tag:blogger.com,1999:blog-91492433880874662342024-03-23T10:14:46.712+00:00Digital technologies for learning mathematicsTom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.comBlogger37125tag:blogger.com,1999:blog-9149243388087466234.post-66559911319961638132023-06-19T11:05:00.001+01:002023-06-19T11:05:54.544+01:00Bring Your Own Device: Equalising learning in A level Mathematics?<p><i>An earlier version of this article appeared in the Mathematical Association Journal: Mathematics in School (November 2020).</i></p><p>In July 2019 I was fortunate to be able to speak at the GeoGebra Global Gathering in Linz, Austria. I gave a presentation about how allowing students to use mathematical software on their own devices, both to support their learning in the classroom and as a tool they can use within assessment, is a potential solution to:</p><p></p><ol style="text-align: left;"><li>Harnessing the power of technology to improve the teaching and learning of mathematics</li><li>Increasing equality of access to technology for mathematics students.</li></ol>This article is a summary of the talk I gave there.<p></p><h2 style="text-align: left;">Why should students use technology in mathematics?</h2><p>When I first started teaching there was an activity that I used with students to give them a strong conceptual understanding of differentiation and what the derivative of a function expresses. The activity was for them to plot the curve for <i>y=x²</i> on graph-paper, draw some tangents at different points, measure the gradient of these tangents, tabulate them and form an expression for the gradient of the tangent at a point. The strength of this activity was that it helped the students construct a strong impression of what the derivative represents in terms of graphical, algebraic and numerical representations of it. This was a much more solid foundation than just meeting differentiation as a rule: “multiply by the power and reduce the power by one”, and one that was a very helpful basis for much future work on calculus.</p><p>As good as this activity was, it had some fairly significant limitations: their tangents were often inaccurately drawn, the students had a limited sense of how the gradient changed as the point varied and, perhaps most frustratingly, it took a long time for students to sketch the curve and the tangents. These factors, in particular the length of time the activity took, meant that in the following years I decided not use it, even though I believed it improved students’ understanding. </p><p>Fortunately there is a solution to the problems with this activity, and many tasks like it: dynamic graphing/geometry. It is now possible for students to use software such as GeoGebra to plot a curve, add a point and a tangent and then measure the slope of the tangent. All of this can be constructed in a minute or two and has the added advantage that the diagram is dynamic: they can move the point and see the gradient change.</p><p> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5bw3kezEHtiARUGQHe0TxF2z93z45x_EK8pi62qagBEslypbgE8IOCnZ5ElgWJUq9yZxI9tuaY_jiXT4Spv5bYEI6W-sGnVXMUcszauHi7Sfijw56FdF7SHue30U-wvYZ86AGw4x_dv3lmlT9TOpSFqerJuvYsBgcH7niLltcI8lruGaQrXWzLEsN1eI/s697/quadratic.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="518" data-original-width="697" height="297" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5bw3kezEHtiARUGQHe0TxF2z93z45x_EK8pi62qagBEslypbgE8IOCnZ5ElgWJUq9yZxI9tuaY_jiXT4Spv5bYEI6W-sGnVXMUcszauHi7Sfijw56FdF7SHue30U-wvYZ86AGw4x_dv3lmlT9TOpSFqerJuvYsBgcH7niLltcI8lruGaQrXWzLEsN1eI/w400-h297/quadratic.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Exploring the gradient on y=x² in GeoGebra</td></tr></tbody></table><br /></p><p>This potential for students to develop an understanding of mathematical relationships through using technology is why the Ofqual A level Mathematics guidance for awarding organisations, published in 2016, contains the statement:</p><blockquote style="border: none; margin: 0 0 0 40px; padding: 0px;"><p style="text-align: left;"><i>“The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A level mathematics.”</i></p></blockquote><p>There is an appreciation from Ofqual of the power of technology to enhance students’ mathematical understanding. The intention of this statement is that part of each student’s experience of learning A level Mathematics should be for them to use mathematical technology, and not just watch a teacher use it.</p><h2 style="text-align: left;">Current use of technology in A level Mathematics classrooms</h2><p>For many A level Mathematics students the only mathematical technology that they will use in their study is a scientific calculator. In the current examinations for A level Mathematics students are allowed to take a graphical calculator into an examination; however, it is not compulsory and many students will just use a scientific calculator. Assessment is a strong driver of classroom practice and because graphical calculator use is not compulsory, scientific calculators are often the only technology used by students in the classroom.</p><p>It is possible to utilise graphing technology in the learning, even if students only use a scientific calculator in the assessment, but alternative devices are needed for this. This can be difficult as the technology infrastructure in schools/colleges is very variable: some are fortunate to have class sets of laptops, Chromebooks or tablets and some issue every student with a tablet. Unfortunately this is not universally the case and, even when class sets of laptops are available, these are often very old and slow. One solution to this problem is to allow students to use their own devices. This approach is often named BYOD: bring your own device.</p><h2 style="text-align: left;">BYOD in learning</h2><p>One of the most frustrating aspects of the lack of hardware in schools/colleges is that many students already have a device that is capable of running mathematical software. Programs such as GeoGebra and Desmos are easily available and free for laptops, Chromebooks, tablets and Android/iOS smartphones. Operating a Bring Your Own Device (BYOD) policy means that schools/colleges do not need to address the lack of hardware issues. </p><p>The power of students using technology has already been discussed but it is important to manage this so that students are supported in how to use it to effectively support their learning. The example provided earlier was about exploring the derivative of <i>y=x²</i>: this was a well-designed activity from the old SMP 16-19 scheme. This task was ideal for me as a novice teacher as it had a clear structure along with teachers’ notes to accompany it: these suggested what I could expect the students to achieve, as well as appropriate questions to ask. A similar level of support is beneficial when teachers are considering using technology in the classroom. For this reason, MEI has produced a series of tasks which describe graphs/constructions for students to enter into the software, along with suggestion for how to explore them and suitable questions for the teacher to ask. These tasks are also accompanied by teacher guidance.</p><p>MEI has developed 50 such tasks for GCSE, A level and Further Mathematics, all of which have a similar structure. They feature:</p><p></p><ul style="text-align: left;"><li><span style="white-space: normal;">Construction steps for the students;</span></li><li><span style="white-space: normal;">Questions for discussion;</span></li><li><span style="white-space: normal;">A mathematical problem (often to be attempted first without technology);</span></li><li><span style="white-space: normal;">Further/extension problems.</span></li></ul><p></p><p>The purpose of these tasks is for students to explore and discuss a mathematical idea, using the power of dynamic graphing to allow them to observe relationships, then to consolidate this learning by attempting a more conventional problem. An example task is given below. In this task students are expected to consider the shape of the curve of the derivative function in relation to the shape of the curve of the original function.