Learning models and trajectories in mathematics

Summary: Various models of learning mathematics suggest in turn different approaches to teaching.

In October 2010, educator Jill Biden, the wife of Vice President Joseph Biden, chaired the first White House summit on community colleges. She restated an idea that is increasingly in the forefront of both political and educational discussions: “The nations that out-educate us today will out-compete us tomorrow.” This idea is considered to be particularly true in the fields of science, technology, engineering, and mathematics (STEM). The philosophy and methods of mathematics education are driven not only by perceived societal needs and events, such as the Industrial Revolution, the Cold War, the civil rights movement, and the coming of the digital age, but by research and theories on the way people learn. Biologists, psychologists, mathematicians, and others have all contributed to the current body of knowledge on how both children and adults learn mathematics, and research in these areas is active and ongoing. In turn, organizations like the Mathematical Association of America and the National Council of Teachers of Mathematics synthesize this knowledge and make recommendations that shape curriculum at all levels.

The Piaget Model

Epistemologist Jean Piaget reportedly believed that what distinguishes human beings from other animals is the ability to reason with abstract symbols. The model of cognitive development that bears his name includes four hierarchical stages that mathematics educators have analyzed with regard to the development of mathematical concepts like spatial skills or abstract reasoning. In Piaget’s model, infants in the first stage can link numbers with objects and may have some understanding of counting. In the second stage, toddlers and young children can recognize the concept of closeness and other topological ideas, as demonstrated in experiments and puzzle-solving activities. However, perceptions at this level are often restricted to one aspect or variable at a time. For example, Piaget poured liquid from one container into a similar container and then into a wider container while children watched. The children failed to recognize that the volume of liquid was the same because the height of the liquid in the new container was lower. In the third stage, elementary school children and early adolescents develop logical operations skills like classification or seriation, the ability to order objects based on a variable like height, and they can analyze many variables at the same time. However, experience and training combine with the cognitive stage to determine the level of advancement, such as how successfully an early adolescent can analyze mental images of rotated objects. Hands-on activities can help students at this stage to connect abstract concepts and symbols with concrete objects. In the fourth stage, adolescents and adults can fully develop abstract arguments using symbolic notation. They can learn to analyze and evaluate logical arguments and to apply deductive reasoning.

Piaget’s theories have had broad impact on mathematics education, but not without criticism. His ideas are used to develop puzzles that are designed for toddlers. Elementary and middle grades students use mathematical manipulatives, like blocks, Pythagorean theorem puzzles, algebra tiles, or tangrams. Abstract courses, like algebra, are usually taught to adolescents because Piaget’s model suggests that abstract reasoning is developed then. Critics believe Piaget’s model may under- or overestimate the abilities of children and adolescents. For instance, children ages 3–5 sometimes notice incorrect counting sequences and can develop ingenious strategies to solve problems related to some higher stage concepts if they have age-appropriate task design and instructions. Middle grade children may not be ready in the way that Piaget asserted, though Piaget recognized that the level and time in a developmental stage varies with each child. As a result, different representations might be more meaningful to some children than to others.

Neurobiology

Right- versus left-brain learning is often discussed in mathematics education. This differentiation appears to have some basis in biology, and many people do exhibit preferences for one style over the other when tested. Critics point to people called “middle-brain” thinkers, who flexibly switch between styles depending on the situation. Some argue that brains, especially those in children, are much more malleable than previously believed, so that anyone can be a flexible learner with training or a variety of methods of engagement. The neurobiology underlying mathematics learning is not yet well understood. In the past, researchers relied largely on verbal descriptions of how people solved problems.

Visualization methods like magnetic resonance imaging allow researchers to make connections between brain components and specific processes, some of which have been found to activate parts of both the left and the right sides of the brain, often in surprising ways. For example, scans of infant brains showed that they seem to detect changes in the number of objects in an array, suggesting they have number sense. Overall, the right brain is commonly associated with holistic, subjective, intuitive learning as well as with artistic skills. Mathematics is often considered to be a left-brain activity, since the left brain is associated with logically and objectively analyzing parts or sequences to understand the whole. People have also researched teaching styles according to right- and left-brain theory. Teachers classified as left-brained more often used highly outlined lectures and discussions in their classes. They also assigned more independent problem solving or research than teachers classified as right-brained, who were more likely to use less-structured, hands-on activities or group projects that included manipulatives, art, visuals, role playing, and music. In many educational settings, teachers are encouraged to consciously consider both the ways they teach and the ways in which their students may learn in order to design a breadth of teaching and assessment methods.

Constructivism

Educational reformers in countries such as the United States, the United Kingdom, Canada, Germany, and Taiwan began strongly promoting constructivism in the late twentieth century. The constructivist framework rejects objective reality; learning is experiential, and the instructor is more of a facilitator than a teacher. The foundations can be traced to Socrates and the term “constructivism” was coined by Giambattista Vico in the eighteenth century, though many consider Piaget to be the first educational constructivist. The United Kingdom mandated constructivism in the 1980s and the National Council of Teachers of Mathematics 1989 Curriculum and Evaluation Standards for School Mathematics endorsed constructivism for U.S. schools. There are many vocal critics of constructivism, sometimes known as the “back to basics” movement. Opponents argue, among other things, that constructivism fails to systematically instill fundamental skills required for true mathematics mastery. Also, constructivist approaches can be very time consuming and difficult to assess fairly, especially in an environment of increasingly common standardized tests. Many constructivists assert that mathematics is a cognitive process shaped by sociocultural context, as well as a sociocultural phenomenon created by the community of active learners. Mathematics learning is therefore seen as a function of prior knowledge; perceptions of what others know; methods of knowledge sharing; norms of participation in the classroom or community of learners; what it means to “do mathematics”; and methods by which mathematical validity is determined.

