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		<title>What are the biggest myths in education today?</title>
		<link>https://technologyforlearners.com/what-are-the-biggest-myths-in-education-today/?utm_source=rss&#038;utm_medium=rss&#038;utm_campaign=what-are-the-biggest-myths-in-education-today</link>
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		<dc:creator><![CDATA[Will Fastiggi]]></dc:creator>
		<pubDate>Sun, 14 Oct 2018 17:02:06 +0000</pubDate>
				<category><![CDATA[Education]]></category>
		<category><![CDATA[21st Century Skills]]></category>
		<category><![CDATA[brain gym]]></category>
		<category><![CDATA[brain training]]></category>
		<category><![CDATA[Daisy Christodoulou]]></category>
		<category><![CDATA[discovery learning]]></category>
		<category><![CDATA[facts]]></category>
		<category><![CDATA[Myths in Education]]></category>
		<category><![CDATA[SAMR]]></category>
		<category><![CDATA[VAK]]></category>
		<category><![CDATA[Vygotsky]]></category>
		<category><![CDATA[zone of proximal development]]></category>
		<guid isPermaLink="false">http://technologyforlearners.com/?p=3046</guid>

					<description><![CDATA[<img width="150" height="150" src="https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-150x150.jpg" class="attachment-thumbnail size-thumbnail wp-post-image" alt="Shakespeare" decoding="async" />Around the world many initiatives are followed in education that have not been carefully scrutinised much less held accountable to empirical evidence.  This first became clear to me when I began my teacher training many years ago, and I was taught about the importance of differentiating lessons according to students&#8217; preferred learning styles (Visual, Auditory, [&#8230;]]]></description>
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									<p>Around the world many initiatives are followed in education that have not been carefully scrutinised much less held accountable to empirical evidence.  This first became clear to me when I began my teacher training many years ago, and I was taught about the importance of differentiating lessons according to students&#8217; preferred learning styles (Visual, Auditory, Kinaesthetic &#8211; VAK).  I was also instructed that &#8216;best practice&#8217; constituted minimising the teaching of facts, implying that facts are much less important than &#8220;understanding&#8221;.  Teacher-led teaching I came to believe, led to passivity in the classroom.  Teachers should, as the rhetoric was put across, serve as no more than facilitators for student learning, to encourage and guide students to discover for themselves.  As I saw what works in the classroom in terms of how humans learn, and I delved into the actual research, it became clear that while such ideas may have enough surface level plausibility to sound convincing, they do not actually have any empirical support.</p><p>This is <strong>not to say</strong> that all of these ideas are completely flawed.  However, I do believe that it is beneficial for the education of our students, to carefully scrutinise some of the most commonly held assumptions in education, by holding them up to the light of empirical analysis.  Based on a thorough enquiry into the myths in education, and following my reading of several excellent books on the subject (referenced below), I present here my notes and personal observations of the most persistent myths in education today:</p><p><strong>Myth 1 &#8211; Teacher-led fact-giving should be avoided in favour of allowing students to &#8220;discover things for themselves&#8221;.</strong></p><p>There is a general rhetoric in education that goes something like this: &#8220;Learning should not be a matter of stuffing a person&#8217;s head full of facts, but rather a process of lighting a fire in people so they have the confidence to learn independently.&#8221;  In other words, the teacher should be a facilitator for student learning rather than a &#8220;fact-giver&#8221;.  Instead of the teacher transmitting knowledge to students, it is widely considered best teaching practice for students to discover things for themselves.  This idea can be traced back to the work of the American psychologist, Jerome Bruner in the 1960s.  Bruner advocated a pedagogy in which the learner interacts with the material to be learned in an active and self-investigator manner, to encourage children to think and solve problems independently.  Inquiry-based learning, problem-based learning and project-based learning can all be seen to have been born from Bruner&#8217;s work.</p><p>There are three main assumptions here, which justify such approaches to teaching and learning:</p><p>1.  Concepts are more fully remembered.</p><p>2.  Learners are more motivated to learn if they have to discover by themselves.</p><p>3.  By discovering for themselves, learners will be more easily able to find solutions in new problem situations.</p><p>But to what extent are each of these assumptions actually true?  Although, as I will explain, some degree of discovery learning can be useful, it should not be used in isolation, and it should only be used in certain situations.  Let&#8217;s look at why, by asking questions of each one of these assumptions:</p><p><em>1.  Are concepts more fully remembered by allowing students to &#8220;discover things for themselves&#8221;?</em></p><p>It all depends on learners&#8217; previous knowledge.  This is because conceptual understanding is only possible by knowing facts &#8211; and usually, lots of facts.  As Daisy Christodoulou (2014, pp. 20) points out, we know that long-term memory is capable of storing thousands of facts, and when these are memorised around a particular topic, these facts together form what is called a &#8216;schema&#8217;.  When we meet new facts about that topic, we assimilate them into that schema &#8211; and if we already have a lot of facts in that particular schema, it is much easier for us to learn facts about that topic.  To put it another way, factual knowledge makes cognitive processes work better.</p><p>Daisy&#8217;s example clarifies this well:</p><p><em>&#8216;Critics of fact-learning will often pull out a completely random fact and say something like: who needs to know the date of the Battle of Waterloo?  What does it matter?  Of course, pulling out one fact like this on its own does seem rather odd.  But the aim of fact-learning is not to learn just one fact &#8211; it is to learn several hundred, which taken together form a schema that helps you to understand the world.  Thus, just learning the date of the Battle of Waterloo will be of limited use.  But learning the dates of 150 historical events from 3000 BC to the present day and learning a couple of key facts about why each event was important will be of immense use, because it will form the fundamental chronological schema that is the basis of all historical understanding.&#8217;  </em></p><p>As Hirsch (2017, pp. 79) states, many educators in education now deplore the &#8220;memorising&#8221; of mere facts.  According to this line of thought: &#8220;We need less memorisation of facts, and more emphasis on critical thinking skills for the 21st century&#8221;.  Hirsch highlights this as a fallacy though, by presenting evidence clearly showing that a well-stocked mind (through knowing lots of facts) is the skill of skills &#8211; essential to critical thinking and to looking things up:</p><p><em>&#8216;Critical thinking does not exist as an independent skill.  Cognitive scientists have shown since the 1940s that human skills are domain specific, and do not transfer readily from one domain to the next.  No matter how widely skilled people may be, as soon as they confront unfamiliar content their skill degenerates.  An unfamiliar topic will quickly degrade both reading and writing.  