If you are an unschooler, you will be nodding your head (or rolling your eyes) here. We have all had this question, and somewhere along the deschooling journey, we have all thought or worried about this too. Will this be enough? In fact, as I read through forums and other blogs and interact with other home educators, it is the one area that I notice people have a reluctance to part with formal learning. ‘We unschool except for maths.’ ‘We do our own thing with a bit of maths each day.’
And to be honest, while I don’t agree with these concerns, I understand where they come from. Adults who have been through a traditional education system are conditioned to view maths education in a linear way that culminates in very abstract concepts you are unlikely to encounter naturally. It has left many of us feeling like this fountain of mathematical knowledge will only be bestowed upon the holder of textbooks and memoriser of principles.
The mathematics curriculum is seen as a pipeline. ‘The layers…correspond to increasingly constricted sections of pipe through which all students must pass if they are to progress in their mathematical education. Any impediment to learning, of which there are many, restricts the flow in the entire pipeline.’ I am sure many of you can relate to this, especially if you are someone who thinks of themselves as ‘bad at maths.’ I am almost certain that a closer examination of these feelings will dig up a time in childhood where you didn’t catch on as quickly as you needed to. Maybe you were sick. Maybe you had a below average teacher. Maybe something was said to undermine your confidence in your ability. Maybe you scored poorly on a test. This blockage in the pipeline served to slow progression forever.
The idea of mathematics being so structured and finite speaks volumes about how maths is communicated and taught. There are so many problems with how we as a society currently view mathematics. Before we can examine these, we need to understand what maths is.
In school, maths has been viewed as a very fragmented study. Most of us view maths in the way it was presented to us. Simple arithmetic, algebra, geometry and then, as if it were the pinnacle of all mathematical knowledge, calculus. All of this is underpinned by manual calculation and formulae. As Paul Lockhart says in his brilliant essay, ‘A Mathematician’s Lament’, ‘the impression we are given is of something very cold and highly technical, that no one could possibly understand— a self-fulfilling prophesy if there ever was one.’
In reality, this is such a narrow view. In life, you don’t start with a formula, you start with a problem, a concept, a question, a theory. Contrary to popular belief, mathematics is defined by creative thinking and problem solving. ‘[It] is an exploratory science that seeks to understand every kind of pattern—patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns.’ Despite the sterile way it is presented in school, mathematics is an art form. Lockhart states that ‘mathematics is the purest of the arts, as well as the most misunderstood…The mathematician’s art [is] asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations.’
It would be impossible in the twelve or thirteen years dedicated to formal schooling to cover its many applications and uncover its beauty. And this shouldn’t be the aim! It serves no purpose in life to have a mass produced population of generalists when what we need to do is support each student’s inherent abilities. Lynn Arthur Steen argues that ‘if mathematics curricula featured multiple parallel strands, each grounded in appropriate childhood experiences…different aspects of mathematical experience will attract children of different interests and talents, each nurtured by challenging ideas that stimulate imagination and promote exploration. The collective effect will be to develop among children diverse mathematical insight in many different roots of mathematics.’ And this is the basis of mathematics in unschooling.
Mathematics should be seen as the cultivating of a very creative and intuitive state of mind. An adventure. It is the art of explanation. ‘If you deny students the opportunity to engage… to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.’
It may seem counterintuitive if your experience of learning maths has been in a traditional classroom. However, a child’s mathematic ability grows when they are ‘exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections.’ In the kaleidoscope of information children encounter in the real world, the patterns that stand out to them, that spark their interest, that spur them to connect more dots, will be unique to them. Sitting them down and telling them which patterns matter, which arrangements to focus on, which patterns they are ready for, regardless of whether they stand out brightly to them in that moment, serves only at best to narrow the realm of possibilities in the mind of the child, and at worst to communicate that they are incapable of seeing and revealing the pattern themselves. Lockhart is more damning in his critique. ‘If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul crushing ideas that constitute contemporary mathematics education.’
So the reason I am not ‘teaching’ my children mathematics is because I want them to love maths. It is fascinating and exciting to encounter a problem and theorise a solution. To seek answers. And I see my children do this every day. Sometimes they ask for my help. Sometimes they want my feedback. Sometimes there is no need. ‘Problems will lead to other problems, technique will be developed as it becomes necessary, and new topics will arise naturally. And if some issue never happens to come up… how interesting or important could it be?’
