalizations about human history, but I that it’s important for me to put this hypothesis into words because I don’t think there’s enough evidence to disprove it.

Fair warning: this may be a lengthy read.

I believe that the reason the curriculum of Geometry specifically references Pythagoras, Euclid, and Descartes is because this emphasis provided Whites another means by which to rationalize slavery. Greek society was a prototype for White society because it used mathematics and philosophy to moralize slavery, and Descartes provided Whites a method of reconciling this position with Christianity.

To understand this, we first need to acknowledge that the entire human history of mathematics is intertwined with human slavery. Both mathematics and slavery emerged in early civilizations after the transition from hunter/gatherer to agricultural society lead to social stratification and the development of written language. In fact, I think mathematics is a necessary precondition for slavery. The very concept of slavery depends on a life being assigned a quantifiable value. Slave labor provided the ruling class with the time and resources to work on furthering mathematical knowledge, and then advances in mathematics were used to increase the efficiency of the slave trade. This vicious cycle quickly elevated the slavers to a god-king status which they then exploited to moralize their slavery.

Our oldest mathematical texts are from Egypt and Babylon, approximately 4000 years ago. Surviving records such as Plimpton 322, the Moscow Mathematical Papyrus, and the Rhind Mathematical Papyrus show that teachers have been giving math worksheets of roughly high school level difficulty since at least 1900 BC. What’s interesting about these three texts is that all three of them show evidence of the Pythagorean Theorem — over a thousand years before Pythagoras was even born! There’s also a strange lack of evidence attributing the theorem to Pythagoras, along with indications that proofs of it may have been circulating in Mesopotamia and India several hundred years prior. The very fact that this theorem bears the name of Pythagoras despite this questionable lack of evidence should alone be sufficient evidence of Geometry’s racism, but Pythagoras is only the first of three dominoes.

When Pythagoras supposedly lived around 500 BC, the Greeks (and Romans) had lively slave trades. However, unlike the god-king theocracies of Babylons and Egypt, the Greeks fashioned themselves as a “Democracy”. The Greeks needed an alternative means of moralizing slavery and whether he intended it or not, Pythagoras’ merger of mathematics and philosophy provided them with all the justification they needed. The Greeks used philosophy and mathematics to separate themselves from the people they enslaved, which served as a prototype for White society. Those that didn’t measure up to Greece’s intellectual standards were labeled as “barbarians” and treated as less than human. To paraphrase Plato, these barbarians were in the dark and needed to be shown the light. Incidentally, the “barbarian tribes” of Northern Europe enslaved by Greeks and Romans were likely a source of genetic markers for light skin, light hair and light eyes that would later be associated with “Whiteness”.

Meanwhile, mathematics probably continued to develop in the Middle East with influence from early civilizations in India and China. However, we don’t know by exactly how much. There’s another strange lack of mathematical records from Babylonian society under the Persian Empire. What we do know is that when Alexander the Great conquered the region in 331 BC, the Greeks assimilated a wealth of astronomical data. It’s quite likely that the Greeks stole whatever mathematical data they could through military conquests and these materials ended up in the Library of Alexandria where Euclid would have had access to them. Euclid’s work is based on a set of stolen mathematical literature that Euclid was privileged to have access to. While it’s impossible to discount the probability that Euclid might have added some original ideas, I do think this fact merits taking a closer look at his work and evaluating whether or not it belongs in our school’s curriculum.

My problem with the presentation of Euclid in Geometry (the course) is the parallel postulate. To be fair, presenting Euclid’s axioms as fundamental truths about the universe made sense in mathematics education up until as recently as 1905. We now know that space is not “flat” as Euclid assumed, but that gravitational forces can bend space around a mass. My understanding is that we teach Euclidean geometry because (a) it’s a useful approximation of reality and (b) it provides an introduction to axiomatic systems. The main issue with Euclid as presented in Geometry is that its axioms are presented without proof and we have solid scientific evidence that one is not always true. How can we ever expect students to think critically about information when we don’t question the assumptions made in mathematics?

