Common Core math was the pinnacle of their evil genius, and low-point of our complacency... It disrupted the minds of an entire generation on how even the most basic things work.

Readin', Writin', & Rithmatic

How far did they get? Let's see.

1895 8th Grade Final Exam, Kansas:

More from Maths

Saharon Shelah and number 4, a thread.

Shelah is an incredibly prolific mathematician working mostly in mathematical logic who has (co-)authored around 1500 papers so far. His other superpower is the ability to discover number 4 where it has absolutely no reason to be. 1/32

Example 1. "There are just 4 second order quantifiers".
In first-order logic, one is allowed to form statements about structures, such as graphs, groups, fields,
etc., with only quantification over elements allowed. 2/32

So you can say "exists x, ..." or "for all x, ..." with x ranging over the elements of the structure M, but you can't say "for all subsets of M, ..." or "for all binary relations on M, ...", etc. Allowing such
quantifiers puts us in the context of _second order logic_. 3/32

It was well-known in logic and computer science that one can express more complicated properties by quantifying over binary relations on M rather than just over the unary ones. But can we say even more quantifying over ternary relations? 4/32

And how does this compare to quantifying over functions, bijections, or any other infinitely many possible types of relations on M one can think of? In his 1973 paper Shelah proved that there are exactly 4 (four) possible types of second order quantifiers. 5/32

$Spencer Greenberg \U0001f50d$

Spencer Greenberg 🔍...
@SpencrGreenberg

Ten components of being compassionate to yourself:

1. Self-friendship: treat yourself at least as well as you would treat a friend who is going through the same situation

[thread]

2. Worthiness: remember that you have value (as all people do), and that your happiness and preferences matter.

3. Self-talk: notice when you are saying something mean to yourself (aloud, or in your head), and rephrase it in a gentler, kinder, more considerate way.

4. Connectedness: reflect on the fact that, whatever challenge you are facing, there are others who are also facing a similar challenge. You are not alone in your pain.

5. Change: note that everything changes with time. If you are dealing with feelings that are difficult to stand at this moment, they will subside. This too shall pass. Things are usually not as important or permanent as they seem to us in the moment that we are focused on them.

6. Self-empathy: orient towards yourself with tenderness, understanding and compassion, especially when you are suffering. Take on the perspective that you would towards a child or loved one that was hurting.

Algebra Etc.
@AlgebraFact

How to mentally compute the cube root of the cube of a two digit number.

Thread (1/10)

In base 10, the last digits of the cubes of digits are distinct.

If d = 0, 1, 4, 5, 6, or 9, then d^3 ends in d.

If d = 2, 3, 7, or 8, then d^3 ends in 10-d.

(2/10)

So if you're given the cube of an integer, you can figure out the last digit of the cube root.

For example, if 50653 is the cube of an integer, it's the cube of a number ending in 7.

(3/10)

Let n = 10a + b be a two-digit number, with a and b single digit numbers.

Then 10a <= n <= 10(a+1), and so n^3 is between (10a)^3 and (10(a+1))^3, i.e. between 1000 a^3 and 1000 (a+1)^3.

(4/10)

So suppose x is the cube of a two-digit number n = 10a + b.

Chop off the last three digits of x.

Then a, the first digit of n, is the largest number whose cube is no greater than what's left of x.

(5/10)

Arthouse-Kuklite
@Miltantvibes

The problems of 3rd grade mathematics work sheets are experienced by many children and their parents are usually frustrated. There is no more, there is information about how to solve the problem of your Mathematics task for children ... My suggestions will help.

Your child needs help with his homework and is useful if you as a parent know how to handle the problems of 3RD mathematics worksheets as follows:

Michael Kinyon
@ProfKinyon

Loops and their multiplication groups

A thread in 15 parts

(0/15)

Recall that a quasigroup (Q,*) is a set Q with a binary operation * such that for each a,b in Q, the equations a*x=b and y*a=b have unique solutions x,y. Groups are quasigroups and this property is usually one of the first things proved in elementary group theory.

(1/15)

Note that we don't assume associativity of *!

A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.

(2/15)

Natural examples of nonassociative loops:
- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples

Rethinking Vector Addition
or
How I Learned to Stop Worrying and Love Nonassociativity

A thread in 29 tweets

(0/28)
— Michael Kinyon (@ProfKinyon) December 1, 2020

For each x in a loop Q, define the left & right translations L_x, R_x : Q->Q by L_x(y)=xy and R_x(y)=yx. These mappings are permutations of Q. The composition L_x L_y of two left translations is not necessarily a left translation because Q is not necessarily associative.

(4/15)

A lot of bootstrappers think they have a marketing problem:

"I need to get better at marketing."