I watch my young children deeply understand fractions, readily multiply, add, subtract, use algebra, geometry. I watch them pose mathematical problems and come up with creative answers. They don’t know they are doing these things. It isn’t a chore. It isn’t boring. For them in that moment it is relevant and interesting and exciting.
Am I worried about my children never learning basic mathematic concepts, or the ominous ‘foundation’? No. Learning these things is a natural consequence of living life and entertaining inquisitive minds as they explore. There was a fascinating study called the Benezet experiment in the 1930s. It was bold. Benezet theorised that ‘greater intellectual powers [could] be secured by warding off material which makes for mental stunting and substituting in its place content in which the children find enjoyment, as well as things common to their understanding, experience, and environment.’ What a revolutionary concept! He took formal mathematic teaching out of the curriculum in Manchester, New Hampshire until sixth grade. In its place, ‘they read, invented, and discussed stories and problems; estimated lengths, heights, and areas; and enjoyed finding and interpreting numbers relevant to their lives.’ Can anyone guess the outcome? The assessment was both qualitative and quantitative. ‘In grade 6, with 4 months of formal training, they caught up to the regular students in algorithmic ability, and were far ahead in general numeracy and in the verbal, semantic, and problem solving skills they had practiced for the five years before.’ How could that be? While the students had undoubtedly missed out on rote learning formal rules and endlessly practising the game of plugging in numbers to a set formula, what they had been doing was thinking. And, it seems that meant a whole lot.
Unfortunately, Benezet was well ahead of his time. Despite the astounding outcomes of his experiment, I can’t find others who have been so brave as to replicate it again in a school setting. What we have instead are a lot of studies showing the alarming results of memorising and applying mathematic principles with little understanding of their application. ‘Mathematics has no meaning for most students; it is a sequence of mysterious steps, which the clever or obedient quickly master to score high on examinations.’ Students become shut off from reality and rational application, focusing instead on what they assume the teacher wants to hear.
In his exploration of the Benezet experiment and its wider application, Sanjoy Mahajan discusses Alan Schoenfeld’s description of a secondary maths exam. The exam was administered to a stratified sample of 45,000 students nationwide. The following question was asked.
‘An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed? Seventy percent of the students who took the exam set up the correct long division and performed it correctly. However, the following are the answers those students gave to the question of ‘how many buses are needed?’: 29% said…31 remainder 12; 18% said…31; 23% said…32, which is correct. (30% did not do the computation correctly). It’s frightening enough that fewer than one-fourth of the students got the right answer. More frightening is that almost one out of three students said that the number of buses needed is ‘31 remainder 12’.
This deep and widespread misunderstanding of the root of the question and applicability of the answer is disturbing, and not unique to secondary school. ‘In his dissertation research, Kurt Reusser… asked 97 first and second grade students the following question: ‘There are 26 sheep and 10 goats on a ship. How old is the captain?’ Seventy-six of the 97 students ‘solved’ the problem, providing a numerical answer by adding 26 and 10.’ Similarly, Radatz tested the problem of rote learning by giving non-questions such as these to students across grade levels and found ‘[t]he percentage of students who answer such non-problems increases consistently from kindergarten through 6th grade.’ The outcomes suggest that the more time students spend rote learning, the more detached they become.
So I guess our approach to mathematics is the same as our approach to everything else. Life throws up its own exciting mix of problems and questions, and we all continue to think and experiment and talk and read and experience together. We are trusting that our children will come across their own frustrations and questions that encourage them to explore further to learn new methods and techniques. We have chosen to step away from the formula of disengaged regurgitation more broadly accepted as the study of maths.
Paul Lockhart, in writing about how teachers can improve mathematics instructions says that ‘[b]y choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject,’ we can truly ‘teach’ maths.’ I couldn’t agree more.
At some point along the way, traditional education has distilled this magnificent use of imagination and creativity into a dull and boring bulk of rote learning. We have chosen something different for our children. We have chosen instead to keep them curious, confident that they will be able to uncover and learn the skills they need as they need them. Maths has long been viewed through a strictly utilitarian lens. What I have discovered as I watch my children and learn more about the process is that maths is really just another way of highly creative thinking. G.H. Hardy has a beautiful description of what it means to be a mathematician. ‘A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.’ My kids don’t know how to make a bar graph yet, but they sure do have a lot of ideas.