Notes on ‘”Uhh, You Know,” Don’t You?: White Racial Bonding in the Narrative of White Pre-Service Teachers’

Fasching-Varner 2013, Educational Foundations

“Uhh, You Know,” Don’t You?: White Racial Bonding in the Narrative of White Pre-Service Teachers

This article is about preparation of White (capitalization is following that in the paper) pre-service teachers to examine their racial identity and its potential impacts on students.  The author looks at attempts at White racial bonding between pre-service teachers via use of the phrase “you know”.  The term “racial bonding” is used to indicate ways White people “show affinity and alliance with each other”.

Educational landscape:
As of ~2011, ~85% of teachers are White and female.  White students were ~55% of public school students, creating a demographic disconnect.  Teachers are increasingly inexperienced, as well.

Critical race theory (CRT) — Whiteness as property:
See Ladson-Billings and Tate 1995 for CRT in education.  Four elements of the value of Whiteness are relevant: benefits (use and enjoyments), a right to exclude ‘others’, rewards for certain behaviors, and status/reputation maintenance.  Part of the value is in never needing to define itself (but

Notes on “Abolitionist Teaching and the Future of Our Schools”


Accessed on June 29, 2020
Bettina Love,
Gholdy Muhammed
Dena Simmons
Brian Jones.

Bettina Love described arriving at college and being pushed onto a “jock track”. She transferred universities, but that was a turning point in her life.

Gholdy Muhammad recalls reading a hadid about ways to respond to oppression. Reading and studying Blackness also helps her hold on to abolitionist teaching.

Dena Simmons was put in the “slow class” in first grade and advocated to switch. Then in high school, leaving the Bronx, she experienced tone policing and hurt. Returning to the Bronx as a teacher, she saw the way the system was set up to fail her students.

Q: (Brian Jones) How does White supremacy show up in schools?
A: (Dena Simmons) Tone policing, tokenization and objectification, over-managing, expectation of being super-people. The emotional labor of asking “am I safe to be Black here” is also work.
(Gholdy Muhammad). It shows up in the curriculum, in the interviewing questions (“how does anti-racism appear in your math curriculum” isn’t one of those questions. The questions focus on meeting “challenges”). It is in tests, in the emphasis on skills, in zero tolerance policies.
(Bettina Love). In education “we manage inequality”. We don’t remove it. Instead of removing barriers, they manage inequity (via directors of equity).

Q: (Brian Jones) White supremacy follow up
A: (Dena Simmons) “We can use anything in education to create harm”. Social-emotional learning (feelings) is often equated to promoting equity. But “if the school is steeped in White supremacy… then the curriculum is a White supremacist curriculum”. Without trauma informed instruction to address the trauma of racism, there is an issue.

Q: (Brian Jones) Abolition is being used about prison and police. What about schools?
A: (Betinna Love) “It is not a radical thing to want to be seen as fully human”. Abolition is a request to start over. “It is a push for everybody’s humanity”. See work by Angela Davis, Ruth Wilson Gilmore. Eradicate the conditions that make it possible to treat children in the way they are treated within school.  The system is oppressive.  Build a community-based curriculum where children learn their history and culture, have the chance to play, and have access to healing.  When you’re an abolitionist, you might not live to see the win, but you work thoughtfully and methodically, to justice.

(Gholdy Muhammad).  “Stop putting fresh coats of paint on the same debilitating structures”.  State learning standards, teacher evaluations, curriculum.  Who authors these?  In the 19th century, Black literary societies came together to strategize to improve social conditions with four learning goals: identity, skills, intellectualism, criticality (understand power).  Teaching these four goals would teach the whole child.

Q: (Brian Jones).  Black women and Black queer folks are important in this.
A: (Bettina Love) For Black queer women, the intersection creates deep knowledge of marginalization.
(Dena Simmons) There is so much erasure of the contributions of Black womyn.

