tag:theconversation.com,2011:/us/topics/arithmetic-19819/articlesArithmetic – The Conversation2024-01-03T20:26:58Ztag:theconversation.com,2011:article/2193202024-01-03T20:26:58Z2024-01-03T20:26:58ZAI is our ‘Promethean fire’: using it wisely means knowing its true nature – and our own minds<p>Future historians may well regard 2023 as a landmark in the advent of artificial intelligence (AI). But whether that future will prove <a href="https://a16z.com/ai-will-save-the-world/">utopian</a>, <a href="https://www.toolify.ai/ai-news/ais-apocalyptic-vision-24116">apocalyptic</a> or <a href="https://www.mckinsey.com/featured-insights/mckinsey-explainers/whats-the-future-of-generative-ai-an-early-view-in-15-charts">somewhere in between</a> is anyone’s guess. </p>
<p>In February, ChatGPT set the record as the fastest app to reach <a href="https://www.reuters.com/technology/chatgpt-sets-record-fastest-growing-user-base-analyst-note-2023-02-01/">100 million users</a>. It was followed by similar “large language” AI models from Google, Amazon, Meta and other big tech firms, which collectively look poised to transform education, healthcare and many other knowledge-intensive fields. </p>
<p>However, AI’s potential for harm was underscored in May by an <a href="https://www.safe.ai/statement-on-ai-risk">ominous statement</a> signed by leading researchers: </p>
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<p>Mitigating the risk of extinction from AI should be a global priority alongside other societal-scale risks such as pandemics and nuclear war. </p>
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<p>In November, responding to the growing concern about AI risk, 27 nations (including the UK, US, India, China and the European Union) pledged cooperation at an inaugural AI Safety Summit at Bletchley Park in England, to ensure the safe development of AI for the <a href="https://www.gov.uk/government/publications/ai-safety-summit-2023-the-bletchley-declaration/the-bletchley-declaration-by-countries-attending-the-ai-safety-summit-1-2-november-2023">benefit of all</a>. </p>
<p>To achieve this, researchers focus on <a href="https://en.wikipedia.org/wiki/AI_alignment">AI alignment</a> – that is, how to make sure AI models are consistent with human values, preferences and goals. But there’s a problem – AI’s so-called “<a href="https://www.technologyreview.com/2017/04/11/5113/the-dark-secret-at-the-heart-of-ai/">dark secret</a>”: large-scale models are so complex they are like a black box, impossible for anyone to fully understand. </p>
<h2>AI’s black box problem</h2>
<p>Although the transparency and explainability of AI systems are <a href="https://www.sciencedirect.com/science/article/pii/S1566253519308103?casa_token=eMCns9rVBmoAAAAA:ZozMhIZEA-Sd4IWnBBWRC6KmXV3THV4lqMYkWKf8-NrwaTxEKHqU2EAw4B-RZP0sCg0wazbml3o">important research goals</a>, such efforts seem unlikely to keep up with the frenetic pace of innovation. </p>
<p>The black box metaphor explains why people’s beliefs about AI are all over the map. Predictions range from utopia to extinction, and many even believe an artificial general intelligence (AGI) will soon <a href="https://www.science.org/content/article/if-ai-becomes-conscious-how-will-we-know">achieve sentience</a>. </p>
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Read more:
<a href="https://theconversation.com/a-year-of-chatgpt-5-ways-the-ai-marvel-has-changed-the-world-218805">A year of ChatGPT: 5 ways the AI marvel has changed the world</a>
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<p>But this uncertainty compounds the problem. AI alignment should be a two-way street: we must not only ensure AI models are consistent with human intentions, but also that our beliefs about AI are accurate. </p>
<p>This is because we are remarkably adept at creating futures that accord with those beliefs, even if we are unaware of them. </p>
<p>So-called “<a href="https://journals.sagepub.com/doi/abs/10.1111/1467-8721.ep10770698">expectancy effects</a>”, or self-fulfilling prophecies, are well known in psychology. And research has shown that manipulating users’ beliefs influences not just how they <a href="https://dl.acm.org/doi/pdf/10.1145/3529225">interact with AI</a>, but how AI <a href="https://dspace.mit.edu/bitstream/handle/1721.1/152316/NMI_AI_beholder_Final-Unformatted%5B85%5D.pdf?sequence=1&isAllowed=y">adapts to the user</a>. </p>
<p>In other words, how our beliefs (conscious or unconscious) affect AI can potentially increase the likelihood of any outcome, including catastrophic ones. </p>
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Read more:
<a href="https://theconversation.com/how-ai-sees-the-world-what-happened-when-we-trained-a-deep-learning-model-to-identify-poverty-217586">How AI 'sees' the world – what happened when we trained a deep learning model to identify poverty</a>
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<h2>AI, computation, logic and arithmetic</h2>
<p>We need to probe more deeply to understand the basis of AI – like Alice in Wonderland, head down the rabbit hole and see where it takes us. </p>
<p>Firstly, what is AI? It runs on computers, and so is automated computation. From its origin as the “<a href="https://idp.springer.com/authorize/casa?redirect_uri=https://link.springer.com/content/pdf/10.1007/BF02478259.pdf&casa_token=Joxu9OnlEd4AAAAA:uCl-_FASTbvXSWBtZAt5bS24ZSRvOMsdufe1PG6PXY1TSdNoU0gL8a5j6I7lGmk4rqrSCbIqE0CQoxd9BnA">perceptron</a>” – an artificial neuron defined mathematically in 1943 by neurophysiologist <a href="https://en.wikipedia.org/wiki/Warren_Sturgis_McCulloch">Warren McCulloch</a> and logician <a href="https://en.wikipedia.org/wiki/Walter_Pitts">Walter Pitts</a> – AI has been intertwined with the cognitive sciences, neuroscience and computer science. </p>
<p>This convergence of <a href="https://www.cs.cmu.edu/afs/cs/academic/class/15883-f21/readings/churchland-1992-ch3.pdf">minds</a>, <a href="https://www.cell.com/fulltext/S0896-6273(17)30509-3">brains</a> and <a href="https://www.nature.com/articles/s41583-020-00395-8">machines</a> has led to the widely-held belief that, because AI is computation by machine, then natural intelligence (the mind) must be computation by the brain.</p>
<p>But what is computation? In the late 19th century, mathematicians <a href="https://en.wikipedia.org/wiki/Richard_Dedekind">Richard Dedekind</a> and <a href="https://en.wikipedia.org/wiki/Giuseppe_Peano">Giuseppe Peano</a> proposed a set of axioms which <a href="https://en.wikipedia.org/wiki/Peano_axioms">defined arithmetic in terms of logic</a>, and inspired attempts to ground all mathematics on a secure <a href="https://en.wikipedia.org/wiki/Hilbert%27s_program">formal basis</a>. </p>
<p>Although the logician <a href="https://en.wikipedia.org/wiki/Kurt_G%C3%B6del">Kurt Gödel</a> later proved this goal was <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">unachievable</a>, his work was the starting point for mathematician (and code-breaker) <a href="https://en.wikipedia.org/wiki/Alan_Turing">Alan Turing</a>. His “<a href="https://en.wikipedia.org/wiki/Turing_machine">Turing machine</a>”, an abstract device capable of <a href="https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis">universal computation</a>, is the foundation of computer science. </p>
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Read more:
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<h2>Deep structure of perception</h2>
<p>So, computation is based on mathematical ideas that trace back to efforts to define arithmetic in logic. But our knowledge of arithmetic exists <a href="https://academic.oup.com/pq/article-abstract/68/273/717/4969397?redirectedFrom=PDF&casa_token=Z-7sIkFvtL0AAAAA:UOuTtFoVh9mpu6guxajdbe44O93oCe6PANK-Uz9yWL_0iX8lo-Lla-pPatTGINKxrAqB-MBpCtmts4cz">prior to logic</a>. If we want to understand the basis of AI, we need to go further and ask where arithmetic itself comes from. </p>
<p>My colleagues and I have recently shown that arithmetic is based on the “<a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">deep structure</a>” of perception. This structure is like coloured glasses that shape our perception in particular ways, so that our experience of the world is ordered and manageable. </p>
<p>Arithmetic consists of a set of elements (numbers) and operations (addition, multiplication) that combine pairs of elements to give another element. We asked: of all possibilities, why are numbers the elements, and addition and multiplication the operations? </p>
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Read more:
<a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">Arithmetic has a biological origin – it's an expression in symbols of the 'deep structure' of our perception</a>
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<p>We showed by <a href="https://psycnet.apa.org/fulltext/2023-84614-001.pdf?sr=1">mathematical proof</a> that when the deep structure of perception was assumed to limit the possibilities, arithmetic was the result. In other words, when our mind views the abstract world through the same “coloured glasses” that shape our experience of the physical world, it “sees” numbers and arithmetic. </p>
<p>Because arithmetic is the foundation for mathematics, the implication is that mathematics is a reflection of the mind – an expression in symbols of its fundamental nature and creativity. </p>
<p>Although the deep structure of perception is shared with other animals and so a product of evolution, only humans have invented mathematics. It is our most intimate creation – and by enabling the development of AI, perhaps our most consequential. </p>
<h2>A Copernican revolution of the mind</h2>
<p>Our account of <a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">arithmetic’s origin</a> is consistent with views of the 18th century philosopher Immanuel Kant. According to him, our knowledge of the world is structured by “pure intuitions” of space and time that exist prior to sense experience – analogous to the coloured glasses we can never remove.</p>
<p>Kant claimed his <a href="https://plato.stanford.edu/entries/kant/?rid=903123293s840c38">philosophy</a> was a “Copernican revolution of the mind”. In the same way ancient astronomers believed the Sun revolved around the Earth because they were unaware of the Earth’s motion, Kant argued, philosophers who believed all knowledge is derived from <a href="https://en.wikipedia.org/wiki/Empiricism">sense experience</a> (John Locke and David Hume, for example) overlooked how the mind shapes perception. </p>
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<p>Although Kant’s views were shaped by the <a href="https://books.google.com/books?hl=en&lr=&id=cBtOFUg4tHAC&oi=fnd&pg=PR11&dq=Kant+and+the+exact+sciences&ots=2Uu_qCDZT-&sig=WKqrfULN9w5qL6TfVc63PEyP5RQ">natural sciences of his day</a>, they have proved <a href="https://www.cell.com/trends/cognitive-sciences/fulltext/S1364-6613(10)00216-0">influential in contemporary psychology</a>. </p>
