Maintaining a focus on concrete representations of mathematical concepts during remote learning.

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.

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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

Insights into the true power of Number Talks

Number Talks are a wonderful way to see where our students are with their mathematical thinking. As a part of a daily routine, a Number Talk promotes number sense and mathematical reasoning. In this post, I revisit what a Number Talk can reveal about our students’ understanding of mathematics, and how they might be used to promote a fresh perspective. In addition, I examine a success criteria for Number Talks that is more expansive and recognises their true power. 

"Number Talks" are an approach to the teaching and learning of Number Sense. Rather than relying on the rote-memorisation of isolated number facts achieved through drills of "table-facts", Number Talks aim to build confident, number fluency, where learners recognise patterns within and between numbers and understand the properties of numbers and operations. Number Talks are a "mind on" learning task that engages students in an active learning process as they search for patterns, decompose and recompose numbers and develop a flexible understanding. It is achieved through direct instruction methods and facilitative dialogue with the teacher or between groups of peers who have had experience with the number talks methodology.  Number talks are all about mathematical reasoning. In place of an emphasis on right answers, we have an emphasis on the rationale for the response. Number talks are most effective when they become one of the routines of a classroom focused on mathematical reasoning and are a great fit with visible thinking strategies. 

Number talks are a valuable classroom routine for developing efficient computational strategies, making sense of math, and communicating mathematical reasoning. A number talk is structured to help students conceptually understand math without memorizing a set of rules and procedures. (Nancy Hughes)

Number talks are:

a brief daily practice where students mentally solve computation problems and talk about their strategies, as a way to dramatically transform teaching and learning in the mathematics classroom. Something wonderful happens when students learn they can make sense of mathematics in their own ways, make mathematically convincing arguments, and critique and build on the ideas of their peers. (Humphreys & Parker)

In a number talk, I am inviting and requiring students to explain their thinking. Mathematical reasoning becomes more important than correct answers. Ask students to solve an addition like 68 + 95 in a number talk and you will know which students understand place value. While participating in the Number Talk students share numerous approaches to each question. They share and hear a range of strategies. Provide students with a whiteboard so they might make their thinking visible and you open new possibilities. Include the option of an extended Number Talk using concrete materials and you allow for diverse representations of mathematical thinking. In each instance, the students are revealing how they understand number and each response offers new insights to the teacher for future learning. Number Talks by design close the gap between student performance and teacher action to address and remediate misunderstandings.

Recently a colleague was engaging her class in a number talk. She had decided to use a “Which one doesn’t belong?” strategy with her class. In this, students are shown a collection of four related items and are asked to nominate one that they feel does not belong in this collection. Thinking is elicited through this strategy when students are asked to justify their choice. She had shared a collection of four numbers: 125, 135, 140, 145 and presented them as in the diagram below. The students were given thinking time and when ready, as indicated by a thumbs-up signal, the teacher invited suggestions. On this day, with this group of learners, suggestions were scarce. After some initial tentative offerings the class began to share with a little more confidence but they never truly picked up on the possibilities. Before the lesson, it had been discussed that the students might identify 140 as the one that did not belong. It was the only multiple of ten, the only odd number, the only number that did not end in a five. Maybe the students would notice that there would be a sequence of numbers increasing by ten from 125 and that the fourth number should be 145. With more mature and capable learners they might have noticed that only 135 is divisible by three. The teacher’s initial reaction was that the Number Talk had not gone well. 

Looked at from the perspective of students demonstrating a high level of mathematical confidence, the session was not great. But this is not the key aim of a Number Talk. A dip into the world of Visible Thinking helps to better understand the value of a Number Talk. In “The Power of Making Thinking Visible”, Ron Ritchhart and Mark Church share the value that thinking routines have and offer the following advice:

Don't judge your success with how smoothly the lesson went. This improves with time. Judge your success by what is revealed about your students' thinking. The question we need to be asking ourselves as teachers after using a thinking routine is: “What have I learned about my students' thinking as a result of doing this routine?” (Ritchhart & Church)

In this instance, the Number Talk revealed much about where the students are with their number sense. Prime amongst the revelations was that they don’t yet have a full understanding of odds and evens and that they are yet to master either multiples of ten or five. The teacher came away with three pieces of useful information and can now develop a strategy to better address each concept. If success is measured by the quality of insights provided by the Number Talk then this was a clear success. 

The success of the Number Talk was further increased by how the teacher reacted, or in this case, chose not to react, as the session revealed gaps in the students’ understanding. The teacher could easily have decided to offer her own suggestions. She might have shared with the students that 140 doesn’t belong because it is an even number or that it is the only one that is a multiple of ten. She could have explained her reasoning to the class, shared some illustrations, referenced a hundreds chart and hoped that the students would have learned the new content. Instead, she allowed the Number Talk to serve its purpose of illuminating for her the current state of her students’ thinking and closed the session. She knew now was not the time to push ahead into direct instruction. She valued the place that thinking plays in Number Talks and refrained from sending a message to the students that in a Number Talk, success is about correct answers. Again, Visible Thinking shines a light on the correctness of the teacher’s choices in this moment. 

Here we must strive to identify when a student's challenge can lead to a productive struggle with the ideas and eventually yield new insights for that student versus when the challenge is overwhelming and likely to cause a student to shut down. (Ritchhart & Church)

In this case, the teacher recognised that the challenge here and the concepts involved were significant and she knows that there will be time in the future to redress these. Just as we create thinking time for our students, we must create thinking time for ourselves. When she does decide to address this content with her students, she will do so strategically. 

The key here is to understand that our teaching can become more powerful when we use the time we have with our learners as opportunities for us to learn more about them as learners. If our singular focus is on teaching content, skills and dispositions we miss the chance to become students of our students, to observe them closely in the acts of thinking and learning and use what we notice to better meet their needs. 

by Nigel Coutts with thanks to Stellina Sim 

Cathy Humphreys & Ruth Parker (2015) Making Number Talks Matter: Developing mathematical practices and deepening understanding. Stenhouse Publishers

Nancy Hughes (2018) Classroom-Ready Number Talks for Third, Fourth and Fifth Grade Teachers. Ulysses Press

Ron Ritchhart & Mark Church. (2020) The Power of Making Thinking Visible: Practices to engage and empower all learners.  Wiley.