tag:theconversation.com,2011:/us/topics/counting-31620/articlesCounting – The Conversation2023-12-26T17:15:34Ztag:theconversation.com,2011:article/2172472023-12-26T17:15:34Z2023-12-26T17:15:34ZHow counting by 10 helps children learn about the meaning of numbers<figure><img src="https://images.theconversation.com/files/564574/original/file-20231208-15-3eojg4.jpg?ixlib=rb-1.1.0&rect=0%2C242%2C4383%2C3017&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Using concrete tools or objects matters for fostering mathematical development – but how can children best learn to count by 10?
</span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><iframe style="width: 100%; height: 100px; border: none; position: relative; z-index: 1;" allowtransparency="" allow="clipboard-read; clipboard-write" src="https://narrations.ad-auris.com/widget/the-conversation-canada/how-counting-by-10-helps-children-learn-about-the-meaning-of-numbers" width="100%" height="400"></iframe>
<p>When children start school, they learn how to recite their numbers (“one, two, three…”) and how to write them (1, 2, 3…). Learning about what those numbers mean is even more challenging, and this becomes trickier yet when numbers have more than one digit — such as 42 and 608. </p>
<p>It turns out that the meaning of such “multidigit” numbers cannot be gleaned from simply looking at them or by performing calculations with them. Our number system has many hidden meanings that are not transparent, making it <a href="https://doi.org/10.1037/dev0001145">difficult for children</a> to comprehend it. </p>
<p>In collaboration with elementary teachers, the Mathematics Teaching and Learning Lab at <a href="https://www.concordia.ca/">Concordia University</a> explores tools that can support young children’s understanding of multidigit numbers.</p>
<p>We investigate the impact of using concrete objects (like bundling straws into groups of 10). We also investigate the use of visual tools, such as number lines and charts, or words to represent numbers (the word for 40 is “forty”) and written notation (for example, 42). </p>
<p>Our recent research examined whether the “hundreds chart” — 10 by 10 grids containing numbers from one to 100, with each row in the chart containing numbers in groups of 10 — could be useful for teaching children about counting by 10, something foundational for understanding how numbers work. </p>
<h2>What’s in a number?</h2>
<p>Most adults know that the placement of the “4” and “2” in 42 means four tens and two ones, respectively. </p>
<p>But when young children start learning about numbers, they do not naturally see 10s and ones in a number like 42. They think the number represents 42 things counted from one to 42 without distinguishing between the meaning of the digits “4” and “2.” Over time, through counting and other activities, children see the four as a <a href="https://doi.org/10.1177/1053451221994827">collection of 40 ones</a>. </p>
<p>This realization is not sufficient, however, for <a href="https://doi.org/10.1111/ssm.12258">learning more advanced topics</a> in math. </p>
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Read more:
<a href="https://theconversation.com/mathematical-thinking-begins-in-the-early-years-with-dialogue-and-real-world-exploration-128282">Mathematical thinking begins in the early years with dialogue and real-world exploration</a>
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<p>An important next step is to see that 42 is made up of four distinct groups of 10 and two ones, and that the four 10s can be counted as if they were ones (for example, 42 is one, two, three, four 10s and one, two, “ones”). </p>
<p>Ultimately, one of the most challenging aspects of understanding numbers is that groups of ten and ones are <a href="https://doi.org/10.17226/12519">different kinds of units</a>. </p>
<h2>Numbers can be arranged in different ways</h2>
<p>The numbers in hundreds charts can be arranged in different ways. A top-down hundreds chart has the digit “1” in the top-left corner and 100 in the bottom-right corner. </p>
<p>The numbers increase by 10 moving downward one row at a time, like going from 24 to 34 using one hop down, for instance. A second type of chart is the “bottom-up” chart, which has the numbers increasing in the opposite direction. </p>
<h2>Counting by 10s</h2>
<p>Children can move from one number to another in the chart to <a href="https://doi.org/10.5951/teacchilmath.24.3.00e1">solve problems</a>. Considering 24 + 20, for example, children could start on 24 and move 20 spaces to land on 44. </p>
<p>Another way would be to move up (or down, depending on the chart) two rows (for example, counting “one,” “two”) until they land on 44. This second method shows a developing understanding of multidigit numbers being composed of distinct groups of 10, which is critical for an advanced knowledge of the number system. </p>
<p>For her master’s research at Concordia University, Vera Wagner, one of the authors of this story, thought children might find it more intuitive to solve problems with the bottom-up chart, where the numbers get larger with upward movement. </p>
<p>After all, plants grow taller and liquid rises in a glass as it is filled. Because of such <a href="https://doi.org/10.1111/tops.12278">familiar experiences</a>, she thought children would move by tens more frequently in the bottom-up chart than in the top-down chart. </p>
<h2>Study with kindergarteners, Grade 1 students</h2>
<p>To examine this hypothesis, we worked with 47 kindergarten and first grade students in Canada and the United States. All the children but one spoke English at home. In addition to English, 14 also spoke French, four spoke Spanish, one spoke Russian, one spoke Arabic, one spoke Mandarin and one communicated to some extent in ASL at home. </p>
<p>We assigned all child participants in the study an online version of <a href="http://mathchart.ca/chart.html#nt">either a top-down</a> or <a href="https://mathchart.ca/chart.html#reversednt">bottom-up</a> hundreds chart, programmed by research assistant André Loiselle, to solve arithmetic word problems. </p>
<p><a href="https://doi.org/10.1111/ssm.12593">What we found surprised us</a>: children counted by tens more often with the top-down chart than the bottom-up one. This was the exact opposite of what we thought they might do!</p>
<p>This finding suggests that the top-down chart fosters children’s counting by tens as if they were ones (that is, up or down one row at a time), an important step in their mathematical development. Children using the bottom-up chart were more likely to confuse the digits and move in the wrong direction. </p>
<h2>Tools can impact learning</h2>
<p>Our research suggests that the types of tools used in the math classroom can impact children’s learning in <a href="https://doi.org/10.1016/j.learninstruc.2008.03.005">different ways</a>. </p>
<p>One advantage of the top-down chart could be the corresponding left-to-right and downward movement that matches the direction in which children learn to read in English and French, the official languages of instruction in the schools in our study. Children who learn to read in a different direction (for example, <a href="https://escholarship.org/uc/item/4tt0k00j">from right to left, as in Arabic</a>) may interact with some math tools differently from children whose first language is English or French. </p>
<p>The role of cultural experiences in math learning opens up questions about the design of teaching tools for the classroom, and the relevance <a href="https://theconversation.com/culturally-responsive-teaching-in-a-globalized-world-109881">of culturally responsive</a> mathematics teaching. Future research could seek to directly examine the relation between reading direction and the use of the hundreds chart.</p><img src="https://counter.theconversation.com/content/217247/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Helena Osana received funding from the Social Sciences and Humanities Research Council of Canada for this research. </span></em></p><p class="fine-print"><em><span>Jairo A. Navarrete-Ulloa receives funding from the National Agency for Research and Development (ANID) in Chile. </span></em></p><p class="fine-print"><em><span>Vera Wagner 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>Findings of a study suggest using a ‘hundreds chart’ showing numbers one through 100, beginning with one in the top-left corner, fosters children’s counting by 10s.Helena Osana, Professor, Principal Investigator of the Mathematics Teaching and Learning Lab, Concordia UniversityJairo A. Navarrete-Ulloa, Adjunct assistant professor, Institute of Education Sciences, Universidad de O’Higgins (Chile)Vera Wagner, Research Assistant, Mathematics Teaching and Learning Lab, Concordia UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1893392022-08-25T20:03:39Z2022-08-25T20:03:39ZCounting from left to right feels ‘natural’ – but new research shows our brains count faster from bottom to top<figure><img src="https://images.theconversation.com/files/481018/original/file-20220825-21-gs7h42.jpeg?ixlib=rb-1.1.0&rect=0%2C0%2C4000%2C6000&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://unsplash.com/photos/4wF66_KWJxA">Gayatri Malhotra / Unsplash</a></span></figcaption></figure><p>When asked to write the numbers from one to ten in a sequence, how do you order them? Horizontally? Vertically? Left to right? Top to bottom? Would you place them randomly?</p>
<p>It has been often been assumed, and taught in schools in Western countries, that the “correct” ordering of numbers is from left to right (1, 2, 3, 4…) rather than right to left (10, 9, 8, 7…). The ordering of numbers along a horizontal dimension is known as a “mental number line” and describes an important way we represent number and quantity in space. </p>
<p>Studies show humans prefer to position larger numbers to the right and smaller numbers to the left. People are usually <a href="https://psycnet.apa.org/record/2009-00781-003">faster and more accurate</a> at comparing numbers when larger ones are to the right and smaller ones are to the left, and people with brain damage that disrupts their spatial processing also show <a href="https://www.nature.com/articles/417138a">similar disruptions</a> in number processing.</p>