</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAr3AhDHj856Axw5rEWlZ_tbjFr8kSARDzdVnjkhxCpeLbZFmQYdXFNk8Zsr1wNWMYlB4tVAzLwuZ1GZKP1Ul6NzzMyMCssT8jdlXbs3H0Q3guFOSbJr5S5VkCzqfAQKjJDrCnk_szHRHL--sbkxxetO8R7vPCEAuyKa6dORKPvfYxQ5Ewn3-3oZ9Ut_o/s892/student-task.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="773" data-original-width="892" height="346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAr3AhDHj856Axw5rEWlZ_tbjFr8kSARDzdVnjkhxCpeLbZFmQYdXFNk8Zsr1wNWMYlB4tVAzLwuZ1GZKP1Ul6NzzMyMCssT8jdlXbs3H0Q3guFOSbJr5S5VkCzqfAQKjJDrCnk_szHRHL--sbkxxetO8R7vPCEAuyKa6dORKPvfYxQ5Ewn3-3oZ9Ut_o/w400-h346/student-task.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Example of a student task</td></tr></tbody></table><br /><p>All the tasks are freely available via the MEI website at: <a href="https://mei.org.uk/resources/?terms=desmos">https://mei.org.uk/resources/?terms=desmos</a> and <a href="https://mei.org.uk/resources/?terms=geogebra">https://mei.org.uk/resources/?terms=geogebra</a></p><h2 style="text-align: left;">Concerns with BYOD in the classroom</h2><p>A BYOD policy often covers students bringing in their own laptops, tablets or smartphones. Most teachers are comfortable with students bringing in laptops or tablets but the use of smartphones in the classroom is a very contentious issue. There are some advocates for completely banning smartphones in schools and a number of schools have enforced such a ban. When I speak to teachers the picture is very mixed: some work in schools/colleges that have banned smartphone use; some don’t ban their use, but don’t make use of them; a few are using them successfully. The experiences of teachers who have used them have been positive: none of the teachers that I’ve met who have used graphing apps on smartphones with A level Mathematics students have reported any classroom-management issues when using them. </p><p>One of the very understandable reservations teachers have about allowing the use of smartphones in class is that students will be using them for distracting or disruptive purposes, such as browsing social media or taking photos and videos. One simple solution that I've heard from a few teachers is to make it compulsory that when phones are used they should be face-up on the students' desks in full view and not held in the students' hands. The strategy is sufficient in many classrooms to ensure they are used appropriately.</p><p>If the "face up, on the desk" strategy isn't sufficient then there is a feature built-in to GeoGebra and Desmos that is a very effective classroom-management tool. This feature, known as exam or test mode, locks the phone down so that the app cannot be exited and other apps cannot be used. It’s straightforward to leave the app but it stops the clock. I would highly recommend the use of exam mode for classroom management to any teacher that has concerns about the potential disruption of using smartphones in class. The feature can be accessed from within the app for GeoGebra or by downloading a separate app for Desmos.</p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGyAws3d4hZ0RCQFoZzMQSoK8B0xXHs5V8hJ3l0pWNFBRDpKR-kI-q7LoqqgNbvao6hDyaJRVAE1_7LM1ipWgQqmxZ0Ka4aq562gl4j5m26Im3Z2YJbiOVbd2gYRWey-kCh7l5tnqiXqCmdOOH6IYBZESrAujctS_KQvUcJ4xCWyvU1z1bH2rXWvBBPVs/s429/desmos-test-mode.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="429" data-original-width="360" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGyAws3d4hZ0RCQFoZzMQSoK8B0xXHs5V8hJ3l0pWNFBRDpKR-kI-q7LoqqgNbvao6hDyaJRVAE1_7LM1ipWgQqmxZ0Ka4aq562gl4j5m26Im3Z2YJbiOVbd2gYRWey-kCh7l5tnqiXqCmdOOH6IYBZESrAujctS_KQvUcJ4xCWyvU1z1bH2rXWvBBPVs/w336-h400/desmos-test-mode.jpg" width="336" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Example of Desmos Test app</td></tr></tbody></table><p>Another concern I’ve heard from teachers is that the screen on a phone is too small for it be an effective tool for mathematics. This is a concern I’ve heard from teachers but not students! As part of MEI’s work on the Advanced Mathematics Support Programme we worked with teachers on using Desmos and GeoGebra in the classroom and recorded some videos on it. The Using Technology in Further Maths video at <a href="https://amsp.org.uk/resource/pd-videos-further-pure">https://amsp.org.uk/resource/pd-videos-further-pure</a> shows the teacher expressing that he thought that the screen would be too small but the students were used to working on a small screen and had no problem with it.</p><p>When implementing BYOD there is the worry that students will not have a device that is suitable for them to bring into lessons. This can be addressed by the school/college providing a small number of tablets for this purpose at a cost that is much less than providing them for every student.</p><p>One final issue raised that I have much less sympathy for is the question “why use this in the classroom when it’s not allowed on the test?”. My response to this is that there are many classroom strategies which teachers use the classroom that are not allowed in tests, such as asking the teacher a question, looking something up in a textbook or discussing ideas with other students! Use of technology in the classroom should be judged using the same fundamental criterion as any other technique: does it help students understand mathematical relationships better? </p><h2 style="text-align: left;">Trying BYOD in the classroom</h2><p>For teachers who are interested in exploring student use of technology in the classroom BYOD offers an opportunity to try tasks without any financial investment. There are many activities available, such as the MEI GeoGebra tasks (<a href="https://mei.org.uk/resources/?terms=geogebra">https://mei.org.uk/resources/?terms=geogebra</a>) and MEI Desmos tasks (<a href="https://mei.org.uk/resources/?terms=desmos">https://mei.org.uk/resources/?terms=desmos</a>). The exam or test modes in Desmos and GeoGebra are highly recommended for teachers who have concerns about the distractions of using smartphones in class.</p>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-4389312831431929242020-10-15T17:01:00.001+01:002020-10-15T17:06:54.511+01:00Maths GIFs: parabolas<p>One of the fantastic features of dynamic graphing software is that you can animate how a graph changes as some aspect of it is varied. I'm particularly keen on making short GIFs of animated graphs that demonstrate interesting properties. </p><p>One advantage of animated GIFs is that they are really easy to distribute via Twitter. There are loads of them if you have a look at the hashtag <a href="https://twitter.com/search?q=%23mathGIF&src=hashtag_click" target="_blank">#mathgif</a>. </p><h2 style="text-align: left;">Parabolas</h2><p style="text-align: left;">I've been posting animated Maths GIFs to Twitter for quite a while now. Here are some of my favourites featuring parabolas:</p><p style="text-align: left;">The midpoint of the points of intersection of <i>y </i>= <i>x</i>² and <i>y </i>= <i>mx </i>+ <i>c </i>lies on the line <i>x = m</i>/2.</p><p style="text-align: left;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgunJysEFhi24YgjqH-A7GQQU6i-2LqDCPrPp86nqofr573hUK3dG_AVAek37nlnStYZY2Szzgp4g9c8N6xaZqtZN0Z38PBVYmThbnBy7EDWCrbK7h0MN0NtwX7EKUNPKIBHL8G7ceACwU/s657/intersect-parabola-line.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="483" data-original-width="657" height="294" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgunJysEFhi24YgjqH-A7GQQU6i-2LqDCPrPp86nqofr573hUK3dG_AVAek37nlnStYZY2Szzgp4g9c8N6xaZqtZN0Z38PBVYmThbnBy7EDWCrbK7h0MN0NtwX7EKUNPKIBHL8G7ceACwU/w400-h294/intersect-parabola-line.gif" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>The tangent at point P to a parabola with vertex Q can be constructed by finding R, the point of intersection of the vertical through P and the horizontal through Q, finding the midpoint, M, of QR and then drawing the line through P and M.