thinking about The Math Myth

[kirby urner]

I only just grabbed this book for my Kindle last night,

and even though I took Evelyn Wood Reading Dynamics

(speed reading) when still in the Philippines, it'd be silly

of me to act like I'd fully digested this new-to-me book.

I've just been diving in, reading at various speeds.  I'm

planning to keep at it.

What I see right away is this title does indeed seem highly
relevant to our discussions on MathFuture of the strategies
different subcultures might pursue.  There's the future of

mathematics, as in mathematics of the future, and there's

the future of its teaching, with which mathematics is inter-

woven.

Remember the screen writers' strike?  When was that again?
A lot of Hollywood screenwriters were growing tired of
depressed wages for what were clearly big moneymaking
programs (rhymes with computer programs) and decided
to boycott writing "source code" (the screen plays).

The current generation of mathematics teachers were

somewhat promised a secure career, including as a stepping

stone for adjuncts and assistants, TAs, perhaps on a student

visa. 

Princeton comes in for special criticism, I think because

the author knows that institution prides itself on bringing top

professors in contact with undergraduates.  That was true in

my case, but then I was sent to Honors Calculus and later

went for Linear Programming (different from linear algebra),

and pioneering enough to merit being taught by one of the

field's pioneers.

However what senior professor wants to selfishly step

in the way of an adjunct hungering to maximize the long

term benefits of a very temporary Princeton assignment.

Having them jump into something customized just for them

is another way Princeton keeps the pipeline moving at a

higher level.  Something like that.

What's the solution?  Maybe specialize *even more* but

then insist senior professors do tours of duty in other

departments, completely alien to their own?  That'd be

like visiting Mars!  Might be good for 'em?  The adjuncts

and assistants will still need their ways to practice, but

if the goal is to give newcomers a front row seat, we

should let the professor really put on a show.  Keep it

niche.  Doesn't mean it can't be comprehensive inside

its scope (e.g. like GST is).

Making math a "hard requirement" has fueled an

ecosystem wherein mathematics teachers enjoyed a

somewhat protected status, even though their discipline

ranks at the top of those failed.

This consensus is now breaking down, the author argues.

People are no longer buying the play, the program, the

thing these screen writers write.  Their wares have eroded

in value to an egregious degree, in the minds of some,

whereas it seems like we'll still enjoy the status quo through

tomorrow at least, to others.

What is this status quo?  The one that makes mathematics,

and not computer science a requirement?  The one that

still permits no coding?

I think we had a window wherein at least in Oregon we

could have legally had innovation and the pioneering of

what I call the lambda calculus track (shop talks vary).

"Discrete Math" or "Computational Thinking" was indeed

math in the eyes of the legislature. 

The problem is, that was already true about three to four
years ago at least, and yet nothing really happened to
suggest the new freedom was being availed of.  The

courseware stayed static with teachers given no time

to innovate, more pressure to teach to the test.  Robot

World went barreling ahead with its brittle criteria for

success.

Lets posit the deadline was missed.  Math did have a

window to get over future shock and adapt, but the deltas
have not been impressive and the signals have already
been sent to the computer science people to come in
and clean up.  Some will liken them to circling vultures

but that's just mean.  They care about kids and not wasting

their time, as much as anyone.  There's no "cornering the

market" when it comes to caring about kids, or cute

animals either (in the eye of the beholder, observer
matters).

"Come take over our ship, we don't know how to change."

And now it's happening.  The window was missed.  That's

what it's starting to look like, at least around here.

I'm anticipating this "Math Myth" school of thought, with
eloquent avatars, is going to cut into the credibility of math's
treasured "required status" even more deeply, and this

will start to dry up much of the promised captive audience
of this particular brand of theater.  Lets check back later

and see if this was on the nose.

The audience is voting with its feet, shifting over to other
tracks, perhaps ungraded and over the Web.  We can
learn about pi versus tau from Khan himself (and Vi). 

Let us have our PWS cubicle and cafeteria and time to catch
up, and we'll learn a lot, say the Global U kids of tomorrow.

Obviously that's just some guy's analysis (i.e. mine). 
Based a lot on tonight's first impressions on reading
The Math Myth and hearing the voices therein.

As a philosophy student with a BA from Princeton and

an abiding interest in the diplomatic side as well (see

earlier autobio), I'm one of those PATH types intersecting

with STEM.  The book looks at taking the A from STEAM

-- i.e. enchanced STEM (Anthro added) -- and making
that A a part of PATH, with Philosophy the first letter. 

Like this (switching to fixed width mode):

   P

STEAM

   T

   H

As if playing Scrabble [tm].  The book doesn't actually

include that diagram I don't think, but it's what I thought

of, in constructing my view of what it says.

You'd think they'd really have to do that, keep that P, as
long as "Ph. D." is in the picture.  That Ph stands for
what exactly?  Acid content? 

So then should I feel secure in my job, as the philo guy
who sees RBF's Synergetics as a namespace per
Wittgenstein?  Is that gonna pay off somehow?  As

the CRO of what pray tell?  Global Data Corporation?

(More about GDC in my blogs).

I'm going to echo Pycon in saying Programming is

Performance Art.  Those passionate about their source

code see themselves as writing something much like
music, for playing on really fast machines. 

Yes, it needs to be precise but we have lots of tools to
catch errors, evenin logic, so it's not has horribly dull as
his book seems to make it ("coding" that is).  I'm all for
making Programming be one meaning for P, as a part
of PATH (the Humanities), as long as we remember
the connection between Programming and Theater.

We should value the work of our screen writers, especially

when the writing is good.  I'm not about to launch into some

program by program judgement at this point.  I've got my
blogs for stuff like that, or some years I read for OSCON.

Thanks again to Peter Farrell for bringing my attention

to this important title.