<p>But so far, there has been little research testing whether the horizontal dimension is the most important one we associate with numbers. In <a href="https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0262559">new research</a> published in PLOS ONE, we found that humans actually process numbers faster when they are displayed vertically – with smaller numbers at the bottom and larger numbers at the top.</p>
<h2>Not just humans</h2>
<p>Our associations between number and space are influenced by <a href="https://doi.org/10.1037/0096-3445.122.3.371">language and culture</a>, but these links are not unique to humans.</p>
<p>Tests on three-day-old chicks show they
<a href="https://www.science.org/doi/full/10.1126/science.aaa1379?casa_token=Ej3SBcsh9zYAAAAA%3AMlxxpmoCsCUzSwuM19VWSzwf2EZ1b6alxcVtfjYiPxNRpTkyznaFR3u2HslD2UHVwc7mCSPDfAxGwkA">seek smaller numbers</a> with a leftwards bias and larger numbers with a rightwards one. <a href="https://psycnet.apa.org/record/2019-41901-001">Pigeons and blue jays</a> seem to have a left-to-right or right-to-left mental number line, depending on the individual. </p>
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<a href="https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A photograph of baby chicks." src="https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/481025/original/file-20220825-25-6rnapv.jpeg?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"></a>
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<span class="caption">Even three-day-old chicks have something like a mental number line.</span>
<span class="attribution"><a class="source" href="https://unsplash.com/photos/Gky4wYp9mvg">Jason Leung / Unsplash</a></span>
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<p>These findings suggest associations between space and numbers may be wired into the brains of humans and other animals. </p>
<p>However, while many studies have examined left-to-right and right-to-left horizontal mental number lines, <a href="https://www.sciencedirect.com/science/article/pii/S0042698916301511">few</a> have explored whether our dominant mental number line is even horizontal at all. </p>
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Read more:
<a href="https://theconversation.com/can-bees-do-maths-yes-new-research-shows-they-can-add-and-subtract-108074">Can bees do maths? Yes – new research shows they can add and subtract</a>
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<h2>How we test for these spatial-numerical associations</h2>
<p>To test how quickly people can process numbers in different arrangements, we set up an experiment where people were shown pairs of numbers from 1 to 9 on a monitor and used a joystick to indicate where the larger number was located. </p>
<p>If the 6 and 8 were shown on the screen, for example, the correct answer would be 8. A participant would indicate this by moving the joystick towards the 8 as fast as possible.</p>
<p>To measure participant response times as accurately as possible, we used fast-refresh 120 Hertz monitors and high-performance zero-lag arcade joysticks.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/480866/original/file-20220824-4813-jiebyn.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/480866/original/file-20220824-4813-jiebyn.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=404&fit=crop&dpr=1 600w, https://images.theconversation.com/files/480866/original/file-20220824-4813-jiebyn.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=404&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/480866/original/file-20220824-4813-jiebyn.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=404&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/480866/original/file-20220824-4813-jiebyn.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=508&fit=crop&dpr=1 754w, https://images.theconversation.com/files/480866/original/file-20220824-4813-jiebyn.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=508&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/480866/original/file-20220824-4813-jiebyn.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=508&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">Testing how participants show preferences for either horizontal or vertical mental number lines by indicating the larger number with a computer gaming joy stick.</span>
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<h2>What we found</h2>
<p>When the numbers were separated both vertically and horizontally, we found only the vertical arrangement affected response time. This suggests that, given the opportunity to use either a horizontal or vertical mental representation of numbers in space, participants only used the vertical representation.</p>
<p>When the larger number was above the smaller number, people responded much more quickly than in any other arrangement of numbers.</p>
<p>This suggests our mental number line actually goes from bottom (small numbers) to top (large numbers). </p>
<h2>Why is this important?</h2>
<p>Numbers affect almost every part of our lives (and our safety). Pharmacists need to correctly measure doses of medicine, engineers need to determine stresses on buildings and structures, pilots need to know their speed and altitude, and all of us need to know what button to press on an elevator. </p>
<p>The way we learn to use numbers, and how designers choose to display numerical information to us, can have important implications for how we make fast and accurate decisions. In fact, in some time-critical decision-making environments, such as aeroplane cockpits and stock market floors, numbers are often displayed vertically. </p>
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Read more:
<a href="https://theconversation.com/numbers-on-the-mind-how-maths-can-help-explain-the-workings-of-our-brain-44405">Numbers on the mind: how maths can help explain the workings of our brain</a>
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<p>Our findings, and another recent <a href="https://www.frontiersin.org/articles/10.3389/fpsyg.2019.00172/full">study</a>, may have implications for designers seeking to help users quickly understand and use numerical information. Modern devices enable very innovative number display options, which could help people use technology more efficiently and safely.</p>
<p>There are also implications for education, suggesting we should teach children using vertical bottom-to-top mental number lines as well as the familiar left-to-right ones. Bottom-to-top appears to be how our brains are wired to be most efficient at using numbers – and that might help getting our heads around how numbers work a little easier.</p><img src="https://counter.theconversation.com/content/189339/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Luke Greenacre receives funding from the NHMRC. He is affiliated with Monash University and the University of South Australia. </span></em></p><p class="fine-print"><em><span>Adrian Dyer receives funding from The Australian Research Council and The Alexander von Humboldt Foundation.</span></em></p><p class="fine-print"><em><span>Scarlett Howard receives funding from Monash University and the Hermon Slade Foundation. She is affiliated with Pint of Science Australia.</span></em></p><p class="fine-print"><em><span>Jair Garcia 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>Horizontal number lines are often the default option – but our brains may process numbers more quickly in a vertical arrangement.Luke Greenacre, Senior lecturer in marketing, Monash UniversityAdrian Dyer, Associate Professor, Monash UniversityJair Garcia, Researcher and analyst, Monash UniversityScarlett Howard, Lecturer, Monash UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1672592021-10-21T14:19:50Z2021-10-21T14:19:50Z4 moves to make math visible with kids, using counters<figure><img src="https://images.theconversation.com/files/425751/original/file-20211011-25-1dnzapk.jpg?ixlib=rb-1.1.0&rect=0%2C135%2C4751%2C3003&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">If we each get to choose four colourful candies, my four includes three orange and one blue. Yours? </span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p>Let’s say you’re a parent helping a Grade 1 child with their math, and they’re subtracting eight from 17, using small items — counters — like Smarties, multicoloured Rocket candies or Lego pieces.</p>
<p>The child counts out 17 items. Then, they count eight of those items to take away. Finally, they start counting the remaining items. </p>
<p>Here is where parents who haven’t revisited math instruction for decades <a href="https://theconversation.com/the-new-math-how-to-support-your-child-in-elementary-school-87479">get confused and want to show their child what they believe</a> is a faster or better way: by picking up a pencil and paper to stack the 17 on top of the eight.</p>
<p>Don’t be surprised if your child brings home a few unfamiliar strategies. Many classrooms today embrace “<a href="http://www.meaningfulmathmoments.com/number-talks.html">number talks</a>” — what math educator Sherry Parish defines as discussions about computation problems “<a href="https://books.google.ca/books/about/Number_Talks.html?id=p4B9F1u2T4kC">designed to elicit specific strategies that focus on number relationships and number theory</a>.” </p>
<h2>Making math thinking visible</h2>
<p>In the example above that involves subtracting eight from 17, parents may be thinking the next step in teaching a child is relaying the rule that you never take a bigger number from a smaller one. </p>
<p>They may want to tell the child: “You can’t take eight away from seven, so you borrow a 10 from the tens column …” That’s when the parent and child both realize they are exactly where they started, subtracting eight from 17. Yikes!</p>