The domain specificity of skills is one of the most important scientific findings of our era for teachers and parents to know about.&#8217; </em></p><p>2.  Are learners more motivated to learn if they have to discover by themselves?</p><p>Not necessarily.  It stands to reason that as long as the subject matter is presented to learners in an engaging way, learners will be just as motivated to learn in a direct-instruction-based lesson as they would be through discovery learning.</p><p>There are two more related problems with these first two assumptions about discovery learning, particularly for younger learners &#8211; and especially in the domain of natural science.  Piaget long ago made clear that younger learners see the world differently from adults (let alone scientists), interpret and understand it differently, and are not capable of carrying out the abstract cognitive transformations necessary for fruitful knowledge construction as it occurs in the sciences (Pedro de Bruyckere et. al, 2015, pp. 50).</p><p>Hattie and Yates (2013, pp. 78) add a further problem, closely related to the importance of prior knowledge:</p><p><em>&#8216;&#8230; several studies have found that low ability students will prefer discovery learning lessons to direct-instruction-based lessons, but learn less from them.  Under conditions of low guidance, the knowledge gap between low and high ability students tend to increase.  The lack of direct guidance has greater damaging effects on learning in low ability students especially when procedures are unclear, feedback is reduced, and misconceptions remain as problems to be resolved rather than errors to be corrected.&#8217;</em></p><p>3.  By discovering for themselves, will learners more easily be able to find solutions in new problem situations?</p><p>As with critical thinking, problem solving is another all-purpose skill that is proffered a lot in education.  According to Hirsch (2017, pp. 84): <em>&#8216;There exists no consistent all-purpose problem-solving skill, independent of domain-specific knowledge.&#8217; </em> There is now plenty of research and evidence to demonstrate that even when people are shown how to solve a problem in one domain, they tend to be baffled by a similar problem in another domain.  Learners, especially the younger they are, need active guidance from the teacher through knowledge transmission.</p><p>Einstein was once reported as having said &#8220;Imagination is more important than knowledge.&#8221;  As Willingham (2009, pp. 46) points out though, if Einstein did actually say this, he was wrong: &#8216;Knowledge is more important, because it&#8217;s a prerequisite for imagination, or at least the sort of imagination that leads to problem solving, decision making, and creativity.&#8217;  It is clear that the cognitive processes, which are most esteemed, are intertwined with knowledge.</p><p><strong>Summary&#8230;</strong></p><p>Evidently, the pure discovery approach to learning is simply not as effective compared to when students are guided by a teacher to the intended learning outcomes.  Even Bruner himself, some years later, replaced the concept of &#8220;discovery learning&#8221; with &#8220;guided discovery&#8221;.  Guided discovery, in which each step of learners&#8217; independent enquiry is scaffolded, should go hand in hand with a certain amount of knowledge transmission from the teacher.  Problem-based learning, for example, is simply inappropriate for acquiring new knowledge or insights.  However, it could be useful for applying and honing existing skills and for making connections between different concepts.</p><p>Pedro de Bruyckere et. al (2015, pp. 51) assert that, as with many other educational initiatives, the effectiveness of discovery learning is dependent on the target group, the objectives and the subjects.  For novice learners, pure discovery learning should never be the method, although it may be a goal.  Experts, on the other hand, possess sophisticated schemas in long-term memory, allowing them to deal differently with problems and solve them in different ways.  The more novice the learner is then, the more important support and guidance are, to the point where, for experts in a domain, discovery learning might well be effective.</p><p><strong>Myth 2 &#8211; Knowledge and skills are distinct.  </strong></p><p>Daisy Christodoulou makes the strong case that knowledge and skills are intertwined, and they are intertwined to such an extent that it is not really possible to separate out skills and teach them on their own.  Skill progression depends upon knowledge accumulation.  For example, building knowledge by committing facts to memory (e.g., the alphabet), allows learners to improve their communication skills.  Likewise, learning the skill of communicating another language requires knowledge of vocabulary and grammar.  Learning all of the 12 times tables, and learning them all so securely that we do not have to think of the answer when the problem is presented, is the basis of mathematical skill (and understanding).  Learning the skill of building websites requires knowledge of HTML and CSS.  Learning the skill of driving requires knowledge of road signs and conditions.  The list of examples can go on and on &#8211; basically, all critical thinking processes are tied to background knowledge.  In all domains, considerable knowledge is found to be an essential prerequisite to expert skill.</p><p>Gibson (1998, pp. 46-47) gives the example of Shakepeare&#8217;s education, which shows just how closely knowledge and skills are connected:</p><figure id="attachment_3055" aria-describedby="caption-attachment-3055" style="width: 600px" class="wp-caption aligncenter"><img fetchpriority="high" decoding="async" class="wp-image-3055" src="https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-799x1024.jpg" alt="Shakespeare " width="600" height="769" srcset="https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-799x1024.jpg 799w, https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-234x300.jpg 234w, https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-330x422.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-690x883.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-1050x1345.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1-452x580.jpg 452w, https://technologyforlearners.com/wp-content/uploads/2018/10/Shakespeare-1-1.jpg 1943w" sizes="(max-width: 600px) 100vw, 600px" /><figcaption id="caption-attachment-3055" class="wp-caption-text">The Chandos portrait (held by the National Portrait Gallery, London). This image is in the public domain due to its age.</figcaption></figure><p><em>&#8216;Shakespeare&#8217;s education at Stratford-upon-Avon Grammar School gave him a thorough grounding in the use of language and classical authors. Although his schooling might seem narrow and severe today (schoolboys learned by heart over 100 figures of rhetoric), it proved an excellent resource for the young playwright. Everything Shakespeare learned in school he used in some ways in his plays. Some of his early plays seem to have a very obvious pattern and regular rhythm, almost mechanical and like clockwork. But having mastered the rules, he was able to break and transform them&#8230; On this evidence, Shakespeare&#8217;s education has been seen as an argument for the value of learning by rote, of constant practice, of strict rule-following. Or, to put it another way, &#8216;discovery favours the well-prepared mind&#8217;. His dramatic imagination was fuelled by what would now be seen as sterile exercises in memorisation and constant practice. What was mechanical became fluid, dramatic language that produced thrilling theatre.&#8217;</em></p><p>Clearly, Shakespeare&#8217;s skill as a playwright came from the way he used the knowledge he had gained.  