tag:theconversation.com,2011:/ca/topics/basic-maths-skills-19320/articlesBasic maths skills – The Conversation2020-12-16T03:32:42Ztag:theconversation.com,2011:article/1502622020-12-16T03:32:42Z2020-12-16T03:32:42ZJump, split or make to the next 10: strategies to teach maths have changed since you were at school<figure><img src="https://images.theconversation.com/files/375248/original/file-20201215-15-1lh0rut.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/portrait-african-girl-writing-solution-sums-1078335890">Shutterstock</a></span></figcaption></figure><p>I’m sure most people can remember trying to master a certain maths rule or procedure in primary or secondary school.</p>
<p>My elderly mother has a story about a time her father was helping her with arithmetic homework. She remembers getting upset because her father did not do it “the school way”. I suspect her father was able to do the calculation mentally rather than the school way, which was to use the vertical algorithm.</p>
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<p>Students are expected to add the numbers in the ones (right) column first, before adding the numbers in the tens (left) column. The task becomes more difficult when the total of the ones column is more than 10 — as you then have to “trade” ten ones for one ten.</p>
<p>Students who give the answer as 713 rather than the correct answer of 83 may well have started with the tens column first. Or they may have written 13 in the ones column rather than trading ten ones for one ten.</p>
<p>The formal school algorithms are still used for larger numbers and decimals but we encourage students to use whichever strategy they prefer for two-digit addition. </p>
<p>The trouble with teaching rules is many students then struggle to remember when to apply the rule because they don’t understand how or why the rule works. </p>
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Read more:
<a href="https://theconversation.com/weapons-of-maths-destruction-are-calculators-killing-our-ability-to-work-it-out-in-our-head-44900">Weapons of maths destruction: are calculators killing our ability to work it out in our head?</a>
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<p>The <a href="https://australiancurriculum.edu.au/f-10-curriculum/mathematics/">Australian Curriculum: Mathematics</a> states that by the end of year 2, students will “perform simple addition and subtraction calculations using a range of strategies”. By the end of year 4, they will “identify and explain strategies for finding unknown quantities in number sentences”. </p>
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<p>We want children to remember how to do these equations in their head, rather than relying on writing down the process. Here are three strategies schools use to teach children how to add and subtract two-digit numbers.</p>
<h2>1. Split strategy</h2>
<p>This is sometimes called the decomposition, partitioning or partial-sums strategy. </p>
<p>You can add or subtract the tens separately to the ones (or units). For example, using the split strategy to add 46 + 23, you would:</p>
<ul>
<li><p>split each number (decompose) into tens and ones: 46 + 23 = 40 + 6 + 20 + 3</p></li>
<li><p>rearrange the tens and ones: 40 + 20 + 6 + 3</p></li>
<li><p>add the tens and then the ones 60 + 9 = 69</p></li>
</ul>
<p>Using the split strategy for addition such as 37 + 65 would be similar, but there would be an extra step:</p>
<ul>
<li><p>split or decompose the numbers into tens and ones: 30 + 7 + 60 + 5</p></li>
<li><p>rearrange the tens and ones: 30 + 60 + 7 + 5</p></li>
<li><p>add the tens and then the ones: 90 + 12 </p></li>
<li><p>split 12 (10 + 2) to give: 90 + 10 + 2 = 100 + 2 = 102</p></li>
</ul>
<p>Many students find the split strategy more difficult for subtraction than addition. This is because there are more steps if performing this strategy mentally. </p>
<p>For a subtraction such as 69 – 46, you would:</p>
<ul>
<li><p>split or decompose each number into tens and ones: 60 + 9 – (40 + 6)</p></li>
<li><p>remove bracket: 60 + 9 – 40 – 6</p></li>
<li><p>rearrange tens and ones: (60 – 40) + (9 – 6)</p></li>
<li><p>subtract the tens, then the ones: 20 + 3 = 23</p></li>
</ul>
<p>Students often make mistakes in the third step. Successful students may say: “I take 40 from 60, then 6 from 9”. Unsuccessful students will say “I take 40 from 60 then add 6 and 9”. </p>
<p>Students who use this strategy successfully are showing they understand place value (the value of each digit in a number) and their knowledge of maths rules needed for algebra. </p>
<h2>2. Jump strategy</h2>
<p>This is sometimes called the sequencing or cumulative sums strategy. The actual steps taken depend on the confidence and ability of the students. </p>
<p>Some students add increments of tens or ones, while others add or subtract multiples of tens then ones. </p>
<p>For example, adding 46 + 23 using the jump strategy might look like this:</p>
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<ul>
<li><p>add two lots of ten to 46: 46 + 10 = 56, then 56 + 10 = 66</p></li>
<li><p>add the remaining 3: 66 + 3 = 69 </p></li>
</ul>
<p>or </p>
<ul>
<li><p>add 20 to 46 which becomes 66</p></li>
<li><p>add the remaining 3: 66 + 3 = 69</p></li>
</ul>
<p>The two versions of this strategy can be shown using an empty number line. Using a blank or empty number line allows student to record their thinking and for teachers to analyse their thinking and determine the strategy they have attempted to use.</p>
<p>Subtracting 69 – 46 with the jump strategy could be done by:</p>
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<ul>
<li><p>subtracting four lots of ten (40) from 69: 69 – 10 = 59; 59 – 10 = 49; 49 – 10 = 39; 39 – 10 = 29 </p></li>
<li><p>then finally subtracting the remaining 6: 29 – 6 = 23 </p></li>
</ul>
<p>or</p>
<ul>
<li><p>subtract 40: 69 – 40 = 29 </p></li>
<li><p>then subtract 6: 29 – 6 = 23</p></li>
</ul>
<h2>3. ‘Make to the next ten’ strategy</h2>
<p>This is sometimes called the compensation or shortcut strategy. It involves adjusting one number to make the task easier to solve. </p>
<p>The “make to the next ten” strategy builds on the “friends of ten” strategy. </p>
<p>Many students in the first years of primary school create all the combinations of two single digit numbers that give a total of ten. </p>
<blockquote>
<p>9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5 … </p>
</blockquote>
<p>These are sometimes called the rainbow facts as the children create rainbows as they connect two numbers together. For instance, 9 may be on one end of a rainbow colour and 1 on the other. </p>
<p>By combining the numbers in this way teachers hope students will realise the answer for 9 + 1 is the same as 1 + 9.</p>