Q: (Brian Jones). What about the shutdown of schools?
A: (Bettina Love).  They canceled standardized tests, handed out computers, relied on the ingenuity of teachers, asked for flexibility and compassion, and asked parents to be partners.  “Why did it take a pandemic” for this?  Expelling children, school shootings, should not be normal.  When schools reopen, that trust in parents and teachers should remain.
(Dena Simmons). Some students are thriving in the shutdown of schools.  Learn from that.  “The school has never known what’s best for us”.
(Gholdy Muhammad).  Be urgent in pedagogy.  Skills and knowledge are not enough.  Students need to be able to agitate as antiracists.
(Dena Simmons).  How will you change how you spend money, etc, grounding in anti-racism?
(Bettina Love).  The work needs to happen in White schools, including anti-racist education.
(Gholdy Muhammad).  Redefining achievement and success is important.  “High performing” schools are not a positive model.
(Bettina Love).  “Black parents care”
(Dena Simmons).  The school can be a place of trauma for the parents.

Q: (Brian Jones) Resources people can turn to?  Black Lives Matter at School movement.  Four demands.  Counselors not cops, hire Black teachers, end zero tolerance discipline, teach Black history and ethnic studies.
A: (Dena Simmons) For “woke” White people, start at home with racist family members.
(Gholdy Muhammad).  See healpedagogies.com.  A goal is to have curricular materials made by Black educators.
(Bettina Love).  There are so many organizations doing the work.  See “Unapologetic” by Charlene Carruthers

Q: (audience Q) Trapped in standardized tests, what do you do on a daily basis?
A: (Dena Simmons).  In one example, as an act of resistant in a 7th grade classroom, talked through with the students about whether to take a standardized test (and the students chose not to).
Community organizing is also an act of resistance.
(Gholdy Muhammad).  “Revise and modify the curriculum that is given to us”.  “How can this unit plan help my students learn about themselves”?  Plan towards advancing student thinking about equity, power, anti-oppression.  “Collect the data you want to collect”.  “If you value it, assess it”.

Q: Unions?
A: (Dena Simmons) Let’s ask our unions to demand these practices.

Q: Do we need to create our own schools?
A: (Gholdy Muhammad).  “Black abolitionists… cultivated their own schools”.  “A lot of power lies with the parents”.
(Bettina Love).  “It is our money”.  We pay the police, teachers, etc.  “Understand the value and power we have as citizens”.

Q: Is abolitionist teaching anti-capitalist?
A: (Bettina Love).  Abolition is anti-capitalist.
(Dena Simmons). The country is “built on stolen Black labor”.

Notes on: “Worldmath Curriculum: Fighting Eurocentrism in Mathematics”

SE Anderson, 1990: “Worldmath Curriculum: Fighting Eurocentrism in Mathematics”.  The Journal of Negro Education

Section one: “A few grim statistics”

The manuscript starts with “A few grim statistics”, looking forward to 2000 and to 2010 and providing numbers about low numbers of Black, Latino, and Native American scholars receiving PhDs in mathematics, physics, and astronomy in the 1980s.

Section two: “The Eurocentric Basis of Mathematics”

The author writes that educators have a duty to build a system “based on the assumption that any person can learn anything”, with the “true beauty of mathematics” coming from the creation of “logical systems that help explain the complexities of Nature”.  (See Joseph 1987 p 22-26).

Eurocentrism:

  1. math is often specifically delinked from materialist concerns
  2. math is confined to an elite group with special gifts
  3. math discovery comes from “deductive axiomatic logic” not from “intuitive or empirical methods”
  4. math results needs to be presented in a particular style, so “new additions to mathematical knowledge” are from a small, special, Eurocentric group.

The narrative of mathematics is presented as going from the Greeks through “European men and their North American descendants”.  “African, Indian, Chinese, or Mayan contributions” receive short mentions in textbooks.

In addition, an unappealing curriculum ends up reinforcing “racist assumptions about people of color”.

Section three: “Six Pedagogical Disasters in Mathematics Education”

  1. “separate arithmetic from algebra”
  2. “teach mathematics without any historical references”
  3. “use textbooks that are elitist and cryptic”
  4. “do work and be tested as an individual” (vs in study groups)
  5. “accept the myth that mathematics is pure abstraction”
  6. “memorize”

1: the author asserts that arithmetic is taught first (for many years), followed by algebra: evidently these subjects are kept separate.  I don’t have any thoughts on this.

2: teaching math as ahistorical: a bunch of European names are attached to math facts and abstractions, without the humans themselves being introduced.

3: cryptic texts: these make math appear unaccessible and not for most people.

4: the individual: math classes are often structured competitively / individually.  this is not how people actually solve problems.

5: real math is abstract: “erudition, abstraction, and compartmentalization”, distance people from creative sources and make math seem unnatural, rather than a natural human act.

