Friday, March 22, 2013

On the Current State of K-12 Mathematics Education

The central claim of this essay is that visualization and computer programming are essential components of a good mathematics classrooms today, and the Common Core’s Standards for Mathematical Practices, while probably a step forward for the average mathematics classroom, fail to properly emphasize these topics. The other "big players" in math education--the text book publishers, classroom technology companies, and curriculum implementation counselors--all seem to have missed the boat on (or chosen to ignore for now) the need for an increased emphasis on visual representation of mathematical concepts and the integration of mathematics with computer programming skills.

Take the Common Core’s Standards for Mathematical Practices, and the suggestion of Johannah Maynor, secondary math consultant for the North Carolina Department of Public Instruction, for teachers to focus on the following three practices:  “Making Sense of Problems and Persevere in Solving Them”, “Constructing Viable Arguments and Critique the Reasoning of Others” and “Attending to Precision”.  (The combination of these three practices is a fairly good summary of the eight practices as a whole.)

However, it is the other two practices--the focus on rigorous arguments and precision--that are out of place when they are applied through compulsory, mass-education to the traditional mathematics curriculum, and are far better suited for the concepts of computer programming, which itself should be increasingly integrated into schools’ mathematics curricula.

The problem with insisting that math educators apply more “rigor and precision” to algebra, geometry, calculus, and statistics is that this approach omits entirely two concepts that are essential for a good understanding of mathematics role in today’s world.

Essential component 1: Machine Logic

AP Computer Science teacher Baker Franke points out in this article, “very few students graduate from high school with CS, because at the moment there simply is no place at the table for CS in American education.”  Computing in the Core, an organization seeking to receive grant money from State legislatures to increase K-12 computer science education, explains,
“roughly two-thirds of the country has few computer science education standards for secondary school education, and 35 states treat secondary school computer science courses as simply an elective, versus an important element of a student’s core education. It counts as a required graduation credit in only nine states.”

But where to fit computer science in students’ already over-booked lives? While the benefits of computers can be effectively utilized in every academic field, I believe that specifically computer programming must eventually come to its natural home within the mathematics curriculum.  In today’s economy it’s not rigor about geometrical arguments or algebraic deductions that matters--this may have been the case when engineers had to do complex computations by hand--but in most fields today, creativity is more important than rigor, with one important exception. This type of strict, formal rigor that we learned from Euclidean geometry still absolutely applies within computer programming.

The program Mathematica, widely used in university math departments, is a perfect example of the natural affinity between mathematics and computer science. (There is coincidentally a free, online conference about using the software program Mathematica in classrooms which is exactly what mathematics education should be trying to do!  Conference registration can be found here: ) As English mathematician Andrew Hodges explains,
What happened to the kind of logical rigour which used to be taught through Euclid’s geometry? It flowed into the logic of numbers, and then through Rusell, Godel, and Turing into computer languages. Computers do exactly what you tell them, not what you mean, and writing programs for them reignites the extreme rigour and mind-game challenge that used to go into Euclid. ... I don’t believe it is too hard. The skill of computer gaming is a good step towards programming.” [One to Nine: The Inner Life of Numbers. p. 291]

Precision and the construction rigorous arguments and critiques are far better suited for the concepts of computer programming, which itself should be increasingly integrated into our mathematics classrooms today.

Does this mean we should quit worrying about mathematics completely and solely focus on computers? Absolutely not! Mathematics is still foundational to understanding scientific investigation, statistics, and additionally is a fascinating language to interpret the world. But in the instruction of these mathematical concepts, the focus needs to be on visualization--on how mathematics is a language for deciphering the sensory, physical universe, whether it’s the 1-dimensional relentless linear progression of time, the 2-dimensional planes of symmetry in nature, and the 3-dimensions our brains can see, or the 4-dimensions Einstein proved are there!

Essential component 2: Visual conceptsVisualization is by far the most common learning style, and always a useful way to stimulate the brain to consider multiple perspectives.

Innovations in education are already beginning to spread online. Dan Meyer (, for example, recommends the use of short videos in the classroom to frame math problems, which is a great start towards a more visual math curriculum. If students understood how to visualize addition and multiplication as movement along a number line, the majority of sign-change and decimal-place errors could be eliminated just like that!

But beyond reducing calculation error, the primary motivation for making mathematics more visual is to connect mathematics with the real, physical world. For too long, the common picture of mathematicians and scientists is that of intellectual geniuses whose thought processes the rest of humanity could not possibly hope to understand.

Mathematics educators should strive to shatter the perception of mathematicians as socially-disconnected geniuses and replace it with an image of mathematics as dynamically connected to the geometry of the sensory world. Towards mathematical concepts (as opposed to computer programming concepts) educators should move away from an insistence on rigor and focus on a creative approach instead, and strive to cultivate their own enthusiasm for the subject.

Mathematics education is big business today. Our country is investing a large amount of money into STEM programs, classroom technologies, and curriculum restructuring, but, needless to say, classroom math teachers are not at the receiving end of much of the funding. But the internet opens all kinds of new doors for teachers themselves taking the initiatives in collaboratively designing the best possible mathematics classroom. The reason I think this project is worthwhile is that the publishing companies and education technology industries are so locked in to the traditional math curriculum that they are failing to look to the future where computer programming will inevitably be an integral part of any mathematics curriculum. I look forward to hearing input and feedback from other math education professionals in order to shift power away from big business and bureaucrats and to educators themselves.

Here is a link to a group to serve as an online hub for further discussion of these issues. -

Monday, March 18, 2013

Aurobindo: excerpt from Savitri

Excerpt from the poem "Savitri" by Sri Aurobindo [quoted from The Essential Mystics by Andrew Harvey, pg. 66]

"Earth shall be made a home of Heaven's light. . . .
Invading matter with the Spirit's ray,
Awakening its silence to immortal thoughts,
Awakening the dumb heart to the living Word.
"The supermind shall claim the world for Light
And thrill with love of God the enamored heart
And place Light's crown on Nature's head
And found Light's reign on her unshaking base.
"A soul shall wake in the Inconscient's house;
The mind shall be God-vision's tabernacle,
The body, intuition's instrument,
And life, a channel for God's visible power.
"The Spirit shall look out through Matter's gaze
And Matter shall reveal the Spirit's face.
The Spirit's tops and Nature's base shall draw
Near to the secret of their separate truth
And know each other as one deity.
Then man and superman shall be at one
And all the earth become a single life."