Teaching fractions is critical. Mastery of fractions is one of the strongest predictors of success in high-school mathematics and helps in areas such as algebra.
Yet, students often struggle to master the concept of fractions. They hold misconceptions about fractions and therefore find it difficult to add, subtract, multiply, divide, compare and order them.
Errors occur at the surface level. Common errors include:
Adding (or subtracting) both the numerator and denominator (wrong)
Not finding a common denominator before adding (or subtracting) fractions (wrong)
Finding a common denominator when multiplying fractions (unnecessary)
Dividing the denominator by the numerator when converting to decimals (wrong)
When teaching fractions, many teachers have seen students make these sorts of common errors. And, many teachers correct their students by telling their students that they are wrong (or that a step is unnecessary) and repeating the correct way to do things.
There is nothing wrong with what these teachers have done. However, many errors are merely surface-level symptoms of an underlying misconception. Research shows that students struggle because they hold deeper misconceptions about fractions.
So, what should you do?
When students hold misconceptions, especially deep misconceptions, you should take advantage of the teaching for conceptual change process. Teaching fractions is an ideal time to use this process.
Understanding the Underlying Misconception
One frequent and deep misconception is rooted in students’ prior knowledge about whole numbers. It is known as whole number bias.
Prior knowledge can often make learning easier, but conceptually, the way whole numbers work is entirely different to the way fractions work. Here are some crucial differences.
While many students represent fractions correctly, they mistakenly (and often subconsciously) use their understanding of whole numbers to try and understand fractions. It just doesn’t work!
For example, the whole number 8 means 8. It is bigger than 7 and smaller than 9. However, the value of 8 in a fraction depends on its relationship with the other number in the fraction. Consider these statements:
In these examples, you cannot figure out the value of 8 by its place in the whole number counting system. It is not that students’ understanding of whole numbers is wrong. Rather, the problem is they cannot make use of this prior understanding when working with fractions.
Numbers in fractions work in a totally different way. 8 can be bigger than 9, and smaller than 7. In a fraction, you cannot know what any number is worth (e.g. 8, 26, 3) without knowing the other number in the fraction.
With whole numbers, the number has a value in its own right. 8 is always 8. It is always less than 9 and more than 4. 28 is always 28, etc. In other words, you can understand the number in isolation.
With fractions, you can only understand each number in relation to the other number. More specifically, you can only understand fractions based on the ratio between the numerator and the denominator.
Many Students Don’t Get This
For most teachers and some students, this new understanding comes quite easily. However, for many students, it doesn’t.
Conceptual change is not likely to happen unless you clearly show the inadequacy of their existing beliefs. Therefore, when teaching fractions, you need to explicitly highlight the fact that:
Their existing way of thinking about the value of numbers doesn’t work (i.e. finding value by looking at numbers in isolation)
A new way of thinking about numbers is needed
Only after this has been done, should you start to teach your students about the fact that the value of a fraction (i.e. how big it is) comes from the relationship between 2 numbers – the numerator and the denominator.
When To Teach About Whole Number Bias
When should this be taught?
After looking closely at the Australian Curriculum, I believe it should be taught in Year 4.
In Year 4, students start:
Working with fraction walls, which you can use to show the inadequacy of whole number thinking
Make connections between fractions and their decimal equivalents, which you can use to show ratio thinking
However, if you teach older year levels, and your students struggle with fractions, it is worth going through this conceptual change process with them.
This is only one example of conceptual change with fractions. You can go through similar processes to dismiss other misconceptions, such as the notion that:
Do you have more things to do than time allows? Take up the #MyTimeMatters challenge.
Excessive teacher workloads are a real issue. Yet, I don’t believe that the system will ever change. There will always be ‘the next’ new initiative and bureaucracies love their paperwork.
Even if a miracle occurred and the system did change, the very nature of teaching means there will always be more that you could be doing. Therefore, it is unavoidable that some things don’t get done, or at best don’t get done well.
This means two things. Teachers need to:
Get comfortable with the idea of saying I’m doing nothing, even when there is more to do.
Prioritise what they attend to, according to an activity’s impact on student learning.
If this sounds like a call for a ‘lazy teacher’ revolution – it’s not. Good teaching needs effort and hard work. But if it is impossible to do ‘everything’, it makes sense to do those things that have the largest effect on your students’ success.
