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I challenge the student of mathematics

It appears to me that most people look on math as something with supernatural qualities. I challenge the student of math to develop and post short essays on Internet discussion forums about those fundamental aspects of math that you think people can and should comprehend.

What follows is something that I have posted regarding my idea of what ordinary citizens should know abut this very fundamental domain of knowledge.

Arithmetic is object collection

It is a hypothesis of SGCS (Second Generation Cognitive Science) that the sensorimotor activity of collecting objects by a child constitute a conceptual metaphor at the neural level leading to a primary metaphor that ‘arithmetic is object collection’. The arithmetic teacher attempting to teach the child at a later time depends upon this already accumulated knowledge. Of course, all of this is known to the child without the symbolization or the conscious awareness of the child.

The pile of objects became ‘bigger’ when the child added more objects and became ‘smaller’ when objects were removed. The child easily recognizes while being taught arithmetic that 5 is bigger than 3 and 3 is littler than 7. The child knows many entailments, many ‘truths’, resulting from playing with objects. The teacher has little difficulty convincing the child that two collections A and B are increased when another collection C is added, or that if A is bigger than B then A+C is bigger than B+C.

At birth an infant has a minimal innate arithmetic ability. This ability to add and subtract small numbers is called subitizing. (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol). Many animals display this subitizing ability.

In addition to subitizing the child, while playing with objects, develops other cognitive capacities such as grouping, ordering, pairing, memory, exhaustion-detection, cardinal-number assignment, and independent order.

Subitizing ability is limited to quantities 1 to 4. As a child grows s/he learns to count beyond 4 objects. This capacity is dependent upon 1) Combinatorial-grouping—a cognitive mechanism that allows you to put together perceived or imagined groups to form larger groups. 2) Symbolizing capacity—capacity to associate physical symbols or words with numbers (quantities).

“Metaphorizing capacity: You need to be able to conceptualize cardinal numbers and arithmetic operations in terms of your experience of various kinds—experiences with groups of objects, with the part-whole structure of objects, with distances, with movement and location, and so on.”

“Conceptual-blending capacity. You need to be able to form correspondences across conceptual domains (e.g., combining subitizing with counting) and put together different conceptual metaphors to form complex metaphors.”

Primary metaphors function somewhat like atoms that can be joined into molecules and these into a compound neural network. On the back cover of “Where Mathematics Comes From” is written “In this acclaimed study of cognitive science of mathematical ideas, renowned linguist George Lakoff pairs with psychologist Rafael Nunez to offer a new understanding of how we conceive and understand mathematical concepts.”

“Abstract ideas, for the most part, arise via conceptual metaphor—a cognitive mechanism that derives abstract thinking from the way we function in the everyday physical world. Conceptual metaphor plays a central and defining role in the formation of mathematical ideas within the cognitive unconscious—from arithmetic and algebra to sets and logic to infinity in all of its forms. The brains mathematics is mathematics, the only mathematics we know or can know.”

We are acculturated to recognize that a useful life is a life with purpose. The complex metaphor ‘A Purposeful Life Is a Journey’ is constructed from primary metaphors: ‘purpose is destination’ and ‘action is motion’; and a cultural belief that ‘people should have a purpose’.

A Purposeful Life Is A Journey Metaphor
A purposeful life is a journey.
A person living a life is a traveler.
Life goals are destinations
A life plan is an itinerary.

This metaphor has strong influence on how we conduct our lives. This influence arises from the complex metaphor’s entailments: A journey, with its accompanying complications, requires planning, and the necessary means.

Primary metaphors ‘ground’ concepts to sensorimotor experience. Is this grounding lost in a complex metaphor? ‘Not by the hair of your chiney-chin-chin’. Complex metaphors are composed of primary metaphors and the whole is grounded by its parts. “The grounding of A Purposeful Life Is A Journey is given by individual groundings of each component primary metaphor.”

The ideas for this post come from Philosophy in the Flesh. The quotes are from Where Mathematics Comes From by Lakoff and Nunez

"It appears to me that most people look on math as something with supernatural qualities."

Being ignorant of mathematics is a point of pride for some.

"I challenge the student of math to develop and post short essays on Internet discussion forums about those fundamental aspects of math that you think people can and should comprehend."

The first question we have to ask is, what is the purpose? What are we trying to achieve from such a program and why? Make better citizens? Make the minds of our citizens beautiful? Preparation from college? Promote self-actualization? Better workers? Any or all of the above (if so, prioritize)? None of the above?