Learning Trajectories

If all learners are unique, then schools must take into account the many ways individuals might learn mathematics. Educators have to consider what it means to know and to do mathematics, both in school and beyond, before they can develop curriculum and select teaching strategies. In some cases there seems to be a natural progression, similar to the way children learn to crawl, then walk, then run. A hypothetical learning trajectory is a hypothesized typical path that students might follow when learning a set of interrelated concepts and skills, including ways in which learning will be facilitated by the instructor. Research suggests that learning trajectories can be effective for early-grades mathematical concepts, such as counting and arithmetic. Additional research is needed on mathematics topics from later in the standard school curriculum, like patterns, as well as for more sophisticated ideas addressed in high school and beyond. Learning trajectories are also empirically linked to teacher development. For example, training in learning trajectories increased knowledge in teachers as well as motivation and achievement in students. Some researchers assert that students should be explicitly included in the formation of learning trajectories to better anticipate individual responses and divergences from the typical path.

A hypothetical learning trajectory begins with the students’ current knowledge and is targeted toward a specific “big idea” or goal, such as the idea that geometric shapes can be analyzed, described, transformed, composed, and decomposed into other shapes. The learning trajectory also includes a sequence of tasks designed to guide students in learning concepts and building upon their previous learning, taking into account that some students may think about ideas in different ways or learn them in a different order. It may also include remediation for students who begin with insufficient knowledge or extensions for students who reach the goals quickly. Consider, for example, counting. Young children first learn the words and sounds associated with numbers. Then they put those words in order, though not always completely, before they begin to associate words with objects on a one-to-one basis. Eventually they can count objects, determine why counting is important and what “how many” means, and finally acquire a true sense of cardinality. A teacher would select tasks, teaching methods, and assessments to address each stage in turn, while working with students to determine whether they are learning and adjusting accordingly. At most grade levels, students are simultaneously involved in multiple learning trajectories.

Scaffolding

Scaffolding is one teaching technique closely associated with learning trajectories. The term “scaffolding” is a metaphor for the teacher’s supporting role with respect to the student. Parents seem naturally to use scaffolding with babies and young children. Research suggests that scaffolding provides individualized instruction that engages and motivates students while also improving learning and retention. However, the method can take a great deal of time to implement in the classroom and relies on trained teachers who have access to appropriate educational materials. Further, teachers must be willing to relinquish some degree of control in order to promote students’ independence and those students must be carefully and differentially assessed at the beginning and throughout the process. In an environment where standardized testing is the norm and teachers are assessed based on student performance, or for teachers used to a more traditional approach to classroom management, the issue of control can be a difficult one to manage. Scaffolding is widely used in business and sports applications. In many settings, peer working groups serve as teachers. From an employer’s point of view, scaffolding seems well suited to promote the lifelong independent learning skills that are needed in the rapidly changing twenty-first-century job market.

In scaffolding, a student learns independently as much as possible. The teacher structures tasks and provides help with concepts or techniques that are just beyond a student’s current capability. Scaffolding usually involves several steps. First the student and teacher agree on the goal. Then the student focuses on the concepts and tasks as a whole, not as a sequence of discrete steps. The teacher is available to provide quick help and feedback. Rapid response is intended to minimize frustration and wasted time while encouraging the student’s self-efficacy. The teacher helps only with immediate needs in areas where it is truly needed—the teacher does not repeat knowledge the student has already mastered and over time intervenes less and less. The teacher may also give an explicit example as an “expert model.” All of this takes into account different student approaches and the student’s current state of knowledge.

Computer software can also include scaffolding to facilitate online or independent learning, though these scaffolds often provide static, versus dynamic, interaction with a teacher. Different sorts of scaffolds that have been explored for software include conceptual scaffolds, which help students organize ideas and connect them to related information; strategic scaffolds, which help students ask more specific questions about concepts and processes; and procedural scaffolds, which clarify tasks. These scaffolds might include suggested readings, templates for presentations or note-taking, journals, and interactive essays.

An Australian dance conference called Moving On 2000 included an interesting application of scaffolding. At an initial workshop, participants created the beginnings of a dance piece. The dancers met again later to further the work. Not every person remembered each step and sequence, so other participants assessed what they did know (prior knowledge) and then modeled the forgotten components as the learners followed along. This was done without explicit direction from anyone. Eventually the students no longer needed the teachers or “experts”; their goal of knowing the whole dance had been met. Then the dance was extended ever further by participants, who later modeled and taught the new moves to others at successive sessions. In this context, people were both teachers and learners in turn.

Conclusion

Many models and theories continue to shape mathematics education. Elements of behaviorism, cognitivism, and humanism appear in some educational approaches. One often-promoted theory is psychologist Howard Gardner’s multiple intelligences, which posits that intelligence is divided into several parts, including logical-mathematical intelligence and spatial intelligence. Italian physician and educator Maria Montessori’s early-twentieth-century philosophies about children’s self-guided, sensory learning also persist. In Montessori schools, shaped and textured beads, sandpaper numbers, and segmented rods help students explore basic mathematical concepts like numbers, place value, operations, geometrical relationships, and algebra, such as the binomial and trinomial theorems.

Bibliography

Lesh, Richard, and Helen Doerr. Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. New York: Routledge, 2003.

Ojose, Bobby. “Applying Piaget’s Theory of Cognitive Development to Mathematics Instruction.” Mathematics Educator 18, no. 1 (2008).

Sarama, Julie, and Douglas Clements. Early Childhood Mathematics Education Research: Learning Trajectories for Young Children. New York: Routledge, 2009.