<p>In the “make to the next ten” strategy, you add or subtract a number larger than the number given (such as the next multiple of ten) and then readjust the number by subtracting what was added or adding what was subtracted.</p>
<p>In the diagrams the relationships are indicated by the use of arrows. </p>
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<p>So, to add 37 + 65, you would</p>
<ul>
<li><p>add 3 to 37 to give 40. </p></li>
<li><p>subtract 3 from 65 to get 62</p></li>
<li><p>this becomes: 40 + 62 = 102.</p></li>
</ul>
<p>If subtracting 102 – 65, you would:</p>
<ul>
<li><p>subtract 2 from 102 to make 100</p></li>
<li><p>subtract 2 from 65 to maintain the balance</p></li>
<li><p>this becomes 100 – 63 = 37.</p></li>
</ul>
<p>Many students using this strategy incorrectly add 2 to 65 instead of subtracting 2.</p>
<h2>Why these strategies?</h2>
<p>Students would have been using all these strategies, or some forms of them, in their head for generations. But for many years, the expectation was that students use the formal written algorithm rather than their own mental strategies. </p>
<p>The introduction of the empty or blank number line allowed students to record their mental strategies, which allowed teachers and parents to see them. Naming these strategies has allowed teachers and students to discuss possible strategies using a common vocabulary.</p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/kids-prefer-maths-when-you-let-them-figure-out-the-answer-for-themselves-44016">Kids prefer maths when you let them figure out the answer for themselves</a>
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<p>Rather than teach rules and procedures, we now need to encourage students to explain their strategies using both concrete materials and diagrams to demonstrate their knowledge of addition and subtraction.</p><img src="https://counter.theconversation.com/content/150262/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Cath Pearn is affiliated with both the Australian Council for Educational Research and the Melbourne Graduate School of Education, The University of Melbourne. </span></em></p>For years you may have been adding and subtracting numbers in your head in a certain way, but these strategies were never formally taught at school. Now they are, and they all have names.Cath Pearn, Senior Research Fellow, Australian Council for Educational ResearchLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/964412018-08-17T12:20:34Z2018-08-17T12:20:34ZMaths: six ways to help your child love it<figure><img src="https://images.theconversation.com/files/230790/original/file-20180806-191028-12mefqt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">shutterstock</span></span></figcaption></figure><p>There is a widespread perception that mathematics is inaccessible, and ultimately boring. Just mentioning it can cause a negative reaction in people, as many mathematicians witness at any social event when the dreaded question arrives: “what is your job?”</p>
<p>For many people, school maths lessons are the time when any interest in the subject turns into disaffection. And eventually maths becomes a topic many people don’t want to engage with <a href="http://www.bsrlm.org.uk/wp-content/uploads/2016/02/BSRLM-IP-27-1-04.pdf">for the rest of their lives</a>. A percentage of the population, at least 17% – possibly much higher depending on <a href="https://www.frontiersin.org/articles/10.3389/fpsyg.2016.00508/full">the metrics applied</a> – develops maths anxiety. This is a debilitating fear of performing any numerical task, which results in chronic underachievement in subjects involving mathematics.</p>
<p>At the opposite end of the spectrum, professional mathematicians see mathematics as <a href="https://www.lms.ac.uk/library/frames-of-mind">fun, engaging, challenging and creative</a>. And as maths fans, we are trying to address this chasm in perception of mathematics, to allow everybody to access its beauty and power. So here are our six ways you can help children fall back in love with mathematics. </p>
<h2>1. Focus on the whys</h2>
<p>The Australian teacher <a href="https://www.youtube.com/channel/UCq0EGvLTyy-LLT1oUSO_0FQ">Eddie Woo</a> has become an internet sensation for his engaging way of presenting mathematics. He starts from the ideas and, using pictures and graphs, develops the theory. </p>
<p>He does not ask his students to do repetitive exercises, but to work with him in developing intuition. And he asks the most powerful question a learner of mathematics can ask: “Why?”. It is possible to hear throughout his classes the “oohs” and “ahhs” of students in the background, when a novel concept is understood. </p>
<h2>2. Make it relevant</h2>
<p>Traditionally (and in particular in the UK) mathematics is taught in a systematic way, <a href="https://eclass.uoa.gr/modules/document/file.php/MATH103/ELENA%20NARDI/NARDI3.pdf">based on rote learning and individual study</a>. Some students thrive in such a system, others, typically more empathetic students – often female – find such an approach to mathematics isolating and disconnected from their values and their reality.</p>
<p>Connecting mathematical concepts with applications in reality can bring meaning to lessons and lectures, and motivate students to put in the necessary effort to understand. For example, derivatives – ways of calculating rates of change – can be introduced as a way to measure slopes, and slopes are experienced in everyday life – think about the skatepark or the big hill you cycle up. </p>
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<span class="caption">Make maths about real life to capture kids imaginations.</span>
<span class="attribution"><span class="source">Pexels</span></span>
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<h2>3. Recognise the challenge</h2>
<p>There is an effort component in learning mathematics. It can be challenging, and understanding it sometimes involves stress, frustration, and struggle over time. This can be an emotionally complex environment for children. But it is one where persistence and perseverance are rewarded when a new concept is understood. </p>
<p>With each success, students gain confidence that they can progress in learning more mathematics. In this way, learning mathematics can be compared to climbing a mountain: plenty of effort, but also some truly blissful moments.</p>
<h2>4. Be a maths role model</h2>
<p>Some people like to climb mountains solo, while others prefer good company to share the effort. Similarly, some people are happy to study mathematics on their own, but others need more help <a href="https://www.nature.com/articles/srep23011">navigating this challenging subject</a>. Research shows that students who are failing in maths tend to be more empathetic than systematising. These are also the students more affected by reactions of people surrounding them: parents, teachers and the media. </p>
<h2>5. Make maths matter</h2>