6: memorization: math problem solving is often turned into the memorization of definitions, theorems, etc.

Section four: “An alternative curriculum and pedagogy”

See Bob Moses and his “algebra project” (learning algebra in elementary school via subway rides).  See Arthur Powell and the “Writing in Math” project.  See Marilyn Frankenstein and “radical math”.  Look for alternatives “to the arithematic-algebra-precalculus-calculus `learning’ sequence that is so pervasive yet so devastating.”  College-level courses should (1) “show the interconnectedness of mathematics and real-world problems” and (2) should “show how people throughout history have created mathematical techniques to solve problems”.

Section five: “How and what I teach”

  1. Via “psychological upliftment”, “emphasizing that ordinary people create mathematical ideas and ‘do’ mathematics”.
  2. By assuming “the role of a confidence builder”.  Letting students know “they all have the intellectual capabilities to understand the material”
  3. Attributing not understanding to “my own or the textbook’s failure to communicate clearly”
  4. Choosing “the quality of mathematics knowledge” over the quantity.  “I may set out to cover six chapters… if they complete only three or four chapters and learn those well, then I am confident they can pick up the rest”

In a typical algebra class: first two classes focus on historical, cultural, sociopolitical.  Relate math to “humanity’s ongoing struggle to understand Nature”.  And to “capitalism’s attempts to control and dominate Nature”.  Emphasize that “some of the very first mathematical/scientific thinkers were African women”.  “Show how early mathematics and science led to the building of the pyramids, the Great Wall of China, and the road to Kathmandu”.  Astronomy, astrology, iron-smelting, surgery, etc.  Name that Euclid “spent 21 years studying and translating mathematical tracts in Egypt” and “Egypt is in Africa and that the people who inhabit the land were and are Africans”.  Pythagoras also studied in Egypt (and perhaps India).  The theorem attributed to him existed 1000 years before him in Babylonian documents.

The “intent… is to shatter the myth that mathematics was or is a `White man’s thing'”.  “I show how certain aspects of European mathematics could not have developed had not the Europeans traded with more advanced societies”.  Example: “The Vatican denounced Hindu-Arabic numerals”.

The research university of Bait al-Hikma is important because of Mohammed ibn-Musa al-Khwarizmi (the name “al-djabr or `algebra’ comes from a text he wrote) and the term “algorithms” is a corruption of his name (dating to his second book, “Algorithmi de Numero Indorum”).  “I also mention… the algebraist Omar Khayyam (c. 1050 – 1122 AD)”.  See Nasir Eddin al-Tusi for non-Euclidean geometry, as well.

In the 1600s, when Europeans were trying to build very large ships to carry African slaves, they used the knowledge of “Gambian, Chinese, and Indian mariners”.  “Developments in hydrodynamics (and its attendant mathematics) contributed … to the horror of .. the slave trade”.

“I further point that that calculus was created to facilitate the study of ballistics” in wars by England and Germany.  Military needs “continue to inspire many mathematicians and scientists to purse the War Machine”.

Structuring class: (1) “About two weeks into a class, I facilitate the creation of study groups” (3-4 people whom students choose).  These groups are used in and out of class.  The groups complete progress reports (no tests).  (2) “I also incorporate a weekly 15- to 20-minute class discussion” of a news article from the Science Times, to emphasize the relationships between math and the social / natural sciences.  (3) students are asked to make notecards of facts that they should bring to class.  (4) computational tools are encouraged.

Section six: “Conclusion”

“a subtle but effective form of educational genocide is taking place”.  “To offer an alternative that is genuinely egalitarian and truthful we must open our eyes to the centrality of the contributions made by the vast majority of the world’s people”.

Python in my dynamical systems class

I have been using Mathematica in my dynamical systems class for a few years. I don’t have a systematic curriculum related to it, though, and need to develop clearer computational learning goals, as well as a pathway for students to develop computational skills.

Ideally, by the end of the semester, students would be able to do an analysis of a one-parameter dynamical system with the aid of computational tools. They would find fixed points, identify stability, create phase portraits and bifurcation diagrams, and perhaps create stability diagrams. I would expect them to be able to identify global bifurcations, as well. For limit cycles, I need to make a decision on my expectations. I suppose I would like for students to be able to create a curve of initial conditions, use “Events” in Mathematica of Matlab integration, and identify the stability / existence of a limit cycle in a 2d system.

