With much of Australia back in lockdown, we are once again facing the challenges of remote learning. One of these is how to make abstract mathematical concepts tangible to our students. One such concept that is routinely challenging to students is fractions. Somehow, despite our best efforts, students develop some substantial misunderstandings about fractions and a profound fear that blocks deeper learning.
In times when we taught face-to-face, some of these challenges would be overcome through the use of concrete materials, at least with younger students. Unfortunately, it is common for the use of concrete materials to decline as students grow older. There is a sense almost that concrete materials benefit only remedial students. As such, any student who demonstrates a preference for concrete materials is likely to be considered amongst the less able learners.
Fortunately, this pattern, and the prejudiced beliefs on which it is founded, are today being questioned. Teachers are beginning to understand that our students develop a deeper understanding of abstract concepts when they are encouraged to represent them with concrete materials. Students benefit from exploring concepts through multiple representations at all stages of mathematical learning and in all areas of mathematics. In mathematics, representation is a tool for thinking; it is the way that we make patterns visible and one of the key ways in which we reason mathematically. If a student truly understands a concept, they should be able to represent it in many forms, and it is through experimenting with multiple representations that students develop a deep understanding.
“We use representational forms to communicate ideas and as tools for reasoning. Therefore, mathematical proficiency hinges on learning how to construct, communicate, and reason with representations.” (Selling. 2016 p191)
Rather than seeing representations such as a graph, diagram or set of concrete materials as a way of communicating a solution, students should see representations as a tool for achieving a solution.
“In mathematics classrooms, representations are often treated solely as the product of mathematics questions. . . This contrasts with authentic mathematical activity in which representations are constructed when useful as tools for thinking and communicating about mathematics.” (Selling. 2016 p191)
“Dreyfus (1991) argues that learning progresses through four stages: using one representation, using multiple representations in parallel, making connections between parallel representations, and finally integrating representations and moving flexibly between them.” (Cited in Selling. 2016 p192)
While it is important to give students the opportunity to engage with open-ended, real-world mathematical tasks, this alone will not be sufficient to build mathematical competence. Students also require experience with the tools of mathematical thinking if they are to thrive in more open, active and authentic learning environments. As teachers, we must “deliberately cultivate mathematical practices, including representation, in classrooms.” (Selling. 2016 p193)
“…students are never too old or too smart to benefit from hands-on materials so never keep them locked away in a cupboard or storeroom. . . Students should feel they can use concrete materials when and if they need them. After all, we want our students to be critical, creative mathematicians, and hands-on materials assist learning, and promote flexibility in thinking and important problem-solving skills.” Dr Catherine Attard - Engaging Maths
Therefore, students need to be shown a variety of representations, have constant access to concrete materials, consider where particular representations may be more or less suitable, experiment with learned and invented representations, and reflectively evaluate the representations they use. The method of representation should not be decided solely by the teacher. Representations should include diagrams, illustrations, charts, tables, graphs, models, concrete materials, numbers and number substitutes (algebra) in both digital and analogue forms as a minimum.
Lockdown brings fresh challenges and opportunities to teachers hoping to develop the mathematical reasoning skills of their students through the use of concrete materials. While access to the cupboards and crates of concrete materials we rely on in school is not currently possible, teachers can find creative ways to transform routine household items into valuable learning resources. Thanks to a colleague, my students used a collection of tape measures to represent decimals between 0 and 1. With the tape measures including millimetre increments, the students could see how one metre is divided into tenths, hundredths and thousandths. Kitchen maths has allowed another colleague to explore a range of mathematical concepts with her students. I routinely share the mathematics used in my workshop with my students, revealing the practical application of the concepts we are investigating.
Many lockdown lessons have been saved by Mathigon’s Polypad. This online platform does an excellent job of replicating and even enhancing the concrete materials we would use if we were face-to-face. The developers are continuing to evolve the site and are regularly adding new features. The image below shows some of the options available. In addition to this, there is a library of well-designed “Files” that include explanations, teaching notes and activities for students.
Click to enlarge.
Having started our Fractions and Decimals unit just after the first lockdown of this term, we were still congratulating ourselves on our excellent timing when the current lockdown was announced. It was nice to have introduced this traditionally tricky concept while the students were in class, and the “Visualise” activities we borrowed from Jo Boaler et al.’s “Mathematical Mindset” series had given us a strong foundation. Moving into remote, we needed to find a way to maintain this momentum and continue building a deep understanding by maintaining an emphasis on physical representations before moving towards the abstract.
Polypad allowed us to do this and more. If you are familiar with using Fraction Bars or blocks, you will find the Fractions options in Polypad very familiar. As you explore the site’s functionality, you find options beyond stacking Fraction Bars to show equivalence or compare fractions. The virtual nature of the manipulatives makes it easy for them to be adjusted. There is an option to rename fraction bars to show equivalent fractions, and this allows the students to see how halves can be transformed into quarters, eighths, twelfths, etc. It enables students to visualise two quarters minus one eighth and why you can’t just do a simple subtraction and claim to have one quarter. Students see and begin to understand why they need to change both the numerator and the denominator and that, indeed, a fraction is a single number that captures a relationship. When it is time to move on to decimals, Polypad allows the fraction bar labels to be converted to either decimals or percentages, making the transfer of knowledge easier. With access to a tool like Polypad, students move beyond rote learning of the processes for working with fractions and instead build an understanding of them.
By Nigel Coutts
Selling, S. (2016) Learning to represent, representing to learn. Journal of Mathematical Behavior. 41 (2016) 191–209