I hope you enjoy this game and its message.
The "Theory of Positional Invariance” states that, "Regardless of the observer’s viewpoint, an object retains its properties." There are many examples of this. Whether observed from the North Pole or the South Pole, the moon has the same mass, craters and rotational speed. Though the moon does appear upsidedown to one observer, a viewer's orientation doesn't change the properties of the object.
Whether you look at a person from the top or back or front, it's still that person with the same colored eyes and personality. I asked my grandchildren, “What’s my name?” Their first thought was, “Oh, no! It's happening to him.” Then they said very tentatively, “Pop.” “Good,” I said. Then I turned around and asked, “Now what's my name?” One said, “It’s still Pop.” The other called to their grandmother, “DeeGee, something’s wrong. We need help in here.”
In support of our theory
A theory starts with corroborating observations, acquires the rule, and then a sophisticated-sounding name to help validate it. Regardless of the observer’s viewpoint, an object retains its properties. This is positional invariance.
Applying this principle, observe that, except for 45° of rotation, the × and the + symbols are the same; so the theory claims that 2 + 2 = 2 × 2. There you are!
Let’s try with some other numbers:
3 + 1.5 = 3 x 1.5
(-4) + 0.8 = (-4) x 0.8
1 + 2 + 3 = 1 x 2 x 3
But, you may ask, what about complex numbers? More evidence that it holds true:
[(-1) + 2i] + [.75 – .25i] = [(-1) + 2i] x [.75 – .25i]
[(-1) + √2i] + [2/3 - √2/6i] x [2/3 - √2/6i]
It turns out, there's an infinite number of corroborating examples. I chose these examples to show that it works with negative numbers, fractions and irrational numbers, but I kept the numbers convenient for your affirmation of the truth of the Theory of Positional Invariance.
The theory is intuitively logical, has a sophisticated name, and is confirmed by data that has an infinite number of cases. Consequently, the claim must be true. I use this truth to support my claim that we shouldn't be wasting time and mental effort by having students memorize both addition and multiplication facts. Addition is all that's needed.
(Yes, this torments DeeGee, a former elementary school teacher, who thinks I have no respect for primary grade objectives. But actually, I have immense respect for teachers, and all that supports what's needed at each level of education, coping with each student's persona, and teaching concepts that are novel and may be difficult for them. But one can also have fun, right?)
Now let’s see if we can relate the Theory of Positional Invariance to engineering practice and a Develop Your Potential column in Control. Just because there's some corroborating evidence and an intuitive basis for a fancily packaged claim, doesn't mean the claim is true. Don’t blindly accept either the technical folklore of your community or your preferred explanation. Don’t seek evidence to support the claim. Seek evidence that could refute it. Data can't prove. Data can only disprove. So, critically shape trials and examples to see if you can disprove the claim.