Kirby

[kirby urner]

On Fri, Jun 3, 2016 at 11:04 PM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>
 

Remember the screen writers' strike?  When was that again?
A lot of Hollywood screenwriters were growing tired of
depressed wages for what were clearly big moneymaking
programs (rhymes with computer programs) and decided
to boycott writing "source code" (the screen plays).

The current generation of mathematics teachers were

somewhat promised a secure career, including as a stepping

stone for adjuncts and assistants, TAs, perhaps on a student

visa. 

Not the greatest grammar...

I should add right away that I didn't finish sharing key

parts of the narrative.  The math teachers still have their

Mandarins i.e. their giants, who have for a long time

proved their worth in defending the pristine purity of

their subject.  They've wanted high school math teachers

to have full fledged math degrees, not education degrees,

as a way of insuring future demand for their subject.

The Mandarins will now need to realign to fend off

this threat from a new angle.  The Math Myth is chipping

away at some of the basic infrastructure.  Going with a

strong lambda calc track might postpone or even deflect

the on-boarding / invasion of Policy 1 (see above).  By

absorbing the new content, instead of conceding

territory to competitors, might be the lesser of two evils?

How it all turns out I don't know.  A lot depends on

what the Mandarins say and do.  I'll be looking for

their spokespeople to step forward.

Kirby

[kirby urner]

I spent a few more hours with The Math Myth yesterday.

The author ends with a pilot course we would say is

Data Science in the code school matrix.  Big Data,

Data Science, and Machine Learning have taken off

as the new buzzwords in Code School World.

I've also been boning up on Clifford Algebra more,

which incorporates both a "dot product" (typically

taught) and a "wedge product" ("exterior product")

rarely taught, into a new kind of complex object per

this framework they call an "algebra" (means there's

closure and associativity -- group properties people

like).

As Andrius likely knows, CA or GA (more general)

is able to carry out both tensor and quaternion based

algorithms thanks to all the machinery it contains.

The QM community thinks it just might be the bees'

knees (English idiom).

Translating into my personal namespace, I'd say

the Math Myth is advocating number-savvy of the

type that allows Comprehending and Critiquing

(two of the three Cs), especially around sifting

through data.  Lets introduce databases for saving

sifted data and the querying thereof, followed by

their presentation in a Jupyter Notebook.

Lambda Calc Track:

after CS-friendly Algebra (one hopes) you may
complete your 3 year (or even 4 year) high school

diploma requirement with:

Data Science and Public Policy (civics spin)

Probability and Simulations (programming goes here)

Graphic Arts and Communications (includes Fuller.4D)

Staying Safe in Cyberspace (computer security)

By CS-friendly Algebra I mean a few things:

(1) not all functions use numbers (lexical domain used)

(2) a REPL is introduced (calculator was a first step)

(3) functions of more than one step are saved and reused

(4) Euclid's method is introduced

(5) includes Euler's function "the totient" of N

(6) introduces number bases other than 10, esp. 2 (binary) & 16 (hex)

(7) friendly to cryptography as a topic, also standard encodings (QR codes)

If you don't know what a QR code is, it's that little square

of funny black squiggly patterns that encodes an alphanumeric

string.  Universal Product Codes also important, bar codes of

any kind (picture classrooms with bar code readers at some

of the workstations -- Montessori-style work-play stations).

We could do the little bit of group theory and operator overloading

right here in Algebra, with understanding both private and public

key cryptography a motivating narrative (comes with a timeline).

Staying Safe in Cyberspace is from a course I'm following on

Wikieducator, which is where my Digital Mathematics is also

filed. 

http://wikieducator.org/User:Vtaylor/Learning_literacies/Protect_it

Digital Mathematics breaks it down into four flavors of
mathematics the innovating teachers may wish to blend:

Historical Dimension:  Neolithic to Space Program

Today's Applications:  Supermarkets to Investment Banking

For example QR Codes would feature in supermarket math as

a way of tracking user bitcoin accounts.

On Fri, Jun 3, 2016 at 11:04 PM, kirby urner <kirby...@gmail.com> wrote:

I only just grabbed this book for my Kindle last night,

and even though I took Evelyn Wood Reading Dynamics

(speed reading) when still in the Philippines, it'd be silly

of me to act like I'd fully digested this new-to-me book.

I've just been diving in, reading at various speeds.  I'm

planning to keep at it.

What I learned from the Evelyn Wood course was to

adjust speed to content type, also to study figures and

diagrams ahead of time, and (most important), close the

text and do a "recall" afterward, using diagrams and

graphs (in the sense of networks) not going for any

verbatim recall.

As a philosophy student with a BA from Princeton and

an abiding interest in the diplomatic side as well (see

earlier autobio), I'm one of those PATH types intersecting

with STEM.  The book looks at taking the A from STEAM

-- i.e. enchanced STEM (Anthro added) -- and making
that A a part of PATH, with Philosophy the first letter. 

Most of those playing on adding A to STEM to make

STEAM are using Art for that purpose.  I find Anthropology,

especially of corporations (such as banks), helps people

learn the ways of the business world more successfully,

whereas Art is just one aspect of artifact-making (what we

look at in Neolithic Math a lot).

However, the core strategy in both cases is to remind

everyone that STEM seems to leave out the Humanities

i.e. all the liberal arts subjects, such as philosophy and

so on.  Was the idea we didn't need them?  No, the idea

was to get the STEM house in order so that it would be

able to bridge to STEAM whatever the A meant.

 

Like this (switching to fixed width mode):

   P

STEAM

   T

   H

As if playing Scrabble [tm].  The book doesn't actually

include that diagram I don't think, but it's what I thought

of, in constructing my view of what it says.

You'd think they'd really have to do that, keep that P, as
long as "Ph. D." is in the picture.  That Ph stands for
what exactly?  Acid content? 

So then should I feel secure in my job, as the philo guy
who sees RBF's Synergetics as a namespace per
Wittgenstein?  Is that gonna pay off somehow?  As

the CRO of what pray tell?  Global Data Corporation?