<p>So given that <a href="https://hpl.uchicago.edu/sites/hpl.uchicago.edu/files/uploads/Maloney%252c%20E.A.%252c%20Schaeffer%252c%20M.W.%252c%20%26%20Beilock%252c%20S.L.%252c%20%25282013%2529.%20Mathematics%20anxiety%20and%20stereotype%20threat.pdf">maths anxiety can spread from one generation</a> to another, parents clearly have a role to play in making sure their children don’t clam up at the very thought of numbers. This is important, because a parent who learns how to avoid passing on mathematical anxiety gives their child a chance to learn a beautiful subject and to access <a href="http://www.bbc.co.uk/news/education-41693230">some of the best paid, most interesting, jobs around</a>. </p>
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<img alt="" src="https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.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">Don’t scared of maths, it could rub off on your child.</span>
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<h2>6. Join the dots</h2>
<p>When it comes to maths, both inside and outside the classroom, the emphasis should shift from solely the numerical aspect to include connected aspects, such as concepts and links with other subjects and everyday applications. This will allow children to see mathematics as a social practice – where discussing mathematical challenges with classmates, teachers and parents becomes the norm.</p><img src="https://counter.theconversation.com/content/96441/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 organization that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Make maths more fun with these tipsSue Johnston-Wilder, Associate Professor, Mathematics Education, University of WarwickDavide Penazzi, Lecturer in Mathematics, University of Central LancashireLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/786602017-06-21T10:31:01Z2017-06-21T10:31:01ZChallenging the status quo in mathematics: Teaching for understanding<figure><img src="https://images.theconversation.com/files/174303/original/file-20170618-28772-1vhqkpw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">How can we change math instruction to meet the needs of today's kids?</span> <span class="attribution"><a class="source" href="https://flic.kr/p/97aGY8">World Bank Photo Collection / flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Despite decades of <a href="http://files.eric.ed.gov/fulltext/ED372969.pdf">reform efforts</a>, mathematics teaching in the U.S. <a href="http://www.jstor.org/stable/20405948">has changed little</a> in the last century. As a result, it seems, American students have been left behind, now ranking <a href="https://nces.ed.gov/pubs2017/2017048.pdf#page=31">40th in the world</a> in math literacy. </p>
<p>Several state and national reform efforts have tried to improve things. The most recent <a href="http://www.corestandards.org/Math/">Common Core standards</a> had a great deal of promise with their focus on how to teach mathematics, but after several years, <a href="http://journals.sagepub.com/doi/full/10.3102/0013189X17711899">changes in teaching practices</a> have been minimal. </p>
<p><iframe id="Grc6N" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/Grc6N/1/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>As an education researcher, I’ve observed teachers trying to implement reforms – often with limited success. They sometimes make changes that are more cosmetic than substantive (e.g., more student discussion and group activity), while failing to get at the heart of the matter: What does it truly mean to teach and learn mathematics?</p>
<h2>Traditional mathematics teaching</h2>
<p>Traditional middle or high school mathematics teaching in the U.S. <a href="http://www.jstor.org/stable/20405948">typically follows this pattern</a>: The teacher demonstrates a set of procedures that can be used to solve a particular kind of problem. A similar problem is then introduced for the class to solve together. Then, the students get a number of exercises to practice on their own.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=686&fit=crop&dpr=1 600w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=686&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=686&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=862&fit=crop&dpr=1 754w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=862&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=862&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The basics of math instruction have changed little since George Eaton taught at Phillips Academy (1880-1930).</span>
<span class="attribution"><a class="source" href="https://flic.kr/p/jKrzFZ">Phillips Academy Archives and Special Collections / flickr</a></span>
</figcaption>
</figure>
<p>For example, when students learn about the area of shapes, they’re given a set of formulas. They put numbers into the correct formula and compute a solution. More complex questions might give the students the area and have them work backwards to find a missing dimension. Students will often learn a different set of formulas each day: perhaps squares and rectangles one day, triangles the next. </p>
<p>Students in these kinds of lessons are learning to follow a rote process to arrive at a solution. This kind of instruction is so common that it’s seldom even questioned. After all, within a particular lesson, it makes the math seem easier, and students who are successful at getting the right answers find this kind of teaching to be very satisfying.</p>
<p>But it turns out that teaching mathematics this way can actually <a href="http://www.jstor.org/stable/3696735">hinder learning</a>. Children can become dependent on <a href="http://www.jstor.org/stable/10.5951/teacchilmath.21.1.0018">tricks and rules</a> that don’t hold true in all situations, making it harder to adapt their knowledge to new situations.</p>
<p>For example, in traditional teaching, children learn that they should distribute a number by multiplying across parentheses and will practice doing so with numerous examples. When they begin learning how to solve equations, they often have trouble realizing that it’s not always needed. To illustrate, take the equation 3(x + 5) = 30. Children are likely to multiply the 3 across the parentheses to make 3x + 15 = 30. They might just as easily have divided both sides by 3 to make x + 5 = 10, but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=320&fit=crop&dpr=1 600w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=320&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=320&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=402&fit=crop&dpr=1 754w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=402&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=402&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Students who learn by rote drilling often have trouble realizing that there are equally valid alternative methods for solving a problem.</span>