<p>Instead of falling into the trap of showing and telling your preferred method for subtracting, mathematics educators recommend <a href="https://theconversation.com/4-things-weve-learned-about-math-success-that-might-surprise-parents-135114">listening to learners and talking about what they already know</a>. </p>
<p>What the child says and does can be a resource for understanding subtraction. </p>
<p>Here are ways parents or teachers could support a learner’s strategy by helping make their thinking visible.</p>
<figure class="align-center ">
<img alt="Six wooden rounds organized into two rows with a wooden figure six on top." src="https://images.theconversation.com/files/426303/original/file-20211013-15-51qber.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/426303/original/file-20211013-15-51qber.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/426303/original/file-20211013-15-51qber.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/426303/original/file-20211013-15-51qber.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/426303/original/file-20211013-15-51qber.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/426303/original/file-20211013-15-51qber.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/426303/original/file-20211013-15-51qber.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">
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<span class="caption">It’s important for kids to learn to quickly see how many without counting, through quick visual recognition.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
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<h2>Move No. 1: Make arrangements</h2>
<p>Talking about numbers involves understanding “how muchness” — for instance: how 17 is 20 less three, or how it’s 15 and two more. Math educators talk about the importance of learning to “subitze” — learning to quickly see how many without counting, through quick visual recognition. </p>
<p>Using counters helps students see quantities. This can be done by making specific arrangements with counters, like grouping counters into circles, rows or clusters of threes, fours and fives. For example, if the number 17 is arranged into subitized images of five, a child might say they see three fives and two remaining single units.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="17 dots arranged into clusters of three fives and a two." src="https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=382&fit=crop&dpr=1 600w, https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=382&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=382&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=480&fit=crop&dpr=1 754w, https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=480&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/425928/original/file-20211012-23-91b1wh.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=480&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">Seventeen seen as five, five, five and two. How would you subtract eight?</span>
<span class="attribution"><span class="source">(Marc Husband)</span></span>
</figcaption>
</figure>
<p>Prompting them to subtract eight might provoke a strategy where the child takes away five, and then another three. Depending on where the child starts taking away (from left to right versus right to left), they could be left with different subitized images: for example, two, five and two; or five and four. </p>
<p>Such processes of “decomposing” numbers — breaking down the eight into five, two and one — builds what’s called <a href="https://www.stenhouse.com/content/number-sense-routines-grades-k-3">number sense</a> (understanding how numbers are related).</p>
<h2>Move No. 2: Colour code</h2>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="17 dots shown with ten of the dots blue, and seven of them in green." src="https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=438&fit=crop&dpr=1 600w, https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=438&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=438&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=550&fit=crop&dpr=1 754w, https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=550&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/425929/original/file-20211012-23-17syhnb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=550&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Using different colours can help kids see numbers within numbers.</span>
<span class="attribution"><span class="source">(Marc Husband)</span></span>
</figcaption>
</figure>
<p>Using two different colour counters can enhance a learner’s ability to see activities <a href="https://doi.org/10.1007/s10857-006-9005-9">that are happening in their minds</a>. </p>
<p>Supposing the child says, in subtracting eight from 17, they want to subtract seven first. You can then lay out seven green counters and 10 blue ones, and then remove seven green and one blue. Coding numbers with colours helps students to see numbers within numbers (that seven and one make eight).</p>
<h2>Move No. 3: Show the action</h2>
<p>When learners subtract, they’re doing a mental action. Seeing this action with counters supports understanding the concept of “taking away.” </p>
<p>Let’s continue with our example of 17 minus eight with the 17 organized into subitized images of five and two. If the child says that they want to subtract the seven first, a parent or teacher can illustrate this by pulling (in a downwards motion) the arrangements of seven, followed by one from the remaining 10. </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Eight dots with arrows are shown moving away from a cluser of 17 dots." src="https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=530&fit=crop&dpr=1 600w, https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=530&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=530&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=666&fit=crop&dpr=1 754w, https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=666&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/425931/original/file-20211012-21-1lrxtez.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=666&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Moving the counters demonstrates what thinking processes are happening.</span>
<span class="attribution"><span class="source">(Marc Husband)</span></span>
</figcaption>
</figure>
<p>Emphasizing the mental activities by doing and recording is beneficial for all learners. It may in particular also be a way for educators to seek to engage Indigenous learners. As math researcher and educator Lisa Lunney Borden writes, in a case <a href="https://eric.ed.gov/?id=EJ961339">study of Mi'kmaw students in Atlantic Canada learning math, most Indigenous languages in Canada have verb-based origins, and language generates world views</a> and cultural ways of knowing. From this perspective, emphasizing process and action in math may help engage Indigenous learners and affirm their identity formation whether or not they currently speak their ancestral language.</p>
<h2>Move No. 4: Check-in repeatedly</h2>
<p>While you’re arranging counters, colour coding and showing actions to make ideas visible, you can support a two-way conversation by checking in multiple times with kids. Consider the following questions or prompts: </p>
<p><strong>“Can you say it again?”</strong>: If you’re unsure what they did, don’t be afraid to ask them to repeat it. Taking time to figure out what they did values their thinking and can be engaging for everyone, including the parent or educator.</p>
<p><strong>Gesture</strong>: When you move counters to make a learner’s thinking visible, gesture by making a circle with a finger over the counters they say they’re seeing. While colour coding, you might ask: “What numbers do you see now? Show me the 10. Where’s the seven? The one?” Take turns gesturing over the counters to seek agreement on what you’re recording. </p>
<p><strong>Go slowly</strong>: Students’ thinking can happen fast, so slowing down what’s happening in their head is a good thing. To support slowing down, ask questions like: “What did you do first?”</p>
<p><strong>Let the learner take the lead</strong>: If the learner says their strategy is: “I subtracted seven first, then one more,” pull seven counters down and ask them which counter (from the remaining 10) should be pulled down to make the eight. The learner might choose a counter different than you. There is no correct answer here — pulling the middle counter down, for example, might make it easier for them to see the remaining nine. </p>
<p>We would love to hear how these moves support talking about math!</p><img src="https://counter.theconversation.com/content/167259/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors 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>Math educators share four ways parents can use counters, like candies or lego pieces, to lead ‘number talks’ that help kids develop an understanding of how numbers are related.Marc Husband, Assistant Professor, School of Education, St. Francis Xavier UniversityHeather Bourrie, PhD Candidate, Mathematics Education, York University, CanadaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1651212021-07-28T14:58:26Z2021-07-28T14:58:26ZWhy animals recognise numbers but only humans can do maths<figure><img src="https://images.theconversation.com/files/413541/original/file-20210728-25-18i3lz8.jpeg?ixlib=rb-1.1.0&rect=1015%2C8%2C4448%2C3628&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/orangutans-posting-thinking-acting-sit-photographer-768691744">Everything I Do/Shutterstock</a></span></figcaption></figure><p>Counting feels utterly effortless to adults, who are unlikely to even remember when or how they picked up this useful, apparently automatic skill. Yet when you think about it, counting is a remarkable invention. It helped early humans to trade, apportion food and organise fledgling civilisations, laying the foundations for life as we know it today.</p>