(More about GDC in my blogs).

 

Getting synchronous and asynchronous (timeline based)

global data displays is a longstanding interest, Google Earth

a reflection of what I mean.  I was an early adopter of

hypertext also, ala Ted Nelson's Computer Lib / Dream Machines,

which led me to propose like Youtube (Videogrammatron).

Nowadays its the refugee camps and displaced human / human
mobility space that has my attention.  They need data displays

reflecting their human condition.  It's not like they don't exist.

Pycon introduced me to a number of people working on getting

more relief to these under-served Global U students, at all age

levels.

Kirby

[Joseph Austin]

On Jun 4, 2016, at 2:04 AM, kirby urner <kirby...@gmail.com> wrote:

yet nothing really happened to 
suggest the new freedom was being availed of.  The 

courseware stayed static with teachers given no time 

to innovate, more pressure to teach to the test.  Robot

World went barreling ahead with its brittle criteria for 

success.

I've been having a discussion with a friend from the corporate world re: changing the education system.

He sent me the following perspective:

Quote:

Let me put it this way. ENIAC was introduced in 1946. In 1986, I was banned from using a calculator while taking an graduate level econometrics exam at ____ University. Introducing a "counting aid" is beyond simplistic compared to changing an educational system. It's IBM on steroids when they were leasing mainframes.

From my experience in revamping training strategies - less than 33% of existing teachers would be able to learn - embrace new skills and processes. Universities would be hard pressed to develop and deliver degree programs preparing new teachers. Roughly 10% of parents would be able to support changes even if they sought to. Society would be totally unprepared for coping with the success should all of the obstacles be met. 

I'm one of the sharpest knives in the drawer - and I have trouble envisioning the integrated effect on our way of life should the majority of people suddenly have thinking skills. Every aspect of industry would have to change - not to mention politics, religion, etc. During the transition, revolution would occur between the thinking and the non-thinking. The discipline and patience required of the thinking to tolerate the non-thinking is significant. 

It's not a matter of changing school curricula - you're talking about changing the fundamental manner by which "mankind" exists. It wouldn't take long for Scripture to be prophetic - with the first coming last syndrome. "Think tanks" and research facilities would be made to look ignorant - or stupid - or both. The level of satisfaction in all things would be under attack. Everyone would be observing all the things occurring around them - and the wastefulness of effort and resources would be more than their good nature could handle. Every bastion of authority would be challenged - and the humility required to surrender habits and known strategies for success would travel much slower than the speed of their obsolescence.

You're talking about chaos on a scale the world has never imagined.

[kirby urner]

On Wed, Jun 8, 2016 at 5:23 PM, Joseph Austin <drtec...@gmail.com> wrote:

<< SNIP >>
 

You're talking about chaos on a scale the world has never imagined.

Interesting quote Joe, your CEO friend is quite eloquent, regardless of whether one subscribes to these views.

Sure does sound apocalyptic, but without weighing against whether his status quo and doing nothing to equalize, would be any less catastrophic.

What if a lot of this future shock is already behind us?

Fuller anticipated a ten year "design science revolution" pulling us back from the brink.

What happened in fact was the free software movement, still rolling ahead, finding and/or carving its channels.

Admittedly we failed to meet the Hunger Project's goal of eliminating death by starvation as a major statistic by the year 2000.

Yet we're poised on the brink of spreading more services to the world's under-served.

I find Apocalyptic talk somewhat wearying, sometimes used as an excuse to not make a coherent plan.

It almost seems as if coherent planning is what those spreading fears of an Apocalypse are most wanting to avoid.

Why?  Is that what spreading thinking skills to everyone would look like?  Better planning?  More self organization?

I'm not persuaded the obstacles are all that serious.  That 33% percent he mentions is a pretty high and encouraging number.

Kirby

 

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[Joseph Austin]

My own unscientific, anecdotal observation is that institutions seldom reform,  

but are often superseded.

"Schools" as we know them will be replaced as other forms of education take hold.

In my locale there is a small but enthusiastic homeschool movement,

including non-trivial but hands-on technical education such as First Robotics and Natural Math.

Accredited for-profit schools and Charter Schools are making a dent the the traditional school landscape.

"Certifications" are replacing "degrees" in some technical careers.

The Internet, particularly YouTube, is a ready source of instruction on most academic and practical subjects.

As the internet eliminates the necessity for physical proximity of student and teacher, the entire premise of "local schools" and "college selection" is being upended. (My state is already wrestling with the implications of on-line courses to local school funding.) The tipping point should come when the YouTube generation  becomes the decision-making generation. 

Planning is prudent, but I wonder, with technology advancing so rapidly,

can we even plan effectively?  The quest for cross-discipline general principles seems a reasonable pursuit. Perhaps the best advice is: try everything; keep what works.

Joe

[kirby urner]

On Thu, Jun 9, 2016 at 12:35 PM, Joseph Austin <drtec...@gmail.com> wrote:

My own unscientific, anecdotal observation is that institutions seldom reform,  

but are often superseded.

"Schools" as we know them will be replaced as other forms of education take hold.

In my locale there is a small but enthusiastic homeschool movement,

including non-trivial but hands-on technical education such as First Robotics and Natural Math.

Accredited for-profit schools and Charter Schools are making a dent the the traditional school landscape.

"Certifications" are replacing "degrees" in some technical careers.

The Internet, particularly YouTube, is a ready source of instruction on most academic and practical subjects.

I can report similar trends in my neck of the woods. 

Privileged kids with their own PWSs (personal workspaces) at home,
with lots of bandwidth, have an ever easier time persuading their
guardians to shop on-line for private tutors. 

Some personality types resist being herded into classrooms, and
with the Internet what it is, they have a good argument to stay
home based, perhaps doing college-level work. 

Why pay the private school that provides no math-through-coding,
when on-line academies are so much farther along?  Just do the math.

