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**NSW Curriculum and Learning Innovation Centre**

Enhancing Mathematics lessons, incorporating language Chris Francis, Leader, Numeracy, Bronwen Camp, Project Officer TEN & Elaine Watkins, SEO Numeracy

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**What does the research tell us?**

“Mathematics truly is a foreign language for most students: it is learned almost entirely at school and is not spoken at home.” (Joan Kenney -Literacy strategies for improving Mathematics Instruction.) The National Numeracy Review Report (2008) acknowledged the significance of language in mathematics learning and recommended that the language of mathematics be explicitly taught. When we attempt to engage students by using real-world examples, we often find the colloquial or “street” language does not always map correctly onto the mathematical syntax. Vocabulary can be confusing because the words mean different things in mathematics and non-mathematics contexts. Slide 2 – Some examples, volume, product, sum In English when you add something it gets bigger, but in Mathematics addition can mean an increase, decrease depending on what number is added or not change, eg 0+0

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**Symbols and “small” words**

Symbols may be confusing because they look alike (eg the division and square root symbols), or because different representations may be used to describe the same process. A double decoding goes on during the process in learning the language of mathematics. A study by Kathryn Sullivan (1982) showed that even a brief 3 week program that centred on helping students distinguish the mathematical usage of “small” words significantly improved students’ mathematics computation results. Slide 3 - Double decoding – When we first encounter written Mathematics words or symbols, they first must be decoded and then connected to a concept. Small words such as find, how many, ten, tens, which, each, same, number, numeral, which, each, all are examples.

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Mathematics texts Mathematics texts contain more concepts per sentence and paragraph than any other type of text. The text often contains a lot of information words as well as numeric and non-numeric symbols to decode. The basic structure of Maths problems differs from other writing. Often the “key ideas” come at the end of a paragraph in the form of a question. Students must learn to read through the problem to work out the main idea and then again to work out which numbers relate to the question. To become proficient in mathematics, students need to participate in mathematical discussion and conversations in the classroom. Slide 4 – In informational writing, often there is a topic sentence or main theme at the beginning followed by more detail in the following sentences. There are often redundant numbers in Maths texts that confuse students. Looking for “key word” should not be used when reading mathematics texts.

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**Language can be confusing! (The syllabus tells us so!)**

Draft NSW Mathematics K-10 syllabus: The term ‘scales’ may be confusing for some students who confuse it with other uses of the word ‘scales’. Difficulties could arise for some students with phrasing in relation to subtraction problems. Some students may hear ‘whole’ in the phrase ‘part of a whole’ and confuse it with the term ‘hole’. The word ‘arm’ has different meanings in different contexts. (Feel free to replace ‘arm’ with ‘volume’, ‘round’, ‘mass’, ‘product’, or ...) …So what do we do? -A teacher was discussing whole numbers and then moved onto odd and even numbers and one student said that 6 and 10 were odd because the “whole numbers were not multiples of 2”. Conversation between the teacher and student did not clarify the difficulties until the student answered that the whole numbers were “6, 8, 9 and 10 and maybe 3.” This student had constructed a mental model of “hole”, that is numbers formed by sticks and “holes”. If a number had one “hole”, like 6, 9 and 10 it was odd, because the number of “holes” was not a multiple of 2. Number 3 had 2 half “holes” that could be combined to make a single one. Issues associated with language and comprehension is evident. This information on its own may not be helpful. Now that we know this, what can we do?

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**Focus on language within planning and teaching Spoken & Written**

WM Outcomes: Communicating WMESI.3 WMSI.3 WMES2.3 WMES3.3 WMES4.3 Describes mathematical language using everyday language, actions, materials and informal recordings Describes mathematical situations and methods using everyday and some mathematical language, actions materials, diagrams and symbols Uses appropriate terminology to describe, and symbols to represent, mathematical ideas Describes and represents a mathematical situation in a variety of ways using mathematical terminology and some conventions Uses mathematical terminology and notation, algebraic symbols, diagrams, text and tables to communicate mathematical ideas If we want students language use, and concept understanding to improve perhaps there is a need for a stronger focus on language when both planning and when teaching. The current syllabus outlines a progression of content and processes. In particular the WM outcome, communicating, focuses on language both written and spoken. Similar references to language for number, measurement, space & geometry. It provides us with a general overview sequencing development and expectations but not necessarily how we support students’ language development.

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**How do we develop students’ ability to talk, to write mathematically?**

Begin with assessment Focus content knowledge Plan instruction Suggested steps. Planning language acquisition in a process similar to that for concept development.

