My patience wore thin so I finally decided to take matters in my own hands and bought some LED lights. I enlisted my son since helping him with a pre-calculus angular velocity problem was what inspired me to try to film this in the first place. The result is a derivative of Dan’s “Three-Act Math Tasks” style questions.
Angular Velocity to Linear Velocity – http://systry.com/angular-velocity-to-linear-velocity/ Suggestions for improvement are welcome and encouraged.

Six, for example, being a composite of 6 units of one is also a single unit of 6^1 = 6 = 6 ● 1. Understanding the numbers as possessing dimension and being able to reason proportionally, more precisely inversely so, creates arrays of equal units of one and varying dimensions. Thus, 6 ● 1 = 1[6(1)] = (⅙)[6●1](6) = [(⅙)6][1(6)] = 1 ● 6, and:

6 ● 1 = 1[6(1)] = 2 (½)[6 ● 1] = [½(6)][1 ● 2] = 3 ● 2
6 ● 1 = 1[6(1)] = 3 (⅓)[6 ● 1] = [⅓(6)][1 ● 3] = 2 ● 3

However, when comparing six to 4, a square number,
4 (¼)[6 ● 1] = [¼(6)][1 ● 4] = 1 ½ ● 4,
six can be understood as 1(4) + ½(4) = 4(1 + ½) = ½(4 (1 + ½))2 = [½(4)][(1+ ½)2] = 2(2 + 1).

These are all important understandings that would aid students in factoring algebraic polynomials and especially quadratics. For example:

x^2 + 7x + 12; let x = 5 → (5)^2 + 7(5) + 12 = 25 + 35 + 12 = 72

Eight of nine is the expression of 72 that compromises whole numbers and presents 72 as close as it can be to a square. When 8(9) is compared with a square of 25 units, as 5 was substituted for x, is (5 + 3)(5 + 4) — the length of 5^2 is extended by three units and the width by 4 units. Thus, the factors/dimensions of x^2 + 7x + 12 are (x + 3)(x + 4).

This also works with negative integers:
x^2 + 7x + 12; let x = -2 → (-2)^2 + 7(-2) + 12 = 4 + -14 + 12 = 2; thus (-2 + 3)(-2 + 4) or (x + 3)(x + 4).

Another example:
2x^2 + 11x + 15; let x = 2 → 2(3)^2 + 11(3) + 15 = 18 +33 +15 = 66

Eleven of six is the expression of 66 that compromises whole numbers and presents 66 as close as it can be to a square. When 11(6) is compared with a square of 9 units, as 3 was substituted for x, is (2(3) + 5)(3 + 3) — 3^2 is doubled and the length extended by 5 units and the width extended by 3 units. Thus, the factors/dimension of x^2 + 11x + 15 are (2x + 5)(x + 3).

I contend that the reason that I cannot presently ascertain many advanced mathematical concepts, or medical, linguistic, psychological, biological, or even theological concepts, at a level of scholarship is due to: (1) my lack of interest at that level and (2) not having had a wealth of specific kinds of experiences – reading of certain literature, discussions, practicums, tutorials, research opportunities, lectures, etc. – that scholars or prospective scholars have had which allow them to access the information at such high levels. I am simply limited due my lack of interest and experiences.

The reason that I decided to teach younger students was due to my experiences teaching older students. I recognized that they had not had certain kinds of experiences in mathematics in their most formative years, which was causing them to have difficulty understanding certain mathematical concepts in their adolescent years. I was spending a great deal of time helping adolescent students gain a more complete understanding of basic mathematical concepts in order to provide them a foundation upon which to build. Thus committing to work with young children, I was ensuring that students would not continue to move through their schooling experience with gaps in their understanding or simply being skilled with using standard algorithms. In doing so, I found it more valuable to teach mathematics as concepts from which there emerged an investment by students toward their development of skills that allow them to convey their thinking.

My experiences will not allow me to view intellect as fixed nor am I able to approach my work with students from a deficit thinking perspective. For what then is the purpose of teaching?

I’d guess most of us here have taught Mathematics for a while. I teach older children, where the limits of their native ability become much more obvious, but it reaches back.

And in my experience, I have found that children — whether ill-prepared, struggling, lacking confidence, and/or unmotivated — are generally quite capable of understanding mathematics deeply.

I reckon I can teach almost any child almost any Maths skill. But getting them to understand it, no way!

I’ve seen it in myself. I cannot follow post-graduate level Mathematics. It’s not that I’m stupid. Nor ill-motivated. Nor badly taught in the basics. But at a certain point I cannot follow sufficiently difficult Mathematics.

For other people their level is lower. Much lower in many cases.

I am definitely not the first to take a more reformed view of Piaget’s ideas. His theory of cognitive development has been strongly contested since the 1960’s,

So about the time when the obvious — that some people are not very clever, by the misfortunes of genetics — became the unsayable?

Nothing about math is concrete. It is an abstract subject matter. Thus, the teaching of mathematics can only be for the purpose of children apprehending that which is abstract. As teachers, we, then, must provide young students with experiences that lead them to abstract: use that which they know or observed in a particular instance and transfer or extend that knowledge to gain greater understanding of the idea or a new one. Certainly, physical and visual aids are part of the experience because of students’ narrow schema. Yet our aim with such aids is to teach students to think, teach them how to abstract.

Again, I do not merely present this as a topic of debate, but as a sharing of my reality. I teach young children mathematics as interrelated concepts. My students are flexible in their thinking of numbers and operations, demonstrating reasoning by challenging me and each other, and confident in using mathematics vocabulary to present their ideas. This success has been due in large part to my deliberate effort to have students understand that mathematics is a study of ideas, calculated exploration of the ideas, and continuous encouragement to make connections between ideas.

I am definitely not the first to take a more reformed view of Piaget’s ideas. His theory of cognitive development has been strongly contested since the 1960’s, and based on my experiences as a practitioner, I have come to understand the reason for reform of his ideas.

Or are born cleverer.

I have to say that I agree with almost none of your analysis Kevin.

I believe (as Piaget did before you “reformed” him to say the opposite) that children cannot think abstractly before they have the appropriate mental equipment. Merely giving more and better training will not improve a skill that is not present.

Although we are no longer really allowed to say these things, everyone knows that some people are born with almost no ability at all for abstract thought. Others get it only very slowly, and even as adults struggle with abstract concepts. Whereas some children merely have to see a concept once and it is embedded.

Teaching to the strugglers can be rewarding, and I often enjoy teaching them Maths — slowly. They can enjoy Maths, provided it is pitched at their level. But I object strongly to being told that it is my failure as a teacher that prevents them from being able to see abstract patterns like more gifted can.

Your analysis effectively causes the blame for lack of progress in Maths to be laid completely at the feet of the teachers — since they have failed to provide the proper instruction. Politicians love this concept, because it allows them to brow-beat teachers for “failure”, when in reality there has been no failure.

We can seek to improve Maths teaching, and I imagine everyone here wants to, but we can’t improve the material we work with — people. No amount of wishing children were blank slates, onto which we can write at will, makes it so.