As Daisy Christodoulou (2015, pp. 22) argues, &#8216;a fact-filled education did not stifle Shakespeare&#8217;s genius; on the contrary, this education allowed that genius to flourish&#8217;.  The same of course can be said to be true of the fact-filled education received by many great writers, scientists, policy-makers, economists and inventors (and thousands of others) &#8211; who have made enormous positive contributions to the world:</p><p>&#8216;By assuming that pupils can develop chronological awareness, write creatively or think like a scientist without learning any facts, we are guaranteeing that they will not develop any of those skills&#8217; (Christodoulou, 2015, pp. 22).  The idea that teaching strategies for analysing or thinking critically will allow our learners to exercise their skills of analysis or critical thinking therefore, is flawed.</p><p><strong>Myth 3 &#8211; New technology <em>always</em> improves students&#8217; learning.   </strong></p><p>This is a myth because it entirely depends on how the <a href="https://learningmole.com/coding-for-kids/" target="_blank" rel="noopener">technology is integrated into teaching and learning</a>.  Clark and Feldon (2014) for example, confirm that the effectiveness of learning is determined primarily by the way the medium is used and by the quality of the instruction accompanying that use.  To take their idea further, it is worth exploring the work of Dr. Ruben Puentedura, who has developed a practical framework, which he calls the SAMR model, to show the impact of technology on teaching and learning.  The model moves through various stages, beginning at a basic level of learning in the substitution phase through to a level where learning is transformational at the redefinition level.</p><p><a href="SAMR%20Model"><img decoding="async" class="aligncenter size-full wp-image-3060" src="https://technologyforlearners.com/wp-content/uploads/2018/10/SAMR-Model.jpeg" alt="SAMR Model" width="683" height="379" srcset="https://technologyforlearners.com/wp-content/uploads/2018/10/SAMR-Model.jpeg 683w, https://technologyforlearners.com/wp-content/uploads/2018/10/SAMR-Model-300x166.jpeg 300w, https://technologyforlearners.com/wp-content/uploads/2018/10/SAMR-Model-330x183.jpeg 330w" sizes="(max-width: 683px) 100vw, 683px" /></a></p><p style="background: white; margin: 0in 0in 13.5pt 0in;"><span style="font-family: 'Lato',serif; color: #444444;">The SAMR model is powerful because it enables us to think about how learning can be extended through the use of technology.  The four stages of the SAMR model are summarised here:</span></p><p style="background: white; margin: 0in 0in 13.5pt 0in;"><strong><span style="font-family: 'Lato',serif; color: #444444;">SUBSTITUTION</span></strong><span style="font-family: 'Lato',serif; color: #444444;"> – Technology acts as a direct tool substitute, with no functional change.  For example, students may type up notes on a word processor instead of writing by hand in an exercise book.</span></p><p style="background: white; margin: 0in 0in 13.5pt 0in;"><strong><span style="font-family: 'Lato',serif; color: #444444;">AUGMENTATION</span></strong><span style="font-family: 'Lato',serif; color: #444444;"> –  Technology still acts as a direct tool substitute, but with functional improvements.  Taking the example of typing on a word processor, augmentation means that the learning process can become more efficient and engaging.  Images can be added, text can be hyperlinked and changes to the text itself can be made quickly.</span></p><p style="background: white; margin: 0in 0in 13.5pt 0in;"><em><span style="font-family: 'Lato',serif; color: #444444;">These first two stages of the SAMR model represent enhancements of existing ways of working.  Digital technology is not necessary in order to carry out the learning task.  The technology simply provides a digital medium for learning to take place, which may enhance learning.</span></em></p><p style="background: white; margin: 0in 0in 13.5pt 0in;"><strong><span style="font-family: 'Lato',serif; color: #444444;">MODIFICATION </span></strong><span style="font-family: 'Lato',serif; color: #444444;">&#8211; By this stage technology not only enhances the learning activity, it also significantly transforms it.  An example might be students setting up a blog in which they open up their work to a worldwide audience.  The blog means that students are much more accountable for the work they present, so will tend to spend more time refining their written work.  In this way, both student learning and literacy improve.</span></p><p style="background: white; margin: 0in 0in 13.5pt 0in;"><strong><span style="font-family: 'Lato',serif; color: #444444;">REDEFINITION </span></strong><span style="font-family: 'Lato',serif; color: #444444;">&#8211; This level requires the teacher to think about learning activities that were previously inconceivable without the use of technology.  This could be for instance, a Google Hangout session that takes place between students from different countries in order for students to swap information about their home countries in real-time. Likewise, the use of </span><a title="Google Docs &amp; Social Constructivist Learning" href="https://technologyforlearners.com/google-docs-social-constructivist-learning/"><span style="font-family: 'Lato',serif; color: black;">Google Docs</span></a><span style="font-family: 'Lato',serif; color: #444444;"> for students in different parts of the world to collaborate on a shared assignment facilitates learning opportunities that would be impossible without such technology.</span></p><p style="background: white; margin: 0in 0in 13.5pt 0in;"><em><span style="font-family: 'Lato',serif; color: #444444;">The modification and redefinition levels represent transformational stages in terms of student learning, as the technology is actively helping to transform the way in which learning can occur.  </span></em></p><p>The SAMR model is essentially a planning tool that helps to design better learning activities for students.  The framework provides pedagogical insight into how technology can and should be used in the classroom.  New technology in and of itself will not necessarily improve students&#8217; learning outcomes.  In order to get the most out of technology for learners, I would make the following recommendations in light of the SAMR model:</p><ol><li>Always consider whether or not the technology improves the learning process.  If the learning process is enhanced through the use of technology, then it’s appropriate to use – if not, more traditional (analogue) methods can work just as well (if not better).</li><li>Collaboration is extremely important, particularly if you are looking at learning from a social constructivist perspective.  Consider how you can use technology to facilitate collaboration.</li><li>Ensure that you use technology to expose students to the outside world.  This not only helps to improve their cultural understanding and international-mindedness, it can be great for building key literacy skills.</li></ol><p><strong>Myth 4 &#8211; Young people read less than they did before there was so much access to technology.</strong></p><p>As Kevin Kelly (2016) states, &#8216;To everyone&#8217;s surprise, ultra thin screens and tablets have launched an epidemic of reading and writing that has continued to swell.  The amount of time people spend reading has almost trebled since 1980.&#8217;  Moreover, according to Pedro de Bruyckere et. al (2015, pp. 151), young people are still doing a lot of reading, and statistics make it clear that a lot of them are reading for pleasure.      <strong>   </strong></p><p><strong>Myth 5 &#8211; Boys benefit if they have male teachers.  </strong></p><p>All the evidence available points to the fact that the gender of the teacher has little or no effect on the learning performance of boys in school (Pedro de Bruyckere et. al, 2015, pp. 183).  