Analogue thinking in maths

David Coulsondtcoulson@gmail.com

Solving math problems in unmathematical ways

Is this a form of maths?

Is this a form of maths?(How would a robot do it?)

(might not be maths but whatever it is, it solves a mathematical problem)

(Birds and dogs can do it too)

Analogue thinking

Primitive brain

Numbers represented as imagined

lengths, distances, etc.

Lengths etc represented as

numbers: manipulate symbols

Higher brain

Digital (Symbolic) thinking

(algebra & arithmetic)(sensory representation)

Analogue thinking

Primitive brain

Numbers represented as imagined

lengths, distances, etc.

Lengths etc represented as

numbers: manipulate symbols

Higher brain

Digital (Symbolic) thinking

(algebra & arithmetic)(sensory representation)

Analogue thinking

Primitive brain

Numbers represented as imagined

lengths, distances, etc.

Lengths etc represented as

numbers: manipulate symbols

Higher brain

Digital (Symbolic) thinking

(algebra & arithmetic)(sensory representation)

Primitive brain Higher brain

Analogue thinking Digital (Symbolic) thinking

(videotape) (CD-ROM)

Medium = senses (usually vision)

Slow but precise

Primitive brain

Fast but rough

(“ballpark thinking”)

Higher brain

Analogue thinking Digital (Symbolic) thinking

(fixed number of steps regardless of size)

Classroom thinking

Primitive brain

Real world thinking

Higher brain

Analogue thinking Digital (Symbolic) thinking

examples

What’s 90 percent of $400 ?

(story)

What’s 90 percent of $400 ?

• how many of you do that? i.e., use size as a guide?

• Would you teach it?

• dirty shortcut?

• Is it legitimate?

• Do we encourage it, ignore it or unintentionally suppress it?

How long is a metre?

What’s the circumference of the pool?

13 m

What’s the circumference of the pool?

13 m

Symbolic thinking: 13 x 3.14 ...... But is the answer right?

Decimal places?

What’s the circumference of the pool?

13 m

Analogue thinking: “Three-and-a-bit diameters ”

forty-something metres

What’s the circumference of the pool?

13 m

How accurate do I need to be? And what am I doing this for?A lot of the time, all we really need is a rough answer.

Should we teach mathematical procedures for quick-and-easy approximations?

If maths is quick and easy, kids will choose to use it.

If there are choices, maths becomes a treasure hunt. Ask “How would YOU solve this problem?”

Is the size of this number important?

Symbolic thinking? No

Analogue thinking? Yes!

Is it big or small?

0.9

0.1

Sin(x) ~ 0.1

Visual representation:

0.9

0.1

O5.7 x

( Correct value 5.48O )

Sin(x) ~ 0.1

Analogue thinking accelerated symbolic thinking!

Is the answer good enough?

If it isn’t then does it guide us towards a better answer?

Should we let kids at school learn to do this kind of thing?

If not, why not?

Quick and easy but imprecise

Accurate but tedious

(requires sensory experience)

(usually visualisation)

Not one or the other but both working together

Who’s good at analogue thinking?

People with rich sensory experience of materials and objects

(sportsmen, builders etc)

Get kids outdoors playing sport and building things

Who’s good at Symbolic thinking?

People who process symbols on a page

( linguists, musicians)

Expose kids to other languages and (especially) music

Therefore to make kids really good at maths...

... teach them lots of things that have nothing to do with maths!

philosophy:

Maths should be one of those things that binds all the other subjects together, not that arcane stuff that sits out there on the edge of education, all by itself.

[END]

dtcoulson@gmail.com

Earthquake experience

Magnitude

. 7 1

. 6 3

. 5.5+ 10

. 5.0+ 47 or more (sway)

In all, 13,000+ earthquakes in ChCh since Sept 2010.

Sensory experience leads to expertise.

Can estimate an earthquake’s size by the seat of the pants.