On the other hand it's not either / or and many personality types
crave a more conventional high school experience, and indeed
the academic component may be entirely secondary (what's meant
by "secondary school" right, studies come second? :-D).

People differ in their foci. 

Some love to study exotic topics and learn way more Quantum
Mechanics than most seventeen year olds find time for.

Others are basking in the latest fashions and designing web site
front ends with all the latest bells and whistles. 

Many permutations, millions and billions really, exist.

Are we saying only one is "right" the others "wrong"?  If
that's the case, then is it our "right / wrong" axis that's flunking
out?  How do we "accommodate diversity" without making

that an ugly euphemism for "enforcing uniformity"?

 

As the internet eliminates the necessity for physical proximity of student and teacher, the entire premise of "local schools" and "college selection" is being upended. (My state is already wrestling with the implications of on-line courses to local school funding.) The tipping point should come when the YouTube generation  becomes the decision-making generation. 

The Internet does not eliminate the need for safe, personal (meaning

also private) workspace conditions, where Internet access is not so

supervised that no learning takes place. 

If all you do is stare at a security camera and watch people with
and without suitcases walk around, you don't really have Internet
access, even if you're using tcp / ip over ethernet and optical fiber.

I'd say "access to an Internet lifestyle" is what eliminates a lot of the

need to bus to some dreary school across town, where study-time
was once in theory made available. 

If you're lucky enough to have an Internet lifestyle, then you can
choose to watch Terence Tao for several hours and not have to
apologize to others or face the charge of "not earning a living"
(a hard charge to refute when you're just trying to pick up some
basic facts, catch up on history -- which is why young adults in
school are "not expected to earn a living" quite yet (that's a lot
of "crazy English" -- as in "hooey" -- if you ask me)).
 

Planning is prudent, but I wonder, with technology advancing so rapidly,

can we even plan effectively?  

I think the two choices are plan or panic.  We like to throw up are hands

and say, "bring it on" without even know what that means.  Or at least

some people seem to enjoy that approach. I call that "inviting panic".

True though, the future never pans out exactly as we expect, so
planning has to include re-planning or re-computing as a perpetual
element. 

It's not like you plan and then sit back, planning done. 

It's that one's life in the present is in the attitude of "one who plans"
since the only inevitability, sometimes, is the onrush of "the future". 

We each get a time tunnel to ride through.  Theme Park metaphor.

 

The quest for cross-discipline general principles seems a reasonable pursuit. Perhaps the best advice is: try everything; keep what works.

Joe

We should keep in mind that even within a largely uniform scene,

where it seems at first glance that everyone is stuck in the same

rut, marching to the same drummer, we'll have exceptions,

alternatives, experiments, new trials.

A key question then becomes, does the ambient culture nurture

or squelch all such deviations from the norm?  If the ambient

culture is brittle and nervous, prone to panic, it's probably a

what I call a Robot World, or maybe call it AI. 

Those worlds have a way of ending suddenly, but not always
in a great cataclysm.  They just blow away as more dust in
the wind, while people move on to some next new thing, ever

forgetful of yesterday's fads.

Planning degenerates into the rigid pursuit of some mindless

agenda if overly isolated from any feedback, and insulated from
an innate willingness to learn.  This was Einstein's definition of
insanity as stated earlier:  continuing with the same
no-longer-working reflexes (habits of thought) and expecting
them to still work (produce useful results).

Sometimes a plan bombs or backfires and yet those who
hatched the plan are so far removed from the action they

hardly know anyone considers their plans to have been

implemented.  They don't recognize the plan as theirs, so

where's the feedback?

With intelligent planning come workflows and ways of
coupling feedback or outcomes, to operations, undertakings,
enterprises. 

This is all the stuff of business management, no need to
reinvent all those wheels. 

In GST (general systems theory) we might speak of the
"be -> do -> have" projects completion cycle, wherein we
dream up and intend for something (be), do what we think
is needed, and either then have or have not the intended
result. 

What keeps it interesting is in adding to what we Have, we
enable new possibilities at the Be end (have -> be).  For more
shoptalk, see www.grunch.net/synergetics/gst1.html

Kirby

[kirby urner]

I feel I should add to this Math Myth thread which I started,
that his book and arguments are hotly debated, and in many
corners roundly denigrated. 

http://www.scientificamerican.com/author/evelyn-lamb/

My mostly sympathetic comments likely put me in a minority,
especially among those with a head for math.

I read Hacker not as saying we need "less math in schools"
but "more kinds of math, more choices".

Obviously that would resonate with me, as a "Bucky disciple" who
preaches incessantly about "A and B modules" (2As + 1B = Mite;
Mite x 8 = Coupler, unit volume space filler and blah blah).

With the advent of the Internet and people free to browse / explore,
it's more difficult to keep everyone "on the same page".  However
what's still quite useful are maps of the terrain, guides, orienting
materials.

My harping on "three kinds of 4D" was along those lines:  giving
a map or guide to three 20th Century lineages that continue into
the 21st.

Kirby

[Daniel Finkel]

Kirby -
There was a thoughtful post on The Math Myth by Keith Devlin: http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html

The main idea is that Hacker has a number of excellent observations to make about the state of math education, but arrives at problematic conclusions.

[kirby urner]

On Fri, Jul 1, 2016 at 10:55 AM, Daniel Finkel <d...@mathforlove.com> wrote:

Kirby -
There was a thoughtful post on The Math Myth by Keith Devlin: http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html

The main idea is that Hacker has a number of excellent observations to make about the state of math education, but arrives at problematic conclusions.

Thank you Daniel, I just finished reading it.

Keith Devlin was also at our Oregon Math Summit back in the day. [1]

My followup question for Keith would be:  since you and Hacker both agree that whatever they're calling Algebra in K-12 today is not the "right stuff", should it nevertheless be a required subject?