Contrary to popular belief, boys do not benefit in any measurable way from having a male teacher.  This is important to note, since according to statistics from the Department for Education, approximately only 1 in 4 teachers are men in England &#8211; accounting for 38% of secondary and 15% of primary schools.  Using data provided by the World Bank, the figures look similar across Europe and North America; female representation in teaching markedly outweighs the number of men in the profession.  There is a significant amount of media attention given to highlight such figures and attract more men into the profession.  As it turns out, this focus on there not being enough men in teaching is unnecessary.</p><p><strong>Myth 6 &#8211; Teaching is best delivered in a format that matches students&#8217; learning preferences.</strong></p><p>There is absolutely no evidence for this statement.  First, evidence from Clark (1982) shows that learners who have reported preferring a particular instructional technique typically derive little benefit from experiencing it.  The second problem deals with the concept of learning styles itself.  The assumption that people can be classified into distinct learning types receives little to no support from objective studies.  As Pedro de Bruyckere et. al (2015, pp. 21) state: &#8216;Most people do not fit one particular style; the information used to assign people to styles is often inadequate; and there are so many different styles that it becomes cumbersome to link particular learners to particular styles.&#8217;  Rohrer and Pashler (2012, pp. 117) summarise it as follows: &#8216;The contrast between the enormous popularity of the learning style approach and the lack of any credible scientific proof is both remarkable and disturbing&#8217;.</p><p><strong>Myth 7 &#8211; Cognitive performance can be enhanced through &#8220;brain training&#8221;.</strong></p><p>In recent years, &#8220;brain training&#8221; has become big business around the world.  As well as brain training being incorporated into the curriculum of many schools, several software companies have been quick to develop Brain Games with the promise of helping to improve everything from problem-solving skills, hand-eye coordination, memory to a whole range of other cognitive abilities.  However, regular practice with the so-called &#8220;brain games&#8221; have not been shown to significantly improve cognitive functioning.  Adrian Owen of the Cognition and Brain Sciences Unit at Cambridge University and his colleagues published their findings in the journal <em>Nature</em>.  Their report concludes: &#8216;&#8230; regular brain training confers no greater benefit than simply answering general knowledge questions using the Internet&#8217;.</p><p>As Pedro de Bruyckere et. al (2015, pp. 110) state:</p><p><em>&#8216;The word &#8220;brain&#8221; is misleading since any training necessarily involves the brain.  There is as yet no evidence at all that brain training that is aimed at improving general cognitive abilities such as fluid intelligence will in any way be effective.</em></p><p><em>In October 2014, 73 psychologists, cognitive scientists and neuroscientists from around the world signed an open letter stating that companies marketing &#8220;brain games&#8221; that are meant to slow or reverse age-related memory decline and enhance other cognitive functions are exploiting customers by making &#8220;exaggerated and misleading claims&#8221; that are not based on sound scientific evidence.&#8217;   </em></p><p>The implication for education is simple &#8211; there is no general training method that can help improve cognitive performance.</p><p><strong>Concluding thoughts&#8230;</strong></p><p>As we have seen, various educational initiatives and theories have gained momentum under the fundamentally false premise that they will improve students&#8217; learning outcomes.  At best, these myths embedded within education, create an unnecessary distraction for students (e.g. &#8220;brain training&#8221;) and at worst, they can lead to significant amounts of teaching time being lost on practices that are simply ineffective (e.g. &#8220;discovery learning&#8221;).</p><p>In particular, there has been a strong focus in recent years on developing very broad skills: &#8220;analytical&#8221;, &#8220;critical thinking&#8221; and &#8220;problem-solving skills&#8221; &#8211; while minimising the importance placed on &#8216;knowledge&#8217; and &#8216;facts&#8217;.  This approach, however, puts the cart before the horse; skills can only be developed in line with knowledge accumulation.</p><p>It makes no sense to explicitly attempt to teach &#8220;21st century skills&#8221;.   Cognitive science has shown that all students will develop their skills naturally, hand in hand with knowledge and hard work.  Learning should be hard if it is to be effective.  As Vygotsky stated in the last century, real learning takes place in a learner&#8217;s zone of proximal development: that narrow region where learning is hard but not too challenging.</p><p>It is worth emphasising that for the most part, teachers already do an excellent job of creating challenging lessons.  In addition, teachers do many other things that we know are not myths: provide timely feedback, get students excited about the subject matter, go over worked examples, review previous learning, provide scaffolds for difficult tasks &#8211; and much more.  All of these things that teachers do on a regular basis have plenty of scientific evidence, which shows they are effective for improving students&#8217; learning.   It is simply important to apply common sense in education, to scrutinise educational practices and hold them up to the light of empirical evidence.</p><p><strong>References: </strong></p><p>Bruyckere, P., Kirschner, P., &amp; Hulshof, C.  (2015).  Urban Myths About Learning &amp; Education.</p><p>Christodoulou, D.  (2013).  Seven Myths About Education.</p><p>Clark, R. E. (1982).  Antagonism between achievement and enjoyment in ATI studies.  <em>Educational Psychologist</em>, 17(2), 92 &#8211; 101.</p><p>Clark, R. E., &amp; Feldon, D. F. (2014). Ten common but questionable principles of multimedia learning.</p><p>Gibson, R.  (1998).  Teaching Shakespeare.</p><p>Hattie, J., &amp; Yates, G. C.  (2013).  Visible learning and the science of how we learn.</p><p>Hirsch, E.D. Jr. (2017).  Why Knowledge Matters &#8211; Rescuing Our children From Failed Educational Theories.</p><p>Kelly, K.  (2016).  The Inevitable &#8211; Understanding the 12 technological forces that will shape our future.</p><p>Puentedura, R. (2014).  <a href="https://sites.google.com/a/msad60.org/technology-is-learning/samr-model  " target="_blank" rel="noopener">https://sites.google.com/a/msad60.org/technology-is-learning/samr-model</a></p><p>Rohrer, D., &amp; Pashler, H. (2012).  Learning styles: Where&#8217;s the evidence?  <em>Medical Education, 46</em>, 630 &#8211; 635.</p><p>Willingham, D.  (2009).  Why Don&#8217;t Students Like School?</p>								</div>
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		<title>Teaching for progression in learners&#8217; understanding of mathematics</title>
		<link>https://technologyforlearners.com/teaching-for-progression-in-learners-understanding-of-mathematics/?utm_source=rss&#038;utm_medium=rss&#038;utm_campaign=teaching-for-progression-in-learners-understanding-of-mathematics</link>
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		<dc:creator><![CDATA[Will Fastiggi]]></dc:creator>
		<pubDate>Sun, 23 Jul 2017 18:09:23 +0000</pubDate>
				<category><![CDATA[Education]]></category>
		<category><![CDATA[calculators]]></category>
		<category><![CDATA[interactive whiteboards]]></category>
		<category><![CDATA[learning styles]]></category>
		<category><![CDATA[mathematics]]></category>