Euclid’s mathematics is not irreplaceable within the mathematics curriculum. We could just as easily teach students about axiomatic systems using first-order logic. Instead of teaching compass and straightedge constructions, we could teach origami. In fact, teaching these alternatives could not only increase the diversity of the mathematics but simultaneously increase the rigor. There is some really great math showing that we can solve problems with paper folding that are impossible to do with a compass and straightedge only. Learning origami would help prepare students for high level physics and computer science. We owe it to our students to do better than Euclid.

Approximately 300 years after Euclid, the Roman Empire was nearing its peak and Christ was born. The death of Christ coincides with the start of two centuries of peace. It’s no wonder that early people considered this a miracle. Early Christians were originally persecuted in the Roman Empire, but by 380 AD it would become the state religion. If you think about it, Christianity offered the Romans a pretty sweet deal considering their involvement in the religion’s origin. The Romans killed Jesus, and through Jesus’ death were forgiven all their sins. And that meant all their sins: slavery included. To suggest that otherwise would be heresy. Before long, Christianity had accumulated so much Power that they rewrote the entire calendar around their own beliefs.

The spread of Christianity throughout Europe is relevant to our discussion because the next major event in mathematical history happens to coincide closely with the birth of Islam. Around 800 AD, Muhammad ibn Musa al-Khwarizmi developed the branch of mathematics we now call “Algebra” and started a new mathematical revolution in the Middle East. This presented Christianity with a threat because geometry was always considered sacred and the followers of Islam were creating new mathematics for another god. The bible says “Thou shalt have no other gods before me“. By 1100 AD, Christianity had come to the conclusion that since they couldn’t disprove the Muslims mathematically, the only other option was through force. The Christians took to war against the Muslims and with each mosque they burned they destroyed part of that mathematical progress. Remnants of this “Holy War” persist through the present day.

It’s because of this religious tension between Muslims and Christians that our story leads to Rene Descartes. Another essential component of the Geometry curriculum is the idea of Cartesian geometry which is an merger of algebraic principles with Euclidean geometry. The very name would have you believe that Descartes invented it, but that honor should probably be attributed to Fermat. No, I think there’s another reason we attribute it to Descartes, and that is because of his work in ontology. Most people are familiar with “I think, therefore I am“, but that statement is really the first axiom in a system that Descartes constructed in order to develop a mathematical argument proving the existence of God. Descartes is one of a few mathematicians that attracts cross-curricular study because of this. We don’t even provide his argument a fair critique. Like his Greek counterparts, we treat Descartes’ work as gospel when in reality he was just repackaging the ideas of others. In this case, Descartes had taken the Muslim-sounding “algebra” and re-packaged it in form that would be more acceptable for Christians.

The philosophical assertion “I think, therefore I am” takes on another meaning when we put it in context of mid-1600s White society. By associating “thinking” with “existing”, Whites were able deny Blacks basic human rights by labeling them as “unthinking savages”. It’s the very same strategy employed by the Greeks to justify slavery, only the vocabulary had shifted. The same idea would later manifest itself through literacy tests, and persists today through standardized testing practices. Whites retain Power by defining a culturally biased set of standards.

A few weeks ago I listened to a podcast from Freakonomics that described America’s Math Curriculum as a “Geometry Sandwich”. Ever since I taught it I’ve been asking myself the same kinds of questions:

  • Why “sandwich” Geometry in the middle of two years of Algebra?
  • Why are Euclid, Pythagoras and Descartes the only three names that students explicitly learn in Geometry?
  • Why is the name al-Khwarizmi not given the same level of treatment in Algebra?

The most probable explanation I can come up with to answer all three of these questions is that the American Math Curriculum is racist. It is probably not the intent of the Curriculum developers to be racist, but racism is not about the intent — it’s about the consequences of the behavior. There is a non-zero probability that cultural biases exist within our Math Curriculum and they produce racist consequences. Mathematics is a just tool, but people decide how that tool is used. We can either continue to use that tool to enable racism, or we can use that tool identify and dismantle it.