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**Focus on language within planning and teaching**

Begin with assessment Know where your students are: Concepts Strategies Verbal language Recordings We are familiar with assessment for learning. In the main, this assessment would focus on content and strategies. This assessment may need to be expanded. Know where your students are: Current level of concept understanding, strategy use, language skills verbal and written

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**Focus on language within planning and teaching**

Focus content knowledge Identify the concept you want to develop and the language you want the students to understand and use. Will you expect that all students will use the same language? Will you expect all students to record in the same way?

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**A subtraction example *1-digit from 2-digit in the range Concept**

2 strategies Concept: Why do you think this is an important concept for students to understand? What year level do you think that this is relevant for? Consider the two strategies presented. What level on the EAS framework do these strategies describe? What language would you want the students to use? What recordings would you want the students to use? *1-digit from 2-digit in the range 1-30 or 1-digit from 1-digit

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**Plan instruction that:**

addresses the concept models language includes explicit teaching about language promotes student discussion, questioning, explanation and reasoning asks the right questions to elicit understanding & address misconceptions links the verbal to the written caters for differing student needs

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**Starting points for discussion**

Begin with assessment What is the purpose of assessing mathematical language development? What do we want to assess? How can we do this? Focus content knowledge Is mathematical language necessary to develop concept understanding? When should we expect students to use mathematical language? How can you support students to develop from using everyday language to using mathematical language? Plan instruction How can student recordings support language development? How can you cater for the diversity of students’ mathematical language development?

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**Mathematics K-6 Syllabus p 12**

“Studies have shown that the causes of student errors on word problems may relate to the literacy components rather than the application of mathematical computations. Mathematics at times uses words from everyday language that have different meanings within a mathematical context. This can create confusion for some students. Clear explanations of these differences will assist students in the acquisition and use of mathematical terminology.”

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**The Language of Mathematics The literacy demands in numeracy lessons.**

All students need support to develop their mathematical language. If you have students from ESL / NESB backgrounds they will need additional support. Teachers need to explicitly model and provide opportunities for students to develop their mathematical language to enable them to be able to understand and apply it. Teachers must provide modelled, guided and independent opportunities for students to develop their mathematical language.

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**An example of a chance lesson using modelled, guided and independent mathematical language**

In this lesson students were provided with the language that I wanted them to use which was explicitly modelled for them at the beginning of the lesson. The language below was displayed on the board at the beginning of the lesson. It was introduced and explained to the students when I was modelling the lesson with them. Students were then told that this is the language that I wanted them to use when communicating within their groups. After groups had practiced the activity using the language provide the students then recorded in their books and shared this with the rest of the class in the lesson reflection. It is likely that ……. It is certain that….. Why do you think that… It is unlikely that……. What’s the chance of….? Predict which colour….. It is more likely that….. It is possible that……. What would happen if… It is equally likely that … It is less likely that …… What might change your prediction?

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**Programming Incorporating the Language of Mathematics**

Teachers should identify relevant mathematical language that they are focussing on and include this in their program. It’s not about cramming as many words into one lesson as you can. Teachers need to choose new or problematic terminology that is relevant to the needs of their class and teach it explicitly. If teachers are not used to doing this or find this difficult, you can run a session with them where they bring along an existing lesson that they have programmed and you can look at embedding the language of mathematics into that lesson together.

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**Effective Lesson Implementation**

Learning intention and success criteria - Be clear about what the purpose of your lesson is and what mathematical language you are focusing on. Tell students what the learning intention is and what your success criteria is. Tell them what language you want them to use. Lesson reflection - It is important to provide opportunities for students to share what strategies they have been using. This helps them to develop a deeper understanding of strategies learnt, provides them with opportunities to speak using mathematical language and it also provides students with a range of different strategies that they can use. (Some students think that there is only one correct way of answering questions.) Making explicit links – Teachers need to make explicit links for students so that they understand the connections between what they are learning to their prior learning, real world links and the mathematical language that they have already learnt. Some students may think that a mathematical word used in an activity is only related to that activity.

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**Effective Lesson Implementation**

Open ended questions Using open ended questions allows students to use a variety of mathematical skills, strategies and language. Hands on learning – Having opportunities to interact with resources and other students assists students to develop their understanding of concepts ,as well as their mathematical language, in a motivating and engaging environment that facilitates investigating and reasoning. Visual literacy – Visual literacy is very important for students but you can’t assume that because you have it up in the classroom students know how to use it. Students need to be taught how read and interpret words, graphs, tables, charts, pictures…. Timing of lessons – Too much teacher talk can be confusing for students. It is important to be explicit and balance your lesson timing carefully.

