In the book The Number Sense: How the Mind Create Mathematics, the professor and researcher Stanislas Dehaene explores how the brain shapes our mathematical abilities. He reviews studies that explore crude calculating capabilities in rats, monkeys and human infants. He also explores the mathematical capabilities and limitations of human adults. A basic finding of the book is that many animals share with humans a basic ability to innately perform simple calculations. In Dehaene’s view, there is a basic calculator-like ability wired into the brains of many animals. He proposes that this ability is essential for survival, enabling animals to collect and store food, avoid predators and track mates. Those animals that were good at this basic calculating ability would be more likely to avoid death and could pass their genetics on to a greater number of offspring.
But Dehaene had another important finding in the book. Just as it is universal that humans have an innate ability to do simple math, we have significant limitations when it comes to more complex calculations and concepts. As an example, we can instantly determine the number of objects presented to us as long as the quantity is 4 or less. Once the quantity reaches 5, we need to start counting. Multiplying requires the memorization of tables as well as a set of rules; it is not an inherent capability of our nervous systems.
Dehaene and other researchers believe that the need to do complex math was not important for survival and reproductive success for the vast part of our evolutionary history. This need to perform complex calculations is only a recent requirement of the modern world. Therefore, it would not be a trait that evolution would have favored. The consequence of not having this complex ability “hard wired” is that we need to study and learn these skills. But for many of us, we either choose to avoid these courses, perform poorly in them or simply forget the lessons and techniques.
So why is any of this important? Can’t we simply rely on calculators, spreadsheets and software to do these calculations for us? The answer is both yes and no. When we need to do a calculation, and can use the appropriate tool we will get a satisfactory solution. But there are many times when we use estimation or intuition to make a mathematical judgement. Not having a strong grasp of complex math can put us at a disadvantage, leading to inappropriate judgements and decisions. Before I provide some examples of these issues, let’s explore a some of these basic human limitations.
An area of mathematics where many people have very limited intuitive capability is combinatorics. For the purposes of this blog post, I am using a more lay definition of combinatorics that involves the calculation of combinations, permutations and exponents. Let’s look at a relatively straightforward example. You’re about to play pool and you’re placing the 15 balls in the rack. How many different unique arrangements of the balls could be made? What did you guess? A few hundred? Several thousand? Would it shock you to find out the answer is 1.3 trillion (calculations at end of post)? In fact, if you could create a new combination of balls every second, it would take over 41,000 years to exhaust every possibility!
Another surprising example involves a story of two merchants. The first merchant needed a loan of $10,000. The second merchant agreed to lend him the money and asked for the following payback scheme. The borrower would pay one penny on the first day, two cents on the second, doubling his daily payments for a month. By the 22nd day, the second merchant had already paid back the double the original loan and was broke. Had he continued, his required payment for the last day would have been over $10 million dollars.
Our natural inability to envision and calculate complex scenarios leads to a number of failings in our everyday lives. One obvious example involves compounding of interest on loans as well as compounding of returns on investments. In the former case, people tend to underestimate the burden and impact of credit cards and other forms of loans. In the latter, people tend to underestimate the value of starting early in life when saving for retirement.
Another classic result of our inherent tendency to underestimate complexity involves project planning. Typically we look at the project as a series of discrete steps or phases that each has some likelihood of successful execution. What we fail to see is the vast number of interconnected relationships undergirding or undermining this success. A large project has hundreds or thousands of unique elements and factors, with complex interrelationships. Think of these unique elements for a typical information technology project. There are team members, internal customers, suppliers, unions, hardware, packaged software, custom software, the weather and endless more sub-elements. These all have unique interrelationships with the possibility of impacting the success of the project.
Let’s look at some examples of how the project could get off track:
- The lead business analyst encounters serious relationship issues with the key customer
- A key supplier runs behind schedule, failing to deliver a critical software patch
- A union strike prevents a carrier from providing network access to a critical user location
- Two pieces of software created by independent teams turn out to be incompatible with each other
- A huge blizzard cripples an entire region, forcing several days of delay
As with the pool balls and pennies examples, as the project gets larger, it rapidly builds up complexity. Soon, you are managing something so enormously complex that it is virtually bound to have some level of failure. Simply working smarter, more carefully or with greater project management discipline can not eliminate the risks. The key to combating this problem is to reduce the complexity as much as practical. While there is no specific rule here, the bias should be towards smaller, more modular efforts. Where there are large scale requirements, it is better to “chunk” these efforts into discrete, measurable phases. Utilizing agile development methodologies can help create a more iterative process, with smaller, more manageable deliverables. These rapid development techniques also deliver progress faster to a firm, ensuring that deliverables aren’t “stale” by the time they reach the customer.
Another issue that is rampant in large enterprises is the tendency to underestimate the complexities of management. As with the project example, managing people, departments and processes is enormously complex. As with our previous examples, this complexity grows geometrically. Unfortunately, traditional top-down management systems naively assume that an all knowing department head can make effective decisions across this complex landscape. But any manager is limited in his ability to master every function and process within his department. He could never exceed the local knowledge and expertise possessed by the individual directly performing the function every day. As the organization grows, the complexity of managing from the top becomes unworkable.
Forward thinking organizations understand this limitation and foster a culture of distributed authority and control. Pushing decision making out to the periphery of the organization reduces the complexity faced by any individual decision maker. The authority of these local decision makers becomes commensurate with their local knowledge and expertise.
People are not natively good at visualizing the rapid growth of complexity from geometric progressions. Being able to recognize that a scenario is subject to this complexity can allow for coping strategies that will result in improved personal and professional decisions.
Math for pool balls example: To determine the total number of unique combinations, the formula is n! where n is the number of items. The exclamation mark stands for the function known as factorial. Factorial is calculated as 15 x 14 x 13 x 12….x 1.