For me, I’ve decided that the probability of the Math Curriculum being racist is too high for me to ignore. I’ve decided on the following course of action:

  • I plan to start substituting “Right Triangle Theorem” in place of “Pythagorean Theorem”.
  • I plan to start substituting “Rectangular Coordinate System” in place of “Cartesian Coordinate System”.
  • I plan to pay closer attention to situations where Euclidean geometry is implicitly assumed and question whether or not this assumption is warranted.
  • I plan to make a consistent effort to present a diverse range of mathematicians in my classrooms.
  • I plan to actively look for other artifacts of racism in the curriculum and address them accordingly.

I realize that my argument here needs work and that I’m personally having difficulty disentangling race and religion. Even if I haven’t yet convinced you that “Geometry is racist”, I think there’s sufficient evidence here to conclude that it’s definitely not anti-racist. That fact alone is reason enough for me to act. If you’re also a teacher of mathematics, I hope I’ve convinced you enough to join me in these actions.

Posted on
by

Recently I’ve read a couple of books (namely, White Fragility and How to be an Antiracist) that have made me reexamine certain aspects of myself through a lens of racial privilege.

For a significant period of my life, I refused to identify my race as “White” on any survey I completed. I’ve since realized that my doing so was an act of racism and I apologize. While I can’t change the past, I hope that sharing my story will serve as a token of my promise to make amends in the future.

The new information which led me to this conclusion was this idea that the white race begins with slavery. I had previously defined “White” as a ~2000 year old construct when in reality it began ~400 years ago. This redefinition caused all sorts of cognitive dissonance until I learned about a defensive mechanism that white people often have called channel-switching, where we redirect discussions about racism to other factors. I discovered I had been subconsciously conflating the “White Race” with “Christianity” because I identified them both with the same structure of “White Power”. I had falsely assumed that since “White Power” was inherently “Christian” that the “White Race” through which “White Power” manifested was also “Christian”.

What bothers me most was that I knew that race and religion were not the same, and was careful not to invoke my atheism as a minority defense. I’d even openly identify as “Caucasian”, but the term “White” triggered in me a storm of rage and “Decline to state” was the calm. I thought that distancing myself from that label made me “not racist”, but I was wrong. The only way to engage in antiracism is through accurate statistical measurements of racial disparities. I wasn’t thinking about the potential harm that mislabeling myself could do, placing my own individuality over the welfare of others, and I’m sorry.

I’m going to try to continue down this path of antiracism, but I need help. I’m trying to look at this as a data analysis problem and realized that I don’t have enough information to properly disaggregate race from religion. I don’t know how to authentically engage in antiracism without also being antireligious. This presents a problem because I’m “in the closet” at work. I carefully avoid revealing my thoughts on religion because I fear that doing so could get me fired. In order to learn how to navigate this space, I need more information about this intersection of race-religion.

The first step I can take on my own. Quite frankly, I need to learn about how and why so many blacks adopted the religion of their oppressor. I believe that the best way for me to acquire this knowledge is through the narratives of black thinkers who are critical of the role religion plays in the white power structure. I plan to seek out black atheists, listen to their stories, and lend whatever weight I can to their voices. I’ve known for a long time that black atheists were underrepresented in the atheist community and it’s time I did something about it.

The second step is going to require feedback. I accept the premise that there is a non-zero probability with which I will commit acts of racism in the future. I need friends of color to call me out when this happens and engage me in an honest discussion about why. If you do this publicly, I will make my best effort to model antiracist behavior in response, but do so with the understanding that my response will likely be influenced by my antireligious views.

If you are reading this and are someone I work with, I hope you can understand the thin line I’m attempting to walk. I propose we establish a hidden signal: invoke each others’ first names when calling out acts of privilege (in whatever form that may take). This will serve as a reminder that what follows is being said as a friend and that we’re both together in this fight for equality as human beings.

Posted on
by

It’s the last day of my summer break, so I thought I’d take some time to reflect on my live coding experiment.   After all, this was meant to be a learning experience and self-reflection is an integral part of the learning cycle.

How’d I do? 