At the moment I’ve been relying on Mathematica and some students have chosen to use Matlab. I would like to move towards Python. Currently the dynamical systems course is the only place students work with Mathematica, while Python is an option across a range of courses.

This means I need to learn how to set up Python for a class. At the moment I’m taking a look at Koehler and Kim, 2018 for some guidance on this. They go in the direction of Jupyter Notebooks so I will explore that for now.

1) Install the Anaconda application on my Mac.
2) Open the Anaconda Navigator: it has seven options when I first open it (Jupyter lab, Jupyter notebook, Qt console, Spyder, Glueviz, Orange 3, RStudio). The first four are already installed and I have the option to Launch them. For the other three, I have the option to install them. I’ll launch Jupyter notebook.

Koehler and Kim 2018: Interactive Classrooms with Jupyter and Python. The Mathematics Teacher. Vol. 111, No. 4 (January/February 2018), pp. 304-308 (5 pages)

Notes on Ljung: System Identification

Reading Ljung.  System Identification: theory for the user.

1: Introduction.

Goal: infer a model from observations.  “Model” refers to the set of relationships between variables in the system.  System identification involves analyzing input and output signals from the system.

Example: assume a linear difference equation relates inputs to outputs.  Use least squares to find parameter values that minimize the least squares error.  This is partly an autoregression: a linear regression “where the regression vector contains old values of the variable to be explained”.

Adding noise: assume the observed data are from a deterministic process with noise.  We’re interested in two expectation values: the parameters and the covariance of the parameter error.

System identification involves: a data set, candidate models, and an assessment rule (see chapter 7).  Then use model validation to check whether the model is good enough.  This process ends up being iterative.

2: Time invariant linear systems.

Impulse response can be used to characterize the system.  If there’s an additive disturbance info is needed on that too (spectrum and pf).

3: Simulation and prediction.

4: Models of linear time-invariant systems.

5: Models for time-varying and nonlinear systems.

Linear time-varying models might be used when we linearize a nonlinear system about some trajectory.

6: Nonparametric time and frequency domain methods.

7: Parameter estimation methods.

8: Convergence and consistency.

9: Asymptotic distribution of parameter estimates.

10: Computing the estimate.

11: Recursive estimation methods.

12: Options and objectives.

 

13: Experiment Design.

14: Preprocessing data.

15: Choice of identification criterion.

16: Model structure selection and model validation.

17: System identification in practice.

 

Notes on Maybeck: Stochastic Models, Estimation, and Control

Notes on Chapter 1 of Maybeck 1979, Stochastic Models, Estimation, and Control.

1.1: why stochastic models, estimation, and control?

A math model isn’t perfect, and parameters are not known absolutely.  Sensors don’t provide perfect data either.  Given uncertainties, you still want to be able to estimate quantities of interest and control the system.

1.2: Overview of the text

skip.

1.3: The Kalman filter: an introduction to concepts

The Kalman filter is an “optimal linear estimator”.  Given (1) knowledge of the system and the measurement device, (2) a statistical description of noise and error, and (3) initial condition info, a Kalman filter can combine the knowledge to create an estimate.  When the system is described by a linear model, and the noise is Gaussian and is white noise, the Kalman filter is the best estimator.

1.4 Basic assumptions

The model is linear, the noise is white noise, and the error is Gaussian/normally distributed.

1.5 A simple example

A static problem: Trying to determine your position, you have a single measurement (z1) and its precision/standard deviation.  This leads to a conditional probability density (conditioned on your measurement).  The best estimate is currently z1.

A friend takes a second measurement, z2, with smaller variance.

The best estimate will now be a combo of these two that takes into account the precision/variance on each.

There is a predictor-corrector structure to the way the estimate is made: we can take the previous best estimate and associated standard deviation and then “correct” it with the new data and new standard deviation.

A dynamic problem: Now add a motion model.  This evolves the pdf forward in time.  (And adds noise so will increase the standard deviation).  This creates a new estimate and variance.  Then take a measurement and use the corrector process from above.