They seem to agree the status quo is unacceptable, but diverge on whether Algebra should stay or go.

Hacker:  Algebra is a confusing mess and should go.

Devlin:  what they're calling Algebra may be a confusing mess, but that's not really Algebra.

OK....  so what's next?

My own answer has been:  pioneer what I call "CS-friendly algebra" which brings in a lot of the coding Devlin also wants to bring in.

Kirby

[kirby urner]

Keith Devlin was also at our Oregon Math Summit back in the day. [1]

Oops, here's that link:

http://worldgame.blogspot.com/2016/06/math-summit-revisited.html

A look-back, with a link to a write-up I did closer to the time (1997).

 

Kirby

[Joseph Austin]

Kriby,

I've been thinking about an answer to this topic which I plan to post in Andrius's "What is Geometry" thread.

Or maybe start my own "Coding IS math" thread.

Stay tuned.

Joe

[Daniel Finkel]

I think the consensus in the math ed community is: we should keep algebra, but begin reforming it, as well as the entire K-12 math sequence. Then there's a huge conversation about what it should actually look like.

To stick with algebra, some people say that the content is relatively good, but you just need to teach it in a more compelling way. I might put Dan Meyer and James Tanton in this group. Others, like Conrad Wolfram, say that you actually need to scrap the content and substitute something else, e.g., a computer-based approach.

There's often a fair amount of overlap between these sides, too.

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[Joseph Austin]

I think the consensus in the math community is irrelevant.

What if we taught only what the students would immediately use?

And who determines that?  Certainly not the teachers, who have a vested interest in teaching what they already know.

I think the fundamental fallacy is that we must "prepare" students with everything they will need to know for the rest of their life by the time they leave school.

But this ignores two fundamental changes in society:

1. the knowledge explosion: too much to know, some will become obsolete, and new things will arise that we don't even know now.  

2. the internet: whatever I need to know, I can find out when I need to know it.

So we should stop teaching subjects and start teaching thinking.

Including quantitative thinking.

Joe Austin

[z...@unizor.com]

We are talking about math education, don't we. And how many times have any of you, not teaching math, but working in some other capacity, had to solve quadratic equation or graph an exponential function?

I doubt that these skills were really helpful in your day-to-day life. My guess is that 90% of people never use anything from the math courses they've studied in schools.

So, what's the purpose to teach our students math? The main purpose is to DEVELOP THEIR MINDS. It includes analytical thinking, logic, creativity, intelligence, inquisitiveness and many similar qualities absolutely necessary for a productive member of a society.

Plain facts and properties memorization won't do it. And we don't have to infinitely widen the spectrum of subjects we learn in school. What we have to do is to deepen their understanding, which includes proving theorems, solving problems, finding common properties in different objects for the purpose to abstract these properties into  a mathematical model, finding interdependencies between entities and operations with them etc.

Zor Shekhtman
Founder of Unizor Education 
Creative Mind through Art of Mathematics
http://www.unizor.com 

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[Anna Roys]

I concur.

How to think about thinking, needs to be modeled so that students may escape from echoing other people's flat uninformed, one-sided thinking.

I think one of the best gifts I offer my students is to share my thinking from multiple perspectives,  so they have the opportunity to develop this kind of thinking on their own. They seem to forget the content unless they keep it fresh.
Anna

[Joseph Austin]

Zor,

"Why?"

That is the question we must answer.

We don't do things unless we have a reason, a motivation.

The best motivation is an immediate need.

As it happens, I discovered the quadratic equation before I knew what it was.  Because I had a need.

As freshman in HS, I wanted a sum of money from my dad.

We were poor, so he proposed to give it to me in installments:

1 cent the first day, 2 cents the second day, 3 cents the third day, etc. until I had the full amount.

I wanted to know how long it would take for me to get the total amount.

By creating a crude "bar chart" by stacking dots, I was able to derive a relation between the base and volume of the triangle.

The equation I came up with involved N squared and N.

Having had only the first few weeks of algebra, I didn't yet know how to solve an equation with N squared.  I didn't even know whether it could be done.

But being painfully shy, I didn't have the courage to ask my math teacher about it.

Since then, I learned enough math to begin graduate studies in physics.  Because as a kid, I became interested in science,

via wiring my electric trains, then building electronic kits, and trying to figure out how things like radios and calculators work.

Then in a later job I re-derived (someone else had done it before, but I couldn't find the paper--before internet) the algorithm for computing a Fast Fourier Transform with 4 points at a time without requiring a post-sort of the values.  

Which is to say, I did actually use math in my job.

But then, I didn't teach "math", unless you concede that Computer Science is Math.

But suppose we taught only what math a student could immediately use.

I've been thinking, a math sequence might go something like this: (off the top of my head--I've probably missed some key topics)

1. Consumer Math: construction math (areas, angles), compound interest, amortization schedule, computing probabilities, interpreting statistics

2. Business Math: making estimates, budgeting, projections, ROI, life-cycle cost, random processes and queues, optimization, etc.

3. Political Math: developing and interpreting statistics and analyzing trends; fiscal policy; monetary policy; trade policy

(Citizens don't typically do these things themselves, but these are the "issues" on which political campaigns are waged,

so citizens should be equipped to evaluate the issues.)

4. Science Math: the math we usually teach everybody, but that only professional scientists and engineers actually use.

The point being that we face decisions all the time that have quantitative components.

We need to teach students to think quantitatively, to ask "how much?" as often as "how" or "why?".

We also need to teach "conservatism" in the literal sense of the word:

that  finiteness of resources constrains possible solutions: if you spend more here, you must spend less there!

The "conservation" laws of motion and energy are the bedrock of science,

but we seldom see "conservation" problems in math.

For example, instead of learning sums as sequences, e.g.: 1+1, 1+2, 1+3, 1+4, 1+5

I would teach them as, for example: what are the ways to make 5?