<span class="attribution"><span class="source">Kaitlyn Chantry</span></span>
</figcaption>
</figure>
<h2>More than a right answer</h2>
<p>A key missing ingredient in these traditional lessons is conceptual understanding. </p>
<p>Concepts are ideas, meaning and relationships. It’s not just about knowing the procedure (like how to compute the area of a triangle) but also the significance behind the procedure (like what area means). How concepts and procedures are related is important as well, such as how the area of a triangle can be considered half the area of a rectangle and how that relationship can be seen in their area formulas. </p>
<p>Teaching for conceptual understanding has <a href="http://math.coe.uga.edu/Olive/EMAT3500f08/instrumental-relational.pdf">several benefits</a>. Less information has to be memorized, and students can translate their knowledge to new situations more easily. For example, understanding what area means and how areas of different shapes are related can help students understand the concept of volume better. And learning the relationship between area and volume can help students understand how to interpret what the volume means once it’s been calculated.</p>
<p>In short, building relationships between <a href="https://doi.org/10.1007/s10648-015-9302-x">how to solve a problem and why it’s solved that way</a> helps students <a href="https://doi.org/10.1037//0022-0663.91.1.175">use what they already know</a> to solve new problems that they face. Students with a truly conceptual understanding can see how methods emerged from <a href="https://doi.org/10.1037/0022-0663.91.1.175">multiple interconnected ideas</a>; their relationship to the solution goes deeper than rote drilling.</p>
<p>Teaching this way is a critical first step if students are to begin recognizing mathematics as meaningful. Conceptual understanding is a key ingredient to helping people think mathematically and use mathematics outside of a classroom.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=412&fit=crop&dpr=1 600w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=412&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=412&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=517&fit=crop&dpr=1 754w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=517&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=517&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Procedural learning promotes memorization instead of critical thinking and problem solving.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/math-study-exam-set-book-pencil-250606378">m.jrn/shutterstock.com</a></span>
</figcaption>
</figure>
<h2>The will to change</h2>
<p>Conceptual understanding in mathematics has been recognized as important for <a href="http://www.nctm.org/uploadedFiles/About/President,_Board_and_Committees/Board_Materials/MLarson-SF-NCTM-4-16.pdf">over a century</a> and widely discussed for decades. So why has it not been incorporated into the curriculum, and why does traditional teaching abound? </p>
<p>Learning conceptually can take longer and be more difficult than just presenting formulas. Teaching this way may require additional time commitments both in and outside the classroom. Students may have never been asked to think this way before.</p>
<p>There are systemic obstacles to face as well. A new teacher may face pressure from fellow teachers who teach in traditional ways. The <a href="https://www.thoughtco.com/high-stakes-testing-overtesting-in-americas-public-schools-3194591">culture of overtesting</a> in the last two decades means that students face more pressure than ever to get right answers on tests. </p>
<p>The results of these tests are also being <a href="https://tcta.org/node/13251-issues_with_test_based_value_added_models_of_teacher_assessment">tied to teacher evaluation systems</a>. Many teachers feel pressure to teach to the test, drilling students so that they can regurgitate information accurately.</p>
<p>If we really want to improve America’s mathematics education, we need to rethink both our education system and our teaching methods, and perhaps to <a href="http://www.nea.org/home/40991.htm">consider how other countries approach mathematics instruction</a>. Research has provided evidence that teaching conceptually has <a href="http://www.ascd.org/publications/educational-leadership/feb04/vol61/num05/Improving-Mathematics-Teaching.aspx">benefits</a> not offered by traditional teaching. And students who learn conceptually typically do <a href="https://doi.org/10.3102/0034654310374880">as well or better</a> on achievement tests. </p>
<p>Renowned education expert <a href="https://pasisahlberg.com/">Pasi Sahlberg</a> is a former mathematics and physics teacher from Finland, which is renowned for its world-class education. He <a href="http://www.smithsonianmag.com/innovation/why-are-finlands-schools-successful-49859555/">sums it up</a> well:</p>
<blockquote>
<p>We prepare children to learn how to learn, not how to take a test.</p>
</blockquote><img src="https://counter.theconversation.com/content/78660/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christopher Rakes receives funding from the National Science Foundation. </span></em></p>Math instruction is stuck in the last century. How can we change teaching methods to move past rote memorization and help students develop a more meaningful understanding – and be better at math?Christopher Rakes, Assistant Professor of Mathematics Education, University of Maryland, Baltimore CountyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/709252017-01-09T20:26:48Z2017-01-09T20:26:48ZSouth Africa can't compete globally without fixing its attitude to maths<figure><img src="https://images.theconversation.com/files/152080/original/image-20170109-23482-uehzdp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Investing in pupils' maths skills is an investment in a country's economy.</span> <span class="attribution"><span class="source">Global Partnership for Education/Flickr</span></span></figcaption></figure><p>South Africa is not producing enough school leavers who are competent in maths and science. This is a fact borne out by international assessments such as the <a href="http://timssandpirls.bc.edu/publications/timss/2015-methods.html">Trends in International Mathematics and Science Study</a> (TIMMS) and the World Economic Forum’s <a href="https://www.weforum.org/reports/the-global-competitiveness-report-2016-2017-1">Global Competitiveness Report</a>. These show that South Africa is not making much headway when it comes to maths and science. </p>
<p>The 2016 Global Competitiveness Report ranked South Africa last among 140 countries for maths and science. This places it behind poorer African countries like Mozambique and Malawi.</p>