I ended up streaming for approximately 16 hours over 8 different sessions.  I then edited the live footage down to about 95 minutes of time-lapsed video.   Truth be told, I had expected to stream more often but kind of got side-tracked after going to see the Overwatch League Grand Finals live in Brooklyn.  I know it might not seem relevant now, but I’ll come back to that in a bit.

Was it “fun”?

It had it’s fun moments, like hearing my first composition come together,  hearing my first randomized chord progressions, and remixing a recorded sample.  There were some moments of genuine excitement recorded here!  It’s amazing when things finally come together to produce something new and different.  However, there were also hours of work going into each of those moments.  Around this moment is where I realized that my “mathematical function” approach wasn’t going to be fast enough in Python to do what I wanted in real time.  I even ended up spending an entire episode just debugging my previous work.  I’m also still kind of uncomfortable with the whole camera situation, so I’m doing all this is a state of perpetual social anxiety.  That part of the experience was slightly less fun…  Overall, I’d say it was about 20%”fun” to 80% “work” and that’s probably what kept me from a more active streaming schedule.

What did I learn?

The first observation I made is that I have a lot of little sub-conscious behaviors I do when I’m deep in thought.   There were lots of “umm”s, whistling, tapping, and beard stroking that I didn’t really notice until I was going back through and editing the videos.  It makes me wonder how often I do these things in the classroom.  I’m pretty sure I’ve been called out on the beard stroking by students before — it helps me think!  I also found it difficult to narrate my thought processes while in this state.  It’s like I needed to separate the act of thinking about the code from the act of describing it verbally.

Math-wise, I discovered that I had a lot to learn about digital signal processing.  I had started exploring these sounds by thinking of them using real valued functions, but eventually learned that the floating point approximations I was actually using were introducing small rounding errors which quickly added up to audible artifacts.  Fixing that issue required me to be more aware of the differences between discrete and analog signals.  I started reading up on DSP in my off-stream time and am now starting to think that there are ways to dramatically simplify some of the things I was original trying to do.  It will probably be some time before that research pays off, but at least now I know where to look.

I also realized that the approach I took to generate my randomized chord progressions is actually a well established procedure called a Markov Chain.  This is one of the amazing things about studying math — you can independently come up with an idea on how to do something only to realize that someone else came up with the same idea a hundred years before you.  Like my DSP observation, this has opened my eyes towards mountains of existing research on this approach.

How will this impact my teaching?

I think this has given me some new respect for creative assignments.  I had been so focused on aligning assignments to standards that perhaps I underestimated the amount of connections between those standards and creative activities.  Furthermore, the announcement that Desmos will be available on the upcoming SOL tests has given me a solid reason to devote substantially more time to using it in class.   I want to make “Desmos artwork assignments” a regular occurrence over the upcoming year.  My hope is that I can construct the assignments in such a way that will generate authentic curiosity about the topics we’ll be learning about in Geometry in the same way that this project inspired me.

Where do I go from here?

I’m not going to abandon the project, but I think there’s some serious “re-conceptualizing” I need to do in order to continue.  Honestly, I wouldn’t want to watch me live code based on the present quality of my stream.   There’s just too much inactive time.  If I’m going to live stream, I want to do it in a way that’s more engaging.   To do so, I think I need to talk some cues from all the Overwatch streams I watched this summer.

I’ve been watching a great deal of Kabaji’s Twitch stream, and am thoroughly impressed by his ability to narrate what he’s doing and why throughout the stream.   Not only that, but he manages to be constantly engaging the members of Twitch chat between fights.   I think those are the two things that I need to work on to make this “LiveCode” more successful in the long run.

First, I think I need to rehearse what’s going to happen in the stream before going live.  This will let me cut down on the number of bugs that I need to track down, the time I spend reading documentation, and will allow me to put more focus on narrating as I code.  Alternatively, I could separate the coding and narration aspects into two different streams — one where I write the code and one where I “cast” over it with commentary later.

Second, I need to come up with a way to involve Twitch chat in the production process.  I’ve already done a bit of research in this direction, and think it’s possible for me to connect my Python session to Twitch chat server.  This would help me from getting tunnel vision and I could use data from the chat to influence the music production.