But, it's more likely that you have a product problem:

"Do customers really want this? Do they care enough about this to switch to a new solution?"

☝️ this is why most of my book, @marketingdevs, is about:

1. Choosing the right market
2. Building something they want

"If your product is remarkable, getting noticed is a lot easier." – @peldi

I was reminded of this concept again this morning while reading @pjrvs' book (Company of One):

"Sales increase when you honestly evaluate what someone needs and then teach them the value of what you're selling."

To succeed, your product has to offer an outcome that is highly desirable to a large group of customers.

The number of sales your product receives is a multiple of these two variables:

1. How big is the target market?
2. How desirable is the outcome you're offering them?

Patrick McKenzie
@patio11

Some people really benefit from hearing advice that everyone knows, for the same reason we keep schools open despite every subject in them having been taught before.

In that spirit, here's some quick Things Many People Find Too Obvious To Have Told You Already.

Your idea is not valuable, at all. All value is in the execution. You think you are an exception; you are not. You should not insist on an NDA to talk about it; nobody serious will engage in contract review over an idea, and this will mark you as clueless.

Technologists tend to severely underestimate the difficulty and expense of creating software, especially at companies which do not have fully staffed industry leading engineering teams ("because software is so easy there, amirite guys?")

Charge more. Charge more still. Go on.

The press is a lossy and biased compression of events in the actual world, and is singularly consumed with its own rituals, status games, and incentives. The news necessarily fails to capture almost everything which happened yesterday. What it says is important usually isn't.

Companies find it incredibly hard to reliably staff positions with hard-working generalists who operate autonomously and have high risk tolerances. This is not the modal employee, including at places which are justifiably proud of the skill/diligence/etc of their employees.

$Billy Bostickson \U0001f3f4\U0001f441&\U0001f441 \U0001f193$

Billy Bostickson 🏴👁&👁 ...
@BillyBostickson

@EricTopol @NBA @StephenKissler @yhgrad B.1.1.7 reveals clearly that SARS-CoV-2 is reverting to its original pre-outbreak condition, i.e. adapted to transgenic hACE2 mice (either Baric's BALB/c ones or others used at WIV labs during chimeric bat coronavirus experiments aimed at developing a pan betacoronavirus vaccine)

@NBA @StephenKissler @yhgrad 1. From Day 1, SARS-COV-2 was very well adapted to humans .....and transgenic hACE2 Mice

1. From Day 1, SARS-COV-2 was very well adapted to humans .....and transgenic hACE2 Mice
"we generated a mouse model expressing hACE2 by using CRISPR/Cas9 knockin technology. In comparison with wild-type C57BL/6 mice, both young & aged hACE2 mice sustained high viral loads... pic.twitter.com/j94XtSkscj
— Billy Bostickson \U0001f3f4\U0001f441&\U0001f441 \U0001f193 (@BillyBostickson) January 30, 2021

@NBA @StephenKissler @yhgrad 2. High Probability of serial passaging in Transgenic Mice expressing hACE2 in genesis of SARS-COV-2

1. High Probability of serial passaging in Transgenic Mice expressing hACE2 in genesis of SARS-COV-2!
2 papers:
Human\u2013viral molecular mimicryhttps://t.co/irfH0Zgrve
Molecular Mimicryhttps://t.co/yLQoUtfS6s https://t.co/lsCv2iMEQz
— Billy Bostickson \U0001f3f4\U0001f441&\U0001f441 \U0001f193 (@BillyBostickson) January 2, 2021

@NBA @StephenKissler @yhgrad B.1.1.7 has an unusually large number of genetic changes, ... found to date in mouse-adapted SARS-CoV2 and is also seen in ferret infections.
https://t.co/9Z4oJmkcKj

@NBA @StephenKissler @yhgrad We adapted a clinical isolate of SARS-CoV-2 by serial passaging in the ... Thus, this mouse-adapted strain and associated challenge model should be ... (B) SARS-CoV-2 genomic RNA loads in mouse lung homogenates at P0 to P6.
https://t.co/I90OOCJg7o

Richard James
@RJSHutton

I think a plausible explanation is that whatever Corbyn says or does, his critics will denounce - no matter how much hypocrisy it necessitates.

One of the oddest features of the Labour tax row is how raising allowances, which the media allowed the LDs to describe as progressive (in spite of evidence to contrary) through the coalition years, is now seen by everyone as very right wing
— Tom Clark (@prospect_clark) November 2, 2018

Corbyn opposes the exploitation of foreign sweatshop-workers - Labour MPs complain he's like Nigel

He speaks up in defence of migrants - Labour MPs whinge that he's not listening to the public's very real concerns about immigration:

He's wrong to prioritise Labour Party members over the public:

He's wrong to prioritise the public over Labour Party