and its Integration with Mathematics Education: 2008 International
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Is meaningful algebra an oxymoron?
But why bother with meaning?
How might we offer meaningful algebra?
Begin with number
Build firm concrete foundations
Introduce graphs carefully
Bring it all together with modelling
Build algebraic meaning using real language
Look for scaffolding opportunities
As a result of research on effective classrooms, on what constitutes quality pedagogical practice, upon learning styles (and the particular needs of students in the middle years), it is possible to make some fairly well-supported and sensible statements at this point in time concerning good teaching and learning, the teaching and learning of mathematics, and of algebra in particular. It is then possible to relate these to the appropriate and effective use of technology for the learning of algebra in a meaningful way.
Students learn best when they are actively engaged in constructing meaning about content that is relevant, worthwhile, integrated and connected to their world.
learn mathematics best when
Students learn algebra best when
If these are the criteria by which we should judge our algebra classrooms, how many of us would be found wanting?
If algebra is to be taught in an effective and meaningful way, then it must be taught differently than has been the case in general to this point. High school algebra is perhaps the clearest example of the malaise which affects almost all of school mathematics. We can scarcely claim to be successful in the teaching and learning of a subject in which the vast majority of students, after studying the subject for at least 11 years, leave school not only being unable to apply much of what they have "learned" in any practical or realistic way to their lives, but with an active and often virulent dislike of the subject. Even many of our "success stories" may be very capable "technicians" but can scarcely claim to have any deep mastery or understanding of this discipline. They can make the moves and perform the manipulations, but do they really understand what they are doing?
By most reasonable measures, it is fair to claim that the teaching of mathematics in schools generally has been less than successful. Some might say spectacularly unsuccessful!
We can identify two significant factors which have contributed to this current state:
So what might be done?
First, look for opportunities to teach school mathematics within contexts that are rich in meaning and significance for students, engaging them and encouraging them to interact both with the mathematics and with their peers in the learning of that mathematics.
Second, reward informal as well as formal approaches to mathematical thinking. Encourage multiple representations and multiple approaches to problems and to solutions. While algorithmic approaches may be considered efficient in reaching a specified solution, the cost of that efficiency has been high, since it robs students of the opportunities to play with the mathematics they are seeking to learn, to make mistakes (and to learn from those mistakes), and to explore individually and with others in a co-operative learning environment.
And the role of technology in the process?
Technology in mathematics and science learning plays two major roles:
Good technology supports students in building skills and concepts by offering multiple pathways for viewing and for approaching worthwhile tasks, and scaffolds them appropriately throughout the learning process.
The potential role for CAS in this domain is problematic. If the biggest single problem with the teaching of algebra in schools has been an over-reliance on a syntactic approach, and computer algebra software is a purely syntactic tool, then it is capable of doing enormous damage, of further reinforcing and rewarding those approaches that have been least successful. CAS is a power tool for learning. Think of it like a chainsaw: place such a tool in the hands of the inexperienced and ignorant, and the results are easily predicted. Similarly with the power tools of learning technology, which may easily do more harm than good unless used carefully, and with caution.
It is easy to find applications of CAS in the senior years with high ability students. But is there a place for it in the early years of high school? I would argue that, CAS alone, especially in the formative years of algebra learning, is a crippled tool; but used together with other representations, and much more is possible. Computer algebra becomes part of a "complete mathematical toolkit", along with the graph plotter, the table of values, interactive geometry and statistics tools. Together these tools have the potential to bring mathematics to life in the hands of students, to offer insights into the power and capabilities of this most beautiful of disciplines, to bring the messy wonder of the real world into the classroom in ways that are capable of informing student understanding - a far cry from the piece of chalk and the textbook.
Within the domain of algebra, it is possible, I
believe, to identify certain "golden rules" for meaningful learning, rules
based upon research, the wisdom of practice and the experiences of capable
and effective educators at all levels who have informed my own pedagogical
1. Begin with Number
Two major limitations may be identified with the use of physical
concrete materials in this context: there is no direct link between
the concrete model and the symbolic form, other than that drawn by
the teacher – students working with cardboard squares and rectangles
must be reminded regularly what these represent.
Of even greater concern, these concrete models promote a static rather than dynamic understanding of the variable concept. Both these limitations may be countered by the use of appropriate technology to scaffold and support the tactile forms of these models.
These basic shapes may be readily extended to model negative values
(color some of the shapes differently and then these “cancel” out their
counterparts) and even to quadratics, using x2
shapes! After even a brief exposure, students will never again confuse
2x with x2 since they are clearly different shapes.
Furthe linking these dynamic concrete models to the graphical
representation and supported by CAS, students have a powerful and
meaningful platform from which to continue their algebraic studies.
Such tools work cohesively together to support students in laying
firm foundations for algebra. Both the syntactic demands of
manipulation and the semantic requirements of meaning are satisfied
in this way. CAS is not a necessary partner in such an approach, but
can offer a useful supplement to the representations which work so
well together here.
5. Build algebraic structure using
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