Sadly, a recent study by ACER showed that too many teachers are letting the wrong things slide.
Over half the teachers surveyed felt they weren’t teaching as well as they could be, let alone catering for the needs of their struggling students. Two-thirds of teachers did not believe they were doing a great job at core activities, such as assessing students’ learning and providing quality feedback.
Your time matters! And, therefore, I invite you to take matters into your own hands and to prioritize those tasks which have the largest impact on your students’ learning.
That is the essence of the #MyTimeMatters challenge.
Type Yes! If you agree and share this with your social networks.
Teachers are always thinking about things they can do to have more of an impact on their students. One popular idea involves matching your style of teaching to different students’ learning styles.
Answer this question in your head.
Do students learn better when they receive information in their preferred learning style?
If you said yes, you are not alone. A recent study revealed that 76% of teachers believe the answer is yes. This finding is consistent with previous research. However, your belief is wrong.
The answer is no. Students do not learn better when they receive information in their preferred learning style.
Research shows that:
Just because a person states or believes they learn a certain way does not mean they actually learn that way.
There is no conclusive evidence that matching your style of teaching to students’ preferred styles of learning has any impact on their learning at all.
Most teachers want to help all their students to succeed. The idea that some students don’t succeed simply because they are not being taught in their preferred style is appealing because it ‘supposedly’ helps all students.
However, teachers have limited time and they are already stretched to the brink. Trying to differentiate teaching to cater for students different learning styles wastes your precious time. Teacher should spend that time in better ways – be that by focusing on evidence based aspects of teaching or achieving a better work-life balance.
Furthermore, Australian research showed that students are often categorized as kinesetic learners leading to limited, inappropriate and ineffective teaching strategies being used with these students.
The Truth About Learning Styles
Students do receive information in different modalities – including through sight (visual) and sound (auditory). However, it does not follow that students actually have preferred learning styles. Nor does it mean that students learn better when you teaching matches their so-called, preferred learning styles. Rather, stronger learning occurs when information is presented in multi-modal ways, such as using both words and visuals.
If you truly want to help more of your students to succeed, do not waste your precious time trying to cater for different learning styles. Rather, present all new information in multi-modal ways and use approaches to teaching that have been proven to work.
Reflection On Learning Styles
Why can’t you take students’ statements about their preferred learning style at face value?
How can trying to cater for different learning styles be harmful?
What is a more useful way that teachers could respond to the fact that students take in information through different modes?
The Conceptual Change Process
You may have thought this article was about learning styles and, on the surface it was. However, it also an example of a conceptual change text, which is one form of potent teaching strategy known as conceptual change.
When you start a new topic, it is important you recognise that your students’ minds are not blank slates. Rather, students come into your classroom with existing ideas and beliefs about the topic at hand.
Connecting new information to students’ prior knowledge is important, but this is not a particularly new idea. Teaching for conceptual change goes further, by:
Acknowledging that some of students’ existing understandings and beliefs may be faulty
Bringing these beliefs to fore, and highlighting their inadequacy
Explaining the correct understanding in an easy to understand way
Helping students to let go of old, faulty ways of thinking and accept new ideas
This is the basic process of conceptual change.
It applies to the contextual change texts that Hattie found had an effect size of d=1.16. In fact, the above is an example of a conceptual change text, as opposed to a traditional text, which would have simply shared information on the importance of presenting information in multi-modal ways.
Yet, you can adapt the same process beyond information and activities in a textbook, to any form of instruction.
Here is an explanation of the conceptual change process in more detail.
Conceptual Change Process Step 1: Identifying Misconceptions
When introducing a new topic, take the time to explore common misconceptions that your students typically have in that area. A simple Google search normally reveals information from a number of trustworthy sources.
In this example, I used the topic of evidence based teaching and found research on a number of prevalent neuromyths in our field, including learning styles, left-brain and right brain learners, and oversimplifying/over identifying dyslexia as seeing reversed letters.
I chose learning styles, as it was the most prevalent (76% of teachers in the UK & the USA believe in catering for learning styles).
However, I then use a question to elicit whether my readers actually believe it themselves.
