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**New Theories / New Maths? (Orders of Magnitude)**

« **on:**14/01/2024 11:07:59 »

An idea for a way of teaching maths that is better related to human senses. This was split off from:

https://www.thenakedscientists.com/forum/index.php?topic=86686.msg719574#msg719574

All comments are welcome! Feel free to pass these ramblings on to real educators!

Most human senses need to span many orders of magnitude in sensitivity, so the response tends to be logarithmic, rather than linear (linear would imply 50% is half-way).

I think that kids would find maths and science more relatable if we talked in units they can feel and relate to - logarithmic units

- Sound intensity is measured in dB (logarithmic), sound frequency is measured in octaves (logarithmic), brightness in astronomy is measured in magnitudes (logarithmic)

- Bicycle gears form a geometric progression

- Hardness is measured on Moh's scale (logarithmic except for diamond)

- Earthquake strength uses the Richter scale, and wind speed uses the Beaufort scale (both logarithmic)

- Electronic component values fall on a logarithmic scale

- I suspect that things like touch, smell and taste would also have a logarithmic response

- Psychologists have even shown that abstract concepts like "value" or "wealth" also have a logarithmic response

- The Metric System and Scientific notation are both based on a logarithmic scale, with units like meters and grams, and their derived units of millimeters, kilometers, kilograms and tonnes.

- 10 of these in a line represents 1 order of magnitude (and 1 dimension)

- 100 of these in a 10x10 plate represents 2 orders of magnitude (and 2 dimensions)

- 1000 of these in a 10x10x10 cube represents 3 orders of magnitude (and 3 dimensions)

This should be supplemented by kinesthetic experiences like:

- Walk a metre: A kilometre is 1000 times farther

- Walk 10m: A kilometre is 100 times farther

- Walk 100m: A kilometre is 10 times farther

- Walk 1 km

- And similar sensory experiences with mass, brightness, sound intensity, frequency, and smell (taste might be a bit more of risk due to potential allergies?)

- Wealth comparisons like kids who survive on $1/day, $10/day, $100/day and $1000/day

- Given random quantities of each, which is it closer to? (estimating orders of magnitude, bearing in mind that 3 is closer to mid-way on a logarithmic scale)

This should be supplemented by kinesthetic experiences like:

- Take a metre of string, compare to 10cm, 1com and 1mm of string

- Take a kilo of sand, compare to 100g, 10g and 1g

- Take a litre of water, compare to 100ml, 10ml, 1ml

- Compare a walk of 1km to 100m, 10m, 1m

- Compare $10, $1, 10c, 1c

- Given random quantities of each, which is it closer to? (estimating orders of magnitude, bearing in mind that 3 is closer to mid-way on a logarithmic scale)

- And similar sensory experiences with brightness, sound intensity, frequency, and smell

- Not forgetting 0 of each, which is a child's ultimate reference point! (and the control value in an experiment, eg no sound, no light, no money)

- When adding 4+4, you quickly run into answers that are beyond subitizing

- This should lead to learning numbers from 0 to 10

- And the more laborious process of counting with 1-to-1 correspondence (which it takes kids a while to achieve)

- Nx 3 (the sensory middle of 1 & 10), 30, 300 [Mathematically 10

- Nx 2 & 5 (break the decade into 3 ranges), 20, 50, 200, 500 [10

- Doubling 1, 2, 4, 8 (especially important in music and computers)

- N x 9, the last digit to be covered (and least useful, being almost imperceptibly different from 10), 90, 900. Immunise against the shopper's fallacy: The gap between 9 & 10 is much smaller than between 1 & 2!

- This means that the next most important numbers are between 11 & 19

- Then fill in the gaps up to 100.

-Give the ideas of units, 10s, 100s in position notation (and digit 0)

Have blocks that increase in powers of 2:

- Unit block, line of 2, plate of 4, cube of 8 = 3 dimensions

- cube of 8, line of 2 cubes of 8, plate of 4 cubes of 8, cube of 8 cubes of 8 = 64

- Count the powers of 2

- Octave 2:1; harmonies 2/3, 3/4, etc

- For lengths, mass, volume, etc, start with 1/10 and progress to other ratios like

- Weights and liquid volumes can be easily obtained

- Computers are easily able to present random groupings of dots, apples, oranges and coins?

- 1-1 correspondence could be helped with touch screens which make the object visibly different after you touch it and count it

- Sound levels and sound frequencies over several orders of magnitude are easily generated on modern computers

- Various strengths of scents tend to be single-use, but I am sure they could be produced faily easily. Ensure you go from weakest to strongest, otherwise the nose will become desensitised!

- Computers are able to simulate simple logical combinations, but they would need to be child-relatable examples, eg assume pet = cat or dog; so an elephant is not a pet?

It's different! The idea of teaching 10, 100 & 1000 before teaching 9 seems radical.

Logarithmic units might be a difficult concept for existing primary school teachers to grasp

Several examples were kindly contributed by alancalverd & vhfpmr

https://www.thenakedscientists.com/forum/index.php?topic=86686.msg719574#msg719574

All comments are welcome! Feel free to pass these ramblings on to real educators!

**Human Senses**Most human senses need to span many orders of magnitude in sensitivity, so the response tends to be logarithmic, rather than linear (linear would imply 50% is half-way).