		<category><![CDATA[National Numeracy Strategy]]></category>
		<category><![CDATA[SEN]]></category>
		<category><![CDATA[setting]]></category>
		<category><![CDATA[VAK]]></category>
		<guid isPermaLink="false">http://technologyforlearners.com/?p=2428</guid>

					<description><![CDATA[<img width="150" height="150" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths6-150x150.jpg" class="attachment-thumbnail size-thumbnail wp-post-image" alt="" decoding="async" />Progression in students&#8217; understanding of mathematics is a fundamental focus for policy in many countries, where the relationship between mathematical standards and the quality of the workforce, which maintains economic competitiveness, forms part of the government&#8217;s rationale for wanting to improve numeracy standards.  Recurring periods of national concern about low standards of numeracy skills shown [&#8230;]]]></description>
										<content:encoded><![CDATA[<img width="150" height="150" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths6-150x150.jpg" class="attachment-thumbnail size-thumbnail wp-post-image" alt="" decoding="async" /><p class="p1">Progression in students&#8217; understanding of mathematics is a fundamental focus for policy in many countries, where the relationship between mathematical standards and the quality of the workforce, which maintains economic competitiveness, forms part of the government&#8217;s rationale for wanting to improve numeracy standards.  Recurring periods of national concern about low standards of numeracy skills shown by Primary school pupils have become more urgent and more political in recent years with the publication of international comparisons of mathematical achievement (Street et al., 2005, pp.1).  In the UK, the National Curriculum offers a model for progression of learning in mathematics, and as with teaching across the developed world, this shows a focus in the way in which children learn as active problem solvers.  Implicitly, this defines the ethos of constructivist&#8217; theories of how children learn and is based on the view that learners need to be actively engaged in constructing their own mathematical knowledge by seeking out meanings and making mental connections in an active manner.</p>
<p class="p1"><img decoding="async" class="aligncenter wp-image-2440" src="https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1382-300x224.jpg" alt="IMG_1382" width="600" height="448" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1382-300x224.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1382-1024x764.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1382-330x246.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1382-690x515.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1382-1050x784.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1382-776x580.jpg 776w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1"><strong><span class="s1">Mathematics is both a body of established, recognised content and a process by which </span>exploration and establishing takes place.</strong>  As Brown (1989, pp. 126) states, there is mathematics to know and mathematics to do.  Atkinson (1992, pp. 16) puts forward the view that the ability to solve problems is at the heart of mathematics.  In the context of these widely accepted definitions of mathematics, progression in children&#8217;s understanding of mathematics requires that in addition to having some knowledge of mathematical facts, concepts and skills, children are able to put their knowledge to work for the purpose of solving problems.  Engaging in the processes of mathematical activity enables children to assimilate the associated skills and knowledge.</p>
<p class="p1">The role of the teacher can be viewed as crucial for enabling children to understand mathematics.  As pointed out by Orton (1995, pp. 13), considerable emphasis in learning mathematics in recent times has been placed on the desirability of understanding rather than on being able to repeat remembered routines and demonstrate particular basic skills.  Progression then, can be seen as the development of a child&#8217;s ability and confidence to tackle the nature and purpose of mathematics whilst being able to explore and utilise a variety of mental strategies.  There are many factors that facilitate progression, including the features of effective teaching and the extent to which pupils need to be accommodated.</p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2444" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths7-300x225.jpg" alt="maths7" width="600" height="450" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/maths7-300x225.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths7-1024x768.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths7-330x247.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths7-690x517.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths7-1050x787.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths7-773x580.jpg 773w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths7.jpg 2048w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1">For teachers to best support children&#8217;s progression in their learning of mathematics, they must first plan lessons appropriately.  Logically, this planning should be underpinned by an understanding of how children think and learn.  As pointed out by Anghileri (1995, pp. 26), the new understandings which derive from all the current research into the processes by which children think and learn have clear and major implications for teaching mathematics to young children.  Above everything else, the main implication is that it is the teacher&#8217;s responsibility to nurture a positive attitude towards the subject of mathematics.  In my own lessons that I taught, I certainly found that the pupils were much more engaged in learning when they enjoyed the activities that were associated with learning.</p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2438" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths3-300x240.jpg" alt="maths3" width="600" height="481" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/maths3-300x240.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths3-1024x820.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths3-330x264.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths3-690x552.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths3-1050x841.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths3-724x580.jpg 724w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1">For example, I would make regular use of the interactive whiteboard for the purpose of virtual activities, which I found were effective in getting the children actively interested in the topic.  Where possible, I made use of games that the children could play in pairs and groups.  For one particular lesson on fractions, decimal &amp; percentage equivalences, I got the children playing a game of dominoes in groups in which the object was to match the correct equivalences. Interestingly, I tended to find that the use of games was not always conducive to forming a good basis for progression in children&#8217;s understanding, since as much time would be spent on understanding the actual game as it would on gleaning understanding of the mathematics involved.  They might well enjoy the activity of playing a game but the learning is often minimal.  In addition, I would say that <strong>the effectiveness of any game or &#8216;enjoyable activity&#8217; is largely determined by the extent to which it places mathematical problems in meaningful real world contexts</strong>.</p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2437" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths4-300x224.jpg" alt="maths4" width="600" height="448" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/maths4-300x224.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths4-1024x764.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths4-330x246.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths4-690x515.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths4-1050x784.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths4-776x580.jpg 776w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1">Indeed, making mathematics enjoyable should go hand in hand with making mathematics meaningful. This is the means by which anyone makes sense of<br />
