Battleground schools response

3 stops:

1. The following quote made me laugh, it's so good:

"Those who like mathematics are (generally male) eggheads, nerds, absent-minded professors, and mad scientists, unable to cope with the world of human interactions, and not fully mentally competent (ranging from the mildly autistic to the completely mad)" (393)

I feel like this is totally how people think of "math people." It's been a really strange experience transitioning gender while in mathematics. These days people are more likely to be surprised when they learn I did a degree in math, and more likely to think I'm crazy than smart. It always feels strange when people get intimidated when they learn I enjoy mathematics (men especially do this), or when people say I must be so smart to do that. I don't think you need to be smarter than anyone else to go far in mathematics, I think you just need to have a passion for mathematics, that's all. And people shouldn't feel unintelligent for not being passionate about math! It's so absurd to me. Imagine a world where people who excel at fiber arts like crocheting and knitting are considered the most intelligent, and people who struggle with such things are made to feel like they are missing an essential cognitive capacity...

2.  The next stop is "Complicated but opaque techniques like the infamous 'rule of three'..." (395)

I've never heard of the rule of three, so I looked it up, and I certainly agree it's complicated and opaque. In my experience algebra is one of the harder math topics to teach to high schoolers given the level of abstraction and symbolic opaque-ness. It's somewhat unintuitive to me that so many students struggle with algebra concepts like variables, constants, and rearranging equations, because I've always found these sorts of things intuitive. I will continue to think of strategies as I progress in my teaching journey.

3. The last stop is the last sentence: "Mathematics education has itself become a focal point for a left versus right political standoff that is being played out on many fronts, and there appears to be little public appetite for concepts like balance and consensus" (400). 

If this was true in 2008 when this was published, it's even more true now. Education, and in particular the questions of what should be taught and who should teach it, seems to be a frontline in the ongoing, exhausting culture war. Most of this culture war talk seems to focus on other subjects, especially what should be taught for history, what books should be read in English class, as well as non-curricular fronts like the recent law Alberta passed about pronouns and chosen names. Despite this, math education is not immune.

To me, it seems like there is a massive gulf in the culture of "Mathematics" that mathematicians do, and the culture of math education. In mathematics proper, there has been a huge corpus of work over the last century that has completely changed the foundations of mathematics and the philosophy of mathematics. What used to be seen as a perfectly consistent, rational, deductive system is now known to be much more relative, flexible, and touchy than previously thought. Sure, math is perfectly deductive, up to a change of topos, relative to the axioms you use, and even then there will always be theorems that are unproveable without introducing additional axioms. There has been extensive work in constructive mathematics recently, much of which is still blasphemous to traditional mathematicians. 

In math education, quite on the contrary, it seems many educators, policymakers, and laypeople have a sort of pseudo-Hilbertism-Platonism-Creationism in their minds, that math is this perfect calculation tool, that everything objective can be made math, that math is timeless and unchanging. It is always ironic to me when people say "2+2=4" to mean the prototypical objective fact - because even something as simple as that depends on how we define "2" and "+" and "=" - none of which are a given in advanced mathematics. It seems that math education always lags behind "Mathematics," not just in content (and I'm not advocating for neo-NewMath here), but in philosophy and culture, too.

Lockhart's Lament

 Lockhart's lament response

To start off, I wholly agree with the overall sentiment behind this essay. I really do think that the way math is taught in high schools is detrimental to students' creativity, inspiration, and even study skills and problem-solving skills. When I was in high school, I didn't particularly enjoy math class. It was only in university that I decided that math was interesting. The first two things I remember being genuinely curious about in university math were,

(1) dy/dx is NOT a fraction but BEHAVES like one.. how strange! I made it my mission to understand WHY this is true. And after learning about differential forms, non-standard analysis, and the abstract algebraic definition of differentials, it makes MORE sense to me, but it's still a little mysterious. I think this is an interesting fundamental concept in advanced math -- an object that behaves "like" something, but isn't technically "that thing." 