I think that kids would find maths and science more relatable if we talked in units they can feel and relate to - logarithmic units

- Sound intensity is measured in dB (logarithmic), sound frequency is measured in octaves (logarithmic), brightness in astronomy is measured in magnitudes (logarithmic)

- Bicycle gears form a geometric progression

- Hardness is measured on Moh's scale (logarithmic except for diamond)

- Earthquake strength uses the Richter scale, and wind speed uses the Beaufort scale (both logarithmic)

- Electronic component values fall on a logarithmic scale

- I suspect that things like touch, smell and taste would also have a logarithmic response

- Psychologists have even shown that abstract concepts like "value" or "wealth" also have a logarithmic response

- The Metric System and Scientific notation are both based on a logarithmic scale, with units like meters and grams, and their derived units of millimeters, kilometers, kilograms and tonnes.

**Number Concepts****Subitizing**: I think that a natural place to start with numbers is with subitizing, the innate ability to count 0 to 4 or 5 objects at a glance. Learning to relate the name and visual representation of the digit to the quantity of objects is important (and 0 is vital, as the thing kids feel most strongly is if someone else gets 1 lolly, and they get 0).**Large Orders of Magnitude**: 0, 1, 10, 100, 1000: When I was a kid, we were given number blocks, where "1" was represented by a small white cube.- 10 of these in a line represents 1 order of magnitude (and 1 dimension)

- 100 of these in a 10x10 plate represents 2 orders of magnitude (and 2 dimensions)

- 1000 of these in a 10x10x10 cube represents 3 orders of magnitude (and 3 dimensions)

This should be supplemented by kinesthetic experiences like:

- Walk a metre: A kilometre is 1000 times farther

- Walk 10m: A kilometre is 100 times farther

- Walk 100m: A kilometre is 10 times farther

- Walk 1 km

- And similar sensory experiences with mass, brightness, sound intensity, frequency, and smell (taste might be a bit more of risk due to potential allergies?)

- Wealth comparisons like kids who survive on $1/day, $10/day, $100/day and $1000/day

- Given random quantities of each, which is it closer to? (estimating orders of magnitude, bearing in mind that 3 is closer to mid-way on a logarithmic scale)

**Small Orders of Magnitude**: 1, 0.1, 0.01, 0.001, 0: Here you could start with the large cube, and the plate and line represent fractions of the whole (whole apple, whole bottle of water, etc)This should be supplemented by kinesthetic experiences like:

- Take a metre of string, compare to 10cm, 1com and 1mm of string

- Take a kilo of sand, compare to 100g, 10g and 1g

- Take a litre of water, compare to 100ml, 10ml, 1ml

- Compare a walk of 1km to 100m, 10m, 1m

- Compare $10, $1, 10c, 1c

- Given random quantities of each, which is it closer to? (estimating orders of magnitude, bearing in mind that 3 is closer to mid-way on a logarithmic scale)

- And similar sensory experiences with brightness, sound intensity, frequency, and smell

- Not forgetting 0 of each, which is a child's ultimate reference point! (and the control value in an experiment, eg no sound, no light, no money)

**Simple Arithmetic and counting further**: Based on Subitization, what happens if you add one, or subtract one, take all away or add another group the same size?- When adding 4+4, you quickly run into answers that are beyond subitizing

- This should lead to learning numbers from 0 to 10

- And the more laborious process of counting with 1-to-1 correspondence (which it takes kids a while to achieve)

**The leading digits**: After the orders of magnitude, the next most important numbers are:- Nx 3 (the sensory middle of 1 & 10), 30, 300 [Mathematically 10

^{1/2}= 3.16, but use 3 as an integer]- Nx 2 & 5 (break the decade into 3 ranges), 20, 50, 200, 500 [10

^{1/3}=2.15 & 10^{2/3}=4.64, but use 2 & 5 as integers]- Doubling 1, 2, 4, 8 (especially important in music and computers)

- N x 9, the last digit to be covered (and least useful, being almost imperceptibly different from 10), 90, 900. Immunise against the shopper's fallacy: The gap between 9 & 10 is much smaller than between 1 & 2!

**Next most important numbers**: According to Benford's law, when looking at numbers in physical laws, finance and election results, about 30% of the time the leading digit will be "1" (which is why we start with the orders of magnitudes).- This means that the next most important numbers are between 11 & 19

- Then fill in the gaps up to 100.

-Give the ideas of units, 10s, 100s in position notation (and digit 0)

**Simple Scientific notation**: Count how many orders of magnitude larger or smaller two quantities are.Have blocks that increase in powers of 2:

- Unit block, line of 2, plate of 4, cube of 8 = 3 dimensions

- cube of 8, line of 2 cubes of 8, plate of 4 cubes of 8, cube of 8 cubes of 8 = 64

- Count the powers of 2

**Logic**: AND, OR,NOT with computer simulation of real examples.**Fractions**: Teach it with music, pies, and apples:- Octave 2:1; harmonies 2/3, 3/4, etc

- For lengths, mass, volume, etc, start with 1/10 and progress to other ratios like

^{1}/_{2},^{1}/_{3},^{2}/_{3},^{1}/_{4},^{2}/_{4}=^{1}/_{2},^{3}/_{4}, etc**Tools Required**- Weights and liquid volumes can be easily obtained

- Computers are easily able to present random groupings of dots, apples, oranges and coins?

- 1-1 correspondence could be helped with touch screens which make the object visibly different after you touch it and count it

- Sound levels and sound frequencies over several orders of magnitude are easily generated on modern computers

- Various strengths of scents tend to be single-use, but I am sure they could be produced faily easily. Ensure you go from weakest to strongest, otherwise the nose will become desensitised!

- Computers are able to simulate simple logical combinations, but they would need to be child-relatable examples, eg assume pet = cat or dog; so an elephant is not a pet?

**Potential Difficulties?**It's different! The idea of teaching 10, 100 & 1000 before teaching 9 seems radical.

Logarithmic units might be a difficult concept for existing primary school teachers to grasp

Several examples were kindly contributed by alancalverd & vhfpmr