<p>But a sensitivity for numbers isn’t uniquely human. Tiny <a href="https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0031923">guppies</a> and <a href="https://royalsocietypublishing.org/doi/10.1098/rsbl.2019.0138">honeybees</a> as well as <a href="https://www.sciencedirect.com/science/article/abs/pii/S0003347211002843">hyenas</a> and <a href="https://link.springer.com/article/10.1007/s10071-002-0140-0">dogs</a> have been found to perceive and act on numerical stimuli. So responding to numbers is an <a href="https://www.cell.com/trends/cognitive-sciences/fulltext/S1364-6613(21)00087-5?dgcid=raven_jbs_aip_email#%20">evolved trait</a> we seem to share with some animals, as well as a skill we’re taught in some of our first lessons. </p>
<p>As a researcher in numerical cognition, I’m interested in how brains process numbers. Humans and animals actually share some remarkable numerical abilities – helping them make smart decisions about where to feed and where to take shelter. But as soon as language enters the picture, humans begin outperforming animals, revealing how words and digits underpin our advanced mathematical world.</p>
<h2>Two number systems</h2>
<p>When we think of counting, we think of “one, two, three”. But that of course relies on numerical language, which young humans and animals do not possess. Instead, they use two distinct number systems.</p>
<p>From as young as <a href="https://www.jstor.org/stable/40063857">ten months old</a>, human infants are already getting to grips with numbers. But there’s a limit to their numerical skills: they can only detect number changes between one and three, as when one apple is removed from a group of three apples. This skill is shared by many <a href="https://www.cell.com/trends/cognitive-sciences/fulltext/S1364-6613(21)00087-5?dgcid=raven_jbs_aip_email#%20">animals</a> with significantly smaller brains, such as fish and bees.</p>
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Read more:
<a href="https://theconversation.com/bees-join-an-elite-group-of-species-that-understands-the-concept-of-zero-as-a-number-97316">Bees join an elite group of species that understands the concept of zero as a number</a>
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<p>This early numerical system, helping infants and animals perceive the number of a small set of objects without having to actually count, <a href="https://jov.arvojournals.org/article.aspx?articleid=2191634">probably relies</a> on an internal <a href="https://pubmed.ncbi.nlm.nih.gov/21679934/">attentional working memory</a> system that is overwhelmed by numbers above around three.</p>
<p>As we grow up, we become able to estimate far higher numbers, again without needing to refer to language. Imagine you’re a hungry hunter-gatherer. You see two bushes, one with 400 redcurrants and the other with 500. It’s preferable to approach the bush with the most fruit, but it’s a big waste of time to count the berries on each bush individually. </p>
<p>So we estimate. And we do this with another internal number system specialised for approximating large numbers imprecisely – the so-called “<a href="https://www.sciencedirect.com/topics/psychology/approximate-number-system">approximate number system</a>”. Given that there’s a clear evolutionary advantage for those who can quickly pick the most bountiful food source, it’s unsurprising that <a href="https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0015232">fish</a>, <a href="https://royalsocietypublishing.org/doi/10.1098/rspb.2016.0083">birds</a>, <a href="https://pubmed.ncbi.nlm.nih.gov/31601685/">bees</a>, <a href="https://www.sciencedirect.com/science/article/abs/pii/S0376635713000284">dolphins</a>, <a href="https://pubmed.ncbi.nlm.nih.gov/22692435/">elephants</a> and <a href="https://dl.acm.org/doi/abs/10.1162/jocn.2008.21032">primates</a> have all been found to possess an approximate number system.</p>
<figure class="align-center ">
<img alt="A crow on a branch" src="https://images.theconversation.com/files/413543/original/file-20210728-17-1w0piyt.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/413543/original/file-20210728-17-1w0piyt.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=394&fit=crop&dpr=1 600w, https://images.theconversation.com/files/413543/original/file-20210728-17-1w0piyt.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=394&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/413543/original/file-20210728-17-1w0piyt.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=394&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/413543/original/file-20210728-17-1w0piyt.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=495&fit=crop&dpr=1 754w, https://images.theconversation.com/files/413543/original/file-20210728-17-1w0piyt.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=495&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/413543/original/file-20210728-17-1w0piyt.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=495&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Crows are able to make numerical estimations, studies have shown.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/carrion-crow-corvus-corone-perched-on-1715185915">Sandra Standbridge/Shutterstock</a></span>
</figcaption>
</figure>
<p>In humans, the precision of this system improves with development. <a href="https://www.pnas.org/content/106/25/10382">Newborns</a> can estimate approximate differences in numbers at a ratio of 1:3, so will be able to tell a bush with 300 berries has more berries than one with 100. Come <a href="https://www.pnas.org/content/109/28/11116">adulthood</a>, this system is honed to a 9:10 ratio. </p>
<p>Even though these two systems appear in a range of animals, including young humans, this doesn’t necessarily mean that the brain systems behind them are the same across all animals. But seeing as so many animal species can extract numerical information, it does appear that a sensitivity to numbers <a href="https://www.sciencedirect.com/science/article/abs/pii/S1364661321000875">evolved</a> in many species a very long time ago.</p>
<h2>Number symbols</h2>
<p>What sets us apart from non-human animals is our ability to represent numbers with symbols. It’s not entirely clear when humans first started to do this, though it has been suggested that <a href="https://www.nature.com/articles/d41586-021-01429-6">marks made on animal bones</a> by our Neanderthal relatives 60,000 years ago are some of the first archaeological examples of symbolic counting.</p>
<p>Externalising the process of counting may have started with our body parts. Fingers are <a href="https://www.sciencedirect.com/science/article/pii/S0010027712000960?casa_token=MdDbDhlaGNsAAAAA:odJ4ktgkn6mipla3JOUJXN6W1Lj1M_RyePIMDoiouOHoxzoOnVvLeFlo72HesKRcbp6loJYhB64">natural counting tools</a>, but are limited to ten. The traditional counting system of the <a href="https://journals.sagepub.com/doi/pdf/10.1177/0022022194251005?casa_token=Q4OFxrNE-mcAAAAA:nPXowBd-xrtP6YAdWbVAIk38ziVChTpxnCx87DQQMUxfrTsWUhOvGe_9sH0gbLdFNj-JcWYptWETOg">Yupno in Papua New Guinea</a> extended this to 33 by counting on additional body parts, starting with the toes, then the ears, eyes, nose, nostrils, nipples, the navel, the testicles and the penis.</p>
<p>But as our appetite for numbers grew, we began using more advanced symbolic systems to represent them. Today, most humans use the <a href="https://www.britannica.com/topic/Hindu-Arabic-numerals">Hindu-Arabic numeral system</a> to count. An amazing invention, it uses just ten symbols (0-9) in a positional system to represent an infinite set of numbers.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Early Arabic numerals" src="https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=90&fit=crop&dpr=1 600w, https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=90&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=90&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=114&fit=crop&dpr=1 754w, https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=114&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/413546/original/file-20210728-17-f58o76.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=114&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">These 1,500 year-old Bakhshali numerals prefigured our present-day numerical system.</span>
<span class="attribution"><a class="source" href="https://upload.wikimedia.org/wikipedia/commons/5/58/Bakhshali_numerals_2.jpg">Augustus Hoernle/wikimedia</a></span>
</figcaption>
</figure>
<p>When children acquire the meaning of numerical digits, they already know number words. Indeed, the words for small numbers are typically within the <a href="https://pubmed.ncbi.nlm.nih.gov/12471971/">first few hundred</a> words that children produce, <a href="https://www.sciencedirect.com/science/article/abs/pii/001002859290008P">reciting sequences</a> like “one-two-three-four-five” with ease. </p>
<p>What’s interesting here is that it takes young children some time to grasp the fact that the last word in the counting sequence doesn’t only describe the order of the object in the count list (the fifth object), but also the number of all objects counted so far (five objects). While this is obvious to the numerate adult, the so-called “<a href="https://link.springer.com/article/10.1007/s11229-015-0854-6">cardinality principle</a>” is a conceptually difficult and important step for children, and takes months to learn.</p>
<p>Number word learning is also shaped by the language environment. The Munduruku, an indigenous tribe in the Amazon, have very few words for exact numbers, and instead use approximate words to denote other quantities, such as “some” and “many”. Outside their exact number word vocabulary, the Munduruku’s <a href="https://science.sciencemag.org/content/306/5695/499.full">calculation performance</a> is always approximate. This shows how different language environments affect people’s accuracy when it comes to naming large exact numbers.</p>
<h2>Counting to calculating</h2>
