Currently, only half the Integer Operation combinations appear accessible to intuitive pedagogies. Whilst the color coding is subjective, it is pretty close to reality. I've come to this conclusion after having reviewed modern-day elementary math books from not only the USA and Australia but also Singapore, Japan, and India. This is in addition to the hundreds of pre-1900 books I have reviewed.
The Laws of Sign relating to (aliquot) division on the Integers ℤ appear incapable of explanation when a positive dividend is paired with a negative divisor. Put simply, there aren't any negative fours in positive twelve and you can't have negative four groups. Such instances of (aliquot) division, (whether partitive or quotitive), can only by understood via multiplication, (finding the missing factor). This means we do not have a standalone 'meta-definition' of division capable of spanning the Naturals ℕ and Integers ℤ. Thus, despite the Laws of Sign having been documented by Brahmagupta in India in 628 CE, the foundations of elementary mathematics education remain incomplete.
It's bad enough when the absolute value of the dividend is greater than the absolute value of the divisor. Yet look what happens when the situation is reversed. Two green cells switch to yellow and two yellow cells switch to red! (Again, this is subjective, yet my opinion is at least informed judgment.)
Laws of Sign relating to division on the Rationals ℚ appear incapable of explanation when one or more terms is negative and the absolute value of the dividend is less than the absolute value of the divisor. Put simply, the standard models fail and geometrically, area models are currently incapable of handling either a negative side length or a negative area. Once again, compliance with the rules can only be a consequence of the missing factor approach. This means we do not have a standalone 'meta-definition' of division capable of spanning the Naturals ℕ, Integers ℤ and Rationals ℚ. Thus, we have further evidence the foundations of elementary mathematics education remain incomplete prior to the introduction of algebra.
PCV is the elementary meta-model resolving such issues. Notably, PCV can probably be introduced in Year 2, even if only by direct instruction at first. If you want to see how simple this can be, have a look at the YouTube video embedded deep within the post at this link https://www.jonathancrabtree.com/mathematics/multiplication-repeated-addition-math-myth-update/
OK, so let's dive a little deeper. For parents reading this, the partitive model is the original META MEANING of division. Before division was called division it was called partition! The partitive model is the intuitive model kids know BEFORE they attend school. Think equal shares and you know what the partitive model involves. It's the 'democratic' model, where everyone gets the same. Now, because multiplication mistakenly morphed into repeated addition, division morphed into repeated subtraction, which is an algorithm, rather than a concept. The repeated subtraction approach to division taught in schools is known as quotitive division. I refer to this as 'dictatorial' division. With the same amount as before, a fixed amount per person is given away regardless of how many people there are. When the quantity is gone it's gone! (Think food distribution at a refugee camp.) What we have here is a reversal of factors. In partitive division the order of the terms is dividend divisor quotient. In quotitive division (which is really just repeated subtraction) it goes dividend quotient divisor.
In the examples below, LOS means Laws of Sign. As should be evident, I also use the superscript – symbol for negative and + for positive. As usual, an unsigned number defaults to positive.
–12 ÷ 4 = –3 because –3 × 4 = –12 via LOS AND –12 into 4 equal parts/groups has –3 in each part/group.
PARTITIVE WORKS & QUOTITIVE FAILS (You can't subtract 4 from –12 to get to zero.)
–12 ÷ 3 = –4 because –4 × 3 = -12 via LOS AND –12 into 3 equal parts/groups has –4 in each part/group.
PARTITIVE WORKS & QUOTITIVE FAILS (You can't subtract 3 from –12 to get to zero.)
–12 ÷ 2 = –6 because –6 × 2 = –12 via LOS AND –12 into 2 equal parts/groups has –6 in each part/group.
PARTITIVE WORKS & QUOTITIVE FAILS (You can't subtract 2 from –12 to get to zero.)
–12 ÷ 1 = –12 because –12 × 1 = –12 via LOS AND –12 into 1 part/group has –12 in that part/group.
PARTITIVE WORKS & QUOTITIVE FAILS (You can't subtract 1 from –12 to get to zero.)
–12 ÷ 0 = undefined to preserve the laws of arithmetic because any number multiplied by 0 = 0
–12 ÷ –1 = 12 because 12 × –1 = –12 via LOS AND from –12 we can subtract –1 a maximum of 12 times.
PARTITIVE FAILS (You can't have –1 groups) & QUOTITIVE WORKS
–12 ÷ –2 = 6 because 6 × –2 = –12 via LOS AND from –12 we can subtract –2 a maximum of 6 times.
PARTITIVE FAILS (You can't have –2 groups) & QUOTITIVE WORKS
–12 ÷ –3 = 4 because 4 × –3 = –12 via LOS AND from –12 we can subtract –3 a maximum of 4 times.
