Stories and legends told by human beings through the ages to explain our abilities and the acquisition of abilities shape our relationships with learning. Whereas entering into the learning process of language, art, or music seems natural (almost unconscious and involuntary), formal teaching and learning is human and has to be organized. The message and meaning of myths about learning evolve. Over the last few centuries, as quantification has entered ordinary life, myths about its learning have become part of life too.
The myths that we inherit about a subject shape our thoughts and our journey in acquiring the knowledge and competence in it. Many times, individuals need courage to get out of those mythical ideas that we have formed to be truly open to learning.
In mathematics learning, as children we come to the subject matter without preconceived ideas, but very quickly we are shaped by the ideas and myths about mathematics learning our caregivers consciously and/or unconsciously share with us. Then we struggle between meaning making from our real experiences about learning math or forming ideas about it and reading these experiences in the shadow of imparted myths. Early on, many students, when they do not have positive experiences or do not have skills to make meaning from their own experiences, succumb to the prevailing myths about math learning.
In spite of many efforts by mathematics educators, psychologists and social reformers, the myths about mathematics learning and achievement persist. These myths color students’ mindsets about mathematics and its learning. Even administrators who are well-meaning and able but not well-versed in mathematics perpetuate these myths by emphasizing gimmicks and easy solutions to the problem of mathematics achievement. Moreover, even the experts in learning assumed that ability to learn (particularly mathematics) was a matter of intelligence and dedicated smarts and therefore did not study the issue. They assumed, it seems, that either people had the skill of learning or they did not. For them, intelligence –and thus the ability to gain mastery—was an immutable trait.
Yes, for learning mathematics, one needs some cognitive abilities, but one also needs to engage in the process of learning. The field of learning is rife with vague terms: studying, practice, know, mastery, etc. For example, does studying mean reading the mathematics textbook? Does it mean doing sample problems? Does it mean memorizing? Does practice mean repeating the same skill over and over again (like memorizing flash-cards, doing mad minutes)? Does practice require detailed feedback? What kind of feedback? Should practice be solving hard problems or easy problems? Should practice be intense or small chunks? There are so many imprecise terms, which feed into the myths a person selects. Effective teachers help students to achieve freedom from these myths.
Mathematics ability is inherent. You have to be born with a mathematical brain.
People, children as well as adults, who are successful in mathematics are not usually born that way. Many individuals with mathematics anxiety tend to believe that you either have the ability or you don’t, rather than assuming that your skills and abilities are the result of study and practice. When students’ mentors—teachers, parents, sports coaches, successful and intelligent people brag about not being good at math, not being numbers people, they reinforce disinterest in mathematics.
Learning math, like learning in general, takes knowledgeable teachers with high expectations, willing students, and, most importantly, a great deal of time and practice that result in success during each session. A growing number of studies shows that learning is a process, a method, a system of understanding. It is an activity that requires focus, planning, and reflection, and when people know how to learn, they acquire mastery in much more effective ways.
Learning math is much like learning a language—both need a great deal of exposure, “gestation” time, and with usage learners get better. Learning mathematics takes time, effective practice, and help in making connections. The symbols and notations make up the rules of grammar and the terminology is the vocabulary. Doing math homework is like practicing the conversation of math. Becoming fluent (and staying fluent) in math requires years of practice and continuous use. That is true about any field. To be good at anything we need to practice. Learning mathematics is a dedicated, engaged process; it is not a spectator sport. It is not about memorizing facts (static data); it is more about what we do with that data (look for patterns) and how we think better (convert patterns into strategies) by the help of that data (learning).
Mathematics is a very difficult subject to learn.
Many believe that only the very few can do mathematics, that it is difficult to learn mathematics. It is not that mathematics learning is difficult. Many times, it is the method of teaching, efficient models, and effective language usage by the teacher and the students that are the key to learning mathematics. Learning and teaching methods affect the outcomes in every field of learning. For example, to make sense of a concept and to make connections with other concepts, the teacher should use efficient language and effective models to conceptualize and should ask a great deal of enabling questions to help internalize the learning.
Effective strategies and practice boost performance from baseball and tennis to balancing equations and proving theorems. On the other hand, when mathematics teaching is approached with an emphasis on procedures and memorization and when concepts and topics are taught in a fragmented manner, students see mathematics as a difficult subject.
Apart from effective teaching, the mathematics curriculum has to be well orchestrated at each grade level. To engage all students and take many more to higher levels, it is important that school systems emphasize (a) articulation and mastery of non-negotiable skills at each grade level, (b) common definition of knowing a concept or procedure by everyone concerned with mathematics education, (c) knowing the trajectory of developmental milestones in mathematics learning, and (d) the most effective and efficient pedagogy that respects the diverse needs of all children.
Many teachers feed into the myth about the difficulty of mathematics when they begin a topic with statements such as: “Fractions are difficult.” “Algebra is not for everyone.” “Irrational numbers are truly irrational, they generally do not make sense.” These kinds of statements make mathematics look difficult and then it becomes truly difficult.