E.g. 5 = 0 + 5 = 1 + 4 = 2 + 3 = 3 + 2 = 4 + 1 = 5 + 0

Joe

[kirby urner]

On Sat, Jul 2, 2016 at 6:26 AM, Joseph Austin <drtec...@gmail.com> wrote:

Zor,

"Why?"

That is the question we must answer.

We don't do things unless we have a reason, a motivation.

The best motivation is an immediate need.

<< good story about the pennies, got me thinking >>

 

But suppose we taught only what math a student could immediately use.

I like the "immediate use" mantra and like to imagine ways this "immediate" blends with longer term needs.  We want students to both "collect the dots" and "connect the dots" over time.

Rather than think in terms of "credits" in only the conventional sense of counting towards a degree or credential, I'd like to wire up systems that provide immediate rewards within "work / study scenarios" in the form of "pennies".  Credits move closer to currency without necessarily becoming "pure cash".

Now that we have "programmable money" in the form of crypto-currencies, a core idea is to issue coins that don't have the same leeway or liquidity as just cash.  

The "programmed money" comes "with strings".  Two examples of common "strings" (constraints): 

(a) it belongs to a specific user such that only that user may spend it

(b) it's only good as a means towards particular ends, i.e. it may be "redeemed" only in specific ways

Say I live in a large camp of people seeking to escape untenable biospheric conditions.  We have millions upon millions of such people today. [1]  

If I study "learning to read" on a app, on my cell or tablet, and I go through some "levels" in so doing (I'm logged in and my progress through the "learning to read game" is auto-tracked), then my wallet receives 0.001 which is actually enough to use against a catalog of items of practical use to people living in such a camp.

The same generic model, of rewarding students with access, including experiences such as field trips, additional courses, catalog items such as microscopes and computers, transfers to other contexts.  

We could design and play similar "language games" in a big city, perhaps newly constructed to accommodate people leaving older ones, perhaps going under water.  

Login and learn about the city transportation system, pass some tests, do some readings, and you're able to get some bus tokens in exchange.  

Just sitting in a coffee shop and going through some reading programs, you'll be able to earn enough bus tokens to get to a next stop.  

You didn't earn cash, you earned something much more specific and tied to your identity in that it's your device that has the wallet. Learning what kinds of "money" may be transferred wallet to wallet is part of the game.  Some can be, some can't be.  

Some money is only good inside that coffee shop, for baked goods and coffee.

I've entertained such fantasies for years, however with the emergence of "programmable money" and micro-payments, I'm seeing the realism increase.  We are in a position to invent any number of "currencies" and experiment with their disbursement in the context of architected "learning experiences" (knit together, we call these schools).

Studying (= working) enough every day to at least be able to swipe one's phone to gain access to a cafeteria should be easy, and you're welcome to bring guests, such as younger siblings.  Learning in your tent is a way to help keep your family fed.

I'm not suggesting some vast global system that top downers extend to us from some nebulous world government in the cloud.  I'm talking about individual communities partnering with vendors and providers to experiment with new ways to "make math relevant" (here I'm using "math" to mean "whatever content" which for simplicity we could say is "all programming" or "stuff" -- like when we say "TV programming" to mean "whatever's on" -- so it's "all math" in that sense, there ain't nothing but).

Although there are science fiction aspects to the above, no new technology is needed that we don't already have today.  We have all the props to stage such theater.  

Also, the model maps easily to what we already call "work" in the sense the adults "do math" (shorthand for "do work") and "get paid". In a sense, we're already all Global U students learning math and trying to survive, one day to the next (also working), thereby.

I'm not suggesting the world adopt this peculiar usage of "math" to mean "everything" (including breathing and sleeping).  That's like saying "it's all data" or "it's all energy" -- these grand sweeping statements may sound "deep" at times, "shallow" at other times, if just left hanging (floating) without a context.

Rather I'm creating a namespace wherein we're free to use "math" to mean "all programming" which helps us think in a more generic way about how we might "channel programming" (sounds like TV again) such that human beings have "immediate needs" met (for food, clothing, shelter, information, guidance).  Their needs are provided for as a result of "learning / doing math" (thereby making it their own).

Kirby

[1]  http://www.independent.co.uk/news/world/europe/refugee-crisis-migrants-world-day-un-a7090986.html

[Joseph Austin]

Devlin's suggestion of teaching Spreadsheet following Arithmetic (instead of going directly to "school algebra")

is probably the most practical means to introduce "math" (and computers) into the everyday and career life of the student.

Furthermore, Spreadsheet is a tool that is ubiquitous in businesses, schools and even the home,

and is available on tablets and handhelds as well as computers 

(though many homes with "Office" apps might not realize what to do with it, or even that it's there.)

For the poor or frugal, there are free apps such as  Open Office.

Spreadsheets have all the power of the most sophisticated calculators.

Besides, spreadsheets introduce concepts of "variable" names, formulas/expressions, row-column indexing, recurrence relations, conditional expressions, and, via Solver, "numerical" or iterative search methods, and even the capability to illustrate approaching limits, and even relational database concepts.  Moreover, spreadsheets offer a variety of charting and plotting capabilities for  "graphing calculator" approaches, and even relational database concepts.  

As a vehicle for introducing both symbolic representation and algorithmic process, the spreadsheet would be a good transition to further study of both Coding and Symbolic Manipulation (algebra).

When I taught spreadsheets, I would encourage students to work problems relevant to their life as students:

formulas they might use in another class or a sport, spreadsheets for a Girl Scout Cookie sale or an athletic competition or a budget or a credit-card payoff schedule.  One spring I sent them to HR to get tax forms and then compute their family income tax.

Joe

[Joseph Austin]

On Jul 2, 2016, at 10:52 AM, kirby urner <kirby...@gmail.com> wrote:

Rather I'm creating a namespace wherein we're free to use "math" to mean "all programming" which helps us think in a more generic way about how we might "channel programming" (sounds like TV again) such that human beings have "immediate needs" met (for food, clothing, shelter, information, guidance).  Their needs are provided for as a result of "learning / doing math" (thereby making it their own).