anything new with which they are faced, by relating it to what is already known.  As explained by Askew (1992, pp. 26) &#8216;children working in Brazilian street markets demonstrated a range of mathematical skills and knowledge, which they had problems demonstrating in formal tests&#8217;.  Arguably, activities used in classrooms need to mirror the sort of situations to which mathematics is actually applied.  This must mean that this way of human learning is vastly inhibited when one is presented with information or experience that does not relate at all to what is known.  Obviously, this has clear implications for the teaching of progression children&#8217;s understanding of mathematics and leads a skilful teacher to place mathematics into a context in which the children are familiar.</p>
<p class="p1">Research, however, has shown that while practical work and &#8216;real&#8217; contexts can be useful, they need to be chosen carefully, and accompanied careful dialogue with the pupils to establish the extent of their understanding (Askew, 1992, pp. 11).  This has been confirmed by my own experience, as it is easy to spend too much time getting resources to fit a particular theme or context but which can detract from the learning outcome.  In order to get pupils really understanding mathematics, it is commonly agreed that pupils need to be supported in giving verbal explanations where they can talk through what they are doing, since this is an essential step in &#8216;going mental&#8217;.  Anghileri (1995, pp. 20) looks at the importance of children giving verbal explanations in terms of it being conducive to developing their cognitive awareness and control &#8211; that is, their ability to learn how to learn.  This means children should be encouraged to be reflective about their own processing, and to adapt and develop strategies, which put them in control.  In fact, this explains to a large extent the ethos behind the &#8216;mental and oral starter&#8217;. <strong> It is clear across all year groups that mathematical language used by the teacher regularly draws on &#8216;open-targeted&#8217; questions, requiring the children lo explain how they solved a problem.</strong>  Insofar as helping children to learn strategies for understanding and applying mathematics, the mathematical language and mental strategies used is, according to Anghileri, important for enabling children to cope with what is known as structural limitation &#8211; that is, their limited memory capacity.  This is achieved through children developing their selective attention, in which they sort out the relevant from the irrelevant, as well as developing their structural knowledge and processing strategies.  In this way, they learn to chunk information together and improve the efficiency with which they assimilate new information.</p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2439" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths2-300x224.jpg" alt="maths2" width="600" height="448" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/maths2-300x224.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths2-1024x764.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths2-330x246.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths2-690x515.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths2-1050x784.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths2-776x580.jpg 776w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1">The guidance given, as defined in the National Numeracy Strategy Framework in the UK, recommends that teachers should use effective questioning<br />
which allow children &#8216;thinking&#8217; time, encourage explanation of methods and reasoning, and probe reasons for incorrect answers&#8217;.  Many studies have shown<br />
that the style of questioning is related to pupil performance (Dillon, 1985,pp.112).  <strong>According to Dillon, the average differences in achievement of those classes whose teachers used demanding questions were, in some cases, equivalent to as much as one year&#8217;s learning.</strong>  Another significant aspect of a teacher&#8217;s questioning style is the time that the teacher pauses after asking a question, before either supplying it themselves, or moving onto a different pupil (Tokin, 1986, pp. 198).  In terms of progression, I have found that in the lower year groups teachers mainly make use of closed and closed targeted questions, as there is less of an emphasis on the method of working but in the upper year groups more use is made of open and open-targeted questions.  This makes sense because as children progress in their understanding of mathematics there becomes less of a need to recite patterns and more of a need to understand the actual working. Likewise, questioning is differentiated according to ability across a class as well as across year groups such that more open-ended style questioning in addition to the usual extension activities tends to be asked of the more able learners, particularly those who are &#8216;gifted and talented&#8217;.  My own observations have also revealed that simple statements made by the teacher help children to develop mental images, which in turn help them answer questions, e.g. &#8216;put that number in your head because that is the first number we are counting&#8230;&#8217;  <strong>Dillon (1985, pp. 114) suggests that when teachers make statements in order to provoke discussion rather than just ask questions, pupils can display more complex thought, deeper involvement, wider participation, greater interconnectedness, and richer inquiry.</strong></p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2433" src="https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1339-300x224.jpg" alt="IMG_1339" width="600" height="448" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1339-300x224.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1339-1024x764.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1339-330x246.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1339-690x515.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1339-1050x784.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/IMG_1339-776x580.jpg 776w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p>It is in the application of this sort of teaching that common misconceptions can be addressed, exposed and discussed.  From my own lesson observations and in fact my own teaching, I observed many misconceptions being demonstrated by pupils during the course of a lesson.  It seems therefore, that to teach in way that avoids creating any misconceptions is not possible, and that many of these misconceptions will remain hidden unless the teacher makes specific efforts to uncover them.  A teacher&#8217;s response to dealing with a child&#8217;s mathematical error demands skill in diagnostic terms: different responses will be required depending on the nature and frequency of the error observed.  For example, in one particular Year 6 lesson I was teaching, I remember there were several pupils who were over-generalising a pattern that is true for whole numbers but not true for decimal numbers such as in the case of 0.12 &lt; 0.117.  Skill is needed on the part of the teacher to correctly ascertain why the children are making such mistakes so that they can be appropriately resolved.  In the case of this particular lesson on Number, I could see the reason the children were making this mistake and was able to aid them in their understanding by speaking of 0.12 as &#8216;one tenth, two hundredths&#8217; and 0.117 as &#8216;one tenth one hundredth &amp; seven thousandths&#8217;.  The language used therefore, clearly is very important but of course, children still have their own individual learning styles and a teacher needs to be aware of this too in order to teach effectively</p>