<p>In 2016 there was a <a href="https://businesstech.co.za/news/government/148875/matric-results-2016-maths-and-science-suffers/">marginal improvement</a> in the maths pass rate, from 49.1% the previous year to 51.1%. The country is moving at a glacial pace in an area that demands urgent attention. After all, science and maths are key to any country’s economic development and its competitiveness in the global economy. </p>
<p>The TIMMS study ranks Singapore, Hong Kong, South Korea and Japan among its top maths and science performers. It’s no coincidence that these countries feature among the <a href="http://www.wipo.int/pressroom/en/articles/2016/article_0008.html">top 20</a> on the Global Innovation Index. Good quality education fuels an economy. South Africa needs to increase its supply of science and technology university graduates, which at the moment constitute the bulk of <a href="http://www.dhet.gov.za/Gazette/Government%20Gazette%20No%2039604,%2019%20January%202016.%20List%20of%20Occupations%20in%20High%20Demand%202015.pdf">scarce skills</a> outlined by the Department of Higher Education and Training. </p>
<p>But instead of chasing improved results the government is lowering the bar for maths at school level. At the end of 2016 it set <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">20% as a passing mark</a> for pupils in grades 7, 8 and 9. This lends credence to the common view of maths as a subject only the “gifted” can comprehend. </p>
<p>It’s time to place a premium on maths and to ensure that pupils – especially those from poorer backgrounds – receive the necessary support to excel at maths. This is critical if South Africa is to produce the human capital needed to drive economic growth and create new industries in the future. </p>
<h2>How maths and science boost economies</h2>
<p>Maths and science are a gateway to new industries. Mastery of them endows an economy with the human capital needed to ride the technological wave. In his work on the industries of the future Alec Ross, who advised Hillary Clinton on innovation during her term as US Secretary of State, <a href="http://www.simonandschuster.com/books/The-Industries-of-the-Future/Alec-Ross/9781476753652">points out</a> that sectors such as robotics, advanced life sciences, codification of money, big data and cybersecurity – all of which require mastery of technology and mathematical skills – are the pillars of the <a href="https://www.weforum.org/agenda/2016/01/the-fourth-industrial-revolution-what-it-means-and-how-to-respond/">fourth industrial revolution</a>. </p>
<p>Simply put, this “revolution” is the age of technology that’s already upon us.</p>
<p>More importantly, a grasp of maths and science boosts confidence and expands career possibilities for pupils. This ultimately gives them an edge in the labour market. </p>
<p>Many students drop out of maths not by choice but because they’re frustrated by a lack of adequate support. I speak from experience: I dropped the subject when I was 14 at the end of what’s now Grade 9 but used to be called Standard 7. Our maths teacher didn’t encourage those he called “slow learners” to continue with the subject and I was one of many intimidated into giving up on maths.</p>
<p>But succeeding in maths, or in any area of skill, isn’t entirely a matter of genetic endowment. Psychologist Anders Ericsson, <a href="http://www.goodreads.com/book/show/26312997-peak">in his book Peak</a>, draws on three decades of research to prove why natural talent and other innate factors have less of an impact than what he calls deliberate or purposeful practice.</p>
<p>He contends that</p>
<blockquote>
<p>a number of successful efforts have shown that pretty much any child can learn math if it is taught in the right way.</p>
</blockquote>
<p>South Africa should be focusing on how to teach maths in the right way rather than buying into the myth that it is an impossible subject. The current approach is robbing the economy of critical human capital.</p>
<h2>Radical interventions</h2>
<p>Some may argue, though, that any improvement or shift is impossible in an education system that’s plagued by weak infrastructure, a lack of teacher development and support and too few qualified maths and science teachers. Even if the numbers of teachers in these subjects were to increase, it’s crucial that the quality of education rises too.</p>
<p>Radical interventions are needed, now – or South Africa will never become a global player in the fourth industrial revolution. </p>
<p>The country must develop new teacher training methods and nurture a supportive environment for teachers. Innovative teaching tools should be introduced in the early phases to demystify maths and science for young pupils. If these subjects are more fun to learn, more pupils may be drawn to them as future career options.</p>
<p>Taking these steps will give South Africa a better chance in the future to harness the talent of its youth to powering the economy, and improve its global competitiveness.</p><img src="https://counter.theconversation.com/content/70925/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mzukisi Qobo does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Good quality education fuels an economy. South Africa needs to increase its supply of science and technology university graduates. But instead it's lowering the bar, especially when it comes to maths.Mzukisi Qobo, Associate Professor at the Institute for Pan African Thought and Conversation, University of JohannesburgLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/465852015-09-09T10:25:07Z2015-09-09T10:25:07ZThe Common Core is today's New Math – which is actually a good thing<figure><img src="https://images.theconversation.com/files/94197/original/image-20150908-4358-zdmhft.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Change can be a good thing – really.</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-182868605/stock-photo-frustrated-father-throws-up-his-hands-in-despair-frustrated-elementary-age-boy-lays-his-head-on.html">Homework image via www.shutterstock.com.</a></span></figcaption></figure><p>Math can’t catch a break. These days, people on both ends of the political spectrum are lining up to deride the <a href="http://www.corestandards.org/">Common Core standards</a>, a set of guidelines for K-12 education in reading and mathematics. The Common Core standards outline what a student should know and be able to do at the end of each grade. States don’t have to adopt the standards, although many did in an effort to receive funds from President Obama’s <a href="http://www2.ed.gov/programs/racetothetop/index.html">Race to the Top</a> initiative.</p>