<p>Many children and adults struggle with mathematics. But are any of these number systems linked to mathematical ability? In <a href="https://psycnet.apa.org/record/2019-63841-001">one study</a>, pre-school children with a more precise approximate number system were found to be more likely to do well in arithmetic in the following year compared to their peers with a less precise approximate number system. But in general, these effects have been <a href="https://pubmed.ncbi.nlm.nih.gov/26768176/">small and controversial</a>. </p>
<p>The ability to <a href="https://www.sciencedirect.com/science/article/pii/S0885201420301210">move</a> from spoken number words (twenty-five) to written number symbols (25) is a more reliable predictor of <a href="https://journals.sagepub.com/doi/abs/10.1177/0956797613516471">arithmetic skills</a> in children in primary school. Again, this shows that language plays a central role in how humans calculate as well as how humans count.</p>
<p>So while animals and humans are routinely extracting numerical information from their environment, it’s language that ultimately sets us apart – helping us not only pick the bush most laden with berries, but perform the kind of calculations upon which civilisation rests.</p><img src="https://counter.theconversation.com/content/165121/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Silke Goebel has received funding from the Economic and Social Research Council (UK), the British Academy and the Experimental Psychology Society. </span></em></p>A wide range of animals seem to have a grasp of numbers – but humans hold the trump card.Silke Goebel, Reader (Associate Professor) in Psychology, University of YorkLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1168202019-05-10T12:41:33Z2019-05-10T12:41:33ZAI develops human-like number sense – taking us a step closer to building machines with general intelligence<figure><img src="https://images.theconversation.com/files/273791/original/file-20190510-183100-1mny09v.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-illustration/3d-rendering-set-robotic-hands-shown-1156860970">Gearstd/Shutterstock</a></span></figcaption></figure><p>Numbers figure pretty high up on the list of what a computer can do well. While humans often struggle to split a restaurant bill, a modern computer can make millions of calculations in a mere second. Humans, however, have an innate and intuitive number sense that helped us, among other things, to build computers in the first place.</p>
<p>Unlike a computer, a human knows when looking at four cats, four apples and the symbol 4 that they all have one thing in common – the abstract concept of “four” – without even having to count them. This illustrates the difference between the human mind and the machine, and helps explain why we are <a href="https://theconversation.com/worried-about-ai-taking-over-the-world-you-may-be-making-some-rather-unscientific-assumptions-103561">not even close</a> to developing AIs with the broad intelligence that humans possess. But now a new study, <a href="https://advances.sciencemag.org/content/5/5/eaav7903">published in Science Advances</a>, reports that an AI has spontaneously developed a human-like number sense. </p>
<p>For a computer to count, we must clearly define what the thing is we want to count. Once we allocate a bit of memory to maintain the counter, we can set it to zero and then add an item each time we find something we want to record. This means that computers can count time (signals from an electronic clock), words (if stored in the computer’s memory) and even objects in a digital image.</p>
<p>This last task, however, is a bit challenging, as we have to tell the computer exactly <a href="https://theconversation.com/breast-cancer-diagnosis-by-ai-now-as-good-as-human-experts-115487">what the objects look like</a> before it can count them. But objects don’t always look the same – variation in lighting, position and pose have an impact, as well as any differences in construction between individual examples. </p>
<p>All the successful computational approaches to detecting objects in images work by building up a kind of statistical picture of an object from many individual examples – a type of learning. This allows the computer to recognise new versions of objects with some degree of confidence. The training involves offering examples that do, or do not, contain the object. The computer then makes a guess as to whether it does, and adjusts its statistical model according to the accuracy of the guess – as judged by a <a href="https://en.wikipedia.org/wiki/Supervised_learning">human supervising the learning</a>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/273794/original/file-20190510-183100-zgy1jo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/273794/original/file-20190510-183100-zgy1jo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/273794/original/file-20190510-183100-zgy1jo.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/273794/original/file-20190510-183100-zgy1jo.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/273794/original/file-20190510-183100-zgy1jo.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/273794/original/file-20190510-183100-zgy1jo.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/273794/original/file-20190510-183100-zgy1jo.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"></span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-illustration/3d-rendering-head-female-robot-breaking-790480444?src=5H_1wZnobUmOHaUmdiKISg-1-19">Sarah Holmlund/Shutterstock</a></span>
</figcaption>
</figure>
<p>Modern AI systems automatically start to being able to detect objects when provided with millions of training images of any sort – just like humans do. These <a href="https://www.mathworks.com/discovery/unsupervised-learning.html">unsupervised learning systems</a> gradually notice parts of the elements in the images that are often present at the same time, and build up layer upon layer of more complicated commonalities.</p>
<p>Take recognising apples as an example. As images containing all manner of shapes are presented to the system, it first begins to notice groups of pixels that make up horizontal and vertical lines, and left and right curves. They’re present in apples, faces, cats and cars, so the commonalities, or abstractions, are found early on. It eventually realises that certain curves and lines are often present together in apples – and develops a new, deeper level abstraction that represents a class of objects: apples, in this case.</p>
<h2>Deep learning</h2>
<p>This natural emergence of high-level abstractions is one of the most exciting results of the machine learning technique called <a href="https://en.wikipedia.org/wiki/Deep_learning">deep neural networks</a>, which in some sense work in a similar way to the human brain. The “depth” comes from the <a href="https://theconversation.com/deep-learning-could-prevent-you-from-drunk-posting-to-facebook-35435">many layers in the network</a> – as the information goes deeper into the network, the commonalities found become more abstract. In this way, networks are created with elements that are strongly active when the input is similar to what it has experienced before. The most abstract things appear at the deepest levels – these are cats, faces and apples rather than vertical lines or circles.</p>
<p>When an AI system can recognise apples, you can then use it to count how many there are. That’s great, yet it’s not quite how you or I would count apples. We have an extremely deep concept of “number” – how many of something there is. Rather than just being active when an object is present, parts of our brain activate depending on the amount of objects present. It means we can look at a bunch of apples and know that there are four without actually counting each one. </p>
<p>In fact, many animals can do this too. That’s because this sense of numerosity is a useful trait for survival and reproduction in a lot of different situations – take for instance judging the size of groups of rivals or prey. </p>
<h2>Emergent properties</h2>
<p>In the new study, a deep neural network that was trained for simple visual object detection spontaneously developed this kind of number sense. The researchers discovered that specific units within the network suddenly “tuned” to an abstract number – just like real neurons in the brain might respond. It realised that a picture of four apples is similar to a picture of four cats – because they have “four” in common.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/273792/original/file-20190510-183106-e6j48g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/273792/original/file-20190510-183106-e6j48g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=327&fit=crop&dpr=1 600w, https://images.theconversation.com/files/273792/original/file-20190510-183106-e6j48g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=327&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/273792/original/file-20190510-183106-e6j48g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=327&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/273792/original/file-20190510-183106-e6j48g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=410&fit=crop&dpr=1 754w, https://images.theconversation.com/files/273792/original/file-20190510-183106-e6j48g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=410&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/273792/original/file-20190510-183106-e6j48g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=410&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Artificial neurons tuned to preferred numbers of dots.</span>
<span class="attribution"><span class="source">Andreas Nieder</span></span>
</figcaption>
</figure>
<p>One really exciting thing about this research is that it shows that our current principles of learning are quite fundamental. Some of the most high-level aspects of thinking that people and animals demonstrate are related deeply to the structure of the world, and our visual experience of that.</p>
