{"id":190,"date":"2021-07-27T08:00:00","date_gmt":"2021-07-27T07:00:00","guid":{"rendered":"https:\/\/blog.lboro.ac.uk\/cmc\/?p=190"},"modified":"2021-07-29T17:03:51","modified_gmt":"2021-07-29T16:03:51","slug":"what-relates-to-understanding-of-equivalence","status":"publish","type":"post","link":"https:\/\/blog.lboro.ac.uk\/cmc\/2021\/07\/27\/what-relates-to-understanding-of-equivalence\/","title":{"rendered":"What Relates to Understanding of Equivalence?"},"content":{"rendered":"\n

Written<\/em> <\/em><\/strong>by Dr Emine Simsek<\/em><\/strong> and edited by Dr Ian Jones and Dr Iro Xenidou-Dervou<\/em>. Emine is a post-doctoral researcher at Loughborough University. Please see <\/em>here<\/em><\/strong><\/a> for more information about Emine and her work.<\/em><\/span><\/p>\n\n\n\n

Understanding of Mathematical Equivalence<\/strong><\/h2>\n\n\n\n

A useful interpretation of the equals sign for students is thinking of it as a symbol which signifies that the two sides of an equation have the same value and are interchangeable. When solving equations, students are expected to assess the equivalence either by calculating the value on both sides of the equation, or by exploiting arithmetic shortcuts to avoid the need for calculation. Let\u2019s call these shortcuts \u201crelational strategies\u201d. For example, for the following equation \u201cFind the missing number in the equation 64 \u2013 __ =  62 \u2013 37\u201d, a relational strategy would be \u201c64 is 2 more than 62, so the answer should be 2 more than 37 which is 39\u201d.<\/p>\n\n\n\n

However, research has shown that primary students often view the equals sign only as a \u201cdo something symbol\u201d or \u201cthe answer to the problem\u201d. They often use incorrect strategies when solving equations that have operations on both sides of the equals sign. Let\u2019s call these strategies \u201coperational strategies\u201d. Two examples of operational strategies used by two Year 5 students are below. <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

How can we improve students\u2019 understanding of equivalence? The change-resistance account<\/a> predicts that too much practice with only a traditional arithmetic format (equations with no operations on the right side, e.g. a + b = c<\/em>) underpins unwanted incorrect strategies. This tells us that we should vary the problem format in the textbooks and other classroom resources. However, this account was based on research conducted in one country (the USA). Also, it did not account for factors such as teacher knowledge which has been shown to be important to support students\u2019 mathematics achievement. Therefore, to explore what relates to students\u2019 understanding of equivalence, we conducted a study across six different countries (China, England, New Zealand, South Korea, Turkey, and USA).<\/p>\n\n\n\n

Our Study<\/strong><\/h2>\n\n\n\n

We looked at whether teacher knowledge, the format of equations as presented in textbooks, or both, relate to students\u2019 understanding of equivalence. Participants were Year 4 and Year 5 students (age range 8 to 12, 72% of the participating children were 10 years old) and their teachers across the countries.<\/p>\n\n\n\n

We asked students to define the equals sign and to solve equations (e.g. 17 + 28 = _ + 27), and we calculated a performance score for each student. <\/p>\n\n\n\n

We asked teachers to predict correct and incorrect strategies that students might use to solve equations. In teachers\u2019 responses, we looked at whether they were knowledgeable about their students\u2019 use of operational and relational strategies. <\/p>\n\n\n\n

Finally, we asked teachers to provide the names of the current year mathematics textbooks that they used. Then, we analysed the textbooks and identified all the equations involving the equals sign on every other page of the textbooks. We then looked at how many traditional equations and non-traditional equations each textbook has. <\/p>\n\n\n\n

Traditional equations <\/strong>
(no operations beyond the equals sign)<\/strong><\/td>		Non-traditional equations<\/strong>
(operations\/operands beyond the equals sign)<\/strong><\/td><\/tr>
a<\/em> + b<\/em>  =  c<\/em>

58
+ 26<\/span><\/td>		a<\/em>  =  a<\/em>
a<\/em>  =  b<\/em> + c<\/em>
a<\/em> + b<\/em>  =  c <\/em>+ d<\/em><\/td><\/tr><\/tbody><\/table>
Examples of traditional and non-traditional equations<\/figcaption><\/figure>\n\n\n\n

We used an advanced statistical modelling technique to analyse the data collected from 2,760 primary students and 108 teachers.<\/p>\n\n\n\n

What we found?<\/strong><\/h2>\n\n\n\n

Teachers provided more relational strategies than operational ones. This evidenced that many teachers lack knowledge about students\u2019 use of incorrect (operational) strategies to solve equations. <\/p>\n\n\n\n

In all the countries in the study, the relevant grade textbooks had more traditional than non-traditional equations.<\/p>\n\n\n\n

Students\u2019 knowledge of definitions of the equals sign related to their equation-solving performance; students who had more sophisticated knowledge of the equals sign performed better on equations.<\/p>\n\n\n\n

The teachers\u2019 knowledge of students\u2019 relational strategies related to the students\u2019 understanding of equivalence but their knowledge of students\u2019 operational strategies did not. From this and based on previous intervention studies, it can be assumed that teachers who are more knowledgeable about (students\u2019) relational strategies support their students\u2019 understanding of equivalence better in the classroom than those who are less knowledgeable of the issue.<\/p>\n\n\n\n

We did not find evidence that the format of equations in the current year textbooks influenced students\u2019 understanding of equivalence.<\/p>\n\n\n\n

Take-home points for teachers <\/strong><\/h2>\n\n\n\n