<p><a href="http://www.usnews.com/news/special-reports/a-guide-to-common-core/articles/2014/02/27/who-is-fighting-against-common-core">Conservatives</a> oppose the guidelines because they generally dislike any suggestion that the federal government might have a role to play in public education at the state and local level; these standards, then, are perceived as a threat to local control.</p>
<p><a href="https://www.laprogressive.com/fighting-common-core/">Liberals</a>, mostly via teachers’ unions, decry the use of the standards and the associated assessments to evaluate classroom instructors.</p>
<p>And parents of all persuasions are panicked by their sudden inability to help their children with their homework. Even <a href="http://www.newyorker.com/news/daily-comment/louis-c-k-against-the-common-core">comedian Louis CK got in on the discussion</a> (via Twitter; he has since deactivated his account). </p>
<blockquote>
<p>My kids used to love math. Now it makes them cry. Thanks standardized testing and common core!
— Louis CK (@louisck) April 28 2014</p>
</blockquote>
<p>In the middle are millions of American schoolchildren who are often confused and frustrated by these “new” ways of teaching mathematics.</p>
<p>Thing is, we’ve been down this path before.</p>
<h2>The old New Math</h2>
<p>When the Soviets launched Sputnik in 1957, the United States went into panic mode. Our schools needed to emphasize math and science so that we wouldn’t fall behind the Soviet Union and its allegedly superior scientists. In 1958, President Eisenhower signed the <a href="http://www.britannica.com/topic/National-Defense-Education-Act">National Defense Education Act</a>, which poured money into the American education system at all levels. </p>
<p>One result of this was the so-called New Math, which <a href="https://en.wikipedia.org/wiki/Secondary_School_Mathematics_Curriculum_Improvement_Study#Curriculum">focused more on conceptual understanding of mathematics</a> over rote memorization of arithmetic. Set theory took a central role, forcing students to think of numbers as sets of objects rather than abstract symbols to be manipulated. This is actually how numbers are constructed logically in an advanced undergraduate mathematics course on real analysis, but it may not necessarily be the best way to communicate ideas like addition to schoolchildren. Arithmetic using number bases other than 10 also entered the scene. This was famously spoofed by <a href="https://en.wikipedia.org/wiki/Tom_Lehrer">Tom Lehrer</a> in his song “New Math.”</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/UIKGV2cTgqA?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">This 60’s song about New Math gives us a glimpse of what the ‘old math’ was like.</span></figcaption>
</figure>
<p>I attended elementary school in the 1970s, so I missed New Math’s implementation, and it was largely gone by the time I got started. But the way Lehrer tries to explain how subtraction “used to be done” made no sense to me at first (I did figure it out after a minute). In fact, the New Math method he ridicules is how children of my generation – and many of the Common Core-protesting parents of today – learned to do it, even if some of us don’t really understand what the whole borrowing thing is conceptually. Clearly some of the New Math ideas took root, and math education is better for it. For example, given the ubiquity of computers in modern life, it’s useful for today’s students to learn to do binary arithmetic – adding and subtracting numbers in base 2 just as a computer does. </p>
<p>The New Math fell into disfavor mostly because of complaints from parents and teachers. Parents were unhappy because they couldn’t understand their children’s homework. Teachers objected because they were often unprepared to instruct their students in the new methods. In short, it was the <em>implementation</em> of these new concepts that led to the failure, more than the curriculum itself.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=427&fit=crop&dpr=1 600w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=427&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=427&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=536&fit=crop&dpr=1 754w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=536&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=536&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Give us our New Math!</span>
</figcaption>
</figure>
<h2>Those who ignore history…</h2>
<p>In 1983, President Reagan’s National Commission on Excellence in Education released its report, <a href="http://www2.ed.gov/pubs/NatAtRisk/index.html">A Nation at Risk</a>, which asserted that American schools were “failing” and suggested various measures to right the ship. Since then, American schoolchildren and their teachers have been bombarded with various reform initiatives, privatization efforts have been launched and charter schools established.</p>
<p>Whether or not the nation’s public schools are actually failing is a matter of serious debate; indeed, many of the claims made in A Nation at Risk were <a href="http://eric.ed.gov/?id=EJ482502">debunked</a> by statisticians at Sandia National Laboratories a few years after the report’s release. But the general notion that our public schools are “bad” persists, especially among politicians and business groups. </p>
<p>Enter Common Core. Launched in 2009 by a consortium of states, the idea sounds reasonable enough – public school learning objectives should be more uniform nationally. That is, what students learn in math or reading at each grade level should not vary state by state. That way, colleges and employers will know what high school graduates have been taught, and it will be easier to compare students from across the country. </p>
<p>The guidelines are just that. There is no set curriculum attached to them; they are merely a list of concepts that students should be expected to master at each grade level. For example, here are the <a href="http://www.corestandards.org/Math/Content/3/NBT/">standards</a> in Grade 3 for Number and Operations in Base Ten:</p>
<ul>
<li><p>Use place value understanding and properties of operations to perform multi-digit arithmetic.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/1/">CCSS.Math.Content.3.NBT.A.1</a>
Use place value understanding to round whole numbers to the nearest 10 or 100.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/2/">CCSS.Math.Content.3.NBT.A.2</a>
Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/3/">CCSS.Math.Content.3.NBT.A.3</a>
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (eg, 9 × 80, 5 × 60) using strategies based on place value and properties of operations.</p></li>
</ul>
<p>There is a footnote that “a range of algorithms may be used” to help students complete these tasks. In other words, teachers can explain various methods to actually accomplish the mathematical task at hand. There is nothing controversial about these topics, and indeed it’s not controversial that they’re things that students should be able to do at that age.</p>
