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Fostering the development of correct mathematical language and terminology

Last Friday, I was working with a group of first class children who were completing some first grade activities on Splash Math, an American maths site. While, on the plus side, the activities on this site are very visual and promote a CPA approach to mathematical instruction, on the down side, the first grade in the US isn’t aligned exactly to the maths curriculum for first class in Ireland, and so we have regularly encountered activities that might have unfamiliar language and terminology.

This was one of those days. We were looking at 2D shapes in the geometry section when a child said quizzically to me, “I’m stuck, miss”. The question was “How many vertices has the shape (a circle): 1, 3, 0, 2?”. I asked the class could anybody remember, from the previous day, what vertices meant? A flurry of hands went up to tell me “corners” at which point the child had no difficulty identifying 0 as the correct answer. Then I asked the children to remind me of all the other American words to do with geometry that we had come across the previous day, which I then recorded on the board for the benefit of all the children (see image below).

It brought home to me how correct mathematical language and terminology is much more prevalent in the primary maths curricula and texts of other countries, and how it is often even introduced much earlier, when compared to Ireland. And, how much of a disservice we do to children in Ireland if we try to shield them from this language in primary school, only to have it all thrust at them in secondary, where some children might wonder if it is the same subject they are doing at all!

It also reminded me of an RSE inservice I attended years ago, which stressed the importance of the children being introduced to the correct terminology for the body parts, so they might be able to properly communicate and report any incidences that might occur. In a similar way, should we not introduce children to the correct mathematical terminology, so as to enable them to communicate their thinking more clearly and to explain the approaches they took and the strategies they used?

That is why Operation Maths has been written as a programme which does not shy away from the correct mathematical language and terminology, rather it specifically uses words like commutative, distributive, associative, dividend, product etc when explaining concepts. Furthermore, when introducing new terminology it is done via concrete and pictorial activities with the back-up of  a range of images that enable the children to not just know the word, but to be able to picture it also, and in that way to truly understand the concept it describes.

As can be seen from the example above, new terminology and language is typically introduced as part of the teaching panels (yellow-coloured sections) and is often in a blue bold font to highlight it as being new/significant. The new term is then explained in simpler words and using visual examples to reinforce its meaning for the children. Since it is envisaged that these teaching panels would be presented/mediated by the teacher, this ensures that the teacher can help explain the vocabulary and that the child is not meeting the new term  in a random section of text.

The questions/exercises for the children that follow these teaching panels have also been specifically chosen to help reinforce the new term and consolidate the concept that it entails. These typically incorporate the use of concrete materials or pictorials representations (as in the case of the 100 dots grid/sheet mentioned above) for further exploration and reinforcement.

With all new terminology, when met again, there is typically some supporting text to remind the child and/or revise the meaning. Furthermore, the child can always consult the colourful glossary at the back of his/her pupil’s book if necessary.

Some of the advantages of using correct mathematical terminology in primary mathematics:

Preparation for second level: The NCCA has published a number of Bridging materials for maths, which encourage continuity between mathematics in primary and post-primary schools. Included in these materials, there is a glossary of terminology that teachers of 5th and 6th classes are encouraged to incorporate, where possible, so that children will be better prepared for second level maths, thus easing the transition from primary. This terminology was deliberately included in the Operation Maths books for 5th and 6th. Furthermore, where useful, some terminology was also incorporated in a simpler way in the Operation Maths books for 3rd and 4th so as to make the introduction more gradual.

Number Sense & Number Talks: The buzz in maths education circles is all about developing number sense. One approach that is being encouraged to support this is to have regular Number Talks to encourage the children to communicate how they mentally solved a calculation and to explore and discuss the various strategies that could  be used. The promotion of the development of number sense is a key principle of the Operation Maths programme, from the use of frames in the junior classes, right up to the use of thinking strategies, bar models and other pictorial structures in the senior classes. Similarly, the strong emphasis on talk and discussion ( eg Talk Time in the pupils books, discussion and questions given in the TRBs) in Operation Maths further supports this process. Ultimately however, this is all dependent on the children having a well developed range of mathematical terminology, by which they can clearly communicate and express their ideas and approaches.

Maths on the internet: Most of the maths we access on the net is american-based, be it You Tube videos, teaching sites, games, drill and practice sites. In the case of the latter, in many schools and homes, the children are encouraged to access teaching, drill and practice sites such as Khan Academy, Manga High, Splash Math, IXL.com etc to complement their core mathematical texts. As a result, Irish children will likely encounter, initially, terminology that is unfamiliar.  However, if they have encountered this terminology in their Operation Maths books, this will better prepare them for these sites. Indeed for those children and classes who have regularly accessed these non-Irish sites, they will probably have developed an understanding of this terminology already and its inclusion in Operation Maths will be unlikely to faze them at all.