<p class="p1">Psychological evidence from as early as the 1960s supports the possibility that individual preferences exist regarding how we like to learn.  In education, learners may sometimes be allocated to one of three types of learning style (Visual, Auditory or Kinaesthetic -VAK) (Kolb, 1984).  <strong>There is a considerable scarcity of quality research to support the value of identifying learning styles (Coffield et al, pp. 42,2004) but to give the notion to benefit of the doubt, one is led to the perspective that presenting learning material in a certain way can improve a pupil&#8217;s learning by suiting his preferred learning style.</strong>  It is for this reason that for the more visual learners in the classroom teachers can make use of a number line so that the children can see visually the place value underpinning numbers.  Likewise, to accommodate kinaesthetic learners number cubes can often be introduced into a lesson, enabling children to tangibly play with and observe the differences between units, tenths and hundredths.  Whatever method is used, <strong>effective teaching should emphasise that there is rarely a single &#8216;right&#8217; method for solving a problem.</strong>  This allows for personalised learning to some extent, as the children can use methods they are most comfortable with.</p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2443" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths9-300x225.jpg" alt="maths9" width="600" height="450" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/maths9-300x225.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths9-1024x768.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths9-330x247.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths9-690x517.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths9-1050x787.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths9-773x580.jpg 773w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths9.jpg 2048w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1">The highest levels of achievement have been found to be in classes where teachers have a good knowledge of pupils&#8217; achievement as well as a wide range of teaching approaches (Peterson et al., 1989, pp. 559).  Herein rests the rationale for assessment, which informs a teacher&#8217;s understanding of where a child is up to in his or her understanding of the subject matter and in so doing is fundamental for children&#8217;s progression.  There are widely recognised differences in the way in which high attainers and low attainers solve problems in mathematics.  <strong>It would seem that successful problem solvers have a range of strategies that they use intensively in problem solving, above and beyond the mathematical content that they know.</strong>  These would include, checking that they have understood the problem, planning their approach and monitoring their progress towards their goal.  Just as competence in early number relies on a blending of &#8216;knowing that&#8217; and &#8216;figuring out&#8217;, improving competence at mathematical problem-solving involves increasing both the fund of mathematical knowledge and a range of general strategic, which have developed together (Askew, 1992, pp. 25).  In the context of the National Curriculum for Mathematics, this would suggest that learning outcomes, which focus on separate attainment targets are less likely to be successful than those that require pupils to integrate ideas from two attainment targets.  In terms of progression, this means that teaching needs to provide activities that focus on the links between different aspects of mathematics rather than treating them as separate topics and this is something I have always strived to achieve m my own teaching.</p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2445" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths8-300x225.jpg" alt="maths8" width="600" height="450" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/maths8-300x225.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths8-1024x768.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths8-330x247.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths8-690x517.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths8-1050x787.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths8-773x580.jpg 773w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths8.jpg 2048w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1">As assessment is able to inform a teacher&#8217;s planning, different children&#8217;s needs are accommodated.  This allows the teacher to adapt teaching according to the individual child and often is accompanied by setting arrangements.  Many schools employ setting for mathematics more than any other subject as an organisational device for coping with the range of competencies within a year group.  The popularity of setting for mathematics reflects recognition by teachers that mathematics spreads pupils out in terms of their competencies more than any subject in the school curriculum.  Furthermore, it is often the case that the lowest set consists of a smaller number of pupils than the others.  Haylock (1992, pp. 10) points out that this itself is recognition of the idiosyncratic nature of low attainment in mathematics and the consequent need for much individual attention by the teacher for those pupils in these lower sets.  <strong>Evidence presented by Allan (1991, pp. 61) states that mathematical attainment groupings can lead to some gains in attainment and although setting can be a contentious issue, there is little evidence to suggest pupils can be harmed by attainment grouping.</strong>  For students with special educational needs (SEN), Stuard (1990,pp. 21) makes the valid point that they should not be put in situations where they fail, but should be given opportunities to display what they can do, without recording or verbalising.  This should help them in their confidence, as they progress to the next level in their understanding of a concept or topic.</p>
<p class="p1">As well as using extra teaching staff to enhance learning objectives, ICT can be extremely effective.  Beardon (2003, pp. 157) refers to examples of how technology as a tool for learning can accommodate different learning styles, since allowing children to use computers as mathematical tools can encourage them to have an input into the direction of inquiries or investigations so that different interests, learning styles and preferences are taken into account.  The evidence presented by Bearden suggests that when the teacher permits real child-centred learning in this way, hidden abilities and creativity can be uncovered and mathematical concept development and thinking often go well beyond the expected outcome.  Additionally, by opening up communications with the wider learning community, the Internet has helped to radically change schools by giving them free access not only to information richer than any school library but also to other schools, expert advice and online resources.</p>