<p>It also hints that we might be on the right track to achieve a more comprehensive, human-level artificial intelligence. Applying this kind of learning to other tasks – perhaps applying it to signals that occur over a period of time rather than over pixels in an image – could yield machines with even <a href="https://theconversation.com/will-ai-ever-understand-human-emotions-70960">more human-like qualities</a>. Things we once thought fundamental to being human – <a href="https://musically.com/2019/04/04/strictly-algo-rhythm-ai-music-is-nothing-to-be-scared-of/">musical rhythm</a> for example, or even a <a href="https://phys.org/news/2018-06-artificial-intelligence-causation.html">sense of causality</a> – are now being examined from this new perspective.</p>
<p>As we continue to discover more about building artificial learning techniques, and find new ways to understand the brains of living organisms, we unlock more of the mysteries of intelligent, adaptive behaviour.</p>
<p>There’s a long way to go, and many other dimensions <a href="https://theconversation.com/why-ai-cant-solve-everything-97022">that we need to explore</a>, but it’s clear that the capability to look at the world and work out its structure from experience is a key part of what makes humans so adaptable. There’s no doubt it will be a necessary component of any AI system that has the potential to perform the variety and complexity of tasks that humans can.</p><img src="https://counter.theconversation.com/content/116820/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adam Stanton 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>A human knows that four cats, four apples and the symbol 4 all have one thing in common – the abstract concept of ‘four’. Now robots are catching up.Adam Stanton, Lecturer in Evolutionary Robotics and Artificial Life, Keele UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1031942018-09-18T10:48:59Z2018-09-18T10:48:59Z5 math skills your child needs to get ready for kindergarten<figure><img src="https://images.theconversation.com/files/236470/original/file-20180914-177953-cvik29.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Getting ready for school.</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/child-learning-count-lego-blocks-620986742?src=e2YWwtQSqdRemA5wteUvYQ-1-5">Studio.g photography/shutterstock</a></span></figcaption></figure><p><em><a href="https://theconversation.com/5-habilidades-matematicas-que-los-ninos-en-edad-preescolar-deben-aprender-enseneselas-de-forma-divertida-103654">Leer en español</a></em>.</p>
<p>Parents <a href="https://doi.org/10.1016/j.jecp.2018.07.005">play a critical role</a> in their children’s early math education. They not only can provide math-related toys and games, but serve as role models demonstrating how math is used in everyday activities. </p>
<p>Children <a href="http://doi.org/10.1155/2012/851657">who see their parents doing everyday math</a> engage more often in math activities. This, in turn, <a href="http://dx.doi.org/10.1080/15427609.2016.1194707">builds early math skills</a>, which <a href="http://doi.org/10.1037/0012-1649.43.6.1428">serve as the foundation</a> for later learning. </p>
<p>As researchers who study children’s math development, we believe there are five math skills that children should have at the start of kindergarten. Opportunities for learning these skills are everywhere – and there are simple, enjoyable activities that parents can lead to foster these skills. </p>
<p>This will help children acquire the age-appropriate vocabulary and skills needed for learning math, while staying engaged and having fun. </p>
<h2>1. Counting and cardinality</h2>
<p>According to <a href="http://mdk12.msde.maryland.gov/instruction/curriculum/mathematics/index.html">the college and career-ready standards in our state, Maryland,</a> children are expected to demonstrate simple counting skills before starting kindergarten. These skills include counting to 20; ordering number cards; identifying without counting how many items are in a small set; and understanding that quantity does not change regardless of how a set of items is arranged. </p>
<p>Children also will need to learn cardinality. That means they should understand that the last item counted represents the number of items in the set. </p>
<p>Counting and cardinality can be easily integrated into daily life. Children can count their toys as they clean up or count how many steps it takes to walk from the kitchen to their bedroom. Parents can point out numbers on a clock or phone. </p>
<p>In the grocery store, parents can ask children to find numbers while shopping. In the car, parents can have children read the numbers on license plates or count passing cars. Parents should ask, “How many?” after a child has counted, to reinforce the idea of cardinality. </p>
<p>Board games like Trouble, Hi Ho Cherry-O and Chutes and Ladders are helpful and fun ways to hone counting and cardinality skills. Have children identify the number on the die or spinner when they take their turn and count aloud when they move their piece. Active games that involve counting aloud – like jump rope, hopscotch or clapping – also foster these skills. </p>
<h2>2. Operations and algebraic thinking</h2>
<p>Kindergartners are expected to solve simple addition and subtraction problems using objects. </p>
<p>Parents can have children do simple math problems during everyday tasks. For example, they can ask children to take out the correct number of plates or utensils when setting the table for dinner. Remember, the math language children hear matters. Parents can ask questions like, “How many more plates do we need?”“</p>
<p>During play, parents can use toys and say things like, "I’m going to give you one of my cars. Let’s count how many cars you have now.” Songs and rhymes that include counting up or counting down, such as Five in the Bed or Teasing Mr. Crocodile, can also be useful for teaching early addition and subtraction. </p>
<h2>3. Numbers and operations in base 10</h2>
<p>Children need to begin to understand that the number “ten” is made up of 10 “ones.” </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/236472/original/file-20180914-96155-55h5v9.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"></a>
<figcaption>
<span class="caption">Playing with coins can help children learn about numbers.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/cute-asian-little-girl-playing-coins-648932986?src=wE7YpB0byAdguC53pVpTyA-1-3">A3pfamily/shutterstock</a></span>
</figcaption>
</figure>
<p>Counting fingers and toes is a great way to emphasize the numbers one through 10. Money, coins in particular, is another great way to emphasize base 10. Parents can play store with their children using pennies and have them “purchase” toys for differing amounts of pennies. During play, they can talk about how many toys they can buy with 10 cents. </p>
<h2>4. Measurement and data</h2>
<p>Kindergartners are expected to sort objects by their features – like shape, color and size – or identify the feature by which objects have been sorted. They also are expected to order objects by some measurable feature, such as from bigger to smaller. </p>
<p>In the kitchen, children can begin experimenting with measurement using spoons or cups. Children can sort utensils, laundry or toys as they put them away. Card and dice games, such as War, are helpful for talking about number magnitude. Additionally, several inexpensive sorting games, such as Ready Sets Go or Ready Set Woof, are commercially available. </p>
<p>Additionally, kindergartners should be able to compare objects and use language like more than or less than, longer or shorter, and heavier or lighter. Parents can help by using these words to emphasize comparisons. When children are helping with tasks, parents can ask questions like, “Can you hand me the biggest bowl?” or “Can you put the smaller forks on the table?” </p>
<h2>5. Geometry</h2>
<p>Early geometry skills include naming and identifying 2D shapes like circles, squares and triangles. Children also need to realize that shapes of different sizes, orientations and dimensions are similar. Children should be able to recognize that a circle is like a sphere and use informal names like “box” and “ball” to identify three-dimensional objects. </p>
<p>Parents can draw children’s attention to shapes found in the environment. On a walk, parents can point out that wheels are circles and then have children find other circles in the environment. Commercially available games like Perfection or Tangrams can help children learn to identify simple and more complex shapes. Puzzles, blocks and Legos are another great way to help build early spatial skills.</p><img src="https://counter.theconversation.com/content/103194/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Susan Sonnenschein received a small grant from the Psychology Department at UMBC for some of the research discussed in this article . </span></em></p><p class="fine-print"><em><span>Rebecca Dowling and Shari Renee Metzger do not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Through games and household tasks, parents can help their children learn basic math skills like counting, geometry and algebraic thinking.Susan Sonnenschein, Professor, Applied Development Psychology, University of Maryland, Baltimore CountyRebecca Dowling, Doctoral Student in Applied Developmental Psychology, University of Maryland, Baltimore CountyShari Renee Metzger, Research Analyst, Prince George's Community CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/780342017-08-20T19:25:33Z2017-08-20T19:25:33ZCurious Kids: Why do we count to 10?<figure><img src="https://images.theconversation.com/files/172582/original/file-20170606-3690-15mo08p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Nature gave us ten fingers, so it makes sense to count to ten. But what happens when we run out of fingers?</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/beth19/4850553530/in/photolist-8oCmzh-6g3FBA-9HiCKn-8UHnRU-9dBUHa-zcNXD-7j2X9j-4z9LEf-7itdqx-dVy7PS-gvFETp-4z9NVY-8rcvSw-4z5xaM-4z9LTQ-82SYHc-9ftJBe-e4qozR-7L7d9T-LmVvz-8tGabm-6Eqw2o-e3xoVV-Tjywea-KdsxK-ctgxj9-aTy7vZ-RnigVE-au4qjr-aThawP-TNY76-UiCrVk-4z5wrn-fFnfC-h9k8Cf-8ho24x-UZUfCr-C98PTz-4z9Kkf-bsZQP9-7jAqNH-DmxSsG-rRVWGH-SfgyrE-d6eNy-8gjNF4-qL4ptZ-cxncg-8xTFJs-LzP1c">Flickr/Bethan</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p><em>This is an article from <a href="https://theconversation.com/au/topics/curious-kids-36782">Curious Kids</a>, a series for children. The Conversation is asking kids to send in questions they’d like an expert to answer. All questions are welcome – serious, weird or wacky!</em> </p>
