Getting the balance right
In this article, we will explore some of the most common misconceptions surrounding the equals sign in the primary classroom, share a useful progression which you may wish to use with your class, and consider a selection of activities which enable young learners to apply their newly-acquired skills in real-life situations.
Let's begin with the problem 8 + 4 = ? + 5.
Although finding the missing number may seem straightforward, researchers and classroom teachers have consistently reported that learners respond with a wide range of responses (Carpenter, Levi & Farnworth, 2000). One of the most frequent answers for the missing number was 12, indicating that those learners may have mistakenly interpreted the equal sign as an instruction or command, leading them to total 8 and 4, then record their answer. Another frequent answer was 17 (the learners having added the 5 digit to the sum of 8 and 4). During follow-up interviews, some learners expressed their view that the calculation couldn't be performed as it was presented, since there was '+5' written where they expected to write their answer. This is a really interesting problem to try out with your own class, or even across a school and compare answers across different year groups, to help identify common misconceptions and provide information about the progression among your classes as they develop their understanding about the equals sign.
Although many learners appear to share common misconceptions about the equals sign, researchers have proposed a three-stage progression which represents the key milestone for learners developing their understanding about equivalence (Molina & Ambrose, 2008). We believe that these three stages may offer a very useful classroom tool for assessing the needs of learners and planning their next steps to become much more fluent in their number work.
Stage One
Learners at Stage One can perform calculations such as 8+1=9 or 5-2=3, but they struggle when they are faced with calculations such as 6=9-3 or 8=2+6. At this early stage the young learners appear to be interpreting the equals sign as an instruction, thinking that they should record their answer immediately after the symbol. Looking back at our example 8+4=?+5, learners at Stage One may suggest the answer 12 for the missing number.
In a classroom situation, researchers recorded the following exchange betwen a teacher and a learner who had suggested that 12 was the missing number in the problem 8+4=?+5:
Teacher: Can you tell me how you got this number [12]?
Learner: Because eight plus four are twelve.
Teacher: And, what happened with this five at the end?
Learner: That is also equal to something else.
Stage Two
At this stage, learners are much more comfortable working with calculations such as 7=5+2 and 8=9-1. In other words, they are displaying an understanding about the symmetrical nature of calculations as they are willing to work either left to right or right to left. However, they may still tend to struggle with more complex number sentences such as 8+4=?+5 and offer incorrect answers which are similar to those suggested by learners at Stage One.
Stage Three
Learners operating at Stage Three tend to give correct answers when presented with calculations such as 8+4=?+5. In other words, they are intrepreting the equal sign as an expression of equivalence rather than an instruction to record an answer. These learners work much more flexibly at this stage, using the symmetrical nature of the equal sign to work left to right, and right to left, as well as appreciating the equals sign as an expression of equivalence.
However, they may need to embed their understanding through regular practice sessions as well as having opportunities to apply their learning in their calculation work. Young learners can improve their calculation fluency by being encouraged to treat each calculation individually and using their understanding of the equal sign and equivalence to become increasingly flexible and more efficient.
Consider the calculation 28+16=?
This calculation can be approached in several different ways, some more efficient than others. For example, some learners may decide to use a formal written method to calculate their answer but others may choose to count on. Both of those approaches are time-consuming and prone to error. However, learners who are encourged to treat each calculation on its individual merits, and who are developing a good understanding of equivalence, may realise that 28+16=30+14, which arguably offers a much simpler calculation for them to perform. This approach is generally known as the 'compensation' method; using a numberline when introducing the compensation method to learners may offer them some visual support before they progress to calculating in their heads.
Researchers have suggested that the following list of calculations not only offers plenty of opportunities for learners to practise applying the compensation method, but also offers opportunties for them to explain their reasoning to others (Carpenter, Levi & Farnsworth, 2000):
74-38
89-45
68-42
53-37
It should be noted that not all learners will make smooth progress through the three stages, they will need sufficient opportunties to enable them to become consistent at each stage. Nevertheless, encouraging them to developing their understanding about equivalence enables young learners to begin the journey from arithmetic towards early algebra, enabling them to think mathematically and become better mathematicians.
References
Carpenter, T. P., Levi, L., & Farnsworth, V. (2000). Building a Foundation for Learning Algebra in the Elementary Grades. In Brief, 1(2), n2.
Molina, M. & Ambrose, R. (2008). From an operational to a relational conception of the equal sign. Thirds graders' developing algebraic thinking. Focus on Learning Problems in Mathematics, 30(1), 61-80.
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