Despite these goals, the new school year is starting soon and that means a lot less time for a side project like this.  If I do continue these live streams, it will probably be a much sparser schedule than I tried to do in July.  Follow me on Twitch for notifications when I go live!

Posted on
by

What’s this math teacher going to do to keep busy during summer break? Get back to my coding roots! That’s what!

What?

Over the course of the summer I’m going to experiment with writing music using Python and broadcast the whole adventure live on Twitch!

When?

In the “evenings”, starting around 7pm EDT.  As often as I’m able to. Follow me on Twitch for notifications or watch for updates on Twitter.  If you miss out on the live coding, don’t worry!  I’ll also be editing and uploading “abridged” versions to YouTube.

Why?

Because I think it will be fun!  And there’s math!  It’s fun with math!

As a teacher, I want get better at encouraging students to “play with math”.  However, over the course of the school year I’ve spent so much time trying to prepare students for SOL tests that I’ve forgotten what that looks like.   The only solution I could think of is for me to “play with math” again, and this is my way of doing that.

But why Python?

Several reasons actually.  First, it’s relatively easy to read.  Second, it’s been growing in popularity as a platform for scientific computing.   Third, I’m terrible at it.

Yes. That’s right.  I’m terrible at it.

I’m not exactly a Python “noob”, but it’s been about a decade since I’ve used it for anything and that was relatively simple scripting.  That’s part of the adventure here!   I’m going to make mistakes, spend hours debugging, and be knee deep in technical documentation.  Furthermore, every stupid mistake is going to be broadcast live on the Internet.  But you know what?  I’m okay with that.  It’s part of the learning process and unavoidable when “playing with math”.

How?

You want to follow along and experiment on your own?  Great!

I’m using Anaconda to manage my Python installation.   In addition to that, I also installed the sounddevice library by using the pip command from an Anaconda shell. Everything after that is all done live!  After each live code sessions, an updated source code file will be posted on GitHub.

Video Archive

After each session, I’ve been going back through the recording and editing to speed up through the boring parts.  The resulting “timelapse” videos are being posted to my YouTube channel.

 

 

 

Posted on
by

As a game developer turned teacher, the one most difficult part of the transition was paper. I’m not even talking about the thousands of copies of handouts and worksheets for students. I’m talking about the paper that compromises the layers and layers of bureaucratic processes that any sufficiently large organization develops over time. The forms. The reports. The mundane paperwork that must be done to uphold the laws that govern the operation of a school.

I get it.

It’s stuff that must be done.

The problem isn’t that it exists, it’s how schools deal with it. The school has limited resources available so it needs to get the most out of the resources it has. That makes sense, right? The school administration has ready access to a large number of highly trained, adaptable, resourceful, and intelligent individuals on hand with a wide range of skills covering every discipline imaginable. It has teachers.

I’m always more than happy to help when needed! I just get frustrated when I’m asked to perform work that could be reduced or eliminated by technology.

In my last post, I talked about school being a game and the need to meta-game it. One of the first issues that I think we need to talk about are “opportunity costs”. Every hour that teachers spend on administrative tasks is an hour that is not being spent on teaching. Furthermore, these costs are recurring. If schools could automate 10 minutes worth of administrative tasks each day from a teacher’s workload, they would save each teacher about 30 hours of work over the course of the year. That’s a lot of time that teachers could reallocate towards improving instruction.

Teacher time is a valuable resource and finite one. Education needs to be engineered to get the most out of that time. Based on my short time as a teacher so far, here are some of the systems that I think could be optimized:

We need a complete “Electronic Individualized Educational Plan Record” system overhaul. The current generation of “Student Information Systems” is grossly insufficient to deal with the complexity of our educational legislation. Schools need to keep documented records of adhering to a student’s legally entitled accommodations, and a significant amount this documentation is still being done on paper. We have the technology to design an educational record system that is secure, fault tolerant, and efficient. It would take an substantial initial effort, but imagine the time that it could save school staff in the long run.