Notes on “Particle filters for high dimensional geoscience applications: a review”. van Leeuwen et al 2019

Notes on

Peter Jan van Leeuwen, Hans R. Künsch, Lars Nerger, Roland Potthast, Sebastian Reich.  Q J R Meteorol Soc. 2019;1–31.  Particle filters for high-dimensional geoscience applications: A review

This paper is focusing on the problem of “weight degeneracy” for the weighting of particles in a particle filter.

Introduction:

“the linear data assimilation problem” is hard in a high dimensions.  Numerical weather prediction has 10^9 state variables and 10^7 observations every 6-12 hours.  Two currently methods are 4DVar, Ensemble Kalman Filter (EnKF).  Hybrids of these evidently do okay but need “ad hoc fixes like localization and inflation”.  These methods are harder to use when there’s underlying advective flow, as well.  The linear problem is evidently hard, but actual problems are nonlinear, and these methods don’t work well.

Variational methods can easily fail when the cost function is multimodal, and are hampered by the assumption that the prior probability density function (pdf) of the state is assumed to be Gaussian.”  EnKFs also have a Gaussian prior.  

Evidently particle filters don’t have assumptions on the prior or the likelihood.  (“Particle filters hold the promise of fully nonlinear data assimilation without any assumption on prior or likelihood, and recent textbooks like Reich and Cotter (2015), Nakamura and Potthast (2015), and van Leeuwen et al. (2015) provide useful introductions to data assimilation in general, and particle filters in particular.”)

There are also MCMC methods that are fully nonlinear.

Description of a particle filter (“standard or bootstrap”):

  • choose N model states (“particles”, x_n-1).  These are sampled from the prior pdf.
  • propagate the particles forward in time to the next observation time using the (nonlinear) model (x_n).  Include a random forcing with the propagation (there’s an assumption that physics is missing from the model and the random forcing compensates for that).
  • there’s an observation (y_n) with random measurement errors at this time (with known characteristics)
  • “assimilate” the observations.  Multiply the prior pdf by the likelihood (how likely the current observation is given a current model state, p(y_n | x_n): this hinges on our measurement error.  Is this measurement of y_n possible given the actual state was x_n? ). The product is proportional to p(x_n | y_n), the posterior pdf (the probability of each x_n given the observation that we have).  This step is using Bayes’ theorem.
  • create a weight (related to the probability) for each particle proportional to p(y_n | x_n_i)
  • resampling is common because weight might concentrate in just a few particles.  “This duplicates high weight particles and abandons low weight particles”.  “for particle tilts to work, we need to ensure that their weights remain similar”.

The review deals with “weight degeneracy”.

  1. “proposal-density freedom” and “equal-weight particle filters”
  2. one-step transformations from particles in the prior to particles in the posterior
  3. use localization
  4. combine particle filters with EnKFs

 

Terms to learn about:

4DVar, ensemble Kalman filter, cost function, likelihood, localization

 

Notes on “No contest: the case against competition”

Kohn, 1986 and 1992, “No Contest: The Case Against Competition”

Chapter 1: “The ‘Number One’ Obsession”.  American life is a succession of contests, so some people fail or lose so that others succeed or win.  There is “structural competition” (an external win/lose framework) and “intentional competition” (a competitive intention on the part of the individual).  A structurally competitive activity has “mutually exclusive goal attainment” (MEGA: a zero-sum game).  Competitive, cooperative, and independent modes are all possible.  In the competitive and cooperative cases, the success of participants is interlinked.  Structural cooperation requires coordinated effort “because I can succeed only if you succeed, and vice versa”.

“The case for competition… has been constructed on four central myths”.  (1) It’s human nature.  (2) It motivates our productivity.  (3) Contests are a good time.  (4) It builds character.

Chapter 2: “Is competition inevitable?”.

Chapter 3: “Is competition more productive?”.

Chapter 4: “Is competition more enjoyable?”.

Chapter 5: “Does competition build character?”.

Chapter 6: “Against each other”.

Chapter 7: “The logic of playing dirty”.

Chapter 8: “Women and competition”.

Chapter 9: “Beyond competition”.

Chapter 10: “Learning together”.
1. Competition promotes anxiety.  2. It contributes to extrinsic motivation.  3. Whether winning or losing, “luck or fixed ability” is often credited.  4.  The presumptive winner is often already known.  5.  Cooperation has emotional benefits.  6.  Academic work becomes a valued activity.  7.  It enhances student enthusiasm.