<p>The recognition that arithmetic is a <a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">natural consequence of our perception</a>, and thus biologically based, suggests a similar Kantian shift in our understanding of computation.</p>
<p>Computation is not “outside” or separate from us in an abstract realm of mathematical truth, but inherent in our mind’s nature. The mind is more than computation; the brain is not a computer. Rather, computation – the basis for AI – is, like mathematics, a symbolic expression of the mind’s nature and creativity. </p>
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<h2>Promethean fire</h2>
<p>What are the implications for AI? Firstly, AI is not a mind and will never become sentient. The idea we can transcend our biological nature and achieve immortality by uploading our minds to the cloud is only <a href="https://medium.com/iva-to/on-achieving-immortality-3ed1d567f7a2">fantasy</a>. </p>
<p>Yet if the principles of mind on which AI is based are shared by all humanity (and likely other living creatures as well), it may be possible to transcend the limitations of our individual minds.</p>
<p>Because computation is universal, we are free to simulate and create any outcome we choose in our increasingly connected virtual and physical worlds. In this way, AI is truly our <a href="https://www.japantimes.co.jp/editorials/2023/11/10/ai-global-governance/">Promethean fire</a>, a gift to humanity stolen from the gods as in <a href="https://en.wikipedia.org/wiki/Prometheus">Greek mythology</a>. </p>
<p>As a global civilisation, we are likely at a turning point. AI will not become sentient and decide to <a href="https://newatlas.com/technology/ai-danger-kill-everyone/">kill us all</a>. But we are very capable of “apocalypsing” ourselves with it – expectation can create reality. </p>
<p>Efforts to ensure AI alignment, safety and security are vitally important, but may not be enough if we lack awareness and collective wisdom. Like Alice, we need to wake up from the dream and recognise the reality and power of our minds.</p><img src="https://counter.theconversation.com/content/219320/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Randolph Grace receives funding from the Royal Society Te Apārangi, Marsden Fund</span></em></p>AI will not become sentient and decide to kill us all. But our own conscious or unconscious beliefs about AI can potentially increase the likelihood of any outcome, including catastrophic ones.Randolph Grace, Professor of Psychology, University of CanterburyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2113372023-08-15T00:32:01Z2023-08-15T00:32:01ZArithmetic has a biological origin – it’s an expression in symbols of the ‘deep structure’ of our perception<p>Everyone knows that arithmetic is true: 2 + 2 = 4. </p>
<p>But surprisingly, we don’t know <em>why</em> it’s true. </p>
<p>By stepping outside the box of our usual way of thinking about numbers, my colleagues and I have recently shown that <a href="https://psycnet.apa.org/fulltext/2023-84614-001.pdf?sr=1">arithmetic has biological roots</a> and is a natural consequence of how perception of the world around us is organised. </p>
<p>Our results explain why arithmetic is true and suggest that mathematics is a realisation in symbols of the fundamental nature and creativity of the mind. </p>
<p>Thus, the miraculous correspondence between mathematics and physical reality that has been a source of wonder from the ancient Greeks to the present — as explored in astrophysicist Mario Livio’s book <a href="https://www.amazon.com/God-Mathematician-Mario-Livio-ebook/dp/B004NNVFW2/ref=sr_1_1?crid=2RR9V2WHD5FK3&keywords=is+god+a+mathematician+by+mario+livio&qid=1691995780&sprefix=Livio+is+god+a+ma%2Caps%2C283&sr=8-1">Is God a mathematician?</a> — suggests the mind and world are part of a common unity.</p>
<h2>Why is arithmetic universally true?</h2>
<p>Humans have been making symbols for numbers for more than 5,500 years. More than 100 distinct notation systems are known to have been used by <a href="https://www.google.co.nz/books/edition/Numerical_Notation/kXZhBAAAQBAJ?hl=en&gbpv=1&dq=Chrisomalis+2010+numeral&pg=PP2&printsec=frontcover">different civilisations</a>, including Babylonian, Egyptian, Etruscan, Mayan and Khmer. </p>
<figure class="align-center ">
<img alt="Several examples of number symbols from different cultures." src="https://images.theconversation.com/files/542447/original/file-20230812-82741-w6lpga.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/542447/original/file-20230812-82741-w6lpga.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=393&fit=crop&dpr=1 600w, https://images.theconversation.com/files/542447/original/file-20230812-82741-w6lpga.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=393&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/542447/original/file-20230812-82741-w6lpga.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=393&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/542447/original/file-20230812-82741-w6lpga.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=494&fit=crop&dpr=1 754w, https://images.theconversation.com/files/542447/original/file-20230812-82741-w6lpga.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=494&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/542447/original/file-20230812-82741-w6lpga.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=494&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">Different cultures have developed their own symbols for numbers, but they all use addition and multiplication.</span>
<span class="attribution"><span class="source">Wikimedia Commons</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>The remarkable fact is that despite the great diversity of symbols and cultures, all are based on addition and multiplication. For example, in our familiar Hindu-Arabic numerals: 1,434 = (1 x 1000) + (4 x 100) + (3 x 10) + (4 x 1). </p>
<p>Why have humans invented the same arithmetic, over and over again? Could arithmetic be a universal truth waiting to be discovered?</p>
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Read more:
<a href="https://theconversation.com/pythagoras-revenge-humans-didnt-invent-mathematics-its-what-the-world-is-made-of-172034">Pythagoras’ revenge: humans didn’t invent mathematics, it’s what the world is made of</a>
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<p>To unravel the mystery, we need to ask why addition and multiplication are its fundamental operations. We recently <a href="https://psycnet.apa.org/fulltext/2023-84614-001.pdf?sr=1">posed this question</a> and found that no satisfactory answer – one that met standards of scientific rigour – was available from philosophy, mathematics or the cognitive sciences.</p>
<p>The fact that we don’t know why arithmetic is true is a critical gap in our knowledge. Arithmetic is the foundation for higher mathematics, which is indispensable for science.</p>
<p>Consider a thought experiment. Physicists in the future have achieved the goal of a “theory of everything” or “<a href="https://www.amazon.com/God-Equation-Quest-Theory-Everything-ebook/dp/B08HRFG7MQ/ref=sr_1_1?crid=RILBTCF8S4IL&keywords=the+god+equation+by+michio+kaku&qid=1691881661&sprefix=Kaku+the+god+equation%2Caps%2C282&sr=8-1">God equation</a>”. Even if such a theory could correctly predict all physical phenomena in the universe, it would not be able to explain where arithmetic itself comes from or why it is universally true. </p>
<p>Answering these questions is necessary for us to fully understand the role of mathematics in science. </p>
<h2>Bees provide a clue</h2>
<p>We proposed a new approach based on the assumption that arithmetic has a biological origin. </p>
<p>Many non-human species, including insects, show an ability for spatial navigation which seems to require the <a href="https://psycnet.apa.org/record/2019-63660-001">equivalent of algebraic computation</a>. For example, bees can take a meandering journey to find nectar but then return by the most direct route, as if they can <a href="https://link.springer.com/article/10.1007/s00359-015-1000-0">calculate the direction and distance home</a>. </p>
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<img alt="A graph that shows a bee's zig-zag flight and the direct route home." src="https://images.theconversation.com/files/542448/original/file-20230812-23-izr2cf.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/542448/original/file-20230812-23-izr2cf.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=628&fit=crop&dpr=1 600w, https://images.theconversation.com/files/542448/original/file-20230812-23-izr2cf.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=628&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/542448/original/file-20230812-23-izr2cf.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=628&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/542448/original/file-20230812-23-izr2cf.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=789&fit=crop&dpr=1 754w, https://images.theconversation.com/files/542448/original/file-20230812-23-izr2cf.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=789&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/542448/original/file-20230812-23-izr2cf.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=789&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">Bees can integrate their zig-zag flight path to calculate the straightest route back to the hive.</span>
<span class="attribution"><span class="source">Nicola J. Morton</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>How their miniature brain (about 960,000 neurons) achieves this is unknown. These calculations might be the non-symbolic precursors of addition and multiplication, honed by natural selection as the optimal solution for navigation. </p>
<p>Arithmetic may be based on biology and special in some way because of evolution’s fine-tuning.</p>
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Read more:
<a href="https://theconversation.com/bees-learn-better-when-they-can-explore-humans-might-work-the-same-way-129439">Bees learn better when they can explore. Humans might work the same way</a>
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<h2>Stepping outside the box</h2>
<p>To probe more deeply into arithmetic, we need to go beyond our habitual, concrete understanding and think in more general and abstract terms. Arithmetic consists of a set of elements and operations that combine two elements to give another element. </p>
<p>In the universe of possibilities, why are the elements represented as numbers and the operations as addition and multiplication? This is a meta-mathematical question – a question about mathematics itself that can be addressed using mathematical methods. </p>
<p>In our <a href="https://psycnet.apa.org/fulltext/2023-84614-001.pdf?sr=1">research</a>, we proved that four assumptions – monotonicity, convexity, continuity and isomorphism – were sufficient to uniquely identify arithmetic (addition and multiplication over the real numbers) from the universe of possibilities. </p>
