As a child, if you were only ever exposed to the family golden retriever, you’d be forgiven for thinking all dogs are retrievers. After all, other dogs are cuddly and have fur, four legs, and floppy ears. 

It’d only become obvious to you what a retriever really is when you discovered the labrador, border collie, and alsatian. You’d then see that they are not retrievers, and the concept of the retriever (and therefore, the dog) would be more fully understood. 

Your Maths students can learn complex skills in the same way - not just through rote learning and regurgitation - but through the use of contrast to deeply understand concepts and be able to apply that knowledge accurately. 

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Discrimination: The key to deeper learning 

As we know, learning is more than just adopting a new idea. To learn, our brains create new schemas in our minds. Schemas are patterns or mental frameworks that impact how we perceive and interpret the world around us. 

The process of creating a schema occurs via both generalising and discriminating information. We connect things we’ve learned before to establish new concepts. And, importantly, we recognise the differences between what we already know and the new things we’re being exposed to. This helps us to build an accurate and reliable picture in our minds. 

So, for example, if we learn what a newspaper is for the first time, our brains will generalise various concepts. Our brains link the layout, text size, font, and subject matter to what a newspaper is. To gain a deeper understanding, though, it’s important to understand that a news homepage, a social media feed, and a magazine, albeit similar concepts, are not the same as a newspaper. 

That means, to create schemas, it’s helpful to understand not only what a concept is, but, importantly, what it’s not. 

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A higher bar for student performance 

In the senior levels of learning, including HSC, VCE, and QCE, we’re seeing trends towards requiring students to show analysis and application of the skills they’ve learned. 

In HSC exams, for example, performance band descriptors require students to ‘critically analyse’ and show ‘critical judgement’ and ‘reasoning’ in their answers. 

That means for students today, it’s not enough to just be able to generalise information they’ve been taught. Doing so doesn’t show a higher-level understanding that will allow them to discriminate between similar concepts accurately. 

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Non-examples for deeper learning 

Examples are no longer enough. Students need non-examples for comparison and contrast. Non-examples can help your students progress from analysis to evaluation––to solidify core concepts and deepen their knowledge for the long run. 

Research has shown, for example, that the use of non-examples can help to deepen learning and improve the mastery of concepts. 

What’s more, non-examples have been shown to help students improve their analysis skills too. 

Research has found that the use of non-examples in Maths can: 

• Enable students to use reasoning skills and justification 
• Help students to compare and contrast similar concepts
• Promote critical thinking 
• Help to clarify ideas
• Enhance student understanding of complicated lessons  

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Moving up the learning hierarchy 

Helping your students to move beyond just analysis and generalisation can also help move them up Bloom’s Taxonomy learning hierarchy. 

At the lower levels of Bloom’s Taxonomy, learning involves remembering, understanding, and applying concepts. Progressing to the higher levels of the learning taxonomy requires having the ability to analyse and evaluate information. 

For students to reach these higher levels, they need to be able to discriminate between similar concepts, which is where non-examples play a crucial role. Using non-examples helps students to deeply understand what they’ve learned so that they can not only analyse, and generalise, but discriminate too. 

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Embedding non-examples in student learning 

In Maths, misconceptions are often embedded in student understanding. Introducing non-examples as a classroom tool can help to clarify what’s being taught, help students overcome misconceptions, and strengthen their overall understanding. 

This goes beyond illustrating a concept to offer comparison and better highlight what something is not. 

Ultimately, by specifically calling out what a mistake is, students can articulate what is wrong in order to learn what is correct 

Some ways to use non-examples in the classroom include: 

• Asking students to identify the incorrect equation in a multi-choice list 
• Providing students with both examples and non-examples so that they can compare and contrast between similar concepts 
• Having students explain why a non-example is incorrect to prove their understanding of the lesson 
• Asking students to provide both an example and a non-example to answer a question 

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A tool for student success 

In a challenging teaching landscape––where many teachers are feeling the pressure––simple tools that help students to succeed are vital. 

Non-examples are a helpful method that can quickly strengthen your student’s learning and boost their skills without significant planning and preparation. 

While the idea may be simple, it can make all the difference to your students and their final scores. 

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