<hr>
<blockquote>
<p><strong>Why do we count to 10? – Quentin, age 5, Randwick.</strong></p>
</blockquote>
<p>Counting is perhaps one of the oldest scientific operations still in use today. </p>
<p>From a young age we learn to count the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are called the digits. But there is a problem with ten: we have to write it differently because we used up all our digits!</p>
<p>What do we do now? </p>
<p>We use two digits. The number 10 has a left digit “1” which has a new meaning. It represents the number of times we ran out of digits. The right digit “0” is the same as before and lets us continue counting again. Mathematicians call this a <a href="https://en.wikipedia.org/wiki/Positional_notation">place-value</a> number system, and counting in tens is called the decimal system. Australia and the UK use the decimal system to count money, distances and lots of other things we need to measure or count. </p>
<p>Machines also count, but not in tens. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/170059/original/file-20170519-12242-fgrp6g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/170059/original/file-20170519-12242-fgrp6g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/170059/original/file-20170519-12242-fgrp6g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/170059/original/file-20170519-12242-fgrp6g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/170059/original/file-20170519-12242-fgrp6g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/170059/original/file-20170519-12242-fgrp6g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/170059/original/file-20170519-12242-fgrp6g.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">What happens when machines run out of numbers?</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/industrial-switch-289201325?src=-fDSnhgPk0yiRcPwRefIKQ-1-2">Riggsby/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Nature gave us ten fingers, and so it is natural for us to count in tens. But machines are built using switches, so it is natural for them to count only off (0) and on (1). This is like counting on one hand that only has one finger.</p>
<p>Machines count bigger numbers in the same way we do: by counting how many times they run out of digits. This system is called binary and the binary number 10 means the machine ran out of digits one time. A human would call this number two.</p>
<p>Today, these are the main ways of counting. But they are just two different ways of doing the same thing.</p>
<p>What about time? </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/170060/original/file-20170519-12217-35fhqm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/170060/original/file-20170519-12217-35fhqm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/170060/original/file-20170519-12217-35fhqm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/170060/original/file-20170519-12217-35fhqm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/170060/original/file-20170519-12217-35fhqm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/170060/original/file-20170519-12217-35fhqm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/170060/original/file-20170519-12217-35fhqm.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">The way we measure and count time comes from the ancient Sumerians who lived thousands of years ago.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/vastfield/2769883027/">vastfield/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>The big hand of a clock has 60 different digits: 0, 1, 2, all the way up to 59. But what happens when we have used up all of those digits? </p>
<p>Like before, we count the number of times we run out digits, and we call each one an hour. Counting in this way is called sexagesimal.</p>
<p>But why do we use a different measurement for time? </p>
<p>We inherited the sexagesimal system from <a href="https://en.wikipedia.org/wiki/Sumer">the Sumerians</a> more than 4,000 years ago. It has lasted for so long because you can easily divide a number into two, three, four, five or six equal parts. Try dividing an hour into three equal parts and you will see there are 20 minutes each. Now try dividing a dollar into three equal parts and you will see there are 33, 33 and 34 cents each.</p>
<p>Our world uses many different place-value number systems, and they are all useful for different reasons.</p>
<hr>
<p><em>Hello, curious kids! Have you got a question you’d like an expert to answer? Ask an adult to send your question to us. You can:</em></p>
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<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
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<p class="fine-print"><em><span>Daniel Mansfield 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>Why are there 60 minutes in an hour, and not 10? Why do we count up to 10, anyway? Quentin, age five, wants to know.Daniel Mansfield, Associate Lecturer in Mathematics, UNSW SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/657102016-09-27T09:56:51Z2016-09-27T09:56:51ZGroup work gets kids more engaged in STEM<figure><img src="https://images.theconversation.com/files/139342/original/image-20160926-31875-1ludi3d.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">What can be done to get more kids interested in STEM?</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-397583227/stock-photo-close-up-of-little-girl-having-fun-focus-is-on-girl-hands.html?src=PCBl5f9Ak13kVf_HPtmzgg-1-1">Child image via www.shutterstock.com</a></span></figcaption></figure><p>Shortage of science, technology, engineering and math (STEM) professionals has been an important <a href="https://www.whitehouse.gov/blog/2016/03/01/supporting-our-youngest-innovators-stem-starts-early">concern in the United States</a>. It is estimated that over the next 10 years, the nation could face a <a href="https://www.whitehouse.gov/sites/default/files/microsites/ostp/fact_sheet_final.pdf">shortage of one million STEM workers</a>. </p>
<p>So, what can we do to get more students interested in STEM?</p>
<h2>STEM starts early</h2>
<p>Research shows that <a href="http://edr.sagepub.com/content/45/1/18.short">science achievement gaps begin very early</a>. Between fourth and eighth grade, the number of children reporting positive attitudes about <a href="http://timss.bc.edu/timss2007/PDF/T07_M_IR_Chapter4.pdf">math</a> and <a href="http://timss.bc.edu/timss2007/PDF/T07_S_IR_Chapter4.pdf">science</a> drops from about 71 percent to about 48 percent.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/139333/original/image-20160926-31856-101nlt5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/139333/original/image-20160926-31856-101nlt5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/139333/original/image-20160926-31856-101nlt5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/139333/original/image-20160926-31856-101nlt5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/139333/original/image-20160926-31856-101nlt5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/139333/original/image-20160926-31856-101nlt5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/139333/original/image-20160926-31856-101nlt5.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">
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<span class="caption">Early math activities such as counting can build STEM skills.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-56610562/stock-photo-little-kid-playing-with-abacus.html?src=6u3PnwRXLCZRjV-UsM1ymA-1-4">Abacus image via www.shutterstock.com</a></span>
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<p>Early math skills are the <a href="http://psycnet.apa.org/journals/dev/43/6/1428/">strongest predictor</a> of later school success. Simple activities like <a href="http://www.sciencedirect.com/science/article/pii/S0022096513002439">counting</a> and <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3289766/">playing with puzzles</a> can <a href="http://groups.psych.northwestern.edu/uttal/vittae/documents/ContentServer.pdf">build children’s STEM skills</a>. </p>
<p>Studies show that preschool children who participated in a 26-week math curriculum had math test scores that <a href="http://aer.sagepub.com/content/45/2/443">improved twice as much</a> as children in a control group with a standard preschool curriculum. Even <a href="http://earlymath.org/earlymath/wp-content/uploads/2014/09/Math-Matters-Report_2ndEd1.pdf">talking more about math</a> and <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1467-7687.2008.00714.x/full">playing board games</a> can boost preschool children’s STEM abilities. </p>
<p>However, these educational activities are also competing with lots of other things for children’s attention. Less than <a href="http://www.usnews.com/news/articles/2016-06-15/op-ed-integrating-stem-learning-in-early-childhood-education">five percent</a> of classroom time in preschool focuses on STEM-related activities. </p>
<p>I am part of a research team at the University of Washington’s <a href="http://staff.washington.edu/almaster/cv.html">Institute for Learning and Brain Sciences</a>. With my colleagues Sapna Cheryan and Andrew Meltzoff, I have been looking for ways to make STEM more engaging for children.</p>
<p>We found an answer: Make it social.</p>
<h2>Here’s what we did</h2>
<p>We <a href="http://psycnet.apa.org/psycinfo/2016-42715-001/">ran an experiment</a> to see whether making STEM social would affect children’s motivation. We brought 141 four-year-old children into our lab. They did two activities, a math game and a puzzle game.</p>