<ul>
<li><p><strong>Monotonicity</strong> is the intuition of “order preserving” and helps us keep track of our place in the world, so that when we approach an object it looms larger but smaller when we move away. </p></li>
<li><p><strong>Convexity</strong> is grounded in intuitions of “betweenness”. For example, the four corners of a football pitch define the playing field even without boundary lines connecting them. </p></li>
<li><p><strong>Continuity</strong> describes the smoothness with which objects seem to move in space and time. </p></li>
<li><p><strong>Isomorphism</strong> is the idea of sameness or analogy. It’s what allows us to recognise that a cat is more similar to a dog than to a rock.</p></li>
</ul>
<p>Thus, arithmetic is special because it is a consequence of these purely qualitative conditions. We argue that these conditions are principles of perceptual organisation that shape how we and other animals experience the world – a kind of “deep structure” in perception with roots in evolutionary history. </p>
<p>In our proof, they act as constraints to eliminate all possibilities except arithmetic – a bit like how a sculptor’s work reveals a statue hidden in a block of stone.</p>
<h2>What is mathematics?</h2>
<p>Taken together, these four principles structure our perception of the world so that our experience is ordered and cognitively manageable. They are like coloured spectacles that shape and constrain our experience in particular ways. </p>
<p>When we peer through these spectacles at the abstract universe of possibilities, we “see” numbers and arithmetic.</p>
<figure class="align-center ">
<img alt="A figure illustrating the four principles of monotonicity, convexity, continuity and isomorphism." src="https://images.theconversation.com/files/542450/original/file-20230812-148555-g5255y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/542450/original/file-20230812-148555-g5255y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=396&fit=crop&dpr=1 600w, https://images.theconversation.com/files/542450/original/file-20230812-148555-g5255y.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=396&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/542450/original/file-20230812-148555-g5255y.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=396&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/542450/original/file-20230812-148555-g5255y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=497&fit=crop&dpr=1 754w, https://images.theconversation.com/files/542450/original/file-20230812-148555-g5255y.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=497&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/542450/original/file-20230812-148555-g5255y.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=497&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">These four principles structure our perception of the world and, collectively, point to arithmetic as an abstract symbol system that reflects that structure.</span>
<span class="attribution"><span class="source">Psychological Review</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Thus, our results show that arithmetic is biologically-based and a natural consequence of how our perception is structured. </p>
<p>Although this structure is shared with other animals, only humans have invented mathematics. It is humanity’s most intimate creation, a realisation in symbols of the fundamental nature and creativity of the mind. </p>
<p>In this sense, mathematics is both invented (uniquely human) and discovered (biologically-based). The seemingly miraculous <a href="https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences">success of mathematics in the physical sciences</a> hints that our mind and the world are not separate, but part of a common unity. </p>
<p>The arc of mathematics and science points toward non-dualism, a philosophical concept that describes how the mind and the universe as a whole are connected, and that any sense of separation is an illusion. This is consistent with many spiritual traditions (Taoism, Buddhism) and Indigenous knowledge systems such as mātauranga Māori.</p><img src="https://counter.theconversation.com/content/211337/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Randolph Grace receives funding from the Marsden Fund, administered by the Royal Society of New Zealand.</span></em></p>Humans have been making symbols for numbers for thousands of years. Different cultures developed their own symbols, but all use addition and multiplication, suggesting arithmetic is a universal truth.Randolph Grace, Professor of Psychology, University of CanterburyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1502622020-12-16T03:32:42Z2020-12-16T03:32:42ZJump, split or make to the next 10: strategies to teach maths have changed since you were at school<figure><img src="https://images.theconversation.com/files/375248/original/file-20201215-15-1lh0rut.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/portrait-african-girl-writing-solution-sums-1078335890">Shutterstock</a></span></figcaption></figure><p>I’m sure most people can remember trying to master a certain maths rule or procedure in primary or secondary school.</p>
<p>My elderly mother has a story about a time her father was helping her with arithmetic homework. She remembers getting upset because her father did not do it “the school way”. I suspect her father was able to do the calculation mentally rather than the school way, which was to use the vertical algorithm.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=166&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=166&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=166&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=208&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=208&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=208&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Students are expected to add the numbers in the ones (right) column first, before adding the numbers in the tens (left) column. The task becomes more difficult when the total of the ones column is more than 10 — as you then have to “trade” ten ones for one ten.</p>
<p>Students who give the answer as 713 rather than the correct answer of 83 may well have started with the tens column first. Or they may have written 13 in the ones column rather than trading ten ones for one ten.</p>
<p>The formal school algorithms are still used for larger numbers and decimals but we encourage students to use whichever strategy they prefer for two-digit addition. </p>
<p>The trouble with teaching rules is many students then struggle to remember when to apply the rule because they don’t understand how or why the rule works. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/weapons-of-maths-destruction-are-calculators-killing-our-ability-to-work-it-out-in-our-head-44900">Weapons of maths destruction: are calculators killing our ability to work it out in our head?</a>
</strong>
</em>
</p>
<hr>
<p>The <a href="https://australiancurriculum.edu.au/f-10-curriculum/mathematics/">Australian Curriculum: Mathematics</a> states that by the end of year 2, students will “perform simple addition and subtraction calculations using a range of strategies”. By the end of year 4, they will “identify and explain strategies for finding unknown quantities in number sentences”. </p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1143666051793788928"}"></div></p>
<p>We want children to remember how to do these equations in their head, rather than relying on writing down the process. Here are three strategies schools use to teach children how to add and subtract two-digit numbers.</p>
<h2>1. Split strategy</h2>
<p>This is sometimes called the decomposition, partitioning or partial-sums strategy. </p>
<p>You can add or subtract the tens separately to the ones (or units). For example, using the split strategy to add 46 + 23, you would:</p>
<ul>
<li><p>split each number (decompose) into tens and ones: 46 + 23 = 40 + 6 + 20 + 3</p></li>
<li><p>rearrange the tens and ones: 40 + 20 + 6 + 3</p></li>
<li><p>add the tens and then the ones 60 + 9 = 69</p></li>
</ul>
<p>Using the split strategy for addition such as 37 + 65 would be similar, but there would be an extra step:</p>
<ul>
<li><p>split or decompose the numbers into tens and ones: 30 + 7 + 60 + 5</p></li>
<li><p>rearrange the tens and ones: 30 + 60 + 7 + 5</p></li>
<li><p>add the tens and then the ones: 90 + 12 </p></li>
<li><p>split 12 (10 + 2) to give: 90 + 10 + 2 = 100 + 2 = 102</p></li>
</ul>
<p>Many students find the split strategy more difficult for subtraction than addition. This is because there are more steps if performing this strategy mentally. </p>
<p>For a subtraction such as 69 – 46, you would:</p>
<ul>
<li><p>split or decompose each number into tens and ones: 60 + 9 – (40 + 6)</p></li>
<li><p>remove bracket: 60 + 9 – 40 – 6</p></li>
<li><p>rearrange tens and ones: (60 – 40) + (9 – 6)</p></li>
<li><p>subtract the tens, then the ones: 20 + 3 = 23</p></li>
</ul>
<p>Students often make mistakes in the third step. Successful students may say: “I take 40 from 60, then 6 from 9”. Unsuccessful students will say “I take 40 from 60 then add 6 and 9”. </p>
<p>Students who use this strategy successfully are showing they understand place value (the value of each digit in a number) and their knowledge of maths rules needed for algebra. </p>
<h2>2. Jump strategy</h2>
<p>This is sometimes called the sequencing or cumulative sums strategy. The actual steps taken depend on the confidence and ability of the students. </p>
<p>Some students add increments of tens or ones, while others add or subtract multiples of tens then ones. </p>
<p>For example, adding 46 + 23 using the jump strategy might look like this:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=158&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=158&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=158&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=199&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=199&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=199&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<ul>
<li><p>add two lots of ten to 46: 46 + 10 = 56, then 56 + 10 = 66</p></li>
<li><p>add the remaining 3: 66 + 3 = 69 </p></li>
</ul>
<p>or </p>
<ul>
<li><p>add 20 to 46 which becomes 66</p></li>
<li><p>add the remaining 3: 66 + 3 = 69</p></li>
</ul>
<p>The two versions of this strategy can be shown using an empty number line. Using a blank or empty number line allows student to record their thinking and for teachers to analyse their thinking and determine the strategy they have attempted to use.</p>