Curricula should be purposeful: (exploration is a purpose)

Not every student has to (or should) follow the same path. Kids develop at different rates, physically and mentally.

Arithmetic, plus
-- K-5 (or 6 or 7) Absolutely mastery of basic arithmetic:
Add, sub, mul, div, exp, decimals fractions, percents
Cartesian system, basics of graphing, equations, inequalities
very elementary geometry, elementary probability
If, given sufficient time, the student doesn't get an A on EVERY SINGLE test of these basic things, then the student has not mastered the subject.
Doing anything else without having done this is a huge waste of effort. This alone is invaluable to students for simple things like "making a budget." It's unconscionable that we have adults in the US who can't do that.

Algebra, plus
-- 6 - 12
mastery required for any HS diploma.
No student should graduate HS who has not passed a comprehensive Algebra 1 test with a grade of A.
Symbolic representation, functions, mathematical logic
more graphing, basic axioms.
If a student learns nothing else in math, this will hold them in good stead. If they go on to other math, mastering algebra beforehand will make their lives easier. Everything else is a waste without this.
As it is, a lot of students get formal logic, if they get it at all, from English teachers (maybe some in geometry). There was a time when maybe this made sense, but no longer. The scary thing is that a lot of them get it wrong. Put it with the math teachers who understand it. (Having English teachers teach logic makes as much sense as having P.E. teachers teach sex ed.)

Geometry
-- optional for most ordinary students, mandatory for advanced track
Students should learn basics of geo in K-7. A full course will not be helpful to most students. The one important thing "formal proofs" are barely done in most geo classes any more. "Formal proofs" could be moved to algebra 1.


Advanced students should have other options available to them (probability, statistics, analysis, etc) and those subjects should only be taught by teachers who are themselves masters of them.

"Advanced students should have other options available to them (probability, statistics, analysis, etc) and those subjects should only be taught by teachers who are themselves masters of them."

Aye, there's the rub. For in most schools we have coaches of sports teaching things like math and social studies. My son's English teacher in High School was a basketball coach, and didn't know his subjects from his prepositional phrases. My son could have taught the English class and done a better job. It isn't only math that should be taught by those who are masters of the subject. I pray that the school system wakes up and smells the coffee. They complain that the students perform poorly on standardized tests, when the students aren't taught standard English to begin with. When the only requisite for teaching a subject is having had two semesters of it in college, the education system suffers.

One suggestion is to explain why math, the science of pattern, can be so helpful in solving problems in the natural sciences but not in the human sciences.

FallableFriend

If the student of math does not know why math is important to know then I certainly cannot help him or her.

"If the student of math does not know why math is important to know then I certainly cannot help him or her."

All students are or should be students of math. And it is the responsibility of teachers - to include children's first teachers, namely their parents, to help them understand that math is useful and encourage them to get excited about it.

But the question I posed was not "Why is math important?"

My question was "What is the state's purpose in teaching mathematics?" It could be that this question has already been answered. My guess would be that there are multiple answers - and this causes disharmonious application of effort.

If we do not measure things, how can we tell if what we propose helps or hinders? Writing down - making explicit - our intent is the beginnings of measuring things. What are we trying to achieve. Where are we and where do we want to be? What are the measures (or "indicators", if you're a soft person).

Learning math is something like learning assembler at the time of high level programming languages.

Yea. Balancing checkbooks is so ... last century.

We are talking about abstract math - not about basic algebra of grammar level.

Therein lay the problem with our (American) educational system.

I am a retired engineer who learned how to "do" math but only after becomeing a self-actualizing self-learner later in life did I begin to uderstand what math was. Our educational system teaches us how to "do" math but gives us no understanding of what math is about.

Therein lay the purpose of my challeneg to math majors. Help people to understand what math is about and why it is useful in some domains of knowledge and not useful in others.

A central problem:
Many K-12 teachers don't know much more math than what they are actually teaching. Some of them barely know that. At the same time, most teachers seem to be recruited into the profession straight out of school. This means that even if they actually majored in math, they have little to no experience applying it to real problems. One thing I and colleagues have done is begun a program of hiring teacher interns in the summer, giving them the opportunity to apply their knowledge of their subjects to important problems. There needs to be a lot more opportunities like this for teachers.