Mindful mathematics

Professor Ellen Langer shares her experience and insight of how we can learn about mathematics and any other discipline in a more mindful way so that we can have the freedom to make it our own.

Illustration by Cayla Basson
Is mathematics mindless?

I think mathematics is a beautiful discipline but I think we need to differentiate something that works in the abstract from something that works in the real world. It has its place in the abstract. If we have one and one in the abstract based on the number system, then one and one is two, period. But if you are in the real world and you add one pile of mud with one pile of mud, then one and one is one. So abstract mathematics is certainly useful but not in all contexts.

In some of your work you said that developing mindfulness is creating new categories, being open to new information, being open to more than one perspective, attention to process (doing) rather than the outcome (results), and trusting of intuition.

That is exactly right and it all comes down to recognizing that you don’t know. So when you don’t know you naturally do these things. And if you were taking 10 items and categorizing them, you need to recognize that it could be categorized differently with some other criteria. Everything we are doing we can be doing in so many different ways. Each time we come up with a different way there are new advantages.

Often in mathematics we are taught to follow a procedure, in a mindless way perhaps. But mathematics can teach us to do all these things that you define as mindfulness mentioned above. Is there a way to teach a more mindful mathematics?

Teaching mathematics should be the same as teaching anything. What I mean by this is that everything should be taught conditionally. Rather than teach this is, teach as this could be. Then you recognize that there could be other ways. And if you are clever enough or interested enough, you find some of those ways. That is the essence of progress, not to take what is as the only way it can be. So if I was to teach one and one, rather than say one and one is two, we say one and one could be two and then the person listening to this would ask “when will it not be two?” and they come up with different alternatives and different understanding. It becomes more fun as they are creating their own knowledge in order to create some of those new ways of understanding.

If you were to teach history you could say for instance “Here are the three reasons for the Civil War”. Instead we should be saying from one perspective and then the person thinks how other people from other countries at other times or other people living at that time could have seen it differently and that again becomes more exciting rather than something you just want to memorize. The way we are taught to learn is to memorize.

Let me tell you a story, I was at a horse event and this man asked me if I could watch his horse because he would go to get his horse a hotdog. But I know that horses are herbivores and horses do not eat meat. He came back with the hotdog and the horse ate it. But then I thought, what does it mean that horses don’t eat meat? How many horses were tested, how many grains were mixed with how much meat, how large is the horse? And I recognized that virtually all our facts are suspect. All our facts are true under some circumstances but not others. And we need to be taught these facts in this way. All we need to know is that those people that are advancing in the academic field are not trapped in the understanding of the past. As a world we tend to solve today’s problems with yesterday’s solutions.

When you are learning the solution you recognize that these solutions are true only some of the time, then I think we will be more willing to test out different options. We need to teach people the information we have discerned over the years. It is fine to know that one and one in some abstract way is two and we should be teaching that. But we also need them to understand that this is only one way of doing things. They should learn that way, but if they come up with another way it is also fine. If you are teaching young kids and you say how much is one and one, the kid say one plus one is one. Teachers are likely to punish him or look at him as he is a fool. In the system I am suggesting the alternative would be to ask the kid “how did you come to that? ” And the kid could proudly tell you that if he takes one clump of chewing gum and if he adds it to one clump of chewing gum then he gets one plus one is one. Then the teacher and the student have learned something.

Illustration by Liani Malherbe
How do marks impact students? 

There is literature on this called over-justification where you see that when you reward people, the reason they are doing it is in order to get the reward rather than learning the intrinsic value of doing it and the fun that one can experience doing the activity. As a side note, in a study I have done on the elderly, people believe that your memory has to become worse as you get older, but I do not think that that is the case, certainly not to the extent that most people assume. Part of the reason you do not remember is that remembering does not matter. That is how it is today for many people if you are inside everyday. What is the difference if it is Tuesday or Thursday. If you ask people what is today, they won’t be sure. So we need to make memories matter in order to have a better memory.

I do not believe that rewards are the best way to lead to engagement. If people are actively noticing what they are doing, then they become engaged and this is its own reward. For instance, if you are painting and you are painting so that you can sell it and make money, it becomes tedious and you become burnt out. In fact when people in medicine and all fields are following these mindless rules, they suffer from burnout. But if when you are painting, when you are seeing a patient, when you are teaching mathematics or listening to mathematics being taught, if you actively engage with the material then it is its own reward.

What I explain to people, no matter what they are doing, is that they can do it the way they are supposed to do it or they can do it their own way. They can fail or they can succeed. The worst case scenario for me personally is to do it your way and to fail. If I do it my way which is utterly different, at least I have enjoyed it and made it my own. The worst is to do it your way and to fail. I do not see any virtue in that. People need to understand that everything that was put there by other people is based on their needs and what was true in the past – it is not true in any absolute sense. You will naturally enjoy something when you understand that you have more freedom to make it your own.

You were the first woman ever to be tenured in psychology at Harvard University. As you know, there are not many female mathematics professors. It is a challenge to increase the number. Do you have any thoughts on the reason for this? How could we inspire more diversity into mathematics?

Part of the problem with diversity, if we look at the criteria for excellence, is that we have to recognize that those criteria were developed by certain kinds of people and we remember that those criteria are not absolute. Let me give an example, let’s say that a woman wants to be a fireman. So she takes the course on how to be a fireman and she does not do as well in all the tests and she cannot be a fireman. So it seems that everything is the way it should be: there is a test and you don’t do well so you do not get admitted. But we need to ask who designed the test. You might say that the fireman needs to be a big strong guy. Ok, that criteria might be necessary for certain fires but if you go to some disaster scenes where you have people who are trapped in small places, it might be that a very small woman who can wedge her way through the hole will be of help. She could get assistance to the person trapped while waiting for the big machine to take the person out.

Basically the criteria for excellence for any field was designed by certain people and met their needs. We need to open up this criteria. It does not mean that we want to take people who are not good at whatever the area is. But we have to question who decided what is good and how might women bring something new to the area of study. And I think that if we teach everybody to be more mindful and we teach conditionally, there will be no reason for as many women as men to rise to the top. If we have a field appealing to people, people are going to join the field. If you have some teachers of mathematics who are not giving women their fair share and time to answer questions, if you are not encouraging them, then there will be fewer women available to become PhDs in that field. People with different experiences in life bring different skills and attributes to whatever the discipline is. Rather than assume that we know exactly what we need, we need to recognize that different talents might lead a field in an exciting new way.

Prof. Ellen Langer

Department of Psychology
HARVARD UNIVERSITY

Prof. Ellen Langer is a professor in the Department of Psychology at Harvard University. Her books written for general and academic readers include Mindfulness and The Power of Mindful Learning, and the forthcoming Mindful Creativity.

Prof. Langer has described her work on the illusion of control, aging, decision-making, and mindfulness theory in over 200 research articles and six academic books. Her work has led to numerous academic honors including a Guggenheim Fellowship, the Award for Distinguished Contributions to Psychology in the Public Interest of the American Psychological Association, the Distinguished Contributions of Basic Science to Applied Psychology award from the American Association of Applied and Preventive Psychology, the James McKeen Cattel Award, and the Gordon Allport Intergroup Relations Prize.

Prof. Langer is a Fellow of The Sloan Foundation; The American Psychological Association, the American Psychological Society, The American Association for the Advancement of Science; Computers and Society; The Society for the Psychological Study of Social Issues; The Society of Experimental Social Psychologists. In addition to other honors, she has been a guest lecturer in Japan, Malaysia, Germany, and Argentina.

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