<p>Subtracting 69 – 46 with the jump strategy could be done by:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=149&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=149&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=149&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=187&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=187&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=187&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<ul>
<li><p>subtracting four lots of ten (40) from 69: 69 – 10 = 59; 59 – 10 = 49; 49 – 10 = 39; 39 – 10 = 29 </p></li>
<li><p>then finally subtracting the remaining 6: 29 – 6 = 23 </p></li>
</ul>
<p>or</p>
<ul>
<li><p>subtract 40: 69 – 40 = 29 </p></li>
<li><p>then subtract 6: 29 – 6 = 23</p></li>
</ul>
<h2>3. ‘Make to the next ten’ strategy</h2>
<p>This is sometimes called the compensation or shortcut strategy. It involves adjusting one number to make the task easier to solve. </p>
<p>The “make to the next ten” strategy builds on the “friends of ten” strategy. </p>
<p>Many students in the first years of primary school create all the combinations of two single digit numbers that give a total of ten. </p>
<blockquote>
<p>9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5 … </p>
</blockquote>
<p>These are sometimes called the rainbow facts as the children create rainbows as they connect two numbers together. For instance, 9 may be on one end of a rainbow colour and 1 on the other. </p>
<p>By combining the numbers in this way teachers hope students will realise the answer for 9 + 1 is the same as 1 + 9.</p>
<p>In the “make to the next ten” strategy, you add or subtract a number larger than the number given (such as the next multiple of ten) and then readjust the number by subtracting what was added or adding what was subtracted.</p>
<p>In the diagrams the relationships are indicated by the use of arrows. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=156&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=156&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=156&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=196&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=196&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=196&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>So, to add 37 + 65, you would</p>
<ul>
<li><p>add 3 to 37 to give 40. </p></li>
<li><p>subtract 3 from 65 to get 62</p></li>
<li><p>this becomes: 40 + 62 = 102.</p></li>
</ul>
<p>If subtracting 102 – 65, you would:</p>
<ul>
<li><p>subtract 2 from 102 to make 100</p></li>
<li><p>subtract 2 from 65 to maintain the balance</p></li>
<li><p>this becomes 100 – 63 = 37.</p></li>
</ul>
<p>Many students using this strategy incorrectly add 2 to 65 instead of subtracting 2.</p>
<h2>Why these strategies?</h2>
<p>Students would have been using all these strategies, or some forms of them, in their head for generations. But for many years, the expectation was that students use the formal written algorithm rather than their own mental strategies. </p>
<p>The introduction of the empty or blank number line allowed students to record their mental strategies, which allowed teachers and parents to see them. Naming these strategies has allowed teachers and students to discuss possible strategies using a common vocabulary.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/kids-prefer-maths-when-you-let-them-figure-out-the-answer-for-themselves-44016">Kids prefer maths when you let them figure out the answer for themselves</a>
</strong>
</em>
</p>
<hr>
<p>Rather than teach rules and procedures, we now need to encourage students to explain their strategies using both concrete materials and diagrams to demonstrate their knowledge of addition and subtraction.</p><img src="https://counter.theconversation.com/content/150262/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Cath Pearn is affiliated with both the Australian Council for Educational Research and the Melbourne Graduate School of Education, The University of Melbourne. </span></em></p>For years you may have been adding and subtracting numbers in your head in a certain way, but these strategies were never formally taught at school. Now they are, and they all have names.Cath Pearn, Senior Research Fellow, Australian Council for Educational ResearchLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1351142020-05-05T13:43:05Z2020-05-05T13:43:05Z4 things we’ve learned about math success that might surprise parents<figure><img src="https://images.theconversation.com/files/332066/original/file-20200501-42942-go7fsv.jpg?ixlib=rb-1.1.0&rect=47%2C228%2C5272%2C3133&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The good news: your child can use their fingers and you can too. </span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p>School <a href="https://www.theguardian.com/world/2020/apr/30/coronavirus-scientists-caution-against-reopening-schools">closures due to conronavirus</a> have put parents in the challenging position of home-schooling their children.</p>
<p>In mathematics education programs for future math teachers, we often discuss the <a href="https://link.springer.com/article/10.1007/s10857-012-9208-1">traditional classroom</a> that those studying to become teachers are familiar with. We’re interested in how their own experiences as students can influence their teaching.</p>
<p>Traditional modes of instruction have emphasized that math is best learned through <a href="https://www.jstor.org/stable/40247978">studying and memorizing alone, with the teacher demonstrating procedures and then checking students’ answers</a>.</p>
<p>If parents grew up with this style of instruction, their ideal home-math classroom might look like strict scheduling, workbooks, a child working alone in silence and parents telling children how to solve problems. But if parents enforce this approach, there could be conflicts and maybe even some crying. </p>
<p>But parents, like future educators, can also learn from newer approaches. Here are some practical tips for a different form of home learning. </p>
<h2>1. Talking about math</h2>
<p>Gone are the days of students sitting quietly while the math teacher does all the talking at the chalkboard. <a href="https://www.jstor.org/stable/749877">Discussion</a> is important in the mathematics classroom. </p>
<p>Parents should be explicit. Tell your child “we learn by sharing ideas and listening to each other.”</p>
<p>Model active listening skills. Show your child that you are listening by asking questions about what they said to clarify your understanding of their idea. Try saying “tell me more …” or asking “how do you know that?”</p>
<p>Try setting aside your own idea(s) so you can listen and build on their ideas. Instead of saying “yes, but …,” use “<a href="https://link.springer.com/chapter/10.1007/978-3-319-78928-6_3">yes, and …</a>” to help children feel that they’re not being judged and their ideas are important.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">In today’s mathematics classrooms, discussion is important.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<h2>2. Attitude</h2>
<p>Researchers have identified <a href="https://link.springer.com/article/10.1007/s10857-009-9134-z?shared-article-renderer">three underlying interconnected aspects of childrens’ relationships</a> with math that impact how they engage with math: emotional disposition (“I like math”), perceived competence (“I am good at math”) and their vision of math: whether math is about problem solving and understanding or math is about memorization and regurgitation.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/mathematics-is-about-wonder-creativity-and-fun-so-lets-teach-it-that-way-120133">Mathematics is about wonder, creativity and fun, so let's teach it that way</a>
</strong>
</em>
</p>
<hr>
<p>Parents can set a positive attitude for children by being mindful not to say things like “I don’t like math” or “I’m not a math person.” Your child might think they don’t have a chance because <a href="https://www.researchgate.net/publication/249054114_A_Quantitative_and_Qualitative_Study_of_Math_Anxiety_Among_Preservice_Teachers">you didn’t pass on a math mind</a>. </p>
<p>Academics have debunked common beliefs about the “<a href="https://www.ams.org/journals/notices/200102/rev-devlin.pdf">math gene</a>” and explain that there’s <a href="https://books.google.ca/books?hl=en&lr=&id=K1Ld7FgOdtoC&oi=fnd&pg=PT17&dq=%22math+gene%22+parent&ots=Bxk5UApbwY&sig=dMLYhCKH%20K7mHhHOvfy8SOEc_es&redir_esc=y#v=onepage&q=%22math%20gene%22%20parent&f=false">lots involved in being good at math</a>. Celebrate the process and not just the final answer. Give high fives for sharing solution strategies, developing a plan to tackle the problem and for not giving up.</p>
<p>Make it clear that <a href="https://books.google.ca/books?hl=en&lr=&id=bOGHDQAAQBAJ&oi=fnd&pg=PR9&dq=dweck+mindset+mistakes&ots=YMX--knDci&sig=y07leb0VLednZ4ZhScAAYsKCkyE#v=onepage&q=dweck%20mindset%20mistakes&f=false">making mistakes</a> is OK and can even be a good thing. Many highly successful people see mistakes as learning opportunities and an indication that learning is happening.</p>
<h2>3. Working in partnership</h2>
<p><a href="https://www.advance-he.ac.uk/knowledge-hub/engagement-through-partnership-students-partners-learning-and-teaching-higher">A partnership</a> is about working together and can include seeing the <a href="https://www.semanticscholar.org/paper/We-are-the-Process%3A-Reflections-on-the-of-Power-in-Kehler-Verwoord/aeecc3e2e8e352474a24ce4ccd407f62629d6f56">teacher as a learner and the student as a teacher</a>. It isn’t about the teacher being “all-knowing” and making all the decisions. </p>
<p>Traditional math teaching, where the teacher assumes an authoritative role, is a major cause of <a href="https://doi.org/10.1177/1365480214521457">math anxiety</a>. Researchers have found that not all <a href="https://doi.org/10.1016/j.cedpsych.2019.101784">math homework help</a> is beneficial. There is a difference between parents being controlling and being supportive.</p>
<p>With this in mind, wait for your child to ask for help. Try not to control everything. Focus on asking questions about their decisions that will help them figure out possible limitations and benefits of their decisions. </p>
<p>Let children fail. Failure can <a href="https://books.google.ca/books?id=q0VZwEZoniUC&lpg=PP1&dq=The%20Optimistic%20Child%3A%20A%20Proven%20Program%20to%20Safeguard%20Children%20Against%20Depression%20and%20Build%20Lifelong%20Resilience&lr&pg=PP1#v=onepage&q&f=false">build confidence</a>. Confidence can come from mastery; mastery can come from <a href="https://www.jstor.org/stable/40248303">practice</a>. Good practice includes analyzing what went wrong and what went right.</p>
<p>Don’t worry about being the expert. Be honest and say “I’m not sure. Let’s figure it out together.” </p>
<p>Start with <a href="https://books.google.ca/books?id=Irq913lEZ1QC&lpg=PR13&lr&pg=PP1#v=onepage&q&f=false">what children already know</a>. When your child is stuck, ask them to talk through what they are doing.</p>
<p>Take turns doing questions and talking about solution strategies.</p>
