Learning - A perspective

“Learning is a basic, adaptive function of humans. More than any other species, people are designed to be flexible learners and active agents of acquiring knowledge and skills. Much of what people learn occurs without formal instruction, but highly systematic and organized information systems – reading, mathematics, the sciences, literature, and the history of a society – require formal training, usually in schools. Over time, science, mathematics, and history have posed new problems for learning because of their growing volume and increasing complexity. The value of the knowledge taught in school also began to be examined for its applicability to situations outside school.

Science now offers new conceptions of the learning process and the development of competent performance. Recent research provides a deep understanding of complex reasoning and performance on problem-solving tasks and how skill and understanding in key subjects are acquired. … ”

My point in citing this and other works on learning research is that learning mathematics is, first and foremost, learning. Our subject is not exempt from what others have learned about learning, and indeed our curricula and pedagogy, to be successful, must be informed by research on learning. Readers of this article will probably not be surprised by any of the findings but may be surprised to learn the strength of the research base underlying the strategies we have come to associate with the words “reform” and “renewal.”

The 1990’s have been described as “The Decade of the Brain,” a period in which the study of live, functioning, normal brains has come into its own through non-invasive technologies, such as positron emission tomography (PET) and functional magnetic resonance imaging (fMRI). This research will continue for many decades, of course. As the NRC study states (p. xv), “What is new, and therefore important for a new science of learning, is the convergence of evidence from a number of scientific fields.” (Emphasis in the original.) That is, the messages from neuroscience are entirely consistent with and supportive of what we have learned from developmental psychology, cognitive psychology, and other areas of research.

There is one sense in which learning mathematics is different from learning many other things, such as speaking our native language, remembering visual and aural images of familiar people and places, and driving a car. The first and most fundamental biological fact about our brains is that they have not evolved significantly from the brains of our hunter-gatherer ancestors. Thus, we are superbly adapted – or would be if it were not for environmental influences – for fight-or-flight decisions and other survival tactics. As Dehaene (1997) has so beautifully documented in The Number Sense, this means that humans (and other species as well) are practically hard-wired to do arithmetic with small integers – but everything else in mathematics is hard, because it doesn’t come to us instinctively. On the other hand, we learn many things that are not instinctive in an evolutionary sense, such as history, philosophy, foreign languages (beyond infancy), music, and neurobiology. One might say the Education is about learning the things that hard to learn – of which mathematics is just one example. [Exercise for the reader: Why is “driving a car” – clearly not an evolutionary adaptation – a relatively easy task for adolescents and adults in a developed society?]

We summarize here some of the key findings (Bransford, et al., 1999, pp. xii-xviii) that are relevant to collegiate education, in particular, to undergraduate mathematics.

♦ Collateral Development of Mind and Brain

• “Learning changes the physical structure of the brain.”

• “Structural changes alter the functional organization of the brain, [i.e.], learning

organizes and reorganizes the brain.”

• “Different parts of the brain may be ready to learn at different times.”

♦ Durability of Learning and Ability to Transfer to New Situations

• “Skills and knowledge must be extended beyond the narrow contexts in which they are first learned.”

• “…a learner [must] develop a sense of when what has been learned can be used …. Failure to transfer is often due to … lack of … conditional knowledge.”

• “Learning must be guided by general principles …. Knowledge learned at the level of rote memory rarely transfers ….”

• “Learners are helped in their independent learning attempts if they have conceptual knowledge. …”

• “Learners are most successful if they are mindful of themselves as learners and thinkers. … self-awareness and appraisal strategies keep learning on target … . … this is how human beings become life-long learners.”

♦ Expert vs. Novice Performance

• “Experts notice … patterns … that are not noticed by novices.”

• “Experts have … [organized] content knowledge …, and their organization … reflects a deep understanding of the subject matter.”

• “Experts’ knowledge cannot be reduced to sets of isolated facts … but, instead, reflects contexts of applicability ….”

• “Experts have varying levels of flexibility in their approaches to new situations.”

• “Though experts know their disciplines thoroughly, this does not guarantee that they are able to instruct others ….”

♦ Designs for Learning Environments

• “Learner-centered environments … Effective instruction begins with what learners bring to the setting … learners use their current knowledge to construct new knowledge … what they know and believe at the moment affects how they interpret new information … Sometimes learners’ current knowledge supports new learning; sometimes it hampers learning.”

• “Knowledge-centered environments The ability to think and solve problems requires knowledge that is accessible and applied appropriately. … Curricula that are a ‘mile wide and an inch deep’ run the risk of developing disconnected rather than connected knowledge.”

• “Assessment to support learning … Assessments must reflect the learning goals …. If the goal is to enhance understanding and applicability of knowledge, it is not sufficient to provide assessments that focus primarily on memory for facts and formulas.”

• “Community-centered environments [An] important perspective on learning environments is the degree to which they promote a sense of community. …”

♦ Effective Teaching

• “Effective teachers need ‘pedagogical content knowledge’ – knowledge about how to teach in [the] particular [discipline], which is different from knowledge of general teaching methods.”

• “Expert teachers know the structure of their disciplines and [have] cognitive roadmaps that guide the assignments they give …, the assessments they use …, and the questions they ask in the … classroom ….”

♦ New Technologies

• “Because many new technologies are interactive, it is now easier to create environments in which students can learn by doing, receive feedback, and continually refine their understanding and build new knowledge.”

• “Technologies can help people visualize difficult-to-understand concepts ….”

• “New technologies provide access to a vast array of information, including digital libraries, real-world data for analysis, and connections to other people who provide information, feedback, and inspiration, all of which can enhance the learning of teachers and administrators as well as students.”

Learning in general and on learning mathematics in particular (with or without technology), together with my teaching and development experiences, lead me to several conclusions:

1. Curricula need to be rethought periodically from the ground up, taking into consideration the tools that are available. It is not enough to think of clever ways to present mathematics as the content was understood in the mid-20th century, when the available tool set was quite different, as was the intended audience.

2. Much of the effort that goes into curriculum design can be squandered if one does not also rethink pedagogical strategies in the light of research showing the effectiveness of active learning strategies and distinguishing between good and bad ways to stimulate deep learning approaches. It is not enough to adopt (or write) a new book or even a new book plus-software package.

3. Our tools for assessing student learning – whether for purposes of assigning grades or for evaluating effectiveness of our curricula – need to be consistent with stated goals for each course and with the learning environments in which we expect students to function. It is not enough to continue giving timed, memory-based, multiple-choice, no-tech examinations.

4. If we are serious about mathematical understanding for everyone with a “need to know” – not just the potential replacements for the mathematics faculty – then we must plan our curricula, pedagogy, and assessments for effective learning of the skill sets and mental disciplines that will be needed by a mathematically and technologically literate public in the 21st century. It is not enough to keep using ourselves as “model learners.”

5. Revision of curricula, pedagogy, assessment tools, and technology tools will accomplish little without concurrent professional development to keep faculty up to date with the required skills, knowledge, attitudes, and beliefs. It is not enough to continue acting as though an advanced degree in mathematics is evidence of adequate preparation to teach.