<p></p><p style="text-align: left;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-bjHBbFbEXxDTAEfACia53ZPtVfYjgcEgEWW5AuQZ_Ypo31uhqXJKjtdf53m7FqWoFceBUpYpuuz4XshPXPTvzWozUYWm4hFHOHJDz83d_rTQzN_cLjsZKXJm8imIfGAVuAqXx1ir5aA/s1330/parabola-tangent.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="730" data-original-width="1330" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-bjHBbFbEXxDTAEfACia53ZPtVfYjgcEgEWW5AuQZ_Ypo31uhqXJKjtdf53m7FqWoFceBUpYpuuz4XshPXPTvzWozUYWm4hFHOHJDz83d_rTQzN_cLjsZKXJm8imIfGAVuAqXx1ir5aA/w400-h220/parabola-tangent.gif" width="400" /></a></div><p style="clear: both; text-align: left;">If the tangents at two points A and B on a parabola are perpendicular then the line AB goes through the focus and the tangents intersect on the directrix.</p><p style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuC0GSdyUjXqkevDn6ZoBpqFXJCLp535JDngPHFINCIw0TVvei6L6RZtn5vlr6MLEWbO-crcz1uaAOu_zOWxWkZA4aFnCz28hx2Ibe4ZoAgKkvmQ1Hx-1uUI3UWr3PrBJXX7H5LzdEuvs/s531/quad-perp.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="445" data-original-width="531" height="335" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuC0GSdyUjXqkevDn6ZoBpqFXJCLp535JDngPHFINCIw0TVvei6L6RZtn5vlr6MLEWbO-crcz1uaAOu_zOWxWkZA4aFnCz28hx2Ibe4ZoAgKkvmQ1Hx-1uUI3UWr3PrBJXX7H5LzdEuvs/w400-h335/quad-perp.gif" width="400" /></a></p><p style="clear: both; text-align: left;">The tangent to <i>y</i> = <i>x² + bx </i>+ <i>c </i>when <i>x</i> = 0 has equation <i>y = bx + c.</i></p><p style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR6T46wPe2HHvbQ5iRKkLOhVT7wjZHpTbAsRoRnIjjYSczu5bYfCs_udbizKs3XJskX2wxXgSkRf7pnBrgih_cPh5lJuXpMRQ_eiBDhWiSII0xmSTkMXR74SUCeLL5f5s0ODQIW7-P0ks/s911/tangent-parabola.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="763" data-original-width="911" height="335" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR6T46wPe2HHvbQ5iRKkLOhVT7wjZHpTbAsRoRnIjjYSczu5bYfCs_udbizKs3XJskX2wxXgSkRf7pnBrgih_cPh5lJuXpMRQ_eiBDhWiSII0xmSTkMXR74SUCeLL5f5s0ODQIW7-P0ks/w400-h335/tangent-parabola.gif" width="400" /></a></p><p style="clear: both; text-align: left;">The tangent to <i>y</i> = <i>x²</i><i> </i>when <i>x</i> = <i>a</i> passes through the <i>y</i>-axis at <i>y = –a</i>²<i> .</i></p><p style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOQiyA_MHZoBnfSfENZBRUXQwyUVvjWHUjbRCYB9VJ8AekmlcllKVmVRG0V4zxyvDX6wxGIQwkT3ExZOg_JUvGa_4y0lXhouwcKrePxzakGqqtS5fi1CTJhNDc7mdQMiOK6nwqhll-Iew/s487/quadratic-tangent.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="453" data-original-width="487" height="373" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOQiyA_MHZoBnfSfENZBRUXQwyUVvjWHUjbRCYB9VJ8AekmlcllKVmVRG0V4zxyvDX6wxGIQwkT3ExZOg_JUvGa_4y0lXhouwcKrePxzakGqqtS5fi1CTJhNDc7mdQMiOK6nwqhll-Iew/w400-h373/quadratic-tangent.gif" width="400" /></a></p><p style="clear: both; text-align: left;">For two points on a parabola A and B the position of the point C on the parabola (between A and B) such that the triangle ABC has the maximum area lies on the tangent to the parabola that is perpendicular to AB.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXUTaDYxMdKuIu-5kRN3cRspSyS3jPliMcO7k6hkpaIwPlsCshS0JRIJugr-lWStvg_D0-zbQADxBXTKIhFtOCixoH8S_t_55RlWj5zoMPAxrozZLfOhamZuzKARMwg2QGeW898zM5OuM/s744/parabola-triangle.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="553" data-original-width="744" height="297" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXUTaDYxMdKuIu-5kRN3cRspSyS3jPliMcO7k6hkpaIwPlsCshS0JRIJugr-lWStvg_D0-zbQADxBXTKIhFtOCixoH8S_t_55RlWj5zoMPAxrozZLfOhamZuzKARMwg2QGeW898zM5OuM/w400-h297/parabola-triangle.gif" width="400" /></a></div><p style="clear: both; text-align: left;">If a pair of perpendicular parabolas have 4 points of intersection these will all lie on the same circle.</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_izNnZ0LqFS56BEq1ymEOK9tp-WWugsO72eoMxJkxnwn7xK57PHH4VTbmVQ0rdq1sP1eSq5Y9Z6-TwJ80WOu3n1EoRjZhFcEySeFX_hoGGQvbtprf99rPGlYwKYNLxgRNSY2VpfO2cqQ/s536/perp-parabolas.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="392" data-original-width="536" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_izNnZ0LqFS56BEq1ymEOK9tp-WWugsO72eoMxJkxnwn7xK57PHH4VTbmVQ0rdq1sP1eSq5Y9Z6-TwJ80WOu3n1EoRjZhFcEySeFX_hoGGQvbtprf99rPGlYwKYNLxgRNSY2VpfO2cqQ/s320/perp-parabolas.gif" width="320" /></a></div><p style="clear: both; text-align: left;">Tracing the parabolas <i>y</i> = <i>x² + bx </i>+ 4<i> </i>as <i>b</i> varies is aesthetically pleasing. </p><p style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjn_FTCjpG3pKFY6Y8kjAHceVD7-tAutDG0-4QACSHNdmF1yAb-Vo0KtRcsoqad2oB15acxi14GPXGemdYVTCNhTYIupWaesUx1beIMXzyISwfWiZuY5ooa7zhB7AmILDoohD_amgtZ2I/s529/quadratic-2.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="423" data-original-width="529" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjn_FTCjpG3pKFY6Y8kjAHceVD7-tAutDG0-4QACSHNdmF1yAb-Vo0KtRcsoqad2oB15acxi14GPXGemdYVTCNhTYIupWaesUx1beIMXzyISwfWiZuY5ooa7zhB7AmILDoohD_amgtZ2I/s320/quadratic-2.gif" width="320" /></a></p><h2 style="text-align: left;">Creating Maths GIFs</h2><p style="text-align: left;">The best tool that I've found for creating Maths GIFs is GeoGebra Classic 5. This can be downloaded from <a href="https://www.geogebra.org/download">https://www.geogebra.org/download</a> </p><p style="text-align: left;">To create one I first construct a file that displays the animation I want based on a single slider. This can be exported as a GIF using use File > Export > Graphics View as Animated GIF.</p><p style="text-align: left;">Some tips that I think are useful:</p><p style="text-align: left;"></p><ul style="text-align: left;"><li>Setting the slider to have about 50 increments and increasing only is usually a good compromise between sufficient for a smooth animation and getting a smallish file size.</li><li>Using a smaller tiled window gives a smaller file size. A ratio of 2:1 displays well on Twitter.</li><li>I like a time between frames of 5ms and the GIFs set to loop.</li></ul><p></p><p></p>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com10tag:blogger.com,1999:blog-9149243388087466234.post-90150619004124607142020-09-30T09:24:00.000+01:002020-09-30T09:24:30.328+01:00Desmos Classroom Activities: An ideal tool for remote/blended learning<p>The past few months have been a very chaotic time for teachers and students. Many schools and colleges are now back to teaching full time but this year is likely to be very challenging and teachers want to know that they can provide opportunities for students to learn, even if the normal day-to-day routines are disrupted. For me, one tool stands out as particularly effective in ensuring that students have interesting and engaging activities that will develop their mathematical understanding: Desmos Classroom.</p><h2 style="text-align: left;">Desmos Classroom Activities </h2><p style="text-align: left;">Desmos Classroom activities are collections of screens that provide students with the opportunity to interact and respond to mathematical questions. They can feature text answers, mathematical answers, graphical features, multi-choice, card sorts and many other types of elements. </p><p style="text-align: left;">There is a large selection of pre-made activities and it's really easy to adapt these or create your own. These activities are great for supporting students in thinking deeply about the maths they are learning but it's worth mentioning that they look great too - the design is really appealing and this makes using them a joy for teachers and students. </p><p style="text-align: left;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQyj5d5OM5WUgnFNF7610x1A6thQbyC6W8aM8e4ZoHv9T5hkUuwDCX-YgCCutMpxmKDvMAWy9oWspzTLZG0L9DlaAnmS2MMwz6U9a2eGbsgNGrDxj3NL2XJA5b7Y8nQqKMn83OjE4hK0s/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="539" data-original-width="1160" height="298" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQyj5d5OM5WUgnFNF7610x1A6thQbyC6W8aM8e4ZoHv9T5hkUuwDCX-YgCCutMpxmKDvMAWy9oWspzTLZG0L9DlaAnmS2MMwz6U9a2eGbsgNGrDxj3NL2XJA5b7Y8nQqKMn83OjE4hK0s/w640-h298/image.png" width="640" /></a></div><br /><br /><p></p><h2 style="text-align: left;">Teacher Dashboard</h2><p style="text-align: left;">Where the activities really come into their own is with the teacher dashboard. This allows you to see a live report of your students' progress, including their responses to any questions and how they have moved dynamic elements such as graphs or card sorts.