<p>Follow your child’s interests <a href="https://theconversation.com/eight-ways-to-keep-your-kids-smart-over-the-summer-break-100132">and ideas</a>. Let them take the lead, even if you think your approach is better.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Focus on asking your child questions that will help them figure out possible limitations and benefits of their decisions.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<h2>4. Basic math skills</h2>
<p>If you grew up with traditional math instruction and haven’t thought about math since your school days, it might surprise you to learn that there are multiple ways to solve problems.</p>
<p>You could ask your child to share their way of solving the problem and also share your way. </p>
<p>For instance: What is 24 x 6? </p>
<p>It’s OK if you’re looking for a pencil to do this: </p>
<p> 24<br>
<u>x 6</u><br>
144</p>
<p>But what are some other ways you might you figure it out? </p>
<p>Multiply 20 x 6 to get 120. Now multiply 4 x 6 to get 24. Add the two figures: 120 + 24 = 144.</p>
<p>Another way would be to focus on 25 x 6 to get 150. Now subtract 6 and you’ve got 144. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/the-new-math-how-to-support-your-child-in-elementary-school-87479">The 'new math': How to support your child in elementary school</a>
</strong>
</em>
</p>
<hr>
<p>In all math problems (including addition or subtraction), your child can use their fingers and you can too. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Author Tina Rapke finds an occasion for everyday math in making cookies.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<p>You can also look for opportunities to highlight math in daily activities. </p>
<p>One fun way is through baking. Arrange three rows of cookie dough with four cookies in each row. Ask how many cookies per batch or how many each family member will get if they share equally. </p>
<p>Being successful at <a href="https://www.jstor.org/stable/10.5951/mathteacher.108.7.0543?casa_token=53fYsdfs758AAAAA:iROqe6Bs17ufC1uUB1x_ToGBlxgh-LgCEmMqSXgYT9cfbcLkdq0BdhWUjkxEfmYM5aLT__nM3eJ2CBiRa7EIwNPcR9W5BhbYspgB1oC4YDJaM2LWdp4#metadata_info_tab_contents">mental math</a> (like the arithmetic you do at the store) happens gradually over time. </p>
<p>Try focusing on basic math skills with your child for 10 minutes or less, every other day. </p>
<h2>The takeaway</h2>
<p>Think of quality over quantity. </p>
<p>If you want to support math learning at home based on math research: talk with your child, see learning as a partnership and make sure to celebrate their ideas. Your child may teach you something new. </p>
<p>We’d love to hear about how math has provoked families to slow down, have fun, go with the flow and connect.</p><img src="https://counter.theconversation.com/content/135114/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Tina Rapke received funding from SSHRC: Partnership Engage Grants. </span></em></p><p class="fine-print"><em><span>Cristina De Simone does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Your cheat sheet for best practices in teaching math at home. Keep it positive and mask your shock when your child tells you there are many ways to multiply numbers.Tina Rapke, Associate Professor of Mathematics Education, York University, CanadaCristina De Simone, Middle School Teacher. PhD Mathematics Education Student, York University, CanadaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1294392020-01-29T19:03:23Z2020-01-29T19:03:23ZBees learn better when they can explore. Humans might work the same way<figure><img src="https://images.theconversation.com/files/312391/original/file-20200129-92959-aj2iah.jpg?ixlib=rb-1.1.0&rect=8%2C2%2C1908%2C1074&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bees learn better when they can explore.</span> <span class="attribution"><span class="source">Author supplied</span>, <span class="license">Author provided</span></span></figcaption></figure><p>Understanding how humans learn is one key to improving teaching practices and advancing education. Does everybody learn in the same way, or do different people need different teaching styles?</p>
<p>The question may sound straightforward, but assessing and interpreting learning performance remains elusive. It is one of the most widely debated educational topics of today, especially for learners who have unique ways of demonstrating their understanding. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/309938/original/file-20200114-151867-jghtao.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/309938/original/file-20200114-151867-jghtao.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=371&fit=crop&dpr=1 600w, https://images.theconversation.com/files/309938/original/file-20200114-151867-jghtao.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=371&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/309938/original/file-20200114-151867-jghtao.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=371&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/309938/original/file-20200114-151867-jghtao.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=466&fit=crop&dpr=1 754w, https://images.theconversation.com/files/309938/original/file-20200114-151867-jghtao.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=466&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/309938/original/file-20200114-151867-jghtao.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=466&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Self-based exploratory behaviour can enhance learning outcomes.</span>
<span class="attribution"><span class="source">Author supplied.</span></span>
</figcaption>
</figure>
<h2>Bees learn</h2>
<p>We looked for answers in what might be an unexpected place: among honeybees. In a <a href="https://doi.org/10.1163/23644583-00401014">new study</a> published in the Video Journal of Education and Pedagogy, we use the bees as a model to understand how different individuals acquire information.</p>
<p>Using animal models to understand learning has a long and proud history. The Nobel Prize winner Ivan Pavlov famously trained dogs to <a href="https://www.simplypsychology.org/pavlov.html">associate a sound with a food reward</a>. Eventually Pavlov demonstrated that the dogs began to salivate at the sound. </p>
<p>Pavlov’s experiment revealed the core theory behind how we understand associative learning in education, society and popular culture. (Think of how <a href="http://maes14.blogspot.com/2011/01/harry-potter-and-classical-conditioning.html">Gringott’s dragon was conditioned</a> in <a href="https://en.wikipedia.org/wiki/Harry_Potter_and_the_Deathly_Hallows">Harry Potter And the Deathly Hallows</a>.) </p>
<p>Much of what we know about the physiology of memory formation comes from the seminal work of the Nobel laureate Eric Kandel. <a href="https://edition.cnn.com/2013/05/14/health/lifeswork-eric-kandel-memory/index.html">Kandel used the simple sea slug</a> (<em>Aplysia californica</em>) to investigate how connections between neurons in the brain enable learning.</p>
<p>Bees are surprisingly good learners and recent research shows individuals can <a href="https://theconversation.com/are-they-watching-you-the-tiny-brains-of-bees-and-wasps-can-recognise-faces-100884">learn faces</a>, <a href="https://theconversation.com/can-bees-do-maths-yes-new-research-shows-they-can-add-and-subtract-108074">add and subtract</a> and even process the <a href="https://theconversation.com/bees-join-an-elite-group-of-species-that-understands-the-concept-of-zero-as-a-number-97316">concept of zero</a>. Bees learn complex tasks through trial and error, where a reward of sugar water is provided for correctly solving a problem.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/309901/original/file-20200114-103951-dybmd0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/309901/original/file-20200114-103951-dybmd0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=676&fit=crop&dpr=1 600w, https://images.theconversation.com/files/309901/original/file-20200114-103951-dybmd0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=676&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/309901/original/file-20200114-103951-dybmd0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=676&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/309901/original/file-20200114-103951-dybmd0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=850&fit=crop&dpr=1 754w, https://images.theconversation.com/files/309901/original/file-20200114-103951-dybmd0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=850&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/309901/original/file-20200114-103951-dybmd0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=850&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A honeybee with a white identification mark learns to discriminate between 3 and 5 item displays that each present the same overall surface area.</span>
<span class="attribution"><span class="source">Author supplied.</span></span>
</figcaption>
</figure>
<h2>Teaching bees arithmetic</h2>
<p>We were very interested to discover whether all individual bees would learn complex tasks in a similar way. Would each individual show similar learning performance throughout training, or would individuals demonstrate different learning strategies?</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/we-taught-bees-a-simple-number-language-and-they-got-it-117816">We taught bees a simple number language – and they got it</a>
</strong>
</em>
</p>
<hr>
<p>One foundation math skill we all learn at about preschool age is how to add and subtract numbers. <a href="https://www.sciencedirect.com/science/article/pii/S1878929316302341">Arithmetic</a> is not a trivial task. It requires long-term memory of rules associated with particular symbols like plus (+) or minus (–), as well as short-term memory of what particular numbers to manipulate in a given instance.</p>
<p>When we <a href="https://www.youtube.com/watch?v=kKqZDbtc6AA">trained bees to add and subtract</a>, we evaluated how many trials it took each bee to acquire the task, and summarised the data examining how individuals learn in a <a href="https://ndownloader.figshare.com/files/18004085/preview/18004085/preview.gif">video</a>.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/kKqZDbtc6AA?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>We were surprised to see that all bees did not learn the task at the same stage of training. Instead, different individuals acquired the capacity to solve the problem after a different number of trials. </p>