Conceptual Change Process Step 2: Mental Disturbance
You then seek to create some form of disturbance in the minds of your students. You do this by explicitly stating that an identified misconception is wrong. Yet, you must do more than this. Misconceptions are hard to dislodge. You must, therefore, explain why it is wrong and you must show how holding onto the misconception is harmful in the students’ world.
In the example of learning styles, I tried to do this here.
Conceptual Change Process Step 3: Explaining Correct Conceptions
You then seek to replace the misconception with the correct conception of the topic at hand. To do this, you must present the correct concept:
Simply and clearly
In relation to prior knowledge
(in this case students do receive information through multiple modalities)
As being useful to students’ needs (in this case, helping more students to succeed)
Conceptual Change Process Step 4: Mental Engagement
Imparting information about the incorrectness of a misconception and about the correct understanding is essential. However, conceptual change is more likely to occur when students actively think through what you have taught them.
What is evidence based teaching? What does evidence based teaching involve? Why should you embrace it?
I answer such questions below.
You must be willing to challenge and sometimes change your beliefs about teaching. Your beliefs are powerful because they underpin the way that you go about your work. Yet, we do not always base our beliefs on objective facts. To a degree, teachers base their beliefs on popular educational theories and fads. Some of these have little if any grounding in evidence at all. We also base our beliefs on our personal experiences. Yet, we tend to notice only those things that reinforce our existing beliefs. For example, if you believe that children are unsettled on windy days, you will notice those students who are unsettled on windy days. I am not suggesting that all your beliefs are wrong. Rather, I am inviting you to consider the fact that some of them might be. It is dangerous to flippantly disregard evidence that challenges your beliefs.
You must want to improve some aspect of your students’ learning, and you must search for ways to help you do that. The only legitimate reason to adopt new teaching practices is to achieve something over and above what you are already achieving. Adopting new approaches without the intention to improve some aspect of your students’ learning is just silly, and it often leads to a senseless increase in teachers’ workload. When you embrace evidence based teaching, you only change the way you teach for good reasons.
You must start by using strategies with the highest chance of success. Research shows that nearly everything teachers do works. At least, to some degree. However, some approaches to teaching have more impact than others. Put another way, some approaches to teaching are more likely to work than others. Researchers have not found “one particular approach to teaching” that works for all students, in all contexts. Yet, evidence based teaching involves you starting by adopting those strategies with the highest chance of success.
You must monitor your students’ progress in the particular aspect of learning that you are trying to improve. And you must do it before it is too late. The reason is not to assign grades, or even to let students know where they are at (although this can be valuable in its own right). Rather, it is to let you know whether your efforts are having the desired effect. It is why experts such as John Hattie see assessment as feedback to teachers and why he encourages teachers to know thy impact. If you want to truly embrace evidence based teaching, you must become a student of your own impact.
You must adjust your approaches to teaching as needed. Monitoring your students’ progress is important. Not for its own sake, but because it enables you to take corrective action when needed. While you may have adopted approaches to teaching that are likely to work, that doesn’t mean they will always do so. When what you are doing is not working, you need to try something else. There is no particular approach that will always work. Embracing evidence based teaching involves acknowledging this. Yet, it also involves adopting a problem-solving approach (as opposed to assigning blame or offering excuses) when things don’t go to plan.
In Short Embracing Evidence Based Teaching Involves
Do you want to help your students to develop a deep understanding of the material you are teaching? It is a noble intention. Yet, many teachers fail to do so. This is not from a lack of trying. Rather, it is because they miss the essential first step.
Foundational Knowledge – The Missing Step In Deep Understanding
Many people see the teaching of knowledge as being the enemy of deeper conceptual understanding. This is not the case. As John Hattie points out, both are needed.
In fact, students cannot develop deep understandings without having a foundational bank of knowledge. Their knowledge base gives them something ‘to think about’. And, it provides the foundation upon which deeper understanding depends. The more facts a student knows, the more capacity they have to glean deeper understandings.
If we want pupils to have good conceptual understanding, they need more facts, not fewer Daisy Christodoulou
Consider this example shared by Dan Willingham:
A Year 4 class was starting to learn about rain forests. Thier teacher asked them to state whether they would like to live in a rain forest and to justify why. Students without sufficient background knowledge gave shallow responses such as:
No, because it would be wet
Yes, because it would be fun
In this case, the teacher asked students to apply a higher-order thinking skill (justifying) without providing them with a sufficient knowledge base. As a result, it did nothing to deepen their understanding. Sadly, we do this a lot in modern classrooms.