PARTITIVE FAILS (You can't have –3 groups) & QUOTITIVE WORKS
–12 ÷ –4 = 3 because 3 × –4 = –12 via LOS AND from –12 we can subtract –4 a maximum of 3 times.
PARTITIVE FAILS (You can't have –4 groups) & QUOTITIVE WORKS
The presence of a negative dividend in all cases above might lead you to believe positive dividends present no problem at all. Yet positive dividends and negative divisors are problematic.
12 ÷ 4 = 3 because 3 × 4 = 12 AND 12 into 4 equal parts/groups has 3 in each part/group AND from 12 we can subtract 4 a maximum of 3 times.
PARTITIVE & QUOTITIVE WORK
12 ÷ 3 = 4 because 4 × 3 = 12 AND 12 into 3 equal parts/groups has 4 in each part/group AND from 12 we can subtract 4 a maximum of 3 times.
PARTITIVE & QUOTITIVE WORK
12 ÷ 2 = 6 because 6 × 2 = 12 AND 12 into 2 equal parts/groups has 6 in each part/group AND from 12 we can subtract 4 a maximum of 3 times.
PARTITIVE & QUOTITIVE WORK
12 ÷ 1 = 12 because 12 × 1 = 12 AND 12 into 1 part/group has 12 in that part/group AND from 12 we can subtract 1 a maximum of 12 times.
PARTITIVE & QUOTITIVE WORK
12 ÷ 0 = undefined because any number divided by 0 is undefined to preserve the laws of arithmetic
12 ÷ –1 = –12 only because –12 × –1 = 12 via LOS * PARTITIVE FAILS & QUOTITIVE FAILS *
12 ÷ –2 = –6 only because –6 × –2 = 12 via LOS * PARTITIVE FAILS & QUOTITIVE FAILS *
12 ÷ –3 = –4 only because –4 × –3 = 12 via LOS * PARTITIVE FAILS & QUOTITIVE FAILS *
12 ÷ –4 = –3 only because –3 × –4 = 12 via LOS * PARTITIVE FAILS & QUOTITIVE FAILS *
At the moment, I am only aware of indirect patterns such as solving for a missing multiplicative factor based on a prior knowledge of the sign laws. Weak proofs by contradiction might also be used to justify the laws of sign for integer division, yet such forced acceptance may not lead to understanding WHY the quotient has a particular sign. The intuitions appear non-existent because you cannot subtract a number a negative number of times and you cannot have a negative number of equal sized groups! Simple set theory and Peano Arithmetic have not helped.
Seeing patterns may result in rules being believed, yet seeing patterns alone, in my opinion, does little for understanding the mathematical rules and laws students must obey! The four binary operations, 0, i.e. (+ – × ÷), applied to a 0 b where |a| > |b| combined with the two integer types (+ve & –ve), generate sixteen distinct expressions in the first Elementary Math Matrix above. Pedagogies, from easy-ish to harder, are available for fifteen cells in the IOT, yet remain absent for Cell B4 pairing a positive dividend with a negative divisor.
After reviewing original writings of Brahmagupta, Leonardo Pisano, Gerolamo Cardano, René Descartes, John Wallis, Isaac Newton, (& others) I now have simple pictorial instantiations of all the cells in both matrices that are mostly green with only a few yellow.
These lessons for the green cells might even be taught in Years 4 and 5 via games in due course. For this to happen, I would also recommend a particular direct instruction approach be adopted in the Russian tradition, perhaps in Year 2 at the earliest or Year 3 at the latest. I am currently writing this up with the goal of submitting to an appropriate journal or conference.
The development of proportional reasoning is said to be a complex operation, yet I disagree. Similarly, fractions do NOT lead to proportion. Proportion leads to fractions! (We followed the Germans when we should have followed the French!) Proportional reasoning is wired into our brains and it is worth noting that magnitude sense precedes number or multitude sense. As things get bigger they get closer. As things get smaller they get further away. This is fight or flight stuff and via the agency of our two eyes and a single point of focus, our brains automatically triangulate in ways that would make Thales proud. Both direct and inverse proportion are experienced before we learn to talk. A child's propensity to cry increases as the distance to a parent increases, etc.
So my approach is to leverage the clichéd right brain thinking in which geometry is ever-present. Matching and correspondence are understood from tasks such as sharing food through to setting the table for a doll's party. My father was one of the first to teach children with Cuisenaire Rods as part of a trial by the Victorian Education Department in Australia. My father did not report kindly about the rods to the Department of Education Inspector. The children loved them too much and never wanted to stop playing with them! Whether it was the colors or the creations made, I also had fond memories of the rods in the 1960s as well.