</p><p style="text-align: left;">The dashboard also allows you to control the screens that the students can see using 'Pacing'. For example, if you just want them to work on the first three screens of a larger activity this is easy to set and change during the lesson. You can also 'Pause' the activity at any time so students can't interact with it. I've heard audible groans in a room when I've paused an activity - which shows how much people enjoy working on them but also highlights that it's good practice to warn before pausing - e.g. "I'm going to pause this activity in 5...4...3...2...1...". With great power comes great responsibility! </p><p style="text-align: left;">There are lots of other features on the dashboard too: you can anonymise it before showing it to a class; you can take snapshots of students' responses and display them; you can give individual feedback; ... and many more than these which are best experienced by exploring it. </p><p style="text-align: left;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSPHKU0EAsmA2_bsosnQWw7uvx6JE3v-K0pDJupoVfJOWYPaCXy16BJrjMAqbXaszCg1vyo_MVRtEzXcAtFqFUoaOsj3w8W2k-CoJ4hTU6GZb0RY1IGSrZOI2-ep0_wLPpa2mHxFFL6VA/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="601" data-original-width="1261" height="306" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSPHKU0EAsmA2_bsosnQWw7uvx6JE3v-K0pDJupoVfJOWYPaCXy16BJrjMAqbXaszCg1vyo_MVRtEzXcAtFqFUoaOsj3w8W2k-CoJ4hTU6GZb0RY1IGSrZOI2-ep0_wLPpa2mHxFFL6VA/w640-h306/image.png" width="640" /></a></div><br /><p></p><h2 style="text-align: left;">Pedagogy behind their development</h2><p style="text-align: left;">It becomes clear very quickly when using classroom activities that a huge amount of thought has gone into the design of the environment. At its heart is a desire for students to be actively engaged in thinking mathematically. This is encouraged by there being a range of ways in which students can interact and an emphasis on making connections. </p><p style="text-align: left;">One of my favourite features is "Explain your answer". It is built-in as a default option for many question types that when students give a response they are asked to explain it. This encourages the sort of mathematical dialogue that is so helpful in developing students' mathematical understanding. I think most teachers have had the experience of finding that they've understood something more deeply when they've had to explain it and this provides mini opportunities for students to have this experience. </p><p style="text-align: left;">The amazing Dan Meyer leads the team developing these activities and the platform. He's published a blog post about building great digital mathematical activities. In this he discusses how the design of the classroom activities is based on a number of principles. Some of the most important ones listed for me are: </p><p style="text-align: left;"></p><ul style="text-align: left;"><li>Ask for informal analysis before formal analysis</li><li>Connect representations</li><li>Create objects that promote mathematical conversations between teachers and students</li><li>Give students opportunities to be right and wrong in different, interesting ways</li></ul><div>However, there are many more in the blog. It's well worth a read: <a href="https://blog.desmos.com/articles/the-desmos-guide-to-building-great-digital-math/">https://blog.desmos.com/articles/the-desmos-guide-to-building-great-digital-math/</a></div><p></p><h2 style="text-align: left;">Supporting remote/blended learning: live online or asynchronous lessons</h2><p style="text-align: left;">Desmos classroom was originally designed to be used in face-to-face teaching environments. However, it has proved to be a fantastic tool for supporting learning in a variety of contexts. The recent upheavals have meant that many teachers have found themselves needing to provide either live online lessons, asynchronous online support, or a combination of these for students. Desmos classroom is particularly effective in supporting this.</p><p style="text-align: left;">The activities partner very well with a live classroom environment such as Zoom or Teams live. You can see what the students are doing live, give feedback and pause when you want them to focus on what you're doing. At AMSP we've been using it in parallel with live online classrooms and it's been really effective. </p><p style="text-align: left;">It also works really well to support asynchronous teaching too. They are constantly updating the platform and one feature they added over the summer was the ability to give individual feedback on a students' responses. This makes it viable to set an activity as a homework task for a student to complete over a week - you can monitor how they are progressing, give them feedback if they are stuck on a question and then pause the activity at the end of the week. </p><h2 style="text-align: left;">Finding out more</h2><p style="text-align: left;">A lot of the activities have been written for a US audience but the majority do transfer really well to teaching in UK schools. Natalie Vernon and I have curated a couple of collections of activities that are aligned to the GCSE and A level curriculums: </p><p style="text-align: left;"></p><ul style="text-align: left;"><li>GCSE collection: <a href="https://teacher.desmos.com/collection/5e827a6e58f1e36e4d220ef8">https://teacher.desmos.com/collection/5e827a6e58f1e36e4d220ef8</a></li><li>A level collection: <a href="https://teacher.desmos.com/collection/5e83d8c2e4f77b7bb43dc438">https://teacher.desmos.com/collection/5e83d8c2e4f77b7bb43dc438</a></li></ul><div>... and if you're wanting somewhere to start I can't recommend Marbleslides enough: <a href="https://teacher.desmos.com/activitybuilder/custom/566b31734e38e1e21a10aac8">https://teacher.desmos.com/activitybuilder/custom/566b31734e38e1e21a10aac8</a> </div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2LpaLvhVt-5s3sqqhkcDI21ZbXNPfPf412FlN8eyCEQNzkQwKuIWtW7oHm5swMygtUFBSYwbg3F-s4t7XVtlt3fFDziJibYMmKH8E0l3-hw2Z_kCjZz3w9uaa-529oKb2OCeCAMtoISk/" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="720" data-original-width="1825" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2LpaLvhVt-5s3sqqhkcDI21ZbXNPfPf412FlN8eyCEQNzkQwKuIWtW7oHm5swMygtUFBSYwbg3F-s4t7XVtlt3fFDziJibYMmKH8E0l3-hw2Z_kCjZz3w9uaa-529oKb2OCeCAMtoISk/w640-h252/image.png" width="640" /></a></div><br /><br /></div><p></p>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com2tag:blogger.com,1999:blog-9149243388087466234.post-58362023497971018452020-05-01T11:30:00.002+01:002020-05-01T13:23:33.028+01:00Effective questions when using dynamic graphsMost graphing tools have a feature to add a variable constant in a way that can be changed dynamically. In Desmos and GeoGebra this is by using sliders; in Autograph it is via the constant controller. When the parameter is changed you can view its effect on the graph and this can be discussed with students. You can even animate the change so you don't need to do it manually!<br />
<br />
This feature of being able to dynamically change a variable constant is a very powerful tool to help students build their understanding; however, to make the best use of it, it is useful to have some questions to ask to direct students' thinking.<br />
<h2>
Effective questions when using dynamic graphs</h2>
<div>
<ul>
<li>When I vary … how does it move and why?</li>
<li>For what value of … will …?</li>
<li>If I vary … how will it move?
