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THE LANGUAGE OF MATHEMATICS. “One should NOT aim at being possible to understand, but at being IMPOSSIBLE to misunderstand.” Quintilian, circa 100 AD Learners need to understand: HOW things are said WHAT is being said WHY it is being said. Consider the following context:.
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“One should NOT aim at being possible to understand, but at being IMPOSSIBLE to misunderstand.” Quintilian, circa 100 AD • Learners need to understand: • HOW things are said • WHAT is being said • WHY it is being said
How do we as educators respond to this problem? • Disbelief • Frustration • Anger • Disempowered • Disinterested What made this word problem difficult? What made this word problem unfair, invalid and lacking in reliability?
What do learners need to be able to do to master a Maths language? • Develop: • an ability to talk about Maths • their personal mathematical graphics • an awareness of number sense • understanding of mathematical symbols • a Maths vocabulary (meta-language)
How can teachers empower learners? • Learners need to be given an opportunity to talk about Maths • “If children already know or are told the method to use, then they are not problem solving” Gifford, S. (2005) • Conversation between adults and children helps to refine their thinking and ideas, as they explore different solutions to a problem • The conversation helps to scaffold children’s thinking • Learners must be encouraged to explain their thinking regarding their approach to a Maths problem
How can teachers empower learners? • This can be done through learners explaining to each other in groups how they arrived at their answer. • If you encourage conversation from the outset, learners will be confident to offer their ideas, talk about their thinking, negotiate and understand meaning • The more they are given an opportunity to do this, the more proficient they become.)
How can teachers empower learners? • Learners need to have the opportunity to develop their personal mathematical graphics • What is mathematical graphics? • Mathematical graphics originated with Carruthers and Worthington (2003) • Refers to children’s own marks and representations that they use to convey their mathematical thinking • Primitive way of conveying their thinking before they become familiar with the symbolic language of Mathematics
How can teachers empower learners? • Why should we encourage learners to use mathematical graphics? • It is a foundation for learners to begin supporting their written calculations • Learners understanding of written mathematics improves if learners are encourage to represent their mathematical thinking when they cannot solve the problem mentally • This enables learners to work out their own strategies for solving problems
How can teachers empower learners? • Recording mathematics offers little opportunity for supporting or extending mathematical thinking • Representing mathematics enables children to use higher-order thinking skills
How can teachers empower learners? • Learners need to become comfortable using the meta-language of Maths (e.g. the sum of; addition) • Learners need to acquire a mathematical vocabulary • Use the correct meta-language from the outset • Differentiate concepts from verbs e.g. the action linked with subtraction is ‘taking away’ • Meta-language must be taught in a meaningful context and not as a list to be remembered • Meta-language must be linked to the symbol e.g. addition (+)
Vocabulary is not fixed and meaning may vary from subject to subject
How can teachers empower learners? (cont.) • Introduce Maths Journals • Journals serve a range of purposes: • Allows for learners to respond to affective, open-ended questions • Learners can write about familiar Mathematical concepts • Write about current Mathematical concepts • For metacognition (reflection) on the learning that has taken place of a particular concept
Affective, open-ended questions • Learners cannot just begin to write about Mathematics in the same way as they write a descriptive paragraph • Begin by scaffolding learners’ responses in their Maths journals • Complete statements based on your lesson today: • I learned that … • I noticed that … • I discovered that … • I was pleased that … • I was concerned that …
Affective, open-ended questions (cont.) • Before you begin this activity model your response to each statement with the learner • Impress upon the class that each learner’s response to these questions will be different based on the learner’s personal experience • Use the learner’s response to the statements to see how the learner is coping with the current concept • Ask each learner to write a ‘Mathography’ This is a paragraph which describes their feelings and experiences about Maths – both past and present
Writing about familiar Mathematical concepts • Use the journals as a way of reinforcing concepts that have already been taught • Foundation Phase: There were 8 birds in the backyard on Monday. On Tuesday there were only 5 birds. Explain in a picture or words how many birds were left • Intermediate Phase A visitor from Venus arrives at your school and is confused by the number on your school door: 574. He asks, “Does 5 plus 7 plus 4 equal 574. Answer his questionand explain your answer
Writing about familiar Mathematical concepts • Senior Phase On November 23, 1942, a British ship sank in the Atlantic Ocean. A sailor from the ship was discovered on a raft along the coast of Brazil 132 days later. On what date was he discovered approximately? Explain your thinking.
Give learners a strategy to decode the language of the problem • Do you understand all the words in the problem? • What are you asked to do? • Write the problem in your own words • Can you think of a picture/diagram/table/mind-map that will help you understand the problem? • Do you have enough information to be able to work out an answer?
Writing about current Mathematical concepts • Use journals as you are teaching current concepts • This enables teachers to follow their learners’ thinking • Learners are able to document and process their thinking which is useful in achieving clarity • Encourage learners to document their thinking in a way that makes most sense for them such as visually (diagrams, graphs, tables, mind-maps etc)
How do we enable children to go “meta”? • Encourage learners to ask the questions: • What do I know? • What don’t I know? • What do I need to know? • Reflective questions are important for learners to: • Assess what they can do and help them to make meaningful connections (self-monitor) • Processing and reflecting enable learners to direct their own learning • Use the learners’ reflection to reflect on your own teaching practice
How should learners organise these Journals? • Avoid dictating a specific format as learners process and think individually • Consider asking them to tab different topics for ease of reference • Encourage learners to ‘own’ their journal • Pages on the left could reflect teacher input – core concepts; notes etc • Pages on the right could show the learner’s processing of information; their engagement with the concepts and their reflection on what they have learnt
Useful resources www.childrensmathematics.net/childrenthinkingmathematically_psrn.pdf “Children thinking mathematically:PSRN essential knowledge for Early Years practitioners Worthington, M. and Carruthers, E. 2003 www.childrensmathematics.net/paper_teachers-practices.pdf “Becoming bi-numerate: a study of teachers’ practices concerning children’s early ‘written’ mathematics
Useful resources floridarti.usf.edu/resources/topic/academic_support/.../classstrategies.pd “Classroom Cognitive and Meta-Cognitive Strategies for Teachers” www.frontiersd.mb.ca/programs/.../MathPrime/JournalWriting.pdf “Journal Writing in Math Class K-8”