Notes on Calculus Blue Volume 1, Chapters 3, 4, 5, 6

More of Calculus Blue by Prof Ghrist Math.

Chapter 3 is on coordinates and distance (see section 12.1 of the 6th edition of Hughes-Hallett).

01.03 (0:36) “Coordinates: intro”.  Review coordinates and see it in data.

01.03.01 (2:16) “coordinates & many dimensions”.  from curves and surfaces we’ll head on.  plane, then 3-space.  what next?  He introduces R-n.  He mentions that the coordinates may have units and that the coordinate is a point in higher dimensional space.  But why care?  (see the next examples).

01.03.02 (2:33) “Example – robot kinematics”.  robot arm with a bunch of joint angles.  There is a configuration space.  He shows a video: the tip of a robot arm is tracing a path in 3-space.

01.03.03 (1:42) “Example – wireless signals & localization”.  the phone can sense a bunch of wireless signals at once and multiple might be nonzero, so the phone has a position in signal space.  the dimension is the number of wireless routers in the building.

01.03.04 (2:04) “Example – customer preferences & profiles”.  create a preference space of how customers feel about a bunch of products.  You’ll want to cluster (group) this dataset.

01.03.05 (7:27) “Distances via coordinates”.  we need to built up algebra and geometry.  start with distance.

Examples: (1) distance from point to line (where the line is parameterized), so the shortest distance; this could be done in single variable calculus: he foreshadows that we’ll learn a better way.  (2) configuration space for four objects on a chess board, so 8 coordinates, and take a distance in that space.  (3) maximal distance between two points in a unit ball in 49-dimensional space: it is 2.  (4) how about a 49-cube?  it is 7, which is weird.

we still need more tools (not quite time for calculus).

01.03 (0:36) “the big picture”.  higher dimensional spaces exist in systems and in data.

Total time: 17:14

Chapter 4 is an intro to vectors (see sections 13.1 and 13.2 of the 6th edition of Hughes-Hallett).

01.04 (0:38) “Vectors: intro”.  This is a tool for organizing variables or data.

01.04.01 (2:38) “Vector components”.  one way to think of them is as a difference between two points.  Another is as two objects that can be added and rescaled.  We’ll work with a coordinate system and interpret vectors as movement in that space.  Stack the components vertically in vectors.  Use an underline to denote a vector.

01.04.02 (6:32) “Basic vector algebra”.  algebra: addition, rescaling, using components and acting term by term.  properties: commutative, identity, subtraction.  geometry: concatenation, so draw u, then v, then the sum.  Define the length of a vector.  Things to prove: triangle inequality, and a couple of others.  Lines and planes can be nicely parameterized using vectors. 1d line, 1 parameter, 1 vector.  2d plane, 2 parameters, 2 vectors.  Nice animation of how the two vector are used to parameterize the plane.

01.04.03 (3:52) “Standard basis vectors”.  vec i, vec j, vec k are introduced, as are vec e_k.

example: (1) take a vector and write it as a linear combination of the vec e_k vectors.  (2) do the same for a vector in 3d using vec i, vec j, vec k.  (3) Take the length of a vector sum.

01.04.04 (1:53) “Caveat & a foreshadowing of fields”.  vectors are actually independent of how you represent them.  where’s the calculus?  we need more background!  At some point, though, we’ll learn a calculus for fields of vectors (foreshadowing of vector fields).

01.04 (0:26) “The big picture”.  Vectors carry both algebraic and geometric data.  This was our intro to them.

Total time: 16:00

Chapter 5 is on dot products (so section 13.3 of the 6th edition of Hughes-Hallett)

01.05 (0:41) “the dot product: intro”.  good data structure: geometry and algebra.

01.05.01 (1:23) “definition of the dot product”.  define it.  properties: commutative, dot with zero is zero, dot product with itself is length.

01.05.02 (3:27) “dot products & orthogonality”.  the angle between two vectors is well-defined.  memorize the geometric definition of the dot product.  use dot products to detect orthogonality.

example: (1) find an angle between two vectors with four components.  (2) can create a pair of vectors that are orthogonal.  (3) the standard basis vectors are all mutually orthogonal.

01.05.03 (3:32) “dot products as orthogonal projection”.  projected length is an important interpretation of the dot product (oriented, projected, length along an axis).  Really great animation / visualization for this!

example: (1) find the component of one vector in the direction of another.

