In your hands is an extraordinary electronic book: an unconventional one, intended for all types of readers who deal or have dealt with mathematics in their life, be it with a sense of bitterness, indifference or delight. If you are a teacher of Mathematics, here you will find many refreshing and inspiring ideas to add to your teaching’s appeal. If you are a student, you get an essential handbook that will support you in your mathematical growth on the exciting journey of your secondary education. For all readers, it is an excellent picture dictionary of algebra that gives a palpable, real sense to the concise, abbreviated algebraic language which sometimes “obscures” the true and precise meaning of its symbolic form.
This book contains works which have been created and refined over the years of my educational practice and represents an anthology of visualisations for teaching and learning algebraic concepts.
Its development began when I started to use alternative strategies to remove students’ algebraic misconceptions. However, after witnessing positive effects beyond the scope of my original intentions, algebraic visualisations became one of the most powerful tools in my classroom. They help accurately present algebraic concepts; students gain proper understanding, which is essential for building confidence and competence in dealing with more demanding mathematical applications. I have found the principle ‘seeing is believing’ to be a proven method to convince many learners, especially visual ones. A playful, sketching approach was quickly accepted by the great majority of my students as an obvious, simple and inexpensive tool which enriched their capacity to reason, communicate and reflect upon their own and other students’ understanding.
Publications covering the history of mathematics show that even the greatest minds used visualisations to conceptualise steps in the process of solving the difficult mathematical problems of their times. Muhammad Ben Musa Al - Khwarizmi (780 - 850 AD) is recognised as the creator of the first algorithm for solving quadratic equations. He solved them using two-dimensional geometrical drawings, visualising every step of his thinking in the process today known as ‘completing the square’. In his time, he did not consider negative numbers as solutions to quadratic equations. Nevertheless, his impressive method (slightly different to how it is done today) literally employed thinking ‘outside the square’ in order to find a solution to the problem. Later on in the development of algebra, other mathematical giants like Michael Stifel (1487 - 1567) and Franziscus Vieta (1540 - 1603) visualised geometrically their algebraic thinking which helped them discover new, more sophisticated algebraic relationships used in today’s secondary education.
All these mathematicians convinced me that geometrical visualisation is the best way to illustrate each step in a student’s learning of algebra because it brings elements of elegance, beauty, illumination and understanding.
Today’s knowledge is so vast that, in the best case, secondary school syllabi contain only skeletons of what is deemed as useful knowledge to be passed on to future generations. Modern societies, reflecting contemporary social imperatives, opt to expose their present student population to the selected discrete, often disconnected and sometimes difficult to understand, fragments of a body of knowledge. Mathematics is not an exception. The unfortunate consequence and undeniable fact is that a great majority of students leave secondary schools with a superficial, algorithmic knowledge of Mathematics. Only fully devoted and dedicated students are able to get a captivating glimpse of what integral Mathematics is like.
When browsing through available secondary mathematics text books it becomes obvious that even the latest publications (even though they are more colourful, glossier and aesthetically pleasing) missed the opportunity to visualise algebraic ideas and make them apparent and comprehensible for the majority of pupils. In general, students find algebra difficult because for many it is a set of abstract, unintuitive and foggy rules. These students are not able to recognise and appreciate the big picture of algebra as a logical continuity and a condensed abstract of number theory. Freedom to overcome the limitations of daily language and move from the concrete to the abstract, and from the specific to the general becomes an inhibition for some learners that completely limits their mathematical growth. My perception is that this happens because textbook authors (and consequently many teachers) emphasise how to do a certain operation (algorithmic approach), rather than spending more time explaining why it is done that way (structural approach).
Philosophers and educational psychologists have been advocating that there are degrees of understanding and that the degree or completeness of the understanding is proportional to the amount of knowledge we have, relevant experience, and to their interconnectedness. Deeper comprehension appears as the natural outcome of meaningful learning and can be improved by selecting appropriate models of explanation which link different aspects of mathematical reality.
