Last night on
Twitter Michael Pershan asked me to weigh in on hands-on math learning. The request
stemmed from a conversation/debate about the various merits of different ways
to learn math.
The minute I read
the question I knew that my answer was going to be more detailed than a response on Twitter
would allow. Here are some of my thoughts on the matter.
1. The discussion reminded me of the "concrete to
abstract" conversations which, to me, seem like an especially frustrating example of recursion. They go round and round but we never really get anywhere new.
I think many connect the
word "concrete" to Piaget and his discussions about children's
thinking moving from the concrete to the abstract. This in turn has led to many
assumptions that take the term "concrete" quite literally. But, as Deborah Ball wrote
in her article Magical Hopes,
“Although kinesthetic experience can enhance perception and thinking, understanding does not travel through the fingertips and up the arm. And children also clearly learn from many other sources—even from highly verbal and abstract, imaginary contexts."
The best treatment
of the concrete/abstract dichotomy comes from Uri Wilensky:
"The more connections we make between an object and other objects, the more concrete it becomes for us. The richer the set of representations of the object, the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that property which measures the degree of our relatedness to the object, (the richness of our representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object."I LOVE this treatment of "concrete" as simply the quality of your relationship to an idea. Seriously, read the whole piece. You'll be glad you did.
2. Professional mathematicians utilize a multi-sensory
approach to their work. Here is some
perspective from researcher Susan Gerofsky:
“Movement, colour, sound, touch and other physical modalities for the exploration of the world of mathematical relationships were scorned ... as primitive, course, noisy and not sufficiently elevated or abstract. This disembodied approach to mathematics education was encouraged despite the documented fact that professional research mathematicians actually do make extensive use of sensory representations (including visual, verbal and sonic imagery and kinesthetic gesture and movement) and sensory models (drawings, physical models and computer models), both in their own research work and in their communication of their findings to colleagues in formal and informal settings. These bodily experiences ground the abstractions of language and mathematical symbolism.”3. Children think and learn through their bodies. We should use children’s bodies in math learning.
Known
in the research world as embodied cognition (thinking and learning with one’s
body) is something we begin developing from birth. Developmental
psychologists have shown that in babies “cognition is literally acquired
from the outside in." This means that
the way babies physically interact with their surroundings “enables the
developing system [the baby!] to educate [herself]—without defined external
tasks or teachers—just by perceiving and acting in the world.” Ultimately, “starting as a baby [as we all
did!] grounded in a physical, social, and linguistic world is crucial to the
development of the flexible and inventive intelligence that characterizes
humankind.”
Understanding what embodied
cognition and embodied learning looks like is the focus of a multidisciplinary
group of cognitive scientists, psychologists, gesture researchers, artificial
intelligence scientists, and math education researchers, all of whom are
working to develop a picture of what it means to think and learn with a moving body.
Their research findings and theory building
over the past few decades have resulted in a general acceptance that it is
impossible to ignore the body’s role in the creation of “mind” and “thought”,
going so far as to agree that that there would likely be no “mind” or
“thinking” or “memory” without the reality of our human form living in and
interacting in the world around us.
4. Finally, instead of sorting out the various merits of individual teaching/learning strategies what we really need to do is look at the bigger picture: Most student learn math best when provided with multiple contexts in which to explore a math idea.
4. Finally, instead of sorting out the various merits of individual teaching/learning strategies what we really need to do is look at the bigger picture: Most student learn math best when provided with multiple contexts in which to explore a math idea.
A learner needs
time and opportunity to experience a math idea in
multiple ways before being able to generalize it and how it can be
applied. An idea, any idea, becomes
“concrete” for the learner when the learner has had an opportunity to get to
know it. Uri Wilensky said it best:
“It is only through use and acquaintance in multiple contexts, through coming into relationship with other words/concepts/experiences, that the word has meaning for the learner and in our sense becomes concrete for him or her.”Pamela Liebeck, author of How Children Learn Mathematics, developed a useful and accessible learning sequence to help bridge the gap between a math idea and a meaningful relationship with that idea. Based on the learning theories of psychologists such as Piaget, Dienes and Bruner, Liebeck’s progression is similar to how babies and young children learn to recognize the meaning of words, begin to speak, and then to first write and then read. It includes four different learning modes in which to interact and express mathematical ideas and includes:
a) experience
with physical objects (hand- or body-based),
b) spoken
language that describes the experience,
c) pictures that
represent the experience and, finally,
d) written
symbols that generalize the experience.
This sequence
illustrates what many math educators already believe, whether or not they use
this exact outline – that elementary students need active and interactive
experiences with math ideas in multiple learning modes to make sense of math.
After a recent and particularly robust online discussion on the many different ways to support primary students in making sense of number lines, including a moving-scale line taped on the floor, Graham Fletcher said, “At the end of the day, it's all about providing [students] the opportunity to make connections.”
Graham's statement points to the importance of focusing on the child's relationship to the math and the environment in which she learns that math. Hopefully it's an environment where many different ways of thinking, expressing and applying mathematics are celebrated and nurtured.
After a recent and particularly robust online discussion on the many different ways to support primary students in making sense of number lines, including a moving-scale line taped on the floor, Graham Fletcher said, “At the end of the day, it's all about providing [students] the opportunity to make connections.”
Graham's statement points to the importance of focusing on the child's relationship to the math and the environment in which she learns that math. Hopefully it's an environment where many different ways of thinking, expressing and applying mathematics are celebrated and nurtured.