<p>However, some of the new methods being taught for doing arithmetic have caused confusion for parents, causing them to take to social media in frustration. Take the 32 - 12 problem, for example:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Just because you didn’t learn it that way doesn’t make it inscrutable or wrong.</span>
</figcaption>
</figure>
<p>Once again, it’s the <em>implementation</em> that’s causing the problem. Most parents (people age 30-45, mostly), remembering the math books of our youth filled with pages of exercises like this, immediately jump to the “Old Fashion” (sic) algorithm shown. The stuff at the bottom looks like gibberish, and given many adults’ <a href="https://theconversation.com/when-parents-with-high-math-anxiety-help-with-homework-children-learn-less-46841">tendency toward math phobia/anxiety</a>, they immediately throw up their hands and claim this is nonsense.</p>
<p>Except that it isn’t. In fact, we all do arithmetic like this in our heads all the time. Say you are buying a scone at a bakery for breakfast and the total price is US$2.60. You hand the cashier a $10 bill. How much change do you get? Now, you do <em>not</em> perform the standard algorithm in your head. You first note that you’d need another 40 cents to get to the next dollar, making $3, and then you’d need $7 to get up to $10, so your change is $7.40. That’s all that’s going on at the bottom of the page in the picture above. Your children can’t explain this to you because they don’t know that you weren’t taught this explicitly, and your child’s teacher can’t send home a primer for you either.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.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">New ways to learn can be better for students – if rolled out appropriately.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/departmentofed/9610695698">US Department of Education</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>Better intuition about math, better problem-solving</h2>
<p>As an instructor of college-level mathematics, I view this focus on conceptual understanding and multiple strategies for solving problems as a welcome change. Doing things this way can help build intuition about the size of answers and help with estimation. College students can compute answers to homework problems to 10 decimal places, but ask them to ballpark something without a calculator and I get blank stares. Ditto for conceptual understanding – for instance, students can evaluate <a href="https://en.wikipedia.org/wiki/Integral">integrals</a> with relative ease, but building one as a limit of <a href="https://en.wikipedia.org/wiki/Riemann_sum">Riemann sums</a> to solve an actual problem is often beyond their reach.</p>
<p>This is frustrating because I know that my colleagues and I focus on these notions when we introduce these topics, but they fade quickly from students’ knowledge base as they shift their attention to solving problems for exams. And, to be fair, since the K-12 math curriculum is chopped up into discrete chunks of individual topics for ease of standardized testing assessment, it’s often difficult for students to develop the problem-solving abilities they need for success in higher-level math, science and engineering work. Emphasizing more conceptual understanding at an early age will hopefully lead to better problem-solving skills later. At least that’s the rationale behind the standards.</p>
<p>Alas, Common Core is in danger of being abandoned. Some states have already <a href="http://academicbenchmarks.com/common-core-state-adoption-map/">dropped the standards</a> (Indiana and South Carolina, for example), looking to replace them with something else. But these actions are largely a result of mistaken conflations: that the standards represent a federal imposition of curriculum on local schools, that the <a href="http://www.parcconline.org/about">standardized tests</a> used to evaluate students <em>are</em> the Common Core rather than a separate initiative.</p>
<p>As the 2016 presidential campaign heats up, support for the Common Core has become a political liability, possibly killing it before it really has a chance. That would be a shame. The standards themselves are fine, and before we throw the baby out with the bathwater, perhaps we should consider efforts to implement them properly. To give the Common Core a fair shot, we need appropriate professional development for teachers and a more phased introduction of new standardized testing attached to the standards.</p>
<p>But, if we do ultimately give in to panic and misinformation, let’s hope any replacement provides proper coherence and rigor. Above all, our children should develop solid mathematical skills that will help them see the beauty and utility of this wonderful subject.</p><img src="https://counter.theconversation.com/content/46585/count.gif" alt="The Conversation" width="1" height="1" />
Both have been much maligned by parents who felt like they couldn’t help their kids with basic math homework. But the Common Core could help with conceptual understanding and math intuition.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/449002015-08-09T20:34:50Z2015-08-09T20:34:50ZWeapons of maths destruction: are calculators killing our ability to work it out in our head?<figure><img src="https://images.theconversation.com/files/91101/original/image-20150807-9952-vo72z3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">from www.shutterstock.com.au</span></span></figcaption></figure><p>Since the 1980s we have had access to calculators of various types. Today, we can include computers and smartphones – which are attached to our hip 24/7. So does this ubiquitous access to calculators affect our ability to do maths in our heads like we used to?</p>
<p>Thirty years ago calculators promised immense opportunity – opportunity, alas, that brought considerable controversy. The sceptics predicted students would not be able to compute even simple calculations mentally or on paper. Multiplication, basic facts, knowledge would disappear. Calculators would become a crutch. </p>
<p>The controversy has not dissipated over time. As recently as 2012, the UK government announced it <a href="http://connection.ebscohost.com/c/opinions/83406522/does-maths-add-up">intended</a> to ban calculators from primary classrooms on the grounds that students use them too much and too soon.</p>
<p>Research conducted in response to this <a href="http://thenferblog.org/2014/11/12/subtracting-calculators-from-maths-tests-doesnt-add-up/">found little difference</a> in performance tests whether students used calculators or not. An earlier US study had <a href="http://www.jstor.org/stable/42802150?seq=1#page_scan_tab_contents">found the same</a>: the calculator had no positive or negative effects on the attainment of basic maths skills.</p>