(2) In a first-year computer science proofs class, the professor did a formal proof that mathematical induction is a valid proof technique. This was fascinating to me, because I was skeptical about mathematical induction at first. This led me down a whole rabbithole about ZF, the axiom of choice, continuum hypothesis, nonstandard axioms, and eventually I learned that foundations / set theory is one of my favourite areas of math. 

Two areas I disagree with Lockhart:

(1) On page 3, he states "there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic as mathematics." On this I wholly disagree. Sure, maybe Lockhart is showing his dramatic flair, but I'd hardly call a discipline "radical" or "subversive" when the majority of its research grants are funded by the military. I'd also argue that there are many things more poetic or psychedelic than math, like poetry or psychedelic art…

(2) On page 4, he says “[math is] fascinating, it’s fun, and it’s free!” I’m not sure if I quite agree. Sure, it’s technically free to do mathematics. All that’s required is your mind, and maybe some way to write things down. People have discovered novel mathematics in very desolate situations – Leray came up with the definition of a spectral sequence, a fundamental tool in algebraic topology, while he was a prisoner of war!

All that said, however, I think there are at least two senses in which mathematics is not free. One is that, unless one is Ramanujan, you need help from other people in order to make significant progress in understanding mathematics. I always maintain that mathematics is a semi-oral tradition. Sure, it’s possible to learn some math from books, but much of the real understanding, the trade secrets, is conveyed orally in classrooms or universities. And even if one is Ramanujan, he still had math class. So a person could pretend to be “Good Will Hunting,” and take out advanced math books from the library, but I think that without university, tutoring, or some other form of oral instruction, one is probably not going to learn much from the books. And oral instruction for math, beyond public school, is not free. 

Another sense in which I think math is not free is if we consider the costs to society. In our highly complex modern industrial society, social roles are so stratified and differentiated that we don’t think much of someone who doesn’t produce anything of value, but in most of history there would need to be very good reason to allow someone to enjoy the benefits of society (food, shelter, community), while spending most of their time thinking about abstract things that are not directly useful for the society. In my opinion, the reason that “professional mathematician who thinks about very abstract things all day that do not immediately provide value to the society” is a real job is a quirk of the military industrial complex more than anything else. The abstract math might provide tangible value to the development of technology in the future, and even if it doesn’t, it acts as costly signalling – by paying an intelligent, motivated person to think of useless things all day, like “Morava p-local sphere spectra,” the government is signaling to rival states that they have so much money and scholars they can afford to fund research that seems to provide no real value to the state. 

Skemp response

 3 stops:

(1) On page 3, the quotation "pupils and teachers whose goals are respectively relational and instrumental understanding" stuck out to me. I'm not sure I agree. Many students are seeking an instrumental understanding, because they may not care about math, but they need to know how to solve math problems in order to get into university, or as preparation for university math courses. Would a relational understanding help them succeed in university math? Absolutely, and I'd argue it's essential for the more advanced courses. But, I think that many students are seeking instrumental understanding, whether this is best for them or not. 

(2) On page 11, Skemp is talking about reasons a teacher may teach instrumental understanding, and one is that "he is a junior teacher in a school where all the other mathematics teaching is instrumental." This stuck out to me because I think this is a very important thing to consider when we are discussing pedagogy and teaching philosophy. Every school, and even departments within a school, has a very different pedagogy culture. Often the career success of a junior teacher is very much dependent on fitting in to the existing norms within the school. Especially during practicum, or in the first year teaching, it's seen as "out of place" for a teacher to do anything very differently than their senior colleagues. 

(3) On page 15: "then the word 'mathematics' is for many children indeed a false friend." On this point, I'd argue that the gap between relational and instrumental understanding is not unique to mathematics. It's particularly obvious in math, for sure, but I think every subject has this same problem. In social studies, people may be good at remembering historical events, laws, or government structures, but may lack an understanding of how to put this knowledge in context or apply it to novel situations. In English, many students know the 5-sentence paragraph structure by heart, but may not understand why most paragraphs in English written work don't fit this mould exactly -- and the students might not understand what circumstances call for a shorter or longer paragraph than usual. 