<p>There was no common learning stage throughout the trials where bees achieved success. Rather the task required bees to try different strategies to see what worked. In particular, the opportunity to learn from mistakes was critical to enabling the bees to learn maths-based problems.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/310714/original/file-20200118-118323-1dc9k0l.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/310714/original/file-20200118-118323-1dc9k0l.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/310714/original/file-20200118-118323-1dc9k0l.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/310714/original/file-20200118-118323-1dc9k0l.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/310714/original/file-20200118-118323-1dc9k0l.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=425&fit=crop&dpr=1 754w, https://images.theconversation.com/files/310714/original/file-20200118-118323-1dc9k0l.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=425&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/310714/original/file-20200118-118323-1dc9k0l.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=425&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Different paths of learning: Performances of three different bees in an arithmetic task. While all three reach success, the path to learning the task is very different.</span>
<span class="attribution"><span class="source">Author supplied.</span></span>
</figcaption>
</figure>
<p>This finding suggests that when brains have to learn multi-stage problems involving different types of memory, an opportunity for exploratory behaviour is what nature prefers. </p>
<h2>What does this mean for education?</h2>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/312398/original/file-20200129-92992-mc38rt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/312398/original/file-20200129-92992-mc38rt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/312398/original/file-20200129-92992-mc38rt.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/312398/original/file-20200129-92992-mc38rt.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/312398/original/file-20200129-92992-mc38rt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/312398/original/file-20200129-92992-mc38rt.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/312398/original/file-20200129-92992-mc38rt.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Learning through experience.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>Humans and bees last shared a common ancestor about 600 million years ago. However, we share a large number of genes and it is likely we have some <a href="https://www.nature.com/articles/443893a">similarities in how we process information</a>. </p>
<p>We know that bees and humans have a common way of processing <a href="https://theconversation.com/bees-can-learn-higher-numbers-than-we-thought-if-we-train-them-the-right-way-124887">numbers from one to four</a>, for instance, suggesting that learning processes may be linked to evolutionary conserved mechanisms. So bees’ improved results when learning maths problems in an individual exploratory fashion suggests this may be how humans too are wired to acquire new skills.</p>
<p>Indeed, some recent research in learning and <a href="https://researchers.mq.edu.au/en/publications/basic-number-processing-deficits-in-developmental-dyscalculia-evi">learning difficulties in children</a> has found evidence that individuals frequently see and learn in different ways <a href="http://refractory.unimelb.edu.au/2015/02/06/dyer-pink/">depending on environmental context</a>.</p>
<p>Our biology may be programmed to encourage exploratory learning, rather than trying to acquire information in a set prescribed way. If so, our education systems should take this into consideration.</p>
<p>This idea may not be new, but may face challenges if computer-based learning is increasingly adopted as there is a risk that limited programming could limit learning styles. </p>
<p>On the other hand, the clever use of exploratory learning environments – digital or physical – may enhance learning outcomes. </p>
<p>We should not shy away from examining how our evolutionary history impacts learning and using this to our advantage. Understanding evolutionary principles could help in designing learning environments best suited to encouraging exploration for optimal learning, for example.</p><img src="https://counter.theconversation.com/content/129439/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adrian Dyer receives funding from The Australian Research Council.</span></em></p><p class="fine-print"><em><span>Elizabeth Jayne White, Jair Garcia, and Scarlett Howard do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Bees may provide surprising insights into the kinds of environments that are best for learning.Adrian Dyer, Associate Professor, RMIT UniversityElizabeth Jayne White, Professor ECE, RMIT UniversityJair Garcia, Research fellow, RMIT UniversityScarlett Howard, Postdoctoral research fellow, Université de Toulouse III – Paul SabatierLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/874792017-11-29T22:41:35Z2017-11-29T22:41:35ZThe ‘new math’: How to support your child in elementary school<figure><img src="https://images.theconversation.com/files/197013/original/file-20171129-12027-1r1wfrk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Parents find new methods for learning math challenging, as they are different. But they work for children, building upon what they have learned about numbers and reinforcing the strategy they use for reading.</span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p>There is likely no topic in Canada at the moment that is more acrimonious than elementary school mathematics education. The entire country, it seems, is divided. </p>
<p>On one side, there are those who are enraged by the so-called “new math” that has been held simultaneously responsible for a) diminished achievement by students and b) frustration among parents who feel helpless in the face of unfamiliar strategies. </p>
<p>On the other side are those who insist that math must make sense to today’s students — children who have grown up in a digital age, are adept with multiple technologies and will likely never be required to perform long division.</p>
<p>As a researcher who is deeply committed to engaging parents as partners in mathematics education, I spend many evenings on the road. I work with school staff and school councils across the province of Ontario to support parents in their efforts to help their children learn and love mathematics. </p>
<p>In communities from Chesterville to Picton, Guelph to Thunder Bay and Courtice to Fort Frances, I have encountered the same question repeatedly: What are you teaching my child?</p>
<h2>Arithmetic from Mexico to Japan</h2>
<p>The question is always sincere. The rationale differs considerably, but in most cases, the question arises because the computational strategies that the child is using to perform multi-digit calculations look very different from those learned by the parents, resulting in confusion and mistrust.</p>
<p>Experience has taught me to give a quick mini-lesson on arithmetic around the world to emphasize that there is no one global set of rules for calculations.</p>
<p>For example, I show a method that was used in Mexico, called “llevamos uno” — <em>we carry one</em>. Instead of noting ones or 10s to be “carried” at the top of the next column, students were taught to note those figures to the right side of the problem.</p>
<p> 1 9 4<br>
<u>+ 3 9¹¹</u><br>
2 3 3</p>
<p>I share a method that I learned from the Philippines, where students use dashes to indicate groups of 10. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/196772/original/file-20171128-28888-1ey6wb9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/196772/original/file-20171128-28888-1ey6wb9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/196772/original/file-20171128-28888-1ey6wb9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/196772/original/file-20171128-28888-1ey6wb9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/196772/original/file-20171128-28888-1ey6wb9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/196772/original/file-20171128-28888-1ey6wb9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/196772/original/file-20171128-28888-1ey6wb9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Elementary students in Baybay City, the Philippines, December 2015.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<p>Finally, I share a Japanese “scratch method” that is similar to the one used in the Philippines, but instead of dashes, overstrikes are used to keep track of groups of 10s. In addition, the leftover amounts are indicated by the use of subscripts.</p>
<p> 2 6<br>
<strike>7</strike>₁ <strike>6</strike>₂<br>
<u>+2 <strike>8</strike>₀</u><br>
1 3 0</p>
<p>Again, we begin at the right, at the top of the column: six plus six is 12, which is 10 (strike through the six) and two is left over (subscript two); two plus eight is 10, (strike through the eight) and zero (subscript zero). Write the zero under the ones column, and carry two groups of 10; two (10s) and two is four, plus seven (10s) is 11. Strike through the seven (to represent 100) and record one (subscript one). One plus two is three. Write the three in the 10s column and carry one group of 100. The answer is 130.</p>
<h2>We read left to right</h2>
<p>Having made the point that there is no universal set of rules to add multi-digit numbers and that all unfamiliar methods (including those used by their children) seem complex and incomprehensible at first glance, I am able to emphasize two important reasons to support new strategies for multi-digit addition.</p>
<p>When I ask parents to reflect on how they read to and with their toddlers, the answer is immediate and consistent: From left to right, using their index finger to trace the direction of the words.</p>
<p>Then I ask them what happens when we introduce children to the task of adding two-digit numbers. The light bulbs go on. We teach them to work right to left. </p>
<p>“Why?” I ask. </p>
<p>Dead silence or: “Because.”</p>
<p>In our number system, the value of a digit depends on its place, or position, in the number. So, for example, the number 4,276 is made up of 4,000 + 200 + 70 + six. Children who understand place value, i.e., that the value of a digit (zero to nine) depends on its position in a number, can easily decompose a number — an important strategy for mental math.</p>
<p>It’s ironic that after months of teaching the importance of place value, a fundamental concept in math, we do not apply that knowledge in practical ways to simplify multi-digit addition. As soon as we introduce questions like …</p>
<p> 8 7<br>
<u>+ 6 5</u></p>
<p>… we instruct students to begin at the right. This is in conflict with everything that children have been taught about reading from left to right and the importance of place value, i.e., that we read numbers from left to right, in order of magnitude. The algorithm, in fact, leads children to “unlearn” everything they know about place value.</p>
<h2>Building on children’s understandings</h2>
<p>Multi-digit arithmetic makes sense when we add from left to right, applying what we know about place value and reading.</p>
<p> 6 7<br>
+<u> 2 4</u><br>
8 0<br>
<u>1 1</u><br>
9 1</p>