</li>
</ul>
<h2>
Examples using the intersection of a line and a parabola</h2>
</div>
<div>
To exemplify this I'm going to use the intersection of the parabola <i>y</i>=<i>x</i><sup>2</sup> with the line <i>y</i>=<i>mx+c</i>. To do this I've plotted:<br />
<br />
<ul>
<li><b><i>y</i>=<i>x</i><sup>2</sup> </b></li>
<li><b><i>y</i>=<i>mx+c</i></b></li>
<li><b>(0,<i>c</i>)</b> - this point isn't strictly necessary but it helps students "see" the motion of the line as vertical when <i>c </i> is varied.</li>
</ul>
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You can play with a dynamic version of this in GeoGebra (<a href="http://www.geogebra.org/m/hrthedxp">www.geogebra.org/m/hrthedxp</a>) or Desmos (<a href="https://www.desmos.com/calculator/qweu6rky6k">www.desmos.com/calculator/qweu6rky6k</a>).<br />
In all of the examples below I want the students to focus on the number of points of intersection of the line and the parabola.<br />
<br />
<h3>
When I vary … how does it move and why?</h3>
<b>Question:</b></div>
<div>
I've set <i>m=</i>2. When I vary <i>c </i>how does the number of points of intersection change and why?<br />
<br /></div>
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For the responses to this I would expect students to be able to observe that the number of points of intersection can be 0, 1 or 2. I would hope that some would be able to link this to solving the simultaneously and using the discriminant of a quadratic equation. This would lead to directing them to thinking about the related equation <i>x</i><sup>2</sup>−<i>mx−c=</i>0.<br />
<br />
<h3>
For what value of … will …?</h3>
<b>Question:</b><br />
I've set <i>m=</i>4. For what value of <i>c</i> will the line and parabola have a single (or repeated) point of intersection?<br />
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This is a less open question but when used as a follow-on from the previous type of question it allows for students to apply their understanding and obtain an answer. The answer can then be verified dynamically.<br />
<h3>
If I vary … how will it move?</h3>
<div>
<b>Question:</b><br />
I've set <i>c=</i>2. If I vary <i>m</i> how many points of intersection will there be between the line and the parabola? Will it be 0, 1 or 2 or will it be always 2?<br />
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<br />
This more demanding style of question requires the students to be familiar with dynamic graphs and should be used after they've experienced answering questions of the first two types a few times. Being able to predict how varying the constant will affect the graph is a very useful stage in supporting students with generalising.</div>
Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com2tag:blogger.com,1999:blog-9149243388087466234.post-11717571741011202122020-02-24T13:43:00.004+00:002020-02-24T13:43:57.773+00:00NCETM article on the use of calculators in A level MathematicsMy article for NCETM: "Ten calculations that A level students should be doing on a calculator: can yours?" can be read at: <a href="https://www.ncetm.org.uk/resources/54231">https://www.ncetm.org.uk/resources/54231</a>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com6tag:blogger.com,1999:blog-9149243388087466234.post-73996771900151913862019-07-17T09:04:00.001+01:002019-07-17T09:08:09.511+01:00Slice of advice 2019: What I have learned this yearI've been asked again this year to contribute to Craig Barton's end of year podcast: Slice of advice - what have I learned this year: <a href="http://www.mrbartonmaths.com/blog/slice-of-advice-2019-what-did-you-learn-this-year/">http://www.mrbartonmaths.com/blog/slice-of-advice-2019-what-did-you-learn-this-year/</a><br />
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The main thing that I've learned this year is how few teachers are familiar with Richard Skemp's ideas on instrumental versus relational understanding. I meet a lot of teachers in my job and I will often have only one or two teachers in the room who have read it. These ideas have been fundamental to my development as a mathematics educator and I see it as the key concept that is most illuminating when considering ideas about teaching and learning in maths. It is also one of the main reasons why I am so passionate about the use of technology in maths.<br />
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<h2>
Instrumental versus relational understanding</h2>
<div>
Skemp published his ideas in the 1970s but they are still available via the ATM website at: <a href="https://www.atm.org.uk/write/MediaUploads/Resources/Richard_Skemp.pdf">https://www.atm.org.uk/write/MediaUploads/Resources/Richard_Skemp.pdf</a></div>
<div>
<br /></div>
<div>
The paper contrasts two meanings to the word "understanding" that are used in relation to maths:</div>
<div>
<ul>
<li>Instrumental is just teaching the "how" of mathematical methods. Most commonly this will be when a method is presented to students, usually by working through an example, and then their activity is to be able to repeat this "recipe" for a number of different examples.</li>
<li>Relational understanding emphasises the "why" of mathematics. Teaching for relational understanding focusses on the students being able to explain why mathematical methods work, especially by making connections, or relations, with other mathematical ideas.</li>
</ul>
<div>
For example, an instrumental approach to learning completing the square would be based on showing the students the mechanics of the method on particular examples and having them repeat this method themselves on some further examples. A relational approach would focus on relationship with the transformed graphs of quadratic equations and, in particular, the special cases of quadratic equations with repeated roots.</div>
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<div>
Instrumental understanding is shallower whereas relational understanding is longer lasting and more easily applied in solving novel problems.</div>
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<div>
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<h2>
Using technology to support relational understanding</h2>
<div>
There are a number of reasons why I believe technology, especially dynamic graphing software such as GeoGebra or Desmos, is so powerful in supporting relational understanding. The main two of these are:</div>
<div>
<ul>
<li>it constantly reinforces the link between algebraic, graphical and numerical representations of mathematical objects;</li>
<li>it gives us the power to create many instances of a situation very quickly so that we can observe and generalise patterns.</li>
</ul>
<h3>
An example: straight lines that intersect a parabola at one point</h3>
</div>
<div>
As an example I am going to think about the following problem:</div>
<div>
<br /></div>
<div style="text-align: center;">
What are the possible values of <i>m</i> and <i>c</i> so that the line <i>y=mx+c </i>has <br />
a single repeated point of intersection with the curve <i>y=x</i>².</div>
<div style="text-align: center;">
<br /></div>
<div>
If you've not met this problem before then you might like to explore it in a graph-plotter yourself.</div>
<div>
<br /></div>
<div>
By putting this into a graph plotter it is fairly quick to find a number of different cases where the line cuts the curve at a single point (i.e. a tangent). These cases can be listed and observed algebraically, graphically and numerically. </div>
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One way of solving this is based on using completing the square (I'm assuming that this is a task being explored by students before they've met calculus). Being able to observe that completing the square is applicable here is difficult for a student that has been taught the method instrumentally compared to a student who has a strong understanding of the relationship between completing the square and quadratic equations with repeated roots:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimdrN6GDJnQfSqDfEai1G3NzDTMmv6mzHjMCP_xh78wj_oiksjFrGqydLI65MD8hFXlK4R3cdwDK_wFD3lsVnmD5ag_iMekr4EELTtpO6Ltmlt7Bgxq7gUoskUmJMADWcCOBPLfB1JUbY/s1600/algebra.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="145" data-original-width="466" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimdrN6GDJnQfSqDfEai1G3NzDTMmv6mzHjMCP_xh78wj_oiksjFrGqydLI65MD8hFXlK4R3cdwDK_wFD3lsVnmD5ag_iMekr4EELTtpO6Ltmlt7Bgxq7gUoskUmJMADWcCOBPLfB1JUbY/s1600/algebra.png" /></a></div>