We need a better “asset management system” for school property used by students. It’s very frustrating to me as a teacher when I need to fill out carbon copied checkout lists for textbooks by hand in the year 2016. When a student doesn’t return the textbook, I’m required to fill out another carbon copy form, manually address an envelop to the student’s home, and put it in the mail bin for processing. Why isn’t this process electronic yet? I should be able to snap a picture on my phone, press a button to assign it to a student or document its return, and everything else should be taken care of by a computer program. We clearly have the technology to do this.

We need a “behavioral intervention tracking and diagnostic system”. The school keeps paper records of certain student behaviors such as tardy slips and misconduct reports — which again are filled out by hand on carbon copy paper. There are also some cases where the teacher is expected to intervene in certain ways such as contacting the parent. The issue is that there are so many different rules that I need to keep track of and responses that I need to take to that data. We need a system that that can track behavior data from multiple sources and suggest interventions based on a statistical models of what has and has not worked for that student.

On top of moving from antiquated “pen and paper” systems, we also need to improve interoperability between the educational software we already use. There’s some good ideas happening with the Tin Can API, but the support from technology providers just isn’t there yet. I love to see new ideas in educational software! The problem is that some of these applications seem to neglect the teacher’s experience with the product. We need to set higher standards for educational software.

Whenever my students complete a learning activity on the computer, it should automatically go into my grade-book. The grade-book should automatically flag any items that need to be manually graded, and the process of providing feedback to the student should be as stream-lined as possible. More detailed information about the student’s performance should be stored into a database for later statistical analysis.

The other problem is the lack of standards regarding assessment items. For example, my students love Kahoot. I would totally use it way more if it were easier for me import multiple choice questions from an existing database. If I could program randomly generated questions in MyOpenMath, export them to a standardized format, and then import them into Kahoot, I would be one happy teacher.

I don’t think any of these technologies are unrealistic. It’s not like I’m asking for facial recognition software to replace hall passes or an artificially intelligent grader (although those would be kinda awesome too). If schools want to instill “21st Century Skills” in their students, they need to lead by example. In the “21st Century”, knowing what processes can be automated by technology is a crucial skill to have. To do otherwise is a disservice to both teachers and students.

Posted on
by

I love games! I love playing them. I love making them. I love theorizing about them. They’re an essential part of who I am as a person.

I used to think schools needed more games.

I was working as a video game developer and was fascinated by “tutorial levels”. You know, that part of the game that is designed to help you learn how to play the game. Some games neglect their tutorial level and it comes off feeling like a dry lecture. Go here. Push button. Repeat. However, I’ve also been completely awed by some games that take their tutorial levels to a completely different level. Games like The Elder Scrolls and Guild Wars for example. The experience is so seamlessly integrated with the “game” that you don’t even realize you’re playing a tutorial. You just play. By the time you’ve completed the tutorial, you were totally immersed in the game and knew exactly what you needed to.

I used to think schools needed more games.

There’s an certain authenticity to this learning that I never really experienced as a student. I thought if I could design the perfect “tutorial level” for math, then everything else would just fall into place. The students would have fun. They’d learn real mathematical concepts in a natural environment. They’d grow and develop as individuals and as a group. I’d be like a “math teacher” and “guild leader” all rolled into one (although I probably wouldn’t run IWAY).

I used to think schools needed more games…

…and then I started teaching.

The problem is not that school doesn’t have enough games, it’s that school has too many games. Now, I’m not talking about the latest web app: Kahoot, Quizizz, Manga High, etc.. Those are certainly a type of game that has a place in school, although perhaps the number of apps is getting overabundant as well, but I’m talking about the games that are school. School itself is like a “Live Action Role Playing Game”. Everyone invents their character, acts out their role, cooperates with some players, competes with others, and are rewarded or punished in accordance with the game master’s rules.