Kirby,

I was thinking in terms of actual needs. 

What I'm saying is, who needs to solve for "x"?  But everybody can relate to solving for "cans of beverage".

We really need to make decisions all the time.  Many, perhaps most, of those decisions involve significant quantitative components: how much money or time or other resources will each option cost vs. benefit?  At what level of satisfaction or reliability or longevity?  To what relative advantage?  How many people will be involved or affected?  What are my chances?

People don't realize they are using "algebra" when solving for cans of beer.  

Partly because when they were first taught how to solve for, well, cookies, 

we called it "word problems" instead of calling it "algebra."

And then, when we finally decided to teach the process we had previously expected them to learn by osmosis,

we called it "algebra" but solved for  "x," 

instead of telling them we were going to show them what they were really doing when they solved for cookies.

Joe

[kirby urner]

On Sat, Jul 2, 2016 at 8:50 AM, Joseph Austin <drtec...@gmail.com> wrote:

Devlin's suggestion of teaching Spreadsheet following Arithmetic (instead of going directly to "school algebra")

is probably the most practical means to introduce "math" (and computers) into the everyday and career life of the student.

I agree with you that spreadsheets are highly practical.  

They create the "rows and columns" we'll need to think about in our work with databases.  As you point out, a spreadsheet is like a database. 

Let's assume the strategy of "spiralling" i.e. we'll be coming back to spreadsheets, over and over.

In the 1980s, part of my job was to teach the "big three" business applications to "older workers", meaning people even younger than me now, now that it's 30 years later.  

The "big three" were:

  

(A) wordprocessing 

(B) spreadsheet 

(C) database.  

In the early days we started with AppleWorks on a IIe.  Those were not prevalent "on the job" and soon we migrated to the PCs, running (A) WordPerfect (B) Lotus 1-2-3 and (C) dBase II.

The database was the most difficult to comprehend and was starting to involve more programming.  The "macros" in Lotus (WordPerfect had macros too) were a jumping off point into dBase.

I was already a dBase programmer by then and also worked for clients, mostly nonprofits, helping them set up their little membership tracking systems and so on.  My wife, whom I met through this job, would offer bookkeeping services.

What both my wife and I would emphasize to the workers we trained, and the clients we took on, was that spreadsheets were not a good place to keep books, because too volatile, to easy to change without leaving a record of those changes.  

Bookkeeping should be done with a database.

Ever since those days (1980s-1990s), I've been a strong advocate for teaching SQL (structured query language) as a part of some Practical Math curriculum, including at the high school level.  I don't think it should be made a "computer science" thing.  Rows and columns are too basic to be tucked away in some rarified elective course.  SQL is mainstream.

So in today's terms, my "spiralling" would treat (A) wordprocessing, as the whole challenge of getting stuff to the web, meaning HTML / CSS would get taken up, as elementary topics.  

[ When I say "elementary topic" I don't always picture little kids in sitting in a classroom, teacher up front.  I think of a dad huddled over a tablet in refugee camp somewhere, trying to get a handle on what's going on.  I.E. "high school level" or "college level" does not necessarily imply students of any particular chronological age ]

(B) spreadsheets would be a surface in which to do a lot of tabular / statistics style math, start working with medium sized data sets, start seeing "records" in tabular form and doing "transformations" on them

(C) databases would help glue (A) and (B) together was we learn how to use a web page to GET and POST to a SQL database in the background.  That's what code schools are all about and it's all math.

However I'm well aware that not all databases are designed around SQL.  As we go around the spiral again and again, we learn about noSQL e.g. graph databases.

Back in the 1980s, if someone were to read all this they'd say "that's all fine and good but how are schools going to afford all that?  A legal copy of Lotus 1-2-3 is like $300 and dbase is even more, schools just don't have that kind of budget."

What happened since was the Free / Open Source revolution, which is not only about making software more affordable and accessible, but about making the guts modifiable, so when you want to customize, you don't have to reinvent the wheel, you can start where someone else left off, provided you have that kind of literacy.

Something like a Jupyter Notebook integrates a lot of the elements of (A)(B) and (C).  I can create a SQL table in a "cell" (spreadsheet terminology) and "run the cell".  I can surround executable cells with writing.  There's a top-to-bottom flow.

My Polyhedrons 101 Jupyter Notebook, linked from this home page:  http://4dsolutions.net/ocn/

shows what it looks like to create and populate a SQL database to store polyhedrons in row-column form, relationally.  I'd call this high school level, well along on the lambda track, with hints of college.

Kirby

[kirby urner]

On Sat, Jul 2, 2016 at 9:14 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Jul 2, 2016, at 10:52 AM, kirby urner <kirby...@gmail.com> wrote:

Rather I'm creating a namespace wherein we're free to use "math" to mean "all programming" which helps us think in a more generic way about how we might "channel programming" (sounds like TV again) such that human beings have "immediate needs" met (for food, clothing, shelter, information, guidance).  Their needs are provided for as a result of "learning / doing math" (thereby making it their own).

Kirby,

I was thinking in terms of actual needs. 

What I'm saying is, who needs to solve for "x"?  But everybody can relate to solving for "cans of beverage".

We really need to make decisions all the time.  Many, perhaps most, of those decisions involve significant quantitative components: how much money or time or other resources will each option cost vs. benefit?  At what level of satisfaction or reliability or longevity?  To what relative advantage?  How many people will be involved or affected?  What are my chances?

We're on the same page Joe.  Actual needs.

Jack and Jill have an actual and immediate need for money (in your story, it was all about pennies from your dad).  I haven't lost sight of that fact.  