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Skemp is still relevant</h2>
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Although these ideas were first published over 40 years ago I think they are just as relevant today. There are increased demands for reasoning and problem solving at both GCSE and A level and teaching for relational understanding is key to supporting students with these. If you've not read Skemp before I'd suggest doing so and if you have I'd suggest a re-read. </div>
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Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com3tag:blogger.com,1999:blog-9149243388087466234.post-56014966896319008512019-05-15T15:18:00.002+01:002019-05-15T15:18:59.709+01:00Masters assignment: Using Dynamic Geometry/Graphing software to set students open-ended mathematics investigationsIn 2011 I completed an MA in ICT and Education through the University of Leeds. My critical study was on "Using Dynamic Geometry/Graphing software to set students open-ended mathematics investigations". It can be downloaded from: <a href="https://drive.google.com/file/d/1pgFvIjkuEQRpzUm4uJ_KZ1mnql6UZfJ9/view?usp=sharing">https://drive.google.com/file/d/1pgFvIjkuEQRpzUm4uJ_KZ1mnql6UZfJ9/view?usp=sharing</a>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com4tag:blogger.com,1999:blog-9149243388087466234.post-85638360440719647682019-03-08T16:52:00.000+00:002019-03-08T16:52:08.230+00:00Neverware CloudReady: turning old laptops into ChromebooksOver the past week I've been exploring CloudReady from Neverware. The idea behind it is that it takes an old laptop and effectively converts it into a Chromebook.<br />
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I took an old Windows laptop (I'm not sure of the exact age but I think it was about 10 years old). It was running so slowly that it was practically unusable. I replaced it with their operating system, which is based on the Chromium OS, to turn it into a Chromebook. The process was pretty straightforward - it required me to make an installer on a USB stick and then I ran the installation. Overall it took less than an hour but it would be quicker if I were to repeat it. It was a full replacement of the OS, but given that the laptop was destined to be disposed of, this wasn't a problem.<br />
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I've now got a fully working Chromebook and I'm really impressed with it. It runs really quickly - the browser is at least as fast as my current Windows laptop and I can watch/stream videos on it clearly. You can see a picture of it below running the GeoGebra app that's available for Chromebooks.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs-scNfV0hYLmGDkPbQqup_VLZOp1nynQ1Dg82aDzsCwkEtmZhHDo0sV3C3AFz7EwvT1wKPu-jyU69hcOLc4Rrqdiv6sgUJNY-3CAnOdoTCtzdyzl8VL-bzLtu0S7ozaOgTwHyZhEV2eE/s1600/chromebook-small.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="434" data-original-width="492" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs-scNfV0hYLmGDkPbQqup_VLZOp1nynQ1Dg82aDzsCwkEtmZhHDo0sV3C3AFz7EwvT1wKPu-jyU69hcOLc4Rrqdiv6sgUJNY-3CAnOdoTCtzdyzl8VL-bzLtu0S7ozaOgTwHyZhEV2eE/s320/chromebook-small.jpg" width="320" /></a></div>
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The only issue I had was that it initially wouldn't connect to WiFi. This wasn't a problem with CloudReady - the WiFi was temperamental on the laptop when it was running Windows. I ended up buying a USB WiFi adapter and it now works fine. I found it difficult to find any information online as to which adapters were compatible with Chromebooks - a couple of forums suggested that I needed 802.11n so I bought this one: <a href="https://www.amazon.co.uk/gp/product/B00EZOQFHO/">https://www.amazon.co.uk/gp/product/B00EZOQFHO/</a> and it worked immediately.<br />
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Overall I think this has the potential to be really useful for schools - if they have old laptops that run really slowly then they can convert these to fast Chromebooks. It's free for home users and £10 per device, annually, for schools. You can see more details at: <a href="https://www.neverware.com/">https://www.neverware.com/</a> <br />
Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com4tag:blogger.com,1999:blog-9149243388087466234.post-387986944310357032018-10-03T11:45:00.001+01:002018-10-03T11:45:08.012+01:00Why Smartphones are a Really Useful Tool in the Maths ClassroomMy article on <i>Why Smartphones are a Really Useful Tool in the Maths Classroom </i>has been published in Teachwire<br /><br />
<a href="https://www.teachwire.net/news/why-smartphones-are-a-really-useful-tool-in-the-maths-classroom">https://www.teachwire.net/news/why-smartphones-are-a-really-useful-tool-in-the-maths-classroom</a>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com7tag:blogger.com,1999:blog-9149243388087466234.post-20263000533094662992018-07-16T18:38:00.002+01:002018-07-16T18:38:33.940+01:00Slice of Advice: What have I learned this year?I recently submitted a short clip for Craig Barton’s podcast on the theme of Slice of Advice: What have I learned this year? In this I suggested two things that had had an impact on me this year: the use of graphing apps on smartphones and diagrams representing functions as mappings between number lines.<br />
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Graphing apps on smartphones</h3>
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I’ve been using GeoGebra for over ten years now and it has had a huge impact on how I think about mathematics and teaching mathematics. I’ve also been massively impressed with Desmos though I don’t know it inside-out in the way I do with GeoGebra. My use of these graphing tools, and Autograph before it, was always based on using them on a computer and this means I have a preference for the computer software over the phone apps: I find them much more usable on a large screen with a proper keyboard and mouse. This works fine for me but, as I’ve commented on here before, I think the greatest impact of technology in the mathematics classroom is as a result of students using it. Consequently I’ve long been an advocate of the use of graphing apps on phones by students, especially in A level classrooms. However, even though I’d promoted the use of phones I had always thought that this was an inferior device for students to be using the software on and that the ideal would be to have them using the software on laptops if it were possible.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcVHQEcD-pS1NuI9Ml_61Aan_UZINk4pJLVut3EVM5lQRTO6HTe7GuDcKMKCKbXNeCW2uCnS0YHo-gO1lNCCrVD4rBUk4GTDtUaWDrfIV79hrj91C4vWXZWzyNRl79S9IBusC9_iOSbrM/s1600/27-learned-this-year-pic1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcVHQEcD-pS1NuI9Ml_61Aan_UZINk4pJLVut3EVM5lQRTO6HTe7GuDcKMKCKbXNeCW2uCnS0YHo-gO1lNCCrVD4rBUk4GTDtUaWDrfIV79hrj91C4vWXZWzyNRl79S9IBusC9_iOSbrM/s320/27-learned-this-year-pic1.png" width="180" /></a></div>
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What has changed for me this year is that I no longer consider phones an inferior device for using the software for students. Today’s generation of students have grown up with small touch-screen devices and find it completely natural to operate software in such an environment. For many students their phone is the device they are happiest using and so the opportunity to situate such fantastic bits of software as GeoGebra or Desmos in this environment is one that we should be capitalising on. The quality of modern phones is such that the apps work beautifully on them. In particular I think the GeoGebra 3D app on Android phones renders the image better, and is more responsive, than the equivalent computer-based version.<br />
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This year we recorded some more PD videos for the Further Mathematics Support Programme – short 5 minute videos that address a particular bit of pedagogy. In one of these we looked at using graphing apps in the Further Maths classroom. In this video you’ll see students using them and hear the teacher comment on how surprised he was that the students didn’t find the small screens an impediment to them using the app effectively. This video, along with questions for reflection, can be seen at <a href="http://furthermaths.org.uk/pd-videos-further-pure#video5">furthermaths.org.uk/pd-videos-further-pure#video5 </a><br />
