Consider the following Mathematical sums:
|(i) 0.04 × 10 = _____||(iii) 0.04 × 100 = ______|
|(ii) 1/2 × 1/3 = _____||(iv) 1/2 × 2/3 = ______|
More than often, students do not have problems deriving the correct answers to these questions, but what follows will leave them dumbfounded:
Is 1/2 × 2/3 the same as 2/3 × 1/2 ? Why?
Some of us might think to ourselves, how on earth am I supposed to ‘explain’ those questions? I have never learnt how to explain them and I have done well in schools! Mathematics education has taken a paradigm shift over the past decades. Being competent in mathematics is no longer about getting the correct answers or relying on memory for formulas. Instead, reasoning and justification is the central goal of studying mathematics.
Mathematical reasoning, new kid on the block?
In year 2000, the National Council for Teachers of Mathematics (NCTM) placed reasoning and proof as one of the process standards through all educational levels in the United States. Similarly, in Singapore, one of the key features of our famous Pentagon Model (Figure 1) stresses the importance of the thinking and reasoning processes.
The Pentagon Model – Singapore’s Mathematics Framework – has been a feature of Singapore mathematics curriculum since 1990 and it is still relevant to date (MOE, 2013) as it reflects the 21st-century competencies. The framework comprises of five important inter-related components with its key focus on mathematical problem solving. Emphasis is placed on conceptual understanding, skills proficiency and mathematical processes, and highlights the importance of attitudes and metacognition.
Why the need for Mathematical reasoning?
Mathematical reasoning refers to the ability to investigate, verify and establish valid and convincing arguments (to oneself and to others). The development of mathematical reasoning in children helps them understand why they are learning mathematics and eventually come to enjoy and love mathematics. It is by reasoning that they are able to:
- make connections
- deepen conceptual understanding
- think creatively
- reason their way out mathematically
Mathematical reasoning is a habit of mind that can be trained and developed through the applications of mathematics in different contextual situations. As parents and educators, there is a need to encourage mathematical learning and reasoning through all levels (not only at higher levels) so that our children may develop and employ their reasoning skills throughout their learning of mathematics. This skill “behooves us to give greater attention and how these vehicles of thinking can foster”children’s mathematical power.
How to help your children hone their mathematical reasoning skills?
Research findings have shown that many of the mathematical misconceptions in children arise because of the reliance on memorising the procedures with little understanding of the concepts that underlie them (Hiebert, & Carpenter, 1992). When children memorise and practise procedures that they do not understand, they have little or no motivation to understand the reasoning that lies behind them (Stigler, & Hiebert, 1999). It is therefore important to develop children’s conceptual understanding and equip them with a strong variety of procedures alongside with their mathematical reasoning ability in preparation for use in different contexts (NRC, 2005, 2012).
How can we develop their mathematical reasoning skills to reverse this effect? Here are 2 ways:
1. Ensure that your child can visualise and make meaning from numbers and number statements
Let’s go back to the mathematical question posed at the beginning:
0.04 × 10 = _____
While most children can easily find the answer to this question, it is important that for them to have a clear understanding of what 0.04 x 10 really means.
|What is 0.04 and what does it look like?|
What does 0.04 x 10 look like?
(40 hundredths is the same as 4 tenths, hence, we write 0.40 or 0.4 as the answer)
When the child understands the fundamental concepts and is able to reason what the requirements of the question are and how to arrive at the final answer with understanding, more than half the battle is won.
2. Give your child opportunities to explore and discover the concept on his own
Reasoning includes the ability to explain and put your point across. Now, let’s consider the following question:
Is 1/2 × 2/3 the same as 2/3 × 1/2 ? Why?
Such questions are commonly seen in today’s mathematics syllabus to encourage and nurture mathematical reasoning.
The use of origami could be an activity for children to explore and discover the concept of fractions and other inter-related concepts, such as like-fractions.
When the child is provided with the opportunity to explore, an opportunity such as the hands-on activities to struggle and discover what it means by one-half of two-thirds and two-thirds of one-half, the child builds on the existing concepts he/she already has. In such a self-discovery learning process, obstacles are inevitable. However, it is the existence of such obstacles that nurture our children to persevere and anticipate the joy of success. They eventually grow up to be more adventurous, more resilient and more confident problem solvers!
This article is contributed by Michelle Choo, co-author of My Pals are Here! Maths.
About the author
Michelle Choo has 25 years of teaching experience. She was a Level Co-ordinator at a local primary school offering the Gifted Education Programme and later took up the position of an Acting Head of Department at another local school. She shifted her focus to authorship and being a full-fledged education trainer. To date, her training experiences include conducting local and international workshops and seminars for educators as part of their professional development programme. Michelle is well-known for crafting creative, activity-based, exercises and assessments that are based on sound Mathematical Pedagogical Content Knowledge.
English, Lyn D. (ed.) (1997). Mathematical Reasoning: Analogies, Metaphors, and Images. L. Erlbaum Associates.
Hiebert, J.; Carpenter, T. (1992). Learning and teaching with understanding. In: Grouws, D.A.,ed. Handbook of research on mathematics teaching and learning, p. 65-97. New York, Macmillan.
Hiebert, J., et al. (1997). Making sense: teaching and learning mathematics with understanding. Portsmouth, NH, Heinemann.
Ministry of Education. (2013). Singapore Mathematics framework.
National Research Council. (2005). How students learn: History, mathematics, and science in the classroom. Washington, DC: National Academies Press.
National Research Council. (2012). Education for life and work: Developing transferable knowledge and skills for the 21st century. Washington, DC: National Academies Press.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York, NY: Free Press.