Some FAQs:

Is this mathematical terminology in-line with the Irish Primary Mathematics Curriculum?

This is taken direct from the curriculum:
Third Class > Number > Operations >
The child should be enabled to explore, understand and apply the zero, commutative and distributive properties of multiplication.

Thus, not only is the terminology in-line with the curriculum, it raises the question how a child could previously have been enabled to “apply the commutative property” without being able to explain what he/she was doing and why, and furthermore how he/she could explain this without using the word “commutative” or “turn-around fact”?

Is is worth noting that the Teacher Guidelines, that accompanies the mathematics curriculum here in Ireland, includes a limited list of symbols, numerals, fractions and certain terminology for each class level (p. 70). However, other more generic terminology (eg product, factor, dividend etc) has not been categorised according to class levels, which contrasts with the curricula of other countries where specific terminology is typically specified for each year level/grade. Therefore, in writing Operation Maths, the authors categorised terminology into certain class levels based on evidence and practice in other countries.

Are the children expected to learn off and define this terminology? 

Of course not. In the same way as a teacher might use such terminology as simile, metaphor, alliteration etc to explain writing concepts in English, it is hoped that the teacher would use and reinforce specific terminology when appropriate, and in this way some of the children might also pick up this vocabulary and use it themselves when communicating their ideas. But it is not suggested or encouraged that these terms be drilled and “learnt off”.

We have a high number of children with dyslexia/English as a second language; should we avoid Operation Maths because of the language?

Actually, quite the opposite. While the teaching panels of Operation Maths may have more mathematical vocabulary that the competitor texts, they also have many more visual images that explain and demonstrate the concepts, and both the teaching panels and the exercises that follow are more concrete-based and pictorial in nature. This will in fact be better for children with limited language or language difficulties, as opposed to texts which are largely just digits and symbols, which themselves can be too abstract, particularly for senior classes. Plus, deliberately avoiding this language in primary only moves the issue on to becoming a bigger one when those children go to second level.
As mentioned previously, all of the Operation Maths programme is based on a CPA approach,  from the Pupils’ Book to the Discovery book, which is dominated by visual, rather than text, activities, to the free place value materials and frames, to the digital resources, eManipulatives and videos all of which place the emphasis on visual representations of content. This makes Operation Maths the most suitable programme for any child who is more of a visual learner.

Further suggestions, hints and tips:

Repetition, repetition, repetition! Whenever a new term is encountered don’t expect the children to know it,  understand it and use it straight away; research suggests that a child will typically need to encounter a word 15-20 times before they will start to use it. This is why it is important to use the term at every suitable opportunity and why in Operation Maths the term will be used repeatedly in various contexts to help this.

Use glossaries: As well as the Operation Maths glossary, use Jenny Eather’s, Maths Dictionary for Kids to look up new terminology and explore the visual and interactive activities that typically accompany each term. Another useful resource are the Math Vocabulary Cards from the Math Learning Centre, available to use online or download as a  free app. However, bear in mind that, while a definition in a glossary is useful, new terms must be also understood from meaningful examples and contexts relevant to the child.

Maths Word Wall: Whenever you encounter new terminology display it on your maths wall for future reference. This can be printed out vocabulary posters from the internet or small flash cards/A4 posters created by the children themselves. Aim to always include a pictorial representation and not just text. There are also lots of printable charts and posters available to download free from Jenny Eather’s, Maths Dictionary for Kids .

Start a personal maths dictionary: This allows children to keep a personal record of the vocabulary they encounter. Operation Maths users can use the vocabulary sections in the Discovery Book, where the children in 3rd and 4th must match the term to a definition and to an example. In Operation Maths 5, the children must provide the term to match the definition and, in Operation Maths 6, the children must provide the definition to match the term, as well as drawing an example in both cases. Thus the activities are getting slightly more difficult at each class level while continuing to emphasise the visual representations.

Use Number Talks: Through the regular use of Number Talks the children will begin to appreciate how having a good grasp of the correct mathematical language can help them explain their thinking in a more accurate and efficient way during number talks. Furthermore, he/she will realise that it is easier to understand the approach of a peer when they use terminology that he/she recognises and understands.

Make it fun: Play games such as matching games or “Just a Minute” word games.

Use matching activities, true or false, always, sometimes, never true etc: These type of language activities are included in the Operations Maths books to reinforce and consolidate the language acquisition. Also included are  oral discussion activities and “Talk Time” activities, to further promote discussion and exploration.

Further reading:

Developing mathematical vocabulary