Overcoming the fear of mathematics in planning education and practice

A vital contradiction in our education system

The former chief scientist of Airbus, Jean Francois Geneste said in a brilliant talk delivered at Skoltech, that when it comes to large and complex systems, 

"We can only master, what we can measure and mathematics is a discipline for measurement -- it is measurement theory."

What he said has great implications for our own field of urban and regional planning too. It is important to measure and to measure correctly, before planning decisions affecting millions of people, thousands of businesses and hundreds of hectares of land-uses of different kinds can be taken. 

Yet, precisely when there is a growing fascination with data and digital technologies, there seems to be a relatively low understanding of the role of mathematics in planning. A substantial part of the problem lies in the fear of the subject itself and the inability to apply it effectively in real situations.

We are all aware, that due to the peculiar limitations of the Indian education system, there is a rigid and unnatural separation between the sciences and the arts. 

This leads to a situation where people trained in engineering techniques are often completely devoid of an awareness of social issues and of creativity; and the people trained in humanities are often clueless when it comes to physics, mathematics etc.

Truth be told, this challenge exists in urban planning education outside India too, though, perhaps, not as severely. As Brian Field and Bryan Macgreggor noted in the preface to their book on forecasting techniques,

"...we had both come to planning from numerate first disciplines and, in planning schools at opposite ends of Britain, had independently concluded that there was an obvious gap in the literature on this particular subject."

The important word here is numerate - which means having a knowledge of mathematics and the ability to work with numbers. It is the mathematical counterpart of the word literate, which is the ability to read and write.

The complex is essentially simple 

The complex is essentially simple, because it is also a function of our learning, experience and skill. To someone who has never stepped into a kitchen, even making a cup of tea may seem like a forbiddingly complex task. However, to most people it is just a regular task -- a simple task. 

The funny thing is that mathematics seems more difficult when it is taught but it seems easier when it is applied !

When it is taught - especially in our schools - its difficulty is cranked up to meet the needs of the engineering entrance exams, whose purpose is to eliminate large numbers of students through a process of cut-throat competition. It is easy to see that this "goal" has nothing to do with solving practical problems of life and society.

Even when students master that gigantic syllabus and get extremely high grades, they may not have internalised the logic behind the topics and may fail to apply them creatively in real life situations. 

However, when one begins to study science subjects because one wants to understand and solve real life tasks, then the relevance and applicability of the topics are automatically evident and the human mind understands and internalises them faster.

Let's try to understand this using an example of a model, where we go from the simple to the complex and then realise that it's essentially simple.

Distance decay and equations that make you run away

Urban models are essentially mathematical equations that describe essential features of an urban system and can enable us to simulate and predict its behaviour.

Consider the following equation from Field and Macgregor's book (I will refer to this book  and David Foot's book on operational urban models many times in these blogs, as they are just brilliant) -

A = f(B, C, D)

This equation basically means that the variable A is a function of (that is, in some way, depends on) three other variables - B, C and D. Therefore, the value of A will change if there are changes in the values of B,C and D.

But what do A,B,C and D stand for ?

Let's assume that we want to understand how many shopping trips are made from various residential areas in the city to commercial areas. We can describe the components of our model in this way --

A = Number of trips made for shopping purposes (what we wish to find out)

B = Population of the residential area 

C = Number of shops in the commercial area

D = Distance between the residential area and commercial area.

But what to do if a city has multiple residential areas and multiple commercial areas and sometimes the residential areas are also commercial destinations and vice-versa ? We would like to express our equation in a way that shows interaction between any number of residential zones and any number of commercial zones.

We do it by introducing two more variables - i and j (where i stands for any residential zone and j stands for any commercial zone) and re-writing our equation thus -

A_ij = f(B_i, C_j, D_ij)

 Where -

A_ij - number of shopping trips made from zone i to zone j

B_i - residential population of zone i

C_j - number of shops in zone j

D_{ij –} distance between zone i and zone j

 

Our model represented by an equation comprising just a few alphabets can now handle a rather complex interaction of trips originating from any number of zones and terminating in any number of zones. 

We can refine the model further by considering the fact that A is directly related to B and C i.e. if B and C increase, chances are that number of shopping trips will also increase and B and C decrease then number of shopping trips would decrease. 

Furthermore, A and D are inversely related. If the shopping area is too far away (distance between i and j is too large) then the tendency to go and shop there would be low. Therefore as D increases, A would decrease. 

 

Therefore A = f(B, D) and A = f(1 / D)

 

Our model now becomes -

We now know how the number of trips would be affected based on the variables B,C and D. But how are they related exactly ? The model tells us that if distance increases from 5 km to 10 km then number of trips should decrease, but by how much exactly ? Do they interact linearly i.e. for every unit increase distance the number of trips decrease by one unit; or in some other way e.g. one unit decrease in trips when the distance becomes a square (from 5 km to 25 km) ? 

 

What we have discussed are the concepts of gravity (how strongly do different zones attract each other) and distance decay (how the intervening barriers between zones e.g. distance, cost, quality of infrastructure etc), which are key concepts in mathematical urban models.

 

When an equation just hits us out of the blue it looks difficult to understand, but when we break it down to its components and understand the logic behind their construction then it starts getting de-mystified. In fact, science is all about de-mystification. Once we get a grip of the logic then it not only does not seem complex, it actually seems very clear and basic. At that point we can begin to play around with it, understand its strengths and limitations and get comfortable with applying it.

 

Constructing mathematical models is a creative process

It is clear that what variables we choose to construct the model depend on us and on our understanding of the reality that we are attempting to study and predict. For example, attraction of commercial areas can be measured by number of shops, but it can also be measured by total floor area of shops in a zone. Distance between zones can be in the form of kilometers but it can also be in the form of the cost of covering distance or the time spent in covering it. 

It is pretty clear that one has to be extremely observant, imaginative and creative if one wants to create a model that is able to capture the essence of various urban phenomena.There are many people who may be experts in mathematics but have no understanding of urban processes or may just apply ready-made models blindly without considering how they ought to be altered or re-constructed given empirical realities.

 

As the Soviet mathematician Elena Wentzel correctly observed,

 

"The trade of model development is an art indeed with the expertise being acquired only gradually, as it would in arts."

In the next blogs we will look deeper into the role of mathematics in planning and explore how to apply it in planning problems we encounter in our day to day professional work.