Mathematics is the integration of language, concepts, and procedures. It is the study of patterns. As Godfrey Harold Hardy (mathematician) said: A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. Statements such as Hardy’s can serve as counter narratives to the myth of the difficulty of mathematics.
Another way to counter the myth is when teachers show students how it took many mathematicians a significant amount of time to develop the formulas and equations that they are attributed with creating and that they are studying now in middle and high schools. Teachers need to show that, as in any other field of endeavor, most mathematicians, even though they made mistakes, persisted in the process.
Students should be encouraged to learn that mistakes are part of learning and creating mathematics. Mistakes and efforts literally improve the cognitive abilities that are needed for any learning and, therefore, mathematics. Students struggle to find an immediate solution when solving problems and they give up so easily, but many mathematicians took many years to solve a single problem. This shows students that math is not about speed but rather devotion and perseverance. A teacher’s selection of a problem and the methods of attacking the problem should be well-thought out so that they begin to take interest.
You cannot be creative and be good at math.
When people asked what I did for living, I would hesitate to admit that I teach or do mathematics. As soon as I announced this to a person at a party, for example, most times the response was: “Math was never my cup of tea. I am right hemispheric person.” “I am a creative person. I like creativity.” “I am humanities and people person.” “Math is not interesting. I wanted to be in an interesting field.”
Can you be an artist, writer, or musician and be good at math too? Yes! Math is found throughout literature, art, music, film, philosophy, and it is essential to many “creative” fields. Although the structure of mathematics is created by man, every aspect of life can be modeled by mathematics principles. It is pervasive in nature, society, and all edifices. Mathematics is constantly making new tools that help all aspects of human endeavor. Its collection of tools shaped the imaginations of Leonardo DaVinci, Mozart, M.C. Escher, and Lewis Carroll. These are just a few of the artists who used math extensively in their works.
Geometry is the right foundation of all painting. – Albrecht Dürer (artist)
I am interested in mathematics only as a creative art. …Mathematics is the study of patterns. – GH Hardy (mathematician)
The mathematician’s best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another. – G Mittag-Leffler (mathematician)
In addition to increasing student interest in learning math, a deeper understanding of how concepts were developed and which mathematicians were responsible for them adds content knowledge. Explaining and showing how the Babylonians and Greeks worked to get more precise values of pi only gives teachers additional credibility as masters of content.
Incorporating the history of math also allows for an interdisciplinary approach to teaching. As we move from a STEM philosophy to a STEAM philosophy (including the arts), the history of math shows us a relationship between music and mathematics, in particular with Pythagoras, as well as between art and mathematics, such as in geometry. There are many ways to incorporate the history of math into your classroom.
Mathematical insight comes instantly if it comes at all.
Most people have an incorrect conception of creativity and insight because they think of them as spontaneous and automatic. But ideas, insights, and creativity in all areas most of the time come after a person devoted time thinking about them. The second wind in running comes only after one has run for sometime. When one has basic skills, thought about an idea, and then continues to think about it, one begins to make connections, see relationships, and get insights. For this reason, we need to move from thinking about why acquire new skills and knowledge to practicing that is more dynamic, productive and efficient.
The learning cycle in mathematics, as in other subjects, is predictable: it begins with information gathering, and we need some basic skills for this. Then, with discussion and exchange of ideas, we convert this information into knowledge. With application and usage in multiple settings (intra-mathematical, interdisciplinary, and extra-curricular applications) we convert our knowledge into insights. And over a period of time, with deeper engagement with the subject, we might even become wise (we may have expert thinking skills or ability to solve unstructured problems) in that area. For tough unstructured problems one begins to look for analogous thinking in other fields. Imagination and creativity become prominent as soon as we cross the threshold of information gathering to conversion into knowledge.
Mathematics is a male domain—women are not as good at math as men.
During the last twenty-five years, a great deal of effort and progress have been made in recruiting a large number of girls and women into mathematics courses and the field of mathematics. However, the myth of mathematics as a male domain still keeps many women out of the field. Even today, young girls may not be encouraged to investigate the world in the same way that boys are. The type of games and toys we give boys and girls to play and the type of language we use with the children is different. Boys are given blocks, science kits, and construction tools and are encouraged to explore, represent, and express their world in more mathematical ways than are girls. If more girls were given the same support and opportunities that boys have to excel at mathematics, there would be many more high-achieving girls and women in mathematics.
I am not going to use mathematics, so I don’t need to learn it.
Most teachers want to answer this question by giving “pie in the sky” kinds of applications of mathematics. Which is not difficult to do. Application of mathematics is the real story of human civilization. However, the message behind this question is: “I have never succeeded in mathematics. I do not know the previous content, so how can I learn this new content.” When teachers are not able to provide success in simple mathematics, they create a population who is going to ask the question: “Where am I going to use it?” every time they introduce a new concept. Interestingly, few successful students ever ask this question, and definitely not in the tone that this question is generally asked. My own approach is to take a simple concept that students find difficult and then provide success in that concept. Then repeat this many times. It is impossible to learn if one does not want to learn, so to gain expertise, we have to see the skills and knowledge as valuable. What is more, we have to create meaning. Learning is a matter of making sense of something.