It is my premise that most people do not really understand how to use mathematics strategically in a concrete world. They don’t think much about what the numbers mean and meaning is everything if you want to know what the numbers are doing. At its heart, math is an abstraction; an idea that is not connected to real world circumstances. (See Steven Strogatz’s NY Times article for a detailed look at math and its misuse in education pedagogy)
The trick to understanding and using math in the real world can often be traced to how we devise the measurements that define the meaning of numbers that are then to be treated mathematically. Let look at some problems relation to the use of numbers and how their meaning is misunderstood.
Problem #1 Educational Testing – Measurement should aways be designed to serve a goal; goals should never be design to fit a measurement protocol. This is why proficiency testing will never help education and the core idea behind a recent New York Times editorial by Susan Engel. Current public school measures do not reflect the capabilities we need to develop in students. It’s not bad that people teach to the test, what’s bad is that the test itself it not worth teaching too.
Our current educational approach — and the testing that is driving it — is completely at odds with what scientists understand about how children develop . . . and has led to a curriculum that is strangling children and teachers alike.
(Curriculum should reflect) a basic precept of modern developmental science: developmental precursors don’t always resemble the skill to which they are leading. For example, saying the alphabet does not particularly help children learn to read. But having extended and complex conversations during toddlerhood does. (What is needed is) to develop ways of thinking and behaving that will lead to valuable knowledge and skills later on.
The problem we see in current testing regimes is that we’re choosing to test for things like alphabet recall for two reasons.
- We base measures on common sense linear thinking like the idea that you must recognize letters, before recognizing words, before using words to build statements. But if fact (as Ms Engel’s article points out) the psychological processes of building complex conversations is the developmental need for students and that is rather unrelated to how thought is considered in schools and how curriculum is developed. Developmental needs should be studied for scientific validity and not left to common sense.
- The current measurement protocols behind proficiency testing is not very good at measuring things like the ability to participate in complex conversations, it simply doesn’t translate well to a multiple choice question. We could develop rubrics to do that, but it would be hard to prove that the rubrics were being interpreted consistently. So instead we test abilities that fit the testing protocol, even if they are rather irrelevant (read invalid) to the capabilities that we really desire to foster.
Problem #2 Business Analytics – Things like analytics and scientific evidence are used in ways that relate mostly to processes and activities that can be standardized. These are ways of doing things where there is clearly a best way to do it that can be scientifically validated and is repeatable. The problem occurs when we try to achieve this level of certainty in everything, even if there is little that science can say about the matter. Math is not about certainty, it’s about numbers.
The problem, says (Roger) Martin, author of a new book, The Design of Business: Why Design Thinking is the Next Competitive Advantage, is that corporations have pushed analytical thinking so far that it’s unproductive. “No idea in the world has been proved in advance with inductive or deductive reasoning,” he says.
The answer? Bring in the folks whose job it is to imagine the future, and who are experts in intuitive thinking. That’s where design thinking comes in, he says.
The problem with things like six sigma and business analytics is that you need to understand what it’s doing mathematically and not just follow a process. If you’er just applying it, and you don’t understanding what it’s doing, you’ll try to do things that make no sense. It not usually a problem with the mathematical procedures, it’s a problem with what the numbers mean. How the numbers are derived and what’s being done as a result of calculations. There is nothing worse than following a procedure without understanding what that procedure is doing or accomplishing. Martin’s basic thought that innovation and proof are incompatible is false. The real problem is a lack of understanding in how mathematics and proof can be use in concrete situations.
Problem #3, Use of the bell curve in annual reviews and performance management.
A recent McKinsey article, (Why You’re Doing Performance Reviews All Wrong, by Kirsten Korosec) generated a lot of negative comments by people force to make their review correspond to a bell curve. In statistic we know that if you take a large enough random sample of anything that can be represented by numbers, the resulting distribution of the represented quality will resemble a bell curve, large in the middle and tapering off at either end. But performance management is about fighting the bell curve; it’s about improving performance and moving the bell curve. If you have to fit your reviews to a bell curve, your making performance look random. That’s exactly what you do not want to do. Once again we see a management practice that uses mathematics without understanding what they are doing.
What’s needed? The valid use of mathematics not the random use
The basic problem is that mathematics is abstract, but human activity is concrete. If we want to bridge these two worlds (and, as Strogatz explains it, they really seem like parallel universes) we must build a bridge of understanding that is called validity. Validity is really the scientific study of how the concrete is made abstract and how the abstract is made concrete. It’s an explicit theory of how the scope of activities can be represented by numbers, laid out so that it can be argued and understood. You can do amazing things with mathematics in the real world, but only if you understand what you are doing, if you understand how the abstract and the concrete are related. You must understand how numbers can represent and are related to the world of human activity.