Now school being RPG isn’t a problem on its own. The problem is that there are a whole bunch of mini-RPGs being played simulatenously, and all of them have conflicting rules. Here is a short list of a few games that might be going on at a given time:

  • The students are playing a game with each other. They compete with each other for social status while cooperating against outside threats to their system.
  • The teachers are playing a game with their students. The teachers are trying to maximize student learning while the students are trying to minimize the work they have to do.
  • The administrators are playing a game with their teachers. The administrators are trying to maximize test scores while minimizing teacher burn-out.
  • The school board is playing a game with their administrators. The school board is trying to maximize community approval while minimizing school funding.
  • The parents are playing a game with their school board. The parents are trying to maximize the quality of education while minimizing the amount of attention paid to local elections.

Within these games, temporary alliances are made to accomplish mutual goals. Teachers and parents might cooperate to get students in for extra tutoring. Administrators and school boards might cooperate for better community awareness. Sometimes these alliances help the system as whole and sometimes they detract from it. It’s one of the most complex network systems I’ve ever seen.

I used to think schools needed more games…

…and now I think schools need to have a closer look at the games that are already being played there.

In most of these games, competition is the dominant strategy. Students that are competing for limited scholarship funds have little incentive to help one another. Schools that receive funding based on standardized test scores have a very strong incentive to focus on instructional strategies that produce short-term results over long-term retention. School boards are underappreciated as a position of political power and tend to just “fly under the radar”. Until we fix the reward systems so that they encourage cooperation, the games will continue to be frustratingly difficult for everyone involved.

We need to start meta-gaming school. We need to look at how the rules affect the relationships between players. We need to look how those rules can be changed to encourage more co-operation and less competition between the parties involved. Until we have these conversations, we’re never going to win.

Posted on
by

Disclosure:  This article is primarily mathematical in nature but the very act of discussing politics makes it difficult to fully remove bias.  I feel obligated to disclose that I’m a member of the Green Party.  While I’m neither a Republican or Democrat, I tend to lean to the north-west section of the Nolan chart.  However, I do intend to try my best to make this analysis as neutral as humanly possible.

During my regular social media browsing the other day, I came across two posts of interest.

The first was a statement from the Green Party of Virginia about why they are not endorsing Bernie Sanders ahead of the primary.  While I had expected this to be the case, there was a section of this statement that really caught my attention: “Whether individual Greens choose to vote for Sanders on March 1st is a choice that will depend on their own calculus of what is best for the country” (emphasis mine).

Since one of the co-chairs of the GPVA is a mathematician, I could reasonably assume that the reference to calculus was intended to mean exactly what it says. The problem is that the general population doesn’t usually look at elections from this perspective.  People tend to vote based on gut feelings rather than mathematical analysis. For this reason, I disagree with the GPVA’s decision. I feel that they have the responsibility to provide party members with information on how to maximize their influence on the election and calculus isn’t a strong point for most voters. If the GPVA refuses to take sides in the primary, then I feel obligated to do so in their place.

The second was a data visualization of how various primary candidates would fare against each other in a general election:

With “Super Tuesday” fast approaching, this was exactly the kind of information that I needed!  This effectively provides a payoff matrix for the primary candidates to which I can apply my “political calculus”.
Continue reading →

Posted on
by

While building up my settlements in Fallout 4, I noticed that there was a “Powered Speaker” and wondered what I could do with it.   Between the “Interval Switch”, “Delayed On Switch” and the “Delayed Off Switch”, I figured I had enough tools to make some some music in my Coastal Cottage.  To start, I decided on remaking one of the first songs I learned on guitar: Nirvana’s Come As You Are.

Despite my best efforts, it’s still not quite perfect.  It seems like the timing on the switches isn’t as exact as I needed it to be for this wiring system, and it seems like some switches will occasionally stay on despite having no power.   I found that the most reliable method was to alternate between two offset interval switches with a slight overlap to form a loop, and use a chain of delays to space out the individual notes.

Resource-wise, I found myself needing tons of copper and ceramic.  The most time consuming part was setting up the delays and notes via terminal.  Below is the diagram I created as a reference so I could set several things at once.

Fallout 4 - Come As You Are

I hope this will inspire some discussion on how to create music in Fallout 4.  If you put one together or have an idea on how to streamline the process, please share in the comments!