It is perhaps little surprise that I advocate for the use of Cuisenaire Rods to introduce similarity of relationships (proportion) via a Done That Do This game. A YouTube video may explain this approach better than words alone. I have been investigating the geometry of ancient Greece and how we might be able to leverage it further, based on what we now know. My ideas have also evolved via a review of some elementary ideas of ancient China and India. The result, after many years of exploration spanning 16 languages*, has led me to 'Done that, Do this!'. (* For example, I am the only person to ever have ever had al-Khwarizmi's explanation of fraction multiplication translated into English.) I believe the pedagogy of 'Done that, Do this!' may have significant advantages in lower level classrooms. The main benefit may be the removal of such false beliefs as 'multiplication is repeated addition', 'multiplication makes more', 'division is repeated subtraction' and 'division makes less'. In short, whatever is done to the unit to make the multiplier, we do to the multiplicand to make the product. Similarly, whatever we do to the divisor to make the unit, we do to the dividend to make the quotient. The words here are for adults. The words are NOT used in the game when first introduced. I have produced two short videos that may help illustrate the Done that, Do this! principle. More videos are planned that will feature fractional multipliers and fractional divisors. The multiplication video can be seen at https://www.jonathancrabtree.com/mathematics/multiplication-repeated-addition-math-myth-update/ At the end of the multiplication video a link will appear for the (aliquot) division version.
Notably, the meta model I call Proportional Covariation (PCV) is based on Euclid's original (proportional) definition of multiplication. Alas, Euclid's definition was incorrectly translated into English in 1570 and it morphed into an illogical repeated addition algorithm. I note this in an article also available via the above URL or directly here. Proportional definitions of multiplication and division existed in Europe more than 500 years ago, long before ideas such as zero and negative numbers began to be accepted. (For most of the past 2000 years the number one was NOT seen to be a number, as it was the UNIT from which numbers are composed. (Like a brick is not a wall.) By making the relationships explicit between the four terms required for binary multiplication and division, (including the unit), I expect children will, in time, more easily comprehend the deeper relationships currently hidden from view.
So eventually, I expect my lesson ideas will adequately cover the Elementary Math Matrix where |a| < |b| ∀ a 0 b mostly with green, and yellow only for Cells F8 – H8. If you have persevered with this post, I am grateful and would be more grateful if you were to recommend people to me who might be able to share their approaches to integer division along with any patterns that reveal rules such as LOS. I would also be interested in any thoughts relating to the current color coding in the Elementary Math Matrices above and any other feedback for that matter. Should those who help me be interested, I will be more than happy to share my new approaches mentioned above.
BACKGROUND & INSIGHTS
Despite being a naive mathematician, (or perhaps because of it), I believe my hard-won ideas are unique. For example, neither Professor Nigel Hitchin, Savilian Professor of Geometry at Oxford University, nor any of his colleagues, knew you can use a circle to calculate products and quotients on the Reals. Click the applet link at http://bit.ly/2kJHSdo or http://bit.ly/RealMult I believe I also understand how al-Khwarizmi FAILED to fully understand Indian mathematics at the time he wrote his 'Algebra' in 820 CE. I believe I also understand why the Chinese would not have documented signed integer arithmetic 2200 years ago. Within an oral tradition, you only document ideas that are NOT obvious. It is of course, difficult to know what is original to me and what is merely lost. For example, whilst I have two simple ways to square the circle (STC) with proofs, (without straightedge), I have no doubt my solutions must have been known centuries ago. I mention STC to show how I explore areas of math that many if not most ignore. That's where the fun is! Whilst the geometry is complex, my goal is to make solutions (almost) accessible to Year 8 children. See http://bit.ly/PodoSquaresCircle and Proof 1: http://bit.ly/squaring-the-circle and Proof 2: http://bit.ly/STCProof
Beyond math and history, I have also investigated aspects of language and learning styles. Thus, my long-term goal is to have my revised foundations taught with specific techniques that accelerate the learning of algorithms and ideas that are simpler than those taught to date. (In a previous life, I wrote two books for Penguin on learning and memory. In a life before that, I was a math tutor.)
After having made it my goal to change the way the western world teaches mathematics in 1983, I have only just started to share what I have discovered. Thus far, my articles on math are limited to three short articles in Vinculum and the longer conference paper at http://bit.ly/LostLogicOfMaths with the presentation slides online via http://jonathancrabtree.com/LLEM/ My website here at https://www.jonathancrabtree.com/mathematics/ also features an assortment of fun ideas. Because I am an 'outsider', most of my research is not online and has only been shared via email.
Thank you again for reading!