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**Effective Lesson Implementation**

Teaching and learning cycle – As with mathematical content, teachers must be aware of what language the students already know and what language they need to learn or further consolidate. Differentiation and scaffolding student learning – Teachers need to differentiate and scaffold the students use of the language of mathematics as well as tasks they are asking them to complete. ICT – There are some fantastic computer programs and interactive whiteboard activities that allow students to engage in mathematics activities whilst also enabling them to further develop their mathematical language.

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**Student Recording in Mathematics**

It is important for students to record what they have been learning in mathematics lessons. This must be modelled to children. Student can record in a wide variety of ways including; Teachers should display student work in mathematics around the classroom. on whiteboards on butchers paper in their books on paper ICT words pictures or illustrations using taking photos of their work create their own questions

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Student Recording in Mathematics**

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**Language of Mathematics Words**

We have all of the language of maths words from the What When How to teach maths folder on flashcards for teachers to use in lessons and displays.

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**Blank Language of Mathematics Chart**

Teachers and students use this chart during maths lessons so that it is interactive rather than ‘wall art’. It is good to model to students that a given word can change in meaning depending on the context it in.

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**Problematic Words in Mathematics**

Face – a person’s face, face of a shape Scale – a fish scale, kitchen scales, drawing a picture to scale Volume – how loud something is, how much something holds Degree – temperature, angle Product – something you buy at the shops, multiply

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**Maths Readers and Maths Big Books**

Each class K-6 has a set of mathematics related readers covering all strands of mathematics. Maths big books We have a large range of mathematics big books in our library.

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Maths Charts All classes have a large set of commercial mathematics charts related to specific mathematics strategies. These are displayed when the strategies are being introduced. All K-2 classes have a commercial set of language of maths posters which provide good visual and language support for students across most strands of maths. A lot of teachers have also been accessing internet based sites to print a range of visual material to support student understanding of the language of mathematics.

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Maths Dictionaries All Yr 2 – 6 classes have a set of maths dictionaries that students can refer to at any time to help them to understand mathematical terminology.

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Passwords To provide students with regular opportunities to reinforce concepts and language taught in class some teachers at our school use a ‘password’ on their door. Each time the students walk in or out of the room they need to say the password. The password can be differentiated to accommodate student learning needs. Password 10 For example, if the password was 10 students could; - say ten if they are learning their numeral identification - say the number after or the number before ten - double or halve the number - provide a number sentence where the answer was 10 The same can be done with other aspects of mathematics. I have seen a teacher use a picture and students had to use positional language to describe something in the picture. You are only limited by your imagination.

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**Newman’s Error Analysis**

It is important to identify the point of difficulty in a child being able to answer a question rather than just saying they got it wrong. Classes K-6 are expected to use Newman’s Error Analysis in their daily classroom practice. Teachers include this in their program. Each class has the Newman’s Error Analysis prompts on laminated cards. Classes in Stages 1,2 and 3 have laminated examples from past NAPLAN and BST papers. These are strand based and used as part of their regular lessons not as NAPLAN practice a week before the test. Teachers also write their own questions using their students names .

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Maths in a Box This series provides a good range of real life links to mathematics across the strands. As there is a lot of visual literacy they provide good opportunities for using the language of mathematics. We purchased sets of these and bagged them in sub-strands so that more teachers could be using them.

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Living Maths Series This series provides a great range of topics linking mathematics to the real world. There are great opportunities for enhancing student’s mathematical language on each page. This series provides a great range of topics linking mathematics to the real world. There is a picture and related questions for all of the topics listed below. Again this provides great opportunities for using the language of mathematics. As well as answering given questions they can be used as a picture talk with students creating their own questions. Book 1 – At home Our street Our house Our pets What’s for breakfast? Washing day Pete the painter In the garden Book 2 – At the theme park Getting there Getting in All about the park The rides Space world zoo Ice cream parlour Collecting litter Book 3 – At the shopping centre Clever clothes At the fruit shop At the hairdresser’s Dave’s cafe Mobile phone shop Bowling At the cinema Book 4 – On holiday Which holiday? How far to the airport? At the airport Money, money, money! At the hotel In the swim! In the gift shop Book 5 – At the supermarket In the car park In the store At the pharmacy Can we help you? Supermarket bill Supermarket petrol station Delivery day Book 6 – At school School plan The students The staff The timetable At the tuckshop In the laboratory Sports day

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**Supporting Parents to Develop their Language of Mathematics**

Maths newsletters These are sent out each term for each grade to inform parents of what we are teaching in maths each term with some visual literacy and language to support their understanding. Parent training sessions We run hands on parent training sessions that cater to parental requests for support in specific areas of mathematics. We have our four Community Language teachers assisting to translate.

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