01.05.04 (2:48) “hyperplanes & machine learning”.  use the dot product to make sense of our implicit equations for lines and planes.  Another nice animation / visualization.  hyperplanes (a “support vector machine”) separate two types of data.  with a “normal vector” to the plane you can tell which side of the plane a data point is on.

01.05.05 (1:54) “dot products and compatibility”.  love: create a preference space with a bunch of opinions.  make two vectors and then take a dot product.  a large positive dot product means two people like similar things.

01.05.06 (1:36) “foreshadowing of Fourier”.  additional math: you could learn to think of functions as infinite-dimensional vectors.  black and white foreshadowing video.

01.05 (0:27) “the big picture”.  the dot product, with algebra, geometry and applications!

Total time:  15:48

Chapter 6 is on cross products (section 13.4 of the 6th edition of Hughes-Hallett).

01.06 (0:41) “The cross product: intro”.  two more products (unique to 3d).

01.06.01 (5:35) “definition of the cross product”.  (only in 3d).  he defines it.  properties.  anti-commutativity, vec u cross vec 0 = vec 0.  cross product with itself is zero.  Look at the geometric meaning of that anti-commutativity.  The cross-product is orthogonal to both of its factors (he proves this).  Then he shows a visualization and defines the right hand rule.  The illustration involved a spinning mill.  🙂

examples: (1) three points, PQR, and find the equation of a plane by finding an orthogonal vector and put it into the formula for a plane.

01.06.02 (2:20) “computing cross products in the standard basis”.  How to remember the cross-product formula?  Use the standard basis vectors, then draw the cyclic diagram

examples: (1) find the cross product by using the standard basis vectors and expanding.

01.06.03 (2:45) “length of the cross product”.  geometric formula for the length of the cross product (a geometry problem).  the vector isn’t just orthogonal to the vec u vec u plane; its length also has a meaning.  use the cross product for a simple formula for a point to a line and the dot product for a formula from the point to a plane.

01.06.04 (3:53) “the scalar triple product”.  this product takes in 3 vectors and returns a scalar.  he gives the algebraic definition.  then he shows the cyclic repeat (5 columns) with diagonal slices for putting together the structure.  properties: there’s a cyclic permutation, anti-symmetry, and geometric meaning.  It’s the volume of a parallelepiped.

01.06.05 (1:12) “bonus: octonians”.  how about a way to multiply vectors together in another dimension?  the octonions work in the 7th dimension.  you can look them up…

01.06 (0:31) “the big picture”.  new products: cross product and scalar triple product.

Total time: 16:57

Notes on “Calculus Blue” Volume 1, Chapter 2

More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by “Prof Ghrist Math”.

Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space.  They also introduce the names and images for all of the quadratic surfaces (so we’ve left the linear world in this chapter).  This is a foreshadowing of the process of the course (start with the linear and then move to the nonlinear).

01.02.00 (0:35) “Curves and surfaces: Intro”

01.02.01 (5:10) “Implicit and parametric curves and surfaces”.  Two ways to define curves.  Implicitly or parametrically.  Implicit: “the solutions to an equation yields a curve in the plane”.  Parametric: “specifying coordinates as a function of a parameter”.  We’ll want to move between these representations.  Surfaces in 3d can also be expressed implicitly or parametrically (requires 2 parameters).

Examples: parameterization for a surface expressed implicitly (he shows the conversion where you set x = s, y = t, and then express z); go from a parameterization to an implicit equation (the example has a square root so he suggests caution).

01.02.02 (0:29) “Some examples please…?”.  it’s important to learn the quadratic surfaces.

01.02.03 (5:33) “Examples of quadratic surfaces”.  start with the sphere, then modify to get the ellipsoids, change a sign to get a hyperboloid, with two negatives signs get a 2-sheeted hyperboloid, then an elliptic paraboloid, hyperbolic paraboloids, cones (degenerate hyperbola), cylinders.

Then the narrator provides a reassurance: this isn’t about memorizing these or drawing pictures of these; it’s just worth recognizing them.  Then the narrator mentions these won’t come back for a while and names surface integrals as an application.  These are in there as a motivation to build up geometry and algebra skills.

01.02 (0:26) “The big picture”.  Previous chapter was lines and planes.  This one was curves and surfaces.  Progression from linear to nonlinear is what will happen in the course.