When asked the same question, students with a larger bank of interconnected knowledge gave answers such as:
No, because the poor soil and constant shade would mean she may have to include meat in her diet, and she was a vegetarian
This is an excellent example of how knowledge builds to allow sophisticated higher-order responses. You can only make that response if you know that:
Rainforests have poor soil and constant shade
This makes it difficult to grow agricultural crops there
When the knowledge base is not in place, pupils struggle to develop an understanding of a topic. Deep thinking (such as justification) is essential. Yet, without the prerequisite knowledge, it cannot be used to nurture deep learning.
Sadly, many teachers ask their students to demonstrate deep understanding using strategies such as:
Yet, they do so without teaching their students the foundational knowledge and skills needed to genuinely learn from such strategies.
This explains why research shows that such strategies often fail to have the desired effect on student learning. It is a matter of when they are used (i.e. after developing foundational knowledge), rather than whether they should be used.
Students Can’t Just Look It Up
Some teachers argue that knowledge is irrelevant. Why? Because students can just look things up as they need to.
It is true that the internet gives us unprecedented access to a wealth of information. It is also true that teaching research skills (especially source evaluation) is essential. However, scanning sites for isolated facts does not automatically lead to students developing a deep understanding of the topic at hand.
It is too easy to confuse information access with genuine knowledge acquisition Gregory Yates
Access to information does not ensure that students understand what they read – even for good readers.
A student’s capacity to make sense of what they are reading is highly dependent upon their background knowledge of the topic at hand.
Comprehension demands background knowledge. Daniel Willingham
When you don’t actively build up students’ knowledge of a topic, they cannot engage in any form of meaningful inquiry. They simply don’t know enough about the broader topic to make sense of what they read. And, they have no hope of discerning what is relevant, let alone turning it all into an answer to the assignment question.
If you want to help students develop essential research skills, teach them foundational facts, as well as the skills themselves. A broad, background knowledge provides the foundation for understanding the specific things that they will read on the internet.
In Closing
You should teach both surface knowledge and develop deeper levels of understanding. This challenges the either–or thinking that we often take when talking about how to teach. There are many examples where changing the word or to the word and makes much more sense. For more on this idea, see One Word That Can Improve Your Teaching.
Teacher clarity is critical. Eminent educators, Dylan Wiliam and John Hattie agree that you must be very clear about what you want your students to learn. You need to know exactly what you want them to understand and what you want them to be able to do. More importantly, you need to ensure that your students are equally clear about what they must learn and how they can prove they have learnt it.
Put another way, clear teaching helps students to progress 9 months further than students whose teachers did not have high levels of teacher clarity.
Teacher Clarity Misunderstood
Both Hattie and Wiliams encourage you to use learning intentions and success criteria to achieve this clarity. However, this has led to a lot of time being spent on plastering walls with learning intentions, mastery ladders and alike with very little benefit. Teachers are doing it for the sake of doing it, rather than to focus their teaching. Yet, it is focusing your teaching that makes the difference.
In fact, the meta-analysis Hattie reviewed does not refer to learning intentions or success criteria per se, but simply to teacher clarity. This doesn’t mean that learning intentions and success criteria are bad, but rather that they are merely one way of achieving teacher clarity. The same applies to other strategies, such as mastery ladders. You can make use of these strategies when they help you achieve clarity what you want your students to learn. However, don’t ever use them for their own sake or worse, in contexts where it wastes your time without affecting your subsequent teaching.
And this leads to my next point.
Teacher Clarity Must Lead To Clear Teaching
Hattie reviewed just one meta-analysis on teacher clarity, which was conducted by Frank Fendick. Fendick found that teacher clarity had a substantial impact on students’ subsequent results. Yet, being clear about what you want your students to learn is just one aspect of Fendick’s definition of teacher clarity. Teacher clarity involves:
Of these dimensions, clearly explaining new content has the largest impact on subsequent results. However, when you attend to the aspects of teacher clarity collectively, your impact on your students’ results nearly doubles.
Students are emotionally driven beings. Thanks to ongoing psychological research, we now know a lot about how our emotions work. Emotionally intelligent teaching involves you using this insight in intelligent ways to improve the impact that you have on your students.