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<br /><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/wT1TmxCpTk4/0.jpg" src="https://www.youtube.com/embed/wT1TmxCpTk4?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
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At MEI we’ve developed a series of classroom tasks that are designed to be compatible with the phone apps for both GeoGebra and Desmos. These can be found at <a href="http://mei.org.uk/geogebra-tasks">mei.org.uk/geogebra-tasks</a> and <a href="http://mei.org.uk/desmos-tasks">mei.org.uk/desmos-tasks</a>.<br />
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<h3>
Representing functions as mappings between number lines</h3>
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At the BCME conference at Warwick in April I attended a talk by Martin Flashman, a visiting professor from the US, about the use of mappings between number lines for graphical representations of functions. This is in contrast to the usual convention of plotting them on a pair of cartesian x-y axes.<br />
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This was an idea I’d seen before but I hadn’t really thought about it in detail. It was a revelation to see how a change of representation could give whole new insights into mathematical ideas I thought I’d understood. In particular I think the representation of composite functions and their inverses is much clearer in this way. I was also amazed at how this could be related to the cartesian system: for example a “linear” function is the set of points that lie on a line; if this is transferred to a mapping between number lines this becomes a set of lines that pass through a single point. What is more, these two representations are the duals of each other!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOiyp2uNi1hQSa85Pl1IkILmE3z9xLOF2JQ7zwB-xzOpk6IYieOOzfsyN9t9U5tSrceVGQZOIqM_jk-6uXHLDD2acHAHIY3O4VOXyj3tTmqhrDfs5ww4ehe4OT3l99iCLECXqnPC1RvVY/s1600/27-learned-mapping.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="462" data-original-width="555" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOiyp2uNi1hQSa85Pl1IkILmE3z9xLOF2JQ7zwB-xzOpk6IYieOOzfsyN9t9U5tSrceVGQZOIqM_jk-6uXHLDD2acHAHIY3O4VOXyj3tTmqhrDfs5ww4ehe4OT3l99iCLECXqnPC1RvVY/s320/27-learned-mapping.JPG" width="320" /></a></div>
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Martin went on to give a lot more detail to how this system could be extended to represent ideas such as calculus and functions of complex variables represented as mappings between two planes in 3D space. There is a lot more information, including presentations from his BCME conference sessions, at: <a href="http://users.humboldt.edu/flashman/">users.humboldt.edu/flashman/</a><br />
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Overall this has had a wider impact on me than just thinking about representation of functions: it has made me consider how the way an idea is represented impacts on my understanding of it. For example I think setting out long multiplication in a grid helps me focus on the two-dimensional nature of a product and I’m now searching for more such examples…Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com7tag:blogger.com,1999:blog-9149243388087466234.post-41552961359969822222018-05-31T11:03:00.000+01:002018-05-31T12:47:18.698+01:00Simultaneous equations: an insight from playing with technology<div class="separator" style="clear: both; text-align: center;">
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I've noticed a couple of tweets recently about the importance of being playful when doing mathematics and this has brought to mind an insight I had about simultaneous linear equations that occurred when I was being playful with them in technology. As a result of this I now have a different method for solving that I prefer to the standard textbook approaches.<br />
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<h3>
Playing with simultaneous equations</h3>
In trying to construct a pair of linear simultaneous equations where the solution went through a given point. I can't remember the point but I'll use (3,2) for the example. I did this by creating a point A at (3,2) then two more points B and C and finding the lines that went through these and A. This gave me the simultaneous equations:<br />
3<i>x</i> + <i>y</i> = 11<br />
<i>x</i> + 5<i>y</i> = 13<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAqEiCfZkKD_C6j9Tu0Pb4c1qt79japo5x2xUS8hi_WDZdh6dMpIQJOnOTBDW2ajkmokCguzjuE-zU3O0O4Mrzn6fv6A-gHvHkrthxziTzdWwI-WSRsy-lpdXqcYnbgrFHvKBLMzYD9AA/s1600/26-pic-1b.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="231" data-original-width="370" height="199" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAqEiCfZkKD_C6j9Tu0Pb4c1qt79japo5x2xUS8hi_WDZdh6dMpIQJOnOTBDW2ajkmokCguzjuE-zU3O0O4Mrzn6fv6A-gHvHkrthxziTzdWwI-WSRsy-lpdXqcYnbgrFHvKBLMzYD9AA/s320/26-pic-1b.png" width="320" /></a></div>
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I then moved the points B and C around to find some different equations that would have the same point of intersection. This is probably best displayed here by showing all the different lines I got in a different colour.</div>
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The equations for these lines are:</div>
4<i>x</i> – <i>y</i> = 10<br />
3<i>x</i> + <i>y</i> = 11<br />
2<i>x</i> + 3<i>y</i> = 12<br />
<i>x</i> + 5<i>y</i> = 13<br />
<i>y</i> = 2<br />
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By playing around with these I noticed some interesting features:<br />
<ul>
<li>There are infinitely many lines through the point A.</li>
<li>The coefficients of these lines are related linearly.</li>
<li>One of these lines will be horizontal and one will be vertical (and hence have a simple equation just in terms of <i>x</i> or <i>y</i>)<i>.</i></li>
</ul>
NB rewriting the final equation as 0<i>x</i> + 7<i>y </i>= 14 makes the linear relationship more obvious.<br />
<i> </i><br />
<h3>
A simple method for solving linear simultaneous equations </h3>
Putting all this together I realised this could be used to form a simple method for solving simultaneous linear equations. They can be solved by moving linearly along the coefficients until one of the coefficients of either <i>x</i> or <i>y</i> is 0. For example:<br />
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3<i>x</i> + <i>y</i> = 11<br />
<i>x</i> + 5<i>y</i> = 13<br />
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The gap from 3<i>x</i> to <i>x </i>is –2<i>x</i>, so if you move half this gap again (–<i>x</i>) from <i>x</i> you'll have 0<i>x</i>.<br />
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<img alt="" src="data:image/png;base64,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" /></div>
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Moving an equivalent amount for the other terms means you need to move half the gap from <i>y</i> to 5<i>y</i>, i.e. +2<i>y</i>, on from 5<i>y </i>to get 7<i>y</i> and half the gap from 11 to 13, i.e. +1, to get 14. </div>
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<img alt="" src="data:image/png;base64,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" /> </div>
<br />
This gives 0<i>x</i> + 7<i>y </i>= 14 and hence <i>y</i> = 2. Then using the first equation 3<i>x</i> = 9 gives <i>x = </i>3.<br />
<br />
I've tried this method with lots of linear simultaneous equations and I think it's quicker and makes more sense to me. It's the method I know use if I need to solve them.<br />
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<h3>
The importance of being playful</h3>
This method only occurred to me because I was being playful with the mathematics in the technology. The dynamic aspect of technology offers a fantastic opportunity to be playful with mathematics as it allows the users to change aspects of a situation and observe the impact on representations (here it was graphical and algebraic but it applies to others such as geometrical or numerical too). The playfulness here is not in the sense of 'lets have fun messing around' but being open to explore a situation and then try to explain what is happening and why.Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com1tag:blogger.com,1999:blog-9149243388087466234.post-29427433777494012892018-03-23T16:59:00.000+00:002018-03-23T16:59:22.919+00:00EEF report: Calculator use has a positive effect on students’ calculation skills EEF (Education Endowment Fund) have published a report today on "Improving Mathematics in Key Stages Two and Three". The report is a meta-analysis of research into teaching and learning strategies and can be access in full at: <a href="https://educationendowmentfoundation.org.uk/evidence-summaries/evidence-reviews/improving-mathematics-in-key-stages-two-and-three/">https://educationendowmentfoundation.org.uk/evidence-summaries/evidence-reviews/improving-mathematics-in-key-stages-two-and-three/</a><br />