<p>In this case, we add the 10s column first, 60 plus 20 to get 80. Next, we add seven to four to get 11. Add 80 and 11 to get the sum. This eliminates the need for “carrying” because the numbers align according to their value. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/196671/original/file-20171128-7465-1xx65e9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/196671/original/file-20171128-7465-1xx65e9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=335&fit=crop&dpr=1 600w, https://images.theconversation.com/files/196671/original/file-20171128-7465-1xx65e9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=335&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/196671/original/file-20171128-7465-1xx65e9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=335&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/196671/original/file-20171128-7465-1xx65e9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=421&fit=crop&dpr=1 754w, https://images.theconversation.com/files/196671/original/file-20171128-7465-1xx65e9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=421&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/196671/original/file-20171128-7465-1xx65e9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=421&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p><br></p>
<p>Children respond positively to this strategy because it makes sense. It builds on their understanding of place value and how numbers are made. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/196670/original/file-20171128-7458-1educix.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/196670/original/file-20171128-7458-1educix.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/196670/original/file-20171128-7458-1educix.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/196670/original/file-20171128-7458-1educix.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/196670/original/file-20171128-7458-1educix.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=565&fit=crop&dpr=1 754w, https://images.theconversation.com/files/196670/original/file-20171128-7458-1educix.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=565&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/196670/original/file-20171128-7458-1educix.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=565&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>Why are parents so resistant to such strategies? The traditional algorithms are used by adults in their peer group and come from adults whom they respect. This may attach an aura to the traditional methods as the “real” or ultimately correct way to compute. </p>
<p>As mathematics education giant <a href="http://www.pearsoned.ca/highered/divisions/hss/vandewalle/index.html">John van de Walle</a> once noted, it’s difficult to ignore the power of adding “the way my dad taught me.”</p>
<p>But it’s time to ask: Are the traditional algorithms really necessary? Or are we holding our children back by our own fears and lack of understanding of the alternatives?</p><img src="https://counter.theconversation.com/content/87479/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Lynda Colgan receives funding from SSHRC, NSERC, The Council of Ontario Directors of Education, The Ministry of Education for the Province of Ontario, The Mathematics Knowledge Network (The Fields Institute for Research in the Mathematical Sciences) </span></em></p>You may not know it, but the elementary math wars are raging. Our expert explains the ‘new math’ - why it works for kids, and how to do it.Lynda Colgan, Professor of Elementary Mathematics, Queen's University, OntarioLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/842322017-09-20T14:14:28Z2017-09-20T14:14:28ZNothing matters: how the invention of zero helped create modern mathematics<figure><img src="https://images.theconversation.com/files/186837/original/file-20170920-16403-yazsqf.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>A small dot on an old piece of birch bark marks one of the biggest events in the history of mathematics. The bark is actually part of an ancient Indian mathematical document known as the Bakhshali manuscript. And the dot is the first known recorded use of the number zero. What’s more, researchers from the University of Oxford <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> the document is 500 years older than was previously estimated, dating to the third or fourth century – a breakthrough discovery.</p>
<p>Today, it’s difficult to imagine how you could have mathematics without zero. In a positional number system, such as the decimal system we use now, the location of a digit is really important. Indeed, the real difference between 100 and 1,000,000 is where the digit 1 is located, with the symbol 0 serving as a punctuation mark.</p>
<p>Yet for thousands of years we did without it. The Sumerians of 5,000BC employed a <a href="http://www.storyofmathematics.com/sumerian.html">positional system</a> but without a 0. In some <a href="https://www.scientificamerican.com/article/what-is-the-origin-of-zer/">rudimentary form</a>, a symbol or a space was used to distinguish between, for example, 204 and 20000004. But that symbol was never used at the end of a number, so the difference between 5 and 500 had to be determined by context.</p>
<p>What’s more, 0 at the end of a number makes multiplying and dividing by 10 easy, as it does with adding numbers like 9 and 1 together. The invention of zero immensely simplified computations, freeing mathematicians to develop vital mathematical disciplines such as algebra and calculus, and eventually the basis for computers. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/pV_gXGTuWxY?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Zero’s late arrival was partly a reflection of the negative views some cultures held for the concept of nothing. Western philosophy is plagued with grave misconceptions about nothingness and the mystical powers of language. The fifth century BC <a href="https://plato.stanford.edu/entries/parmenides/">Greek thinker Parmenides</a> proclaimed that nothing cannot exist, since to speak of something is to speak of something that exists. This Parmenidean approach kept prominent <a href="http://www.nothing.com/Heath.html">historical figures</a> busy for a long while. </p>
<p>After the advent of Christianity, religious leaders in Europe argued that since God is in everything that exists, anything that represents nothing must be satanic. In an attempt to save humanity from the devil, they <a href="http://yaleglobal.yale.edu/history-zero">promptly banished</a> zero from existence, though merchants continued secretly to use it.</p>
<p>By contrast, in Buddhism the concept of nothingness is not only devoid of any demonic possessions but is actually a central idea worthy of much study en route to nirvana. <a href="http://www.huffingtonpost.com/lewis-richmond/emptiness-most-misunderstood-word-in-buddhism_b_2769189.html">With such a mindset</a>, having a mathematical representation for nothing was, well, nothing to fret over. In fact, the English word “zero” is <a href="http://www.etymonline.com/index.php?term=zero">originally derived</a> from the Hindi “sunyata”, which means nothingness and is a central concept in Buddhism.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=299&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=299&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=299&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=376&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=376&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=376&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Bakhshali manuscript.</span>
<span class="attribution"><span class="source">Bodleian Libraries</span></span>
</figcaption>
</figure>
<p>So after zero finally emerged in ancient India, it took almost 1,000 years to set root in Europe, much longer than in China or the Middle East. In 1200 AD, the Italian mathematician Fibonacci, who brought the decimal system to Europe, <a href="http://www.springer.com/gb/book/9780387407371">wrote that</a>: </p>
<blockquote>
<p>The method of the Indians surpasses any known method to compute. It’s a marvellous method. They do their computations using nine figures and the symbol zero.</p>
</blockquote>
<p>This superior method of computation, clearly reminiscent of our modern one, freed mathematicians from tediously simple calculations, and enabled them to tackle more complicated problems and study the general properties of numbers. For example, it led to the work of the seventh century Indian <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">mathematician and astronomer Brahmagupta</a>, considered to be the beginning of modern algebra.</p>
<h2>Algorithms and calculus</h2>
<p>The Indian method is so powerful because it means you can draw up simple rules for doing calculations. Just imagine trying to explain long addition without a symbol for zero. There would be too many exceptions to any rule. The ninth century Persian mathematician Al-Khwarizmi was the first to meticulously note and exploit these arithmetic instructions, which <a href="https://books.google.co.uk/books?id=zTQrDwAAQBAJ&pg=PA47&lpg=PA47&dq=al+khwarizmi+abacus&source=bl&ots=PakFxbCVwk&sig=FWjwHlnppHAU9i_zgAficOcw4ug&hl=en&sa=X&ved=0ahUKEwii-46257PWAhUhBcAKHaWtCRcQ6AEIajAP#v=onepage&q=al%20khwarizmi%20abacus&f=false">would eventually</a> make the abacus obsolete.</p>
<p>Such mechanical sets of instructions illustrated that portions of mathematics could be automated. And this would eventually lead to the development of modern computers. In fact, the word “algorithm” to describe a set of simple instructions <a href="https://en.oxforddictionaries.com/definition/algorithm">is derived</a> from the name “Al-Khwarizmi”.</p>
<p>The invention of zero also created a new, more accurate way to describe fractions. Adding zeros at the end of a number increases its magnitude, with the help of a decimal point, adding zeros at the beginning decreases its magnitude. Placing infinitely many digits to the right of the decimal point corresponds to <a href="https://www.youtube.com/watch?v=JmyLeESQWGw&list=PLYoCqdGsxmn9HOU84Ln2PhPKpxRfaEO9h&index=17">infinite precision</a>. That kind of precision was exactly what 17th century thinkers Isaac Newton and Gottfried Leibniz needed to develop calculus, the study of continuous change.</p>
