# THINGS THAT MAKE US SAY NOOOOO!!!

## Teaching incorrect mathematical meta-language

Among the many awesome and accurate maths resources, there are some that makes us say ‘nooo!!!’, and teaching students incorrect mathematical metalanguage is one of them! One that is oft repeated is calling three-dimensional objects, ‘shapes’. A shape by definition has 2 dimensions. The 3 dimensions are left to right, front to back, up and down. Another that we’ve come across is calling one of the vertices on a pyramid, an apex. An apex is simply the highest point on an object. If the pyramid is not standing on its base, it no longer has an apex. The point where sides meet are vertices (singular = vertex), regardless of the number of sides that meet there. In an interesting exchange, a provider of maths resources defended the two mathematical inaccuracies, saying that, in many Maths books the language is widely-used and highly acceptable. We argue that widely-used and highly acceptable does not equal correct! Indeed many maths textbooks are just plain wrong. Why teach students correct grammatical terms (nouns, verbs, adjectives), then teach the same students incorrect mathematical terms? Is it because we were also not taught correct mathematical terms, and so think they are somehow too difficult for a student to develop as metalanguage? On posters available on some pay resource sites, we’ve seen shapes described as having ‘straight sides’. Again, sides, by definition are straight! If a line on a shape is not straight, it is just a curved line! Unfortunately, these are just a few examples. Current learning research emphasises the importance of using correct mathematical metalanguage to explain understanding. As William L Schaaf observed, ‘Maths is a linguistic activity; its ultimate area is preciseness of communication’. With number talks currently experiencing great popularity, it is vital that teachers and students alike, ensure they are using mathematical language...

read more## Ignoring the meaning of the equals sign

Among the many awesome and accurate maths resources available on-line, there are some that makes us say ‘nooo!!!’, and ignoring the meaning of the equals sign is one of them! A fabulous investigation involving groups of students creating number sentences using post-it notes with numbers and symbols, was marred when groups recorded mathematical incorrect number sentences like: 2 x 6 = 12 + 12 = 24 The incorrect number sentence was displayed on the internet as an example of students demonstrating their knowledge of multiplication properties. And was followed by an exclamation that the students loved it. So how could this investigation have resulted in deep relational understanding rather than shallow incorrect maths? It was actually a great investigation, just implemented shallowly. When students recorded mathematically incorrect number sentences, like the one above, a golden teachable moment was born… and then died. What’s wrong with the number sentence? It ignores the meaning of the equals sign – that both sides of the equals sign are equal. On the left of the first equals sign we have 2 x 6 and on the right of the first equals sign we have 12 + 12. Clearly 2 x 6 is not equal to 12 + 12. Because the number sentence has 2 equals signs, It also means that the left of the first equals sign is also equal to the right of the second equals sign. On the left of the first equals sign is 2 x 6 and on the right of the second equals sign is 24. Again, clearly 2 x 6 is not equal to 24. The students needed to create 2 number sentences: 2 x 6 = 12 12 + 12 = 24 When students recorded mathematically incorrect number sentences, like the one above, the class teacher was provided with an excellent opportunity to deepen understanding of the equals sign. Using questioning to allow students to explain why the number sentence was incorrect, for example, what does the equals sign mean? are both sides of the first equals sign equal? are both sides of the second equals sign equal? what does it mean when there are more than one equals sign in a number sentence? is the left of the first equals sign equal to the right of the second equal sign? how could we record this so it is mathematically correctly? ‘Activities’ abound on the Internet. But nothing beats teacher understanding and quality...

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