Maths questions include numerical information that when combined with a mathematical symbol (or the text version of this), prompts students to use what they already know to help them make inferences about what they need to do with the information to find the answer. Having had a rich discussion about this with my friend and colleague Sheree Hodge (Ranui Schoo In-school Maths CoL teacher), I decided to dig a little deeper to strengthen my own understanding.
Inference helps students comprehend text. It is the skill of using what you already know to work out what you don't know based on the clues given to help you visualise what is happening. Simply put, inferencing involves using what you know to make a guess about what you don't know. There are two types of inference, default inference (automatic assumptions) and reasoned inference (a conclusion made based on the information available). 'Once students have identified the premises on which they've based their inferences, they can engage in the most powerful part of the process—examining the validity of their thinking.' Marzano (2010).
'Inferences are made when information the author assumes can be logically made are left out,' Carr (1983). We make inferences every day, and often because this is automatic don't realise what we have inferred wasn't included in the information we were given. If we are driving and suddenly the flow of the traffic slows down we infer that there is a problem ahead by exploring possibilities (maybe the traffic lights are short phasing, or someone has broken down or maybe there has been an accident) to make sense of the situation. No one gave us these details but as the traffic flow has slowed we know there is a problem and automatically draw on what we know to be possible reasons to help us explain the situation. In reading we 'read between the lines' to use the information given about a character or situation to visualise what is happening then use what we know to be possible behaviours or outcomes to help us draw conclusions about what might happen next or explain why something happened the way it did. In maths this is no different because written maths is inference. Students have to infer from the language in the written problem what numbers are involved, the number knowledge they require and whether or not they're trying to find a sum or a missing addend.
From a word problem our learners need to be able to work out what the mathematical problem is first and in order to do this they need to understand the literacy of maths. If you think about this in relation to English, think about how as a teacher you use word studies to grow vocabulary knowledge. We help our learners explore the different synonyms for a word by unpacking the definition and talking about the different ways that word can be used in a sentence. We then scribe the student generated examples to allow our learners to make a visual connection to what this looks like in context. There is no difference in maths. If addition is used as the example, we need to talk about what addition looks like eg: A + B = C or B + A = C and explore the synonyms of 'add' to grow vocabulary knowledge in context. Once we have this knowledge we then need to use it help us make the inferences needed to find the sum (eg: A + B = ____), or the missing addend (eg: A + ___ = C). What I mean by that is once the students have identified the key mathematical words and the numbers they need to work with, they must then use this knowledge to infer what it is they are being asked to do with the information they have.
My next step is to find out if my learners know and understand what inferring means in a reading context, then see if they are able to transfer this knowledge to a maths context.
'Inferences are made when information the author assumes can be logically made are left out,' Carr (1983). We make inferences every day, and often because this is automatic don't realise what we have inferred wasn't included in the information we were given. If we are driving and suddenly the flow of the traffic slows down we infer that there is a problem ahead by exploring possibilities (maybe the traffic lights are short phasing, or someone has broken down or maybe there has been an accident) to make sense of the situation. No one gave us these details but as the traffic flow has slowed we know there is a problem and automatically draw on what we know to be possible reasons to help us explain the situation. In reading we 'read between the lines' to use the information given about a character or situation to visualise what is happening then use what we know to be possible behaviours or outcomes to help us draw conclusions about what might happen next or explain why something happened the way it did. In maths this is no different because written maths is inference. Students have to infer from the language in the written problem what numbers are involved, the number knowledge they require and whether or not they're trying to find a sum or a missing addend.
From a word problem our learners need to be able to work out what the mathematical problem is first and in order to do this they need to understand the literacy of maths. If you think about this in relation to English, think about how as a teacher you use word studies to grow vocabulary knowledge. We help our learners explore the different synonyms for a word by unpacking the definition and talking about the different ways that word can be used in a sentence. We then scribe the student generated examples to allow our learners to make a visual connection to what this looks like in context. There is no difference in maths. If addition is used as the example, we need to talk about what addition looks like eg: A + B = C or B + A = C and explore the synonyms of 'add' to grow vocabulary knowledge in context. Once we have this knowledge we then need to use it help us make the inferences needed to find the sum (eg: A + B = ____), or the missing addend (eg: A + ___ = C). What I mean by that is once the students have identified the key mathematical words and the numbers they need to work with, they must then use this knowledge to infer what it is they are being asked to do with the information they have.
My next step is to find out if my learners know and understand what inferring means in a reading context, then see if they are able to transfer this knowledge to a maths context.
Readings to support my learning:
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