<p>For one of these activities, children were made to believe that they were part of a group. Children were required to do the other game all by themselves. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/138352/original/image-20160919-11103-j0wv7r.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/138352/original/image-20160919-11103-j0wv7r.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=447&fit=crop&dpr=1 600w, https://images.theconversation.com/files/138352/original/image-20160919-11103-j0wv7r.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=447&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/138352/original/image-20160919-11103-j0wv7r.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=447&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/138352/original/image-20160919-11103-j0wv7r.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=562&fit=crop&dpr=1 754w, https://images.theconversation.com/files/138352/original/image-20160919-11103-j0wv7r.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=562&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/138352/original/image-20160919-11103-j0wv7r.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=562&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A child about to work on a STEM task in the ‘green’ group.</span>
<span class="attribution"><span class="source">Allison Master</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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<p>Each group had a special color. For example, children in the green group put on a green t-shirt. Then they sat at a green table with a green flag, and took the group’s activity out of a green box. In reality, all children actually completed both activities alone. All that they saw was a poster that showed pictures of other children in the group, all wearing a green t-shirt. </p>
<p>For the other nongroup task, children were also provided with t-shirts of a certain color. There was a poster on the wall with pictures of other children. However, that poster showed children wearing different colored shirts that did not match theirs. For example, if children wore a yellow shirt for the nongroup task, then none of the children on the poster would have a yellow t-shirt. This helped emphasize their solo status. We also reminded them that none of the children on the poster did the same activity as them.</p>
<p>We made the group imaginary because children’s groups in real life can be complicated. With imaginary groups, we could make the experience exactly the same for all children and test how the idea of being part of a group is motivating.</p>
<h2>Impact on motivation</h2>
<p>Our reason for having children believe they were part of a group was based on a simple idea: You are then part of something bigger than yourself. Other people are <a href="https://www.cambridge.org/core/journals/behavioral-and-brain-sciences/article/understanding-and-sharing-intentions-the-origins-of-cultural-cognition/F9C40BF73A68B30B8EB713F2F947F7E2">working toward the same goal as you</a>. <a href="http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0152001">Even young children</a> understand that working together unites people in meaningful ways. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/139331/original/image-20160926-31853-1cj51cs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/139331/original/image-20160926-31853-1cj51cs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/139331/original/image-20160926-31853-1cj51cs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/139331/original/image-20160926-31853-1cj51cs.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/139331/original/image-20160926-31853-1cj51cs.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/139331/original/image-20160926-31853-1cj51cs.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/139331/original/image-20160926-31853-1cj51cs.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">Children feel more motivated when they are part of a group.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-342072323/stock-photo-group-of-pre-school-children-answering-question-in-classroom.html?src=od4j7oDck9_EOzOlCsTcyw-1-0">Children image via www.shutterstock.com</a></span>
</figcaption>
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<p>Even with an imaginary group, children showed greater motivation for the group task compared to the individual task. Children worked for more time before choosing to quit the group task, and correctly placed more pieces for that task. </p>
<p>Afterwards, we asked children to rate how fun each task was, and how good they were at each one. On average, children rated the group task as more fun and said that they felt like they were better at it. When we asked children to pick which task they liked better, about half the children chose the group task, about one third chose the individual task, and a few had no preference.</p>
<h2>Similar findings from other studies</h2>
<p>Could this effect be due to <a href="http://pps.sagepub.com/content/10/6/886.full">something like random chance</a>? We are confident that this is a real effect for a couple of reasons.</p>
<p>First, in our study, all the children did two different tasks, one in a “group” and the other as an individual. Some children did the math task as their group task and others did it as their individual task. Same for the puzzle task: Some children did it as their group task and others did it as their individual task. </p>
<p>We found children showed greater motivation for whichever task they did as part of a group. On average, they showed greater motivation for the group task about 40 percent of the time, equal motivation on both tasks about 32 percent of the time and greater motivation for the individual task about 28 percent of the time. </p>
<p>So, it’s not something about the particular group task or the children who were put in a group. The exact same children were more motivated, on average, by being in a group than by working as an individual.</p>
<p>Second, <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1467-8624.2012.01867.x/abstract">several studies</a> that I conducted with <a href="http://gregorywalton-stanford.weebly.com">Professor Greg Walton</a> at Stanford University came up with similar findings. Those studies also looked at children’s motivation and learning when they were part of a group. In those studies, we found that children persisted longer on a puzzle when they were part of a puzzle group, and learned more new words when they were part of a word-learning group. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/139335/original/image-20160926-31847-s75zlx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/139335/original/image-20160926-31847-s75zlx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=411&fit=crop&dpr=1 600w, https://images.theconversation.com/files/139335/original/image-20160926-31847-s75zlx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=411&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/139335/original/image-20160926-31847-s75zlx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=411&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/139335/original/image-20160926-31847-s75zlx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=516&fit=crop&dpr=1 754w, https://images.theconversation.com/files/139335/original/image-20160926-31847-s75zlx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=516&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/139335/original/image-20160926-31847-s75zlx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=516&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Children persist longer on a puzzle when they are part of a group.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/criminalintent/14401901954/in/photolist-nWDvGo-e5TdEF-eF6fEB-bv8gWK-fdi1sv-bnt93F-ebAUvZ-aXBWJk-6pypBR-bJphaX-dm5Q3x-bm3W7W-adMQGX-bntgHk-9qRLWE-bKkNkp-e3a8Dj-aZS77e-a2MUY8-acMU5H-ebEMbb-b8LFGe-kHuhep-9uYa1k-bDcCjx-9TukK9-aZXo2x-r41AvX-eavyD4-b5enCv-qLJrmi-9zC2sW-6PRmcJ-iMSTfV-dRRK7S-ef2XJp-a2XcC1-6pyiwZ-f9ei57-fiDzAv-e9DBgK-6pypTD-dS3o9y-6pCy21-9UeF3e-7jU2EN-f4Hfgq-qJ5V65-eeJTag-qYnCby">Lars Plougmann</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>And if we combine our results with those studies into a meta-analysis – a study that analyses multiple previous studies – the effect is even stronger, even with imaginary groups and minimal information about the group.</p>
<h2>What parents, teachers can do</h2>
<p>Children spend a large portion of classroom time working independently. For example, one study found that American eighth graders worked individually <a href="http://nces.ed.gov/pubs2003/2003013.pdf">80 percent of the time</a> in math class. </p>
<p>So what does this mean for teachers trying to get students excited about math? For parents trying to get children passionate about puzzles? We have a couple of ideas about how parents and teachers can use these findings to talk about STEM. We haven’t tested these yet, but they send the message that STEM is social.</p>
<p>For example, parents and teachers can use social language such as, “Let’s figure this puzzle out together.” Teachers can also create classroom-wide groups to make sure everyone feels included: “Our whole class does math together.”</p>
<p>Children need to be engaged in STEM before they start <a href="http://psycnet.apa.org/psycinfo/2015-37516-001/">to lose interest</a>. The <a href="http://mindsetscholarsnetwork.org/wp-content/uploads/2015/09/Reduce-Gender-Gaps-in-pSTEM.pdf">image of STEM</a> as <a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2015.00049/full">solitary and isolating</a> is strong in our culture. If we make STEM social, we can help inspire more students to discover their interest in STEM.</p><img src="https://counter.theconversation.com/content/65710/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Allison Master has received funding from the National Science Foundation and the Bezos Family Foundation.</span></em></p>A study with pre-school children found that their motivation and interest improved when they believed they were part of a group.Allison Master, Research Scientist, University of WashingtonLicensed as Creative Commons – attribution, no derivatives.