Emotionally Intelligent Teaching – Tip 1
One of the most potent findings regarding emotional intelligence is the presence of mirror neurons in our brain. Mirror neurons activate in our brain when we interact with another person. They enable us to feel what the other person feels, which explains why the emotions of others are often contagious.
But why is this important?
Students are likely to catch the emotions you are feeling. Sadly, research shows that most students are ambivalent to school. However, they are far more likely to do well when they are interested and passionate about what they are learning.
Teachers need to be engaged in the passion of teaching and learning. John Hattie
You can help students do better at school simply by being passionate about the work you do. It is one key aspect of emotionally intelligent teaching.
Emotionally Intelligent Teaching – Tip 2
Another key finding in the emotional intelligence literature centres on the importance of empathy. In short, empathy involves recognising how other people feel and then experiencing their feelings yourself. This creates a sense of a shared connection, which in turn allows you to deal with the situation (whatever it may be) together. At a broader level, empathy involves seeing the situation through another person’s eyes.
A meta-analysis of available research shows that students do substantially better at school when you show them empathy and warmth. Take an interest in their lives, pay attention to how they are feeling and feel with them.
Emotionally Intelligent Teaching – Tip 3
This leads into my third tip, which is to remain calm and objective when dealing with students’ misbehaviour. All too often, we let anger and disappointment control how we respond to a student’s misbehaviour. In emotional intelligence terms, this is known as an emotional hijack.
However, research reviewed by Robert Marzano reveals that the most effective behaviour management technique teachers can adopt is a particular mindset. This mindset has two core components:
A heightened sense of awareness regarding what is going on in your classroom
A capacity to stay calm and objective
The essential point is that we want students to behave well so that they can learn. Losing your cool (especially over minor misbehaviour) disrupts both teaching and learning.
Practising feeling empathy will help you to remain calm. It also helps to remember not to take things personally.
One of the most potent things a teacher can do to improve student learning is to monitor the progress of each of their students, and to adjust their teaching accordingly – or, in John Hattie’s words, to know thy impact.
When you know how far each of your students has progressed, you also know the impact your teaching has had.
How To Measure Impact
To know thy impact involves measuring progress, not A-E achievement. To do this you need to use the basic pre-test/post-test format. You can use this format with any measurable assessment tasks (e.g. early reader text levels, draft writing sample scored with a numerical rubric (see attached sample) or actual tests (teacher written or commercial).
A student may score 4/12 on their first attempt at using persuasive devices in a text, and then 7/12 second attempt. They may be reading a level 3 text in week one, and a level 7 text at the end of the term. Or, they may get 24/50 on a random pre-test of tables and 30/50 on a subsequent test.
How To Know If Thy Impact Is Sufficient
In the above examples, it is easy to see progress – the harder question to answer is whether that progress is sufficient. This is where Hattie’s effect size research comes in really handy. Effect sizes range from 0 to infinity, but most range between 0 and 1.
After reviewing thousands of research articles, Hattie found that the typical effect of teaching 0.4. That is, after teaching a unit on fractions, a pre-test/post-test should show progress of 0.4. If it doesn’t, the strategies being used by the teacher are not effective.
The next question is how do you work out effect sizes, especially when you are a busy teacher. There are statistical formulas you can use. However, thankfully there is an Excel spreadsheet called the Progress Achievement tool that does it all for you. It only works though if you punch in data for a whole class or at least a group. Open it up and typing some dummy data get the idea – note just enter their score, not their score out of (e.g. 8 not 8/12).
Know Thy Impact & Adjust Teaching As Necessary
I now get to the main point. What happens when you have done all this?
If the effect size for the whole class is lower than 0.4 you need to problem solve new approaches to teaching the next lot of similar content. This is best done in with a trusted colleague/s.
You then repeat the exercise for any individual students who have progressed less than 0.4.
Every time the effect is higher than 0.4 is a time for congratulations and celebration, as your approaches to teaching are working.
When your students view you as a credible teacher, they are more likely to do well in school. According to John Hattie’s latest results (2016), teacher credibility has a massive impact (d = 0.9) on the subsequent learning that happens in the classroom. To put this in perspective, teacher credibility has more than twice the impact of student motivation. This doesn’t mean that student motivation is not important – it is. Rather, it simply shows that teacher credibility is even more important.