In making "math" mean "all programming", I get to think outside the box and talk about hygiene and sanitation:

Clearly these are very generic concepts, intentionally.  In application, we might focus on a Global U student in a tent, call him Jack, call her Jill (is she a girl scout?).  Jack is learning about germs i.e. bacteria and viruses, while Jill is learning about environmental contaminants.  They're siblings and definitely compare notes on what they're learning as these topics overlap. 

These are also topics of vital everyday interest as the camp struggles to improve.  Jack and Jill both want to make a difference in terms of contributing to higher living standards for the camp.  One way to do that is to stay healthy.  Another way is to help others stay healthy.  A series of bike lanes has been set up and Jack spends some of the day delivering specific meds to specific tents.

That's from a blog post of this morning, linking back to this thread:

http://controlroom.blogspot.com/2016/07/camp-life.html

 

People don't realize they are using "algebra" when solving for cans of beer.  

Partly because when they were first taught how to solve for, well, cookies, 

we called it "word problems" instead of calling it "algebra."

Lets call it all "math stuff".

It's one big computation after all, a lot it done by "meat" with no need for brains.  Cells and silicon, all computing together in one big glorious ball.

This was the subject of one of Keith Devlin's books, where he talked about birds and ants being "mathematicians".

https://www.amazon.com/dp/B001T4ZAYW/
The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs)

That's all you do, from cradle to grave (actually from before cradle to after grave):  math math math.... sounds like hell, I realize. :-D Or maybe heaven if math is your thing.

 

And then, when we finally decided to teach the process we had previously expected them to learn by osmosis,

we called it "algebra" but solved for  "x," 

instead of telling them we were going to show them what they were really doing when they solved for cookies.

Joe

The virus often comes with an icosahedral structure.  Very stable, very strong.  

The virus is doing math math math.... (very practically, from one moment to the next).

Kirby

[Andrius Kulikauskas]

Zor, Joseph,

Another way to say this is that we want to teach "mathematical
thinking", which is qualitative, not so much quantitative, though
dependent on mastery of the quantitative.

A whole new world opens up when we realize that we can model our world
with different kinds of models, different kinds of functions, when we
can distinguish between linear, quadratic, polynomial, exponential,
reciprocal, periodic, etc. And, most importantly, when we appreciate
that all such models are only tentative, they break down at some point.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

>> On Jul 1, 2016, at 7:51 PM, z...@unizor.com <mailto:z...@unizor.com> wrote:
>>
>> We are talking about math education, don't we. And how many times
>> have any of you, not teaching math, but working in some other
>> capacity, had to solve quadratic equation or graph an exponential
>> function?
>> I doubt that these skills were really helpful in your day-to-day
>> life. My guess is that 90% of people never use anything from the math
>> courses they've studied in schools.
>>
>> So, what's the purpose to teach our students math? The main purpose
>> is to DEVELOP THEIR MINDS. It includes analytical thinking, logic,
>> creativity, intelligence, inquisitiveness and many similar qualities
>> absolutely necessary for a productive member of a society.
>>
>> Plain facts and properties memorization won't do it. And we don't
>> have to infinitely widen the spectrum of subjects we learn in school.
>> What we have to do is to deepen their understanding, which includes
>> proving theorems, solving problems, finding common properties in
>> different objects for the purpose to abstract these properties into
>> a mathematical model, finding interdependencies between entities and
>> operations with them etc.
>>

>> *Zor Shekhtman
>> *Founder of Unizor Education <http://www.unizor.com/>
>> /Creative Mind through Art of Mathematics/
>> http://www.unizor.com <http://www.unizor.com/>

>>
>>
>> -------- Original Message --------
>> Subject: Re: [Math Future] thinking about The Math Myth
>> From: Joseph Austin <drtec...@gmail.com
>> <mailto:drtec...@gmail.com>>
>> Date: Fri, July 01, 2016 7:28 pm
>> To: mathf...@googlegroups.com <mailto:mathf...@googlegroups.com>
>>
>> I think the consensus in the math community is irrelevant.
>>
>> What if we taught only what the students would immediately use?
>> And who determines that? Certainly not the teachers, who have a
>> vested interest in teaching what they already know.
>>
>> I think the fundamental fallacy is that we must "prepare"
>> students with everything they will need to know for the rest of
>> their life by the time they leave school.
>>
>> But this ignores two fundamental changes in society:
>>
>> 1. the knowledge explosion: too much to know, some will become
>> obsolete, and new things will arise that we don't even know now.
>> 2. the internet: whatever I need to know, I can find out when I
>> need to know it.
>>
>> So we should stop teaching subjects and start teaching thinking.
>> Including quantitative thinking.
>>
>> Joe Austin
>>
>>> On Jul 1, 2016, at 2:38 PM, Daniel Finkel <finke...@gmail.com

>>> <mailto:finke...@gmail.com>> wrote:
>>>
>>> I think the consensus in the math ed community is:
>>
>>
>>
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[Peter Farrell]

There are people who think there's a list of things we need to learn, and we should just make a checklist and learn them one by one. Remember Hirsch's Dictionary of Cultural Literacy? It had very brief entries on all kinds of literary references an educated person might come across. Should we make a similar math dictionary and go through it page by page? Otherwise you risk leaving something out! And you can't understand Calculus without limits. A professor wrote that, conveniently forgetting the fact that scientists made superb use of the tools of Calculus for a century and a half before Cauchy "set Calculus on a rigorous foundation" by inventing limits. I wonder who set limits on a rigorous foundation. 

Everybody who really understands something (or pretends to) has to admit they didn't learn it by making a list. They came to it by doing project after project on it, getting more and more success with it, all the while spiraling around and getting a little deeper into the nuts and bolts of it every time. They didn't start with axioms.

In case I sound too negative I'll end with a quote (Kirby will have to confirm if it's accurate or not):

"Don't fight reality. Build a new model that makes the existing model obsolete." -- R. Buckminster Fuller