My opinion: I more-or-less agree with Skemp, that relational understanding is under-taught and important. There were several points throughout the article where I didn't quite follow Skemp's reasoning, or where I disagreed somewhat. I do think though, that the distinction between relational and instrumental understanding is not as clear-cut as Skemp makes it out to be. 

Curricular microteaching lesson plan

 Teachers: Maxine, Jason and Jimena

Curricular Micro-teaching.

Here are the subjects included in the content we chose:

• two-variable continuous linear relations; includes rational coordinates

• horizontal and vertical lines

• graphing relations and analyzing

• interpolating and extrapolating approximate values

• spirit canoe journey predictions and daily checks

We chose graphing relations and analyzing for our 15 minutes lesson

Lesson Plan for a Micro session about understanding Linear Relations: 15 minutes

Should we do sth like this:

https://the-world-is-my-classroom.weebly.com/uploads/1/8/3/8/18384137/grade_9_-_unit_4_notes.pdf

or some context to the activities? or do we jump into the activities right away?

1.  Human Coordinate Plane- Same as we saw it during our First School Visit

Goal: Experience graphing physically and collaboratively.

Setup:

• Use tape or chalk to draw a large coordinate plane on the floor.

•  Assign students as points (x, y) from given relations (e.g., y = 2x + 1, or y = |x|).

• Students move into position, then discuss what they notice about the pattern, symmetry, slope, etc.
  
  

Critical Thinking: Visualizing and reasoning spatially.
Teamwork: They must organize themselves accurately and communicate effectively.

2. MUSICAL CHAIRS GAME

Setup:

• Place all in two rows back to back.

• Make the students walk around while the music is playing.

• When the music stops, the students sit down in the closest chair.

• In the second round, the teacher takes 2 chairs out and plays again.

Discuss what happened. Tell the students to talk about the variables involved and how they relate to each other.

3. REAL WORLD SCENARIOS - UBER FARES

Setup:

• Post real UBER fares in Vancouver.

• Groups answer some questions, and groups respond to another group’s inference.

Questions: 

• What does it mean low demand, and rush hour?

• How much does it cost to travel 10 km in rush hour?

• How much does it cost to travel 10 km in low demand?

• EDIT: (new question) Does traffic seem to have more of an impact on price for low-kms trips or high-kms trips? 

• EDIT: (new question) We model a line with the equation y=mx+b. Does increased traffic change m, or b? Slope, or y-intercept? Why?

Assessment: Teacher notes on the use of mathematical vocabulary during discussions
Metacognition: How did other groups’ ideas challenge or support your thinking?”

EDIT:

ADAPTATION: We may not have enough time for part 2, musical chairs. 

Materials needed: 

• Tape for floor

• Speaker for music

• Slide with Uber fare graph

Connections to curriculum: 

• Two new questions added for section 3.

Assessment questions for section 1: 

• Do the students understand the different roles x and y play?

• Are students beginning to understand the effects scaling and shifting have on equations?

• Can the students collaborate with each other effectively, to help students who may have trouble finding their spot on the graph?

• Can students guess where they go based on where their friends go (interpolation)?

Microteaching lesson plan

Lesson plan: how to buy a used motorcycle. 

Key concepts: 

- What kind of motorcycle is right for your use-case?

- Brief mechanical overview

- Negotiation considerations

Goals: students will feel more comfortable buying a used motorcycle on facebook marketplace or craigslist / kijiji, and some advice also applies to buying other used vehicles or items. 

Activities / structure:

- Brief talk about the different kinds of motorcycle (sport bike, cruiser, etc) and their uses (1min)

- Ask students if they have any experience buying a used vehicle (1min)

- Activity: I show the students bikes on facebook marketplace and we decide as a group whether we think it’s a good buy (3min)

- Brief talk about brands, mechanical considerations, and other expenses to consider. (1min)

- Brief talk about negotiation techniques (1min)

- Activity: have students negotiate with each other over a bike (3min)

- Reflection

If needed to shorten: Remove the facebook marketplace activity. 

If need to lengthen: give more time for activities.