I have found that, when teaching algebra, geometric illustrations represent missing stepping stones which effectively visualise generalisations of numerical concepts and make algebraic representations unambiguous and comprehensible even for slower learners. They are valuable complements in teaching algebra at curriculum levels 4 and 5, but once accepted and assimilated as a powerful classroom asset, they become irreplaceable tools for expressing, explaining and integrating new concepts at curriculum levels 6, 7 and 8. By using basic information technology tools, I managed to create, modify and systematise visualised algebraic concepts for students’ and teachers’ utilisation, further investigation and exploration. Initially the resources were made for slower learners to help them establish the missing links in their algebraic understanding. With time, I have seen that even excellent students adopted this geometrical way as their preferable tool for expressing their algebraic thinking since this visual approach uncovers the structure behind the façade and helps them check and confirm the validity of their answers or locate and discover their or someone else’s mistakes. These well accepted resources also offer materials for the gifted and talented students, giving extra opportunities for devoted learners, willing and being able to explore mathematical world wider and deeper. They stimulate learners’ curiosity and encourage their personal aspirations for further mathematical learning and intellectual development.
When I was a student, only pupils with cognitive limitations were recognised as individuals who might have learning difficulties. Today it is widely acknowledged that there is a whole range of learners who have different learning styles. Teachers are expected to provide a variety of tools and teaching techniques that make the learning process interesting, engaging and meaningful for all. This set of resources is another attempt to scaffold educational activities and enrich a spectrum of possible strategies when delivering algebra as a meaningful learning experience.
Many years ago, within one educational publication, I found an elaborated idea that as soon as we start teaching basic algebra, we either “capture or lose” our students’ attention because drill and futile practice of never ending repetition in hope for a moment of enlightenment cannot substitute the comprehension of algebraic structures and processes. Understanding cannot be acquired by more simplifying, more factorising, more solving… If not properly delivered, “mysterious” algebraic language becomes a definitely confirmed mystery. Students’ learning needs are either met by a suitable choice of explanation, appropriate for our learners, or they remain unsatisfied!
This insight, which fully resonated with me, completely articulated my convictions during years of initial teaching practice. It compelled me to start developing visual teaching resources for my diverse students on both sides of the normal distribution curve. After years of utilisation and refining (based on the feedback from many learners and educators), I have decided to make them available for other students and teachers. Algebra is beautiful and students are not only willing to explore it but they become attracted to it provided they are made ready for each next step on an algebraic journey by skilful and equipped presenters.
The latest changes in the national, and consequently school curricula, accentuate the need to shift students’ thinking skills from unistructural to multistructural (relational and extended abstract). Results have shown that students who possess only procedural (algorithmic or operational) knowledge are not able to reach such goals. Students who are helped to obtain structural (or conceptual) knowledge are able to reach excellent results which demand skills for solving complex and unfamiliar mathematical situations.
This set of visualised algebraic resources is prepared for methodical, incremental building of the big picture of algebra at secondary level. The materials are presented in an increasing level of complexity in order to prepare students well for the stumbling stones on their bumpy mathematical journey. The resources uncover concealed algebraic structures and emphasise the value of the processes employed when doing algebra. Although the algebraic visualisations are primarily made as eco - friendly materials, to be used on a computer screen, data projector, smart boards or electronic tablets, they could be conventionally utilised as separate printouts to teach from, or as a copy for a relieving teacher or absent students.
The contemporary characteristic of these materials is their interactive feature. We achieve different outcomes with different learners. However, with the help of these algebraic visualisations we can softly and imperceptibly push forward the learning intentions for all learners. Instead of using a single frame which explains a certain learning objective, by going through a set of situations we can obtain longitudinal augmentation or we can have a cascade that expands learning outcomes in their depth (transversal enrichment). A well - paced combination yields outcomes which previously were inconceivable. Each new step brings on the surface new insights which help making the fine comparisons and hence assist learners to see what remains the same and what and how it has been changed. This links the logic of cause and effect - very important factor in students’ mathematical reasoning.
Algebraic visualisations present different levels of mathematical reality. You, or your students, may find them useful in different ways, but from my own experience I know that the barriers which automatically pop up when mentioning algebra to some school students can be, using the principles explained here, gradually dissolved. Algebra not only can be, but is, comprehensible and beautiful. Let’s experience it!