<p>And so algebra, algorithms, and calculus, three pillars of modern mathematics, are all the result of a notation for nothing. Mathematics is a science of invisible entities that we can only understand by writing them down. India, by adding zero to the positional number system, unleashed the true power of numbers, advancing mathematics from infancy to adolescence, and from rudimentary toward its current sophistication.</p><img src="https://counter.theconversation.com/content/84232/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ittay Weiss does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Turning zero from a punctuation mark into a number paved the way for everything from algebra to algorithms.Ittay Weiss, Teaching Fellow, Department of Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/460032015-08-27T07:50:00Z2015-08-27T07:50:00ZBack to school? A crucial time for kids’ social and emotional development<figure><img src="https://images.theconversation.com/files/92931/original/image-20150825-15923-xfrs2w.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Back-to-school time comes with rich, teachable moments.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/wwworks/4114157284/in/photolist-7gy8Ch-bAZeg1-8rRSFY-8rNM66-9sKchx-8CKUy5-5AGwzb-afez5b-afbMx4-6neX7k-4p4xqX-cQCyBU-8w6AzK-dPkbC8-9k7KdF-8w6zFg-8w9yuf-8w6vHk-dPkbA4-det3NF-desUMB-desSCf-desSnm-det1Rn-det3Ur-det1tH-desVGL-desZv7-det2su-desUFn-desS6d-desZk9-desVHk-desSwA-desYXS-det41X-96Lzht-a61cx-7djt5E-74uSHk-bBEw7Z-bBEw1c-dGG8aN-5DjpQQ-8boTzZ-8xb15n-8xe2j3-oDMgsy-7b17gZ-5dph7g">PROwoodleywonderworks</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>It’s that time of the year. Summer vacations are almost over. </p>
<p>For most kids, this time of summer has been about finishing the readings and completing the packets that were handed out to them as summer work. As a result, school often conjures up ideas about reading, writing and arithmetic (the “three R’s”). </p>
<p>But this approach is both <a href="http://downloads.ncss.org/legislative/AcademicAtrophy.pdf">problematic</a> and myopic. As pressures to meet standards in the three R’s increase, other areas fall off the radar. Having an answer to a question becomes more important than knowing how to think about it.</p>
<p>As a psychoanalyst in private practice and a classroom teacher, I know that the time of transitioning back to school is crucial for both parents and children. This can also be a time to support the emotional development of children. </p>
<p>Sometimes just one hour of “emotional tutoring” – attending to social and emotional development – can be more efficient than spending hours tutoring. It can remove blocks to learning and open up energy for higher-order thinking. </p>
<p>So what can we do as parents and educators to get there? </p>
<h2>The pressure of the reading, writing, arithmetic</h2>
<p>There are <a href="https://nces.ed.gov/surveys/sass/tables/sass0708_035_s1s.asp">180 days of school, on average, in US public schools</a>. Ask a teacher how many days are spent administering tests and preparing for said tests, and you might wonder how anything else gets covered. I was supervising a school-based clinician who worked on Thursdays and couldn’t see his clients for six straight weeks because of test prep and actual testing. </p>
<p>Peter Taubman, a professor of education at Brooklyn College, in his book <a href="https://www.routledge.com/products/9780415890519">Disavowed Knowledge</a>, describes how today’s students get treated as though they were just animals that need to be trained and told what to repeat, as opposed to building their “curiosity, attunement, analysis, and a focus on creating conditions such that the …student can generate material for further elaboration… .” </p>
<p>Even the <a href="http://downloads.ncss.org/legislative/AcademicAtrophy.pdf">Council on Basic Education (CBE) report</a> notes:</p>
<blockquote>
<p>Of particular concern… are signs that the growing attention to mathematics, reading, writing, and science may well be coming at the expense of other academic subjects, including the arts and foreign language.“</p>
</blockquote>
<p>The fact is, we are feeling creatures first. Pressures and obsessions to perform in one area of learning, growth and development can lead to neglect in others. </p>
<p>The start of school is a time when mixed emotions need to be processed. A commercial from the 1990s for an office supply store captures this. With background music from a popular singer – <a href="http://www.biography.com/people/andy-williams-162966">Andy Williams’</a> It’s the Most Wonderful Time of the Year – the scene showed two sad children next to a jubilant dad pushing a shopping cart around a store collecting school supplies.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/4DComGO8JYo?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>The commercial serves as a reminder that the start of a school year has as much to do with emotions as it does with other subjects. </p>
<h2>The importance of social and emotional learning</h2>
<p>Trying to only accentuate the positive or force learning before a child is emotionally ready ignores what neuroscience and psychology tell us about the brain: we are ruled by emotions and not reason. Briefly, all information to our rational brains must pass through our limbic system, which is our feeling brain. </p>
<p>If we become <a href="http://www.choice4change.co.uk/emotional-hijacking">emotionally hijacked</a> – that is, if our feelings become unregulated – even the best-made rational plan can be inaccessible. The stress of daily coursework can lead to such emotional hijacking. </p>
<p>That is where Social and Emotional Learning (<a href="http://www.edutopia.org/social-emotional-learning">SEL</a>) can come into play. SEL is learning how to understand ones own feelings and making good decisions to get what you want out of life. Both children and adults can benefit from the <a href="http://www.casel.org/">various</a> SEL <a href="http://ei.yale.edu/ruler/">programs</a>. </p>
<p>They differ in some ways, but when distilled down, SEL programs share three common elements: self-awareness, talking or putting your thoughts and feelings into words, and providing some structure or a "holding” environment. These elements help reduce “emotional hijacking.” </p>
<p>Any student who is identified as lacking some element of an SEL competency, for instance, <a href="http://www.casel.org/social-and-emotional-learning/core-competencies">self-awareness, self-management, social awareness, relationship skills and responsible decision-making</a> can benefit. These SEL competencies make reading, writing and arithmetic easier as a student’s emotional world is validated. </p>
<p>If a child’s social and emotional world is left unattended to, he or she can be driven to distraction, acting out, and various unregulated emotional states. </p>
<h2>How you can support the transition</h2>
<p>Back-to-school time comes with rich, teachable moments to support the social and emotional development of children. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/92932/original/image-20150825-15920-1l6kscb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/92932/original/image-20150825-15920-1l6kscb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=444&fit=crop&dpr=1 600w, https://images.theconversation.com/files/92932/original/image-20150825-15920-1l6kscb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=444&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/92932/original/image-20150825-15920-1l6kscb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=444&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/92932/original/image-20150825-15920-1l6kscb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=558&fit=crop&dpr=1 754w, https://images.theconversation.com/files/92932/original/image-20150825-15920-1l6kscb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=558&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/92932/original/image-20150825-15920-1l6kscb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=558&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">What can you do as a parent?</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/rocksinmyhead/8028638014/in/photolist-desSnm-det1Rn-det3Ur-det1tH-desVGL-desZv7-det2su-desUFn-desS6d-desZk9-desVHk-desSwA-desYXS-det41X-96Lzht-a61cx-7djt5E-74uSHk-bBEw7Z-bBEw1c-dGG8aN-5DjpQQ-8boTzZ-8xb15n-8xe2j3-oDMgsy-7b17gZ-5dph7g-7MU3mT-7b16NK-bqHayp-8FgcPX-atpGAU-8QH9Ev-aYw9ct-4G1wxF-desUCG-aBhict-aewZRz-desRWE-81WD5J-det2Ah-det2Wb-desZ9p-desUvF-desZJK-det22W-desRCu-det1VC-det1NY">RocksInMyHead</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>So, if you are a parent, here is what you need to do:</p>
<ul>
<li><p>Be self-aware! Focusing entirely on your child’s transition overlooks your own feelings about transitions. They are tough for everyone. Knowing what you are bringing to the mix is important. Once you know it, you can work on helping your child develop his or her self-awareness.</p></li>
<li><p>Talking helps. A simple question like, “Any thoughts or feelings about school starting?” is an opening. Let the question hang there. You don’t need to clarify or say more. Give them space to think and feel. If you have concerns about transitioning your child from summer to school mode, talk to your friends, a pediatrician or your child’s therapist. </p></li>
<li><p>Structure time. Depending on your child’s age, you can start implementing bedtimes and times for breakfast in the morning that slowly move back to school timetables. This is easy enough with children, but tough to do with adolescents who just have notoriously poor sleep hygiene. </p></li>
<li><p>And don’t forget, if there is summer homework that needs to be completed (summer reading, projects, etc), then you may want to help schedule that for your child. </p></li>
</ul>
<h2>“Education is life itself”</h2>
<p>As an academic, I love and hate the back-to-school time. I love teaching. I am a sucker for back-to-school sales and love the smell of stationery and the feel of opening a new notebook. </p>
<p>But I also love the summertime, without my students and schedules. I enjoy reading through the stack of both pleasure and subject-area books I have accumulated over the academic year. My summer is “me” time. Fall, winter and spring are “we” time. These are the elements of my own “self-awareness.” </p>
<p>Back-to-school time brings mixed feelings, as do most important events in life. Our jobs as parents and educators should be to help with the social and emotional development of those in our care so that they can more easily do the reading, writing and arithmetic that they need as well, not the other way around. </p>
<p>In an age where answers are only a smartphone away, knowing how to think and critically evaluate information should be the focus of education. </p>
<p>In many ways, this is what <a href="http://www.biography.com/people/john-dewey-9273497">John Dewey</a>, the noted academic, philosopher and educator meant when he wrote:</p>
<blockquote>
<p>Education is not preparation for life, education is life itself.</p>
</blockquote><img src="https://counter.theconversation.com/content/46003/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>William Sharp does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The time of transitioning back to school is crucial for both parents and children. Here’s what you can do.William Sharp, Instructor, Psychology and Human Development, Wheelock CollegeLicensed as Creative Commons – attribution, no derivatives.