<p class="p1"><img loading="lazy" decoding="async" class="aligncenter wp-image-2442" src="https://technologyforlearners.com/wp-content/uploads/2017/07/maths10-300x224.jpg" alt="maths10" width="600" height="448" srcset="https://technologyforlearners.com/wp-content/uploads/2017/07/maths10-300x224.jpg 300w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths10-1024x764.jpg 1024w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths10-330x246.jpg 330w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths10-690x515.jpg 690w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths10-1050x784.jpg 1050w, https://technologyforlearners.com/wp-content/uploads/2017/07/maths10-776x580.jpg 776w" sizes="(max-width: 600px) 100vw, 600px" /></p>
<p class="p1">In my personal experience of teaching, I frequently used the Internet as a means for obtaining useful ideas, worksheets and resources in which to aid children&#8217;s progression in their understanding of mathematics.  Any mention of technology within the context of mathematics should not overlook the impact of calculators.  <strong>The National Curriculum in the UK requires the use of the calculator in Key Stage 1 and Key Stage 2, as there are many benefits to the use of calculators.</strong>  For example, children of 6 or 7 can explore the &#8216;constant&#8217; facility of the calculator, and derive patterns by counting on in twos or threes.  Stuard (1990, pp. 10) also makes the point that the use of calculates encourages children to work with large numbers, negative numbers and decimals, and to discuss these ideas with other children and adults.  I remember in a Year 5 lesson for example, giving pupils the chance to learn about the calculator&#8217;s memory function, which implicitly is highly useful for providing an effective means for enabling children to cope with the notion of &#8216;structural limitation&#8217;, discussed earlier. Calculators therefore provide a fundamental tool with which to both encourage and help children develop in their mathematical understanding.</p>
<p>Once resources such as ICT, calculators and teaching staff are appropriately in place, considerable effort is required by the teacher to then structure the class into groups, which will work effectively together.  This is because many studies have shown that pupils may be working in groups but not working as a group (Askew, 1992, pp.&#8217;36), which has serious implications for children&#8217;s progression.  Research presented by Webb (1984, pp. 34) suggests that the optimum grouping would comprise of &#8216;near&#8217; mixed ability groups such as high and middle attainers or middle and low attainers as well as mixed sex groups, which include balanced numbers of boys and girls.  Particular focus on grouping arrangements needs to be made if there are children with English as an additional language (EAL) in the classroom.  This is because it is often the case according to Askew (1992, Pp. 61) that poor language skills &#8211; reading, writing, and speaking &#8211; are associated with low attainment in mathematics.  Given the point made by Brown (1989, pp. 40) that through carrying out work in a social setting, group work promotes learning, it is easy to see how children with EAL would benefit from working in mixed groups of native English speaking children.</p>
<p class="p1"><strong>Concluding thoughts&#8230;</strong></p>
<p class="p1">To be involved in mathematics, at whatever level, requires both content and process, in order for the subject to be experienced in a balanced way.  For<br />
children&#8217;s progression in their understanding of mathematics to be realised therefore, effective teaching is needed, which accommodates pupils with different needs and abilities.  This requires good planning, making use of exercises and activities children enjoy.  Enjoyment is most likely to be achieved if mathematics is placed in &#8216;real&#8217; contexts that the children would have come across in their everyday experience and in this respect, it is good discipline for teachers of mathematics to evaluate tasks given to pupils against the criterion of usefulness.  However, a productive dialogue between the teacher and pupils in which the teacher uses statements and questions, which, as children progress, increasingly requires them to verbalise understanding of their working, should underpin this criteria.  It is in this way that misconceptions can be more frequently addressed, enabling children to progress in their conceptual development.  Both formative and summative assessment data should be used to record these misconceptions and inform planning in terms of where children are up to in their understanding of a topic or concept.  Teaching can then be appropriately differentiated to offer more personalised learning that meets a diverse range of needs and abilities, including&#8217; children with SEN and children with EAL as well as Gifted &amp; Talented children.  Careful attention needs to be paid by the teacher to the grouping arrangements to allow for maximum learning, since it is by such processes of social interaction and dialogue with more experienced learners that children learn to be reflective about their own processing and so begin to learn how to learn.  Likewise, effective uses of resources, including technology and teaching assistants is important for helping children further develop this meta-awareness and control, which ultimately, is fundamental in enabling children to progress in becoming increasingly independent learners.</p>
<p class="p1"><strong>References</strong></p>
<p class="p1">Askew, M &amp; William, D. (1995). Recent Research in Mathematics Education 5-16. OFSTED Reviews of Research.<br />
Atkinson, S. (1992). Mathematics with Reason. Hodder &amp; Stoughton.<br />
Anghileri, J. (1995). Children&#8217;s Mathematical Thinking in the Primary Years. Cassell.<br />
Brown, J. &amp; Staines, J. (1989). Mathematics and the Primary Curriculum. NCET.<br />
Campione, J. C.; Brown, A. L. &amp; Connell, M. L. (1988). Metacognition: on the importance of understanding what you are doing. In R. I. Charles &amp; E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving: pp. 93-114.<br />
Cockbum, A. (1999). Teaching Mathematics with Insight. Palmer Press.<br />
Coffield et al, (2004) Learning styles and pedagogy in post-16 learning<br />
Dillon, J T (1985). Using questions to foil discussion. Teaching and Teacher Education, 1 (2), pp. 109-121.<br />
Hansen, A. (2005). Children&#8217;s Errors in Mathematics. Learning Matters.<br />
Haylock, D (1991). Teaching Mathematics to Low Attainers, 8-12. Paul Chapman Publishing Ltd.<br />
Hembree, R., Experiments and relational studies in problem solving: a meta-analysis. Journal for Research in Mathematics Education, 1992. 23(3): pp. 242-273.<br />
Hughes, M (1986). Children and number: difficulties in learning mathematics. Oxford: Basil Blackwell.<br />
Kolb, D.A. (1984), Expert ential Learning: Experience as the Source of Learning and Development<br />
Orton, A., Frobisher, L. (1996). Insights into Teaching Mathematics. Cassell.<br />
Peterson, P L; Carpenter, T &amp; Fennema, E (1989). Teachers knowledge of students&#8217; knowledge of mathematics problem solving: correlation and case analyses. Journal of Educational Psychology, 81(4), pp. 558-569.<br />
Shuard, H., Walsh, A., Goodwin, J., Worcester, V. (1990). Children, Mathematics and Learning. Simon &amp; Schuster.<br />
Street, B., Baker, D., Tomlin, A. (2005). Navigating Numeracies. Springer.<br />
TokJn, K (1986). Effects of teacher wait time on discourse characteristics in mathematics and art classes. American Educational Research Journal, 23(2), pp. 191-200.<br />
Way, J. and Beardon, T. (2003) tICT as a Tool for Learning &#8211; Where are we going?&#8217;, in Way, J. and<br />
Beardon, T. (Eds.) ICT and Primary Mathematics. OU Press.<br />
Webb, N. M. (1991). Task-related verbal interaction and mathematics learning in small groups.<br />
Journal for Research in Mathematics Education, 22(5): pp. 366-389.</p>
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