Analysis: too often we learn by memorising without understanding, but the new primary school maths curriculum changes this
By Seán Delaney, Marino Institute of Education
Maya, who is in third class, looked puzzled. Her teacher noticed that Maya hadn't written anything for the last few minutes so he walked towards Maya’s desk.
"This doesn’t make sense" Maya said to her teacher pointing towards the fraction 5/4. "Why not?" her teacher responded, "It’s a special kind of fraction called an improper fraction, where the numerator is greater than the denominator. It is more than one."
"But you told us that a fraction is a part of a whole unit. So it’s impossible for a fraction to be more than one," replied Maya.
Maya’s teacher realised how something he had said when teaching fractions was becoming a barrier for Maya as her mathematical experience expanded.
A new primary mathematics curriculum has been introduced for Irish schools. Children will become proficient in mathematics by being creative, solving problems and learning in playful ways. Children are expected to "make sense" when learning mathematics. Making sense means that teachers teach so that children will understand what they learn, and children learn better when they understand something. This connection between sense making and learning is underpinned by much research on mathematics education.
Although the need to understand in order to learn may seem obvious, many of us learned mathematics without understanding it. Think about learning off tables and rules that didn't make sense to us: Why do you put a 0 on the right-hand side of the second row of a long multiplication calculation? Why does "invert the divisor and multiply" work when dividing fractions? Too often we learned by memorising without understanding.
It doesn’t have to be this way. Children enjoy trying out different ways of solving problems and reasoning about what they did. Even when children make an error, such as in this calculation, the error contains some logic, which can be discussed.
4 2
-1 7
__
3 5
In this example, the child tried to subtract 7 from 2 but could not do so and instead decided to take 2 from 7 to get 5, then the child subtracted 1 from 4 and got 3. When a child gets used to making sense in mathematics, they become open to learning new ideas and moving from naïve to more sophisticated understanding.
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Despite the importance of making sense when teaching mathematics, many occasions arise where making sense can be difficult. Children sometimes learn ideas such as the following:
- You cannot subtract a larger number from a smaller number.
- Multiplying a number makes the number bigger.
- Dividing a number makes the number smaller.
- Squares and rectangles are different kinds of shape.
Such concepts are challenging because each contains a partial truth, but it is not the full story. Indeed each of the statements above is false but because children experience them as frequently true, especially when beginning to learn mathematics, the misconceptions can be difficult to shake off as you move through primary school mathematics and into post-primary school.
When we begin learning subtraction, we work with natural numbers {1, 2, 3, 4,…} and whole numbers {0, 1, 2, 3, 4,…}. And in those domains of number it is true that you cannot subtract a larger number from a smaller number. If you have 3 colouring pencils on the table, you cannot take 5 colouring pencils away and bring them to school. However, this changes in senior primary school classes where mathematics becomes more abstract and you can take 5 from 3 and the result is -2 in the domain of integers. The rule that previously made sense needs to be reconsidered in light of new knowledge.
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When children begin learning multiplication they typically work with whole numbers. In most cases when we multiply one number by another number, the number gets larger. Multiply 3 by 5 and the product, 15, is larger than 3 (and 5) or multiply 6 by 6 and the product, 36, is larger than 6. Thus children come to accept the notion that multiplication makes a number bigger and this is consistent with society's general view, possibly tracing itself back to the Book of Genesis, which advised people to be fruitful, multiply and fill the earth (Genesis, 1:28).
However as you gain more experience in mathematics, you’ll find that multiplying does not always make a number bigger. If you multiply a number by 1, it remains the same size. If you multiply a whole number by 0, it gets smaller. If you multiply a positive integer (say 5) by a negative integer (e.g. -3) the product, -15, is smaller than both numbers that are multiplied. When you advance to rational numbers and multiply a whole number, say 2, by a unit fraction (¼, perhaps), the product is ½, which is smaller than 2.
When we teach children division, we often do this by asking them to share 8 objects among 4 children. Or we may ask them how many times we can subtract 4 from 8. Consequently, through repeated practice, children learn that dividing a number results in an answer (quotient) that is smaller than the number you started with and that holds true for many examples. However, when you introduce rational numbers, 3 ÷ ¼ for example (as might happen if you want to figure out how many quarter hours in three hours), the answer, 12, is greater than 3. Or the task "It takes a scuba diver 4 minutes to dive 8 metres below sea level, how far does she travel in 1 minute if she travels at a constant speed?" could be calculated as -8 ÷ 4 = -2. Here the quotient (-2) is larger not smaller than the dividend (-8).
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Finally, when children are taught about two-dimensional shapes, squares and rectangles are introduced as if they are different shapes. However, most mathematical definitions of rectangle categorise a square as a special kind of rectangle where all sides are of equal length.
So, as we welcome a new primary mathematics curriculum that focuses on making sense, we can look forward to teaching and learning for understanding being a feature of all classrooms. This will be evident in children and teachers discussing mathematical ideas together, explaining how they solve problems, justifying their approach, and reasoning about how solutions and methods did or did not work. Teachers will be more explicit and nuanced about generalisations that can lead to misconceptions. Errors will be used as resources for the entire class to learn and everyone will use language more precisely to ensure that others understand their thinking.
Dr Seán Delaney is Registrar and Vice President (Academic Affairs) at Marino Institute of Education.
The views expressed here are those of the author and do not represent or reflect the views of RTÉ