<p>Researchers recommended moving the conversation on. What types of tasks and activities suit calculators? How can calculators complement and reinforce mental and written methods of arithmetic in maths?</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/91093/original/image-20150807-18724-1aiqnoz.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">Studies have found the use of calculators doesn’t reduce the ability to compute in our heads.</span>
<span class="attribution"><span class="source">from www.shutterstock.com</span></span>
</figcaption>
</figure>
<h2>Using calculators to extend maths</h2>
<p>Teachers had high hopes that calculators would be used in enhancing and extending the learning of mathematics. While standard procedures for the four operations (+, -, x, ÷) would still be taught and the basic facts of arithmetic would still need to be mastered, calculators could facilitate the study of number patterns and the absence of tedious calculations would free students up to pose, model and solve interesting and relevant problems.</p>
<p>Rather than replacing mental computation, calculators actually make calculating more efficient. Even the simple four-function calculator is a powerful instrument for investigating a range of concepts that previously were not so easily accessed by young children independently. </p>
<p>Counting, skip counting, negative numbers, relationships between common and decimal fractions and other number patterns all open up. The calculator lets students investigate and generalise patterns in numbers that they have previously not had access to. </p>
<p>The “constant” function means young children can explore numbers to infinity, if they fancy, without being restricted to charts or number lines. Skip-counting is also possible using the constant function. </p>
<p>Multiplication tables are no longer limited to 12 x 12. In the diagram below the child is exploring the pattern made by entering 11+11 and continuing to press the Equals sign to see what happens to the pattern once you count beyond 99 by elevens.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=923&fit=crop&dpr=1 600w, https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=923&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=923&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1160&fit=crop&dpr=1 754w, https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1160&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/90538/original/image-20150803-17164-ucxli9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1160&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Lots of number patterns are possible.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Calculators have great potential in concept development. For example, what happens when you multiply or divide a number by 10 or 100? These generalisations are spectacularly demonstrated and discovered with a calculator, which frees students to ask more questions about number patterns.</p>
<p><a href="http://search.informit.com.au/documentSummary;dn=462445434680297;res=IELHSS">In a 1997 review</a> of the extent to which calculators were being used in schools, numerous studies were examined that indicated the use of calculators at primary levels had no detrimental effects on students’ arithmetic abilities.</p>
<p>Unfortunately, the research indicated that calculators were still being used for trivial things like checking answers and were making little difference in mathematics education. </p>
<p>Although teachers indicated their support for the use of calculators across all levels of primary school, there was little evidence that these ideas were being taken up and implemented. Parental disapproval of the use of calculators was cited as a possible cause of the limited take-up.</p>
<h2>Calculators’ potential is not being achieved</h2>
<p><a href="http://espace.library.uq.edu.au/view/UQ:176139">In a 2008 study</a> this finding was reiterated. Researchers reported that despite educators’ high hopes for digital technologies to transform maths education, the uptake, both internationally and in Australia, had been disappointing. </p>
<p>Influential in this has been the lack of professional development to assist teachers in planning and implementing teaching approaches that take advantage of the technology. British technologist Conrad Wolfram said in his <a href="http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en">TED talk</a>:</p>
<blockquote>
<p>From rockets to stock markets, many of humanity’s most thrilling creations are powered by math. So why do kids lose interest in it?</p>
</blockquote>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=423&fit=crop&dpr=1 600w, https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=423&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=423&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=532&fit=crop&dpr=1 754w, https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=532&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/90390/original/image-20150731-18735-o27k24.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=532&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">We’re not harnessing calculators’ full potential.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/dominicspics/3915942881/">Dominic Alves/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Wolfram pointed out that students in mathematics classes across the globe spend up to 80% of their time learning and practising mathematical procedures. This time could be spent more productively if the digital technology already in the classrooms was used more effectively and efficiently. </p>
<p>While mathematics is popular, challenging and useful in the real world, kids are rapidly losing interest in the subject in schools. Wolfram blames teaching that focuses on calculation by hand: it’s tedious and mostly irrelevant to real mathematics and the real world.</p>
<p>Sadly, the potential for calculators to transform school mathematics and enhance our facility with mental arithmetic is not being achieved. We are not being provided with opportunities to solve real and interesting mathematical problems in the most effective ways. </p>
<p>So to answer whether calculators are affecting our mental arithmetic: not as much as we would like them to.</p><img src="https://counter.theconversation.com/content/44900/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jeanne Carroll does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Smartphones double as calculators and are attached to our hip 24/7. Does the ubiquitous access to calculators affect our ability to do maths in our heads like we used to?Jeanne Carroll, Senior Lecturer, College of Education, Victoria UniversityLicensed as Creative Commons – attribution, no derivatives.