Naked Science Forum
On the Lighter Side => New Theories => Topic started by: evan_au on 14/01/2024 11:07:59
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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!
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 (https://en.wikipedia.org/wiki/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 (https://en.wikipedia.org/wiki/Cuisenaire_rods), 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 101/2 = 3.16, but use 3 as an integer]
- Nx 2 & 5 (break the decade into 3 ranges), 20, 50, 200, 500 [101/3=2.15 & 102/3=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 (https://en.wikipedia.org/wiki/Benford%27s_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
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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 ?, ?, etc
Minor niggle: these "harmonies" are cultural rather than innate/. The western scale of 12 semitones is distinctly crude compared with some Chinese scales of up to 45 notes per octave, though appreciation of the octave (except a traditional Chinese musician wouldn't call it that!) does seem to have some physiological basis.
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Once you have a basic idea of orders of magnitude, you can view the universe at many different scales, introducing ideas of maps, the world, astronomy, as well as biology, chemistry and nuclear physics...
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Producing test scents can be done fairly easily with the homeopathic method of dilutions:
- Place 1 drop of water-soluble scent in 1litre of water, and mix.
- Pour 100ml of this into another 1litre container, fill with tap water, and mix.
- Repeat process a few times
- As always, the control is a bottle containing just tap water. After a few dilutions, the faint scent will be indistinguishable from no scent...
- After seeing how many orders of magnitude to get to the atomic scale, you could work out how many dilutions will probably end up with 0 scent molecules in a litre of water...
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Hi.
Good article, thanks for posting it. You'll probably get plenty of comments about how well the ideas will work. So you might as well have one comment that puts a little counter-balance to that.
Teaching mathematics may be done in various ways and using kinaesthetic approaches can be good.
However, you have to consider whether you want to teach just some mathematical methods (like arithmetic) or if what we're really doing when you teach and learn mathematics is developing a more abstract set of skills and mathematics is just a cover or background in which that can be done.
The number 3 is just an abstract thing, it is not three dots or a "middle sized feeling" you have. It's very useful to have a grasp of arithmetic and some way of understanding how 3 + 1 may be 4 but the arithmetic itself is basically irrelevant.
You can and often do get better results if you teach mathematics with a mixed approach (for example there's a VARK acronym that supposedly encapsulates the main different styes of learning) and accept that the students have very diverse methods of learning. However, if you leave the students perceiving numbers as a number of things or a feeling they have, then you have let them down. They can be abstract.
We need to be careful not to make the mistake of thinking that Mathematics lessons were there to teach you how to do long division. If you left the lessons thinking mathematics was a bit abstract, then you actually learnt something important. Be carefull not to deprive a child of the opportunity to make that connection by basing everything on real world phenomena or how the magnitude of some number may feel, smell or be brighter than some other number.
Don't just consider Mathematics, consider English Literature or some other subject. Making your students good at spelling does not make them engage with the human condition or capable of describing the weight of some loss in words.
What's missing in the article (the Original Post) is some appreciation of the skills, ideas or concepts you would actually want to develop in a student of mathematics. For example, one important concept in mathematics is that things can have a different appearance but still have the same value.
x2 - 4 does not look the same as (x-2)(x+2) and indeed they are not the same in any sensible way. However, we can write x2 - 4 = (x-2)(x+2) and accept that they have the same value despite having a long list of ways in which they are so very different. Once you grasp the concept of keeping the same value but changing appearance then you have algebra. It's equally sensible go the other way around, use algebra as a background to teach the abstract concept.
Forget about just academia for a moment. Does an employer want you (a teacher) to produce students who can enagage with abstract concepts or do they need half-a-dozen calculators? If it's the former then, at some points, you have got to engage with abstract concepts and not relate everything to the real world and sensory experience.
Best Wishes.
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After a few dilutions, the faint scent will be indistinguishable from no scent...
Unless you are a shark. One drop of blood in a swimming pool will attract a Mako.
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