But don’t most teachers like to see themselves as credible? They probably do, but it is not teachers’ perception of themselves that matters. Rather, it is a student’s perception that is important. Yet, it is not simply a matter of whether they “like” you, but a matter of whether they think you are a good teacher. Students are very accurate at judging which teachers are good at their jobs.
If a teacher is not perceived as credible, the students just turn off. John Hattie
There are three core aspects that are important to students’ judgments about teacher credibility:
Trusting Relationships
Competence
Passion
(NB some academics include a fourth aspect, “immediacy” that I have amalgamated into trusting relationships).
If you want to be seen as credible, you must form trusting relationships with your students. Such relationships are based on care. You must care about your students as both:
People
Learners
Credible teachers care about the results that each of their students achieves. They are not happy to let a struggling student fail anymore and than they are to let a bright student coast along. Yet, they do not expect their students to do it on their own. Rather, they are there for their students every step of the way – and their students know it.
Students don’t care how much you know until they know how much you care.
Adapted from Theodore Roosevelt
Credible teachers are also there for their students as people. You need to make sure that your students know that they are important to you. Teacher-student relationships flourish when you accept and like each of your students for who they are as an individual. Go out of your way to talk to them about their lives and their interests outside of class. Share their excitement, empathize with their sadness/fears, and be mentally present with them when you do.
So, if you want to increase your teacher credibility, try nurturing even better relationships with your students.
Teacher Credibility Aspect 2: Competence
If you want to be seen as credible, your students must believe that you:
Know your stuff
Are good at helping them learn it
Can manage their behaviour
You need to know the content that you must teach and you need to know it well. That way you can be clear about what it is you want your students to learn – including what they must know and what they must be able to do to succeed. Teachers with a deep understanding of the material they are teaching can also help students achieve conceptual change when needed.
While knowing your stuff is important, you also must be good at teaching it. My daughter had a teacher whose knowledge of her subject was remarkable. However, according to my daughter, she was not very good at sharing that knowledge in ways that help to students to understand it. Of course, my daughter may be wrong. Yet, according to Hattie, students are very good at making such judgments.
You may know your stuff well and be good at teaching it. However, if your students don’t behave well they will actually judge you and form beliefs about your competence. Credible teachers are good at managing their students’ behaviour. Of course, it is true that good teaching nurtures good behaviour. Yet it is also true that teachers who manage behaviour effectively are then able to teach well. You need to be both firm and fair. Related: Top 10 Behaviour Management Strategies; My 5 Favourite On-the-Spot Strategies To Use.
Teacher Credibility Aspect 3: Passion
Students enjoy being taught by teachers who are passionate about their work. Furthermore, they learn more. Therefore, it is not surprising that students see passionate teachers as being more credible than other teachers. Passion and teacher credibility go hand-in-hand.
Some teachers are passionate about what they teach. Personally, I’m extremely passionate about history – both modern and ancient. Passionate teachers are infectious, causing the students to engage more fully with whatever content is being taught. I don’t normally get excited by mathematics despite the fact I find it relatively easy. Yet I once went to an in-service when the national curriculum first came in. The in-service was presented by a university lecturer whose passion for mathematics emanated from her very being. As a result, I was fully engaged for two days despite not being personally excited by the material.
Students appreciated being taught by knowledgeable and passionate teachers. John Hattie
However, sometimes you must teach subjects you are not passionate about. In such cases, faking it just doesn’t work. Thankfully, students still pick up on your passion to help them learn the material, even if you aren’t passionate about the material itself. Students regard teachers who love their work and who relished the challenge of teaching as being more credible than other teachers.
You must also be careful not to confuse being passionate with being easy. You may be passionate about a subject. You may be passionate about your job. And you may be passionate about helping students to make real progress. In turn, you make it easier to engage your students. Yet, your students will still need to do the hard work necessary for success.
In Short
There is a strong link between teacher credibility and student achievement. It is your students’ views about your credibility that matter – not your own views. You can increase your perceived credibility by:
Forging trusting relationships with your students
Knowing your stuff, teaching it well and effectively managing your students’ behaviour
Being passionate about what you teach, about being a teacher, and about helping each of your students succeed