<br />
The report has been widely publicised, mainly for what it has to say about calculator use. Although Key Stages Two and Three are outside my area of expertise I think there are useful reflections that can be made with reference to the use of technology, including calculators as well as other tools, in GCSE and A level Maths.<br />
<br />
<h3>
Calculator use has a positive effect on students’ calculation skills </h3>
The conclusion in calculator use states: <i>"When calculators are used as an integral part of testing
and teaching, their use appears to have a positive effect on students’
calculation skills. ... When integrated into the teaching of mental and other calculation
approaches, calculators can be very effective for developing non-calculator
computation skills; students become better at arithmetic in general and are
likely to self-regulate their use of calculators, consequently making less (but
better) use of them." </i><br />
<br />
This emphasises the importance of calculator use being integrated into the teaching and learning (and assessment). As with any technology simply adding in a technology without changing the teaching or activities is unlikely to have positive impact on students' understanding. However, if they are used purposefully, with appropriately designed tasks, this is suggesting that students' calculation skills will improve. The challenge is to design appropriate tasks that take advantage of this opportunity. This point is further highlighted by a suggestion that such tasks can enhance students' problem solving skills.<br />
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I think much of this will apply at GCSE and A level too. Some classroom activities that can be tried are:<br />
<ul>
<li>attempting the same problem with and without a calculator and comparing;</li>
<li>using a calculator to investigate a function numerically, such as sin(<i>x</i>) or ln(<i>x</i>). </li>
</ul>
<h3>
Technology: technological tools and computer-assisted instruction</h3>
A separate section of the report discusses technology tools other than calculators. These are split into two categories: technological tools and computer-assisted instruction. I am pleased that this split has been made. I think technology tools for learning mathematics have a much greater potential than computer-assisted instruction.<br />
<br />
The analysis looked at three types of tools: dynamic geometry software, exploratory computer environments and educational games. Here the report suggests that dynamic geometry software has huge potential but stresses the importance of how these tools are used if they are to have a positive impact and that there is a need for professional development for teachers to keep pace with this change.<br />
<br />
I think there are parallels with the use of technology in GCSE and A level Maths; however, the technology tool that is most widely used (apart from calculators) is graphing software. This is probably due to the nature of the GCSE/A level content: there is a much greater emphasis on the behaviour of functions and understanding them through a combination of graphical and algebraic techniques. For graphing tools it is still essential to use them carefully, and in a structured way, if they are going to have the most impact. This is similar to the use of calculators in that care should be taken to design and use tasks that take advantage of the technology and not just replicate what is done with pen and paper in the software. Detailed consideration is required about how the features of the software, such as sliders, can be used to illuminate mathematical concepts and this then needs to expressed through appropriate tasks (and professional development for teachers).<br />
<br />
I would be interested to see a similar analysis of graphing software at GCSE and A level. I have definitely found it more difficult to design classroom tasks for dynamic geometry software than I have for graphing software, though this possibly warrants a different blog post.<br />
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<h3>
It's all about the maths!</h3>
Reading through the technology sections of this report I'm left with a strong impression that calculators and other technology tools are useful for doing and learning maths and so should be integrated into students' experiences of the subject. This can then have a positive impact on their understanding and skills. Calculators and technology tools are just one of many strategies that can be employed and they should be judged in the same way as any other strategy - can they help students understand maths better? Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com1tag:blogger.com,1999:blog-9149243388087466234.post-45968922366012071412018-01-26T13:49:00.001+00:002018-01-26T13:49:53.733+00:00Using graphing software for multiple representations (FMSP PD Video)In March of 2017 I was involved in creating a set of videos looking at aspects of using technology in A level Maths teaching. The full list of videos can be found at: <a href="http://furthermaths.org.uk/pd-videos-technology">http://furthermaths.org.uk/pd-videos-technology</a><br />
<br />
The first of these videos is on using graphing software for multiple representations.<br />
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<div class="separator" style="clear: both; text-align: center;">
<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Sz6FmeirFH4/0.jpg" src="https://www.youtube.com/embed/Sz6FmeirFH4?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
<br />
The video features an example of me using graphing software to highlight the link between graphical, algebraic and numerical representations so students can understand the ideas behind differentiation.<br />
<h2>
Questions for reflection </h2>
The video suggests three questions for reflection:<br />
<ul>
<li>What topics would you use graphing software for?</li>
<li>What are the advantages of using prepared files?</li>
<li>What questioning strategies are effective when using graphing software?</li>
</ul>
Here are my responses to these questions:<br />
<br />
<h3>
What topics would you use graphing software for?</h3>
When teaching A level Maths, especially Pure, I can't think of a single topic where using software to display multiple representations won't enhance students' understanding. This video shows how graphical, and numerical representations can be used to show connections for differentiation but similar could be used for coordinate geometry, trigonometry, series, integration, vectors and almost all other topics.<br />
<br />
When using multiple representations the numerical representation is often considered as much as the link between algebraic and graphical but it is a very powerful representation for students to hang their understanding on. A table of values is very concrete and easily understandable for students. When first meeting graphs or functions this will often be the representation that students start with and so referring back to it in more advanced topics will help them appreciate how these topics have be constructed by extending ideas they are familiar with.<br />
<br />
<h3>
What are the advantages of using prepared files?</h3>
For teachers who are not as confident with using software it can be very reassuring to use a file that has been prepared in advanced. If you know that there are only one or two sliders or points on the screen that you need to move, and everything else will be displayed, then there is less chance that it will go wrong or that the software will display something strange. Although I am now confident to produce files "on the fly" with software, building them up as I go along, I did not start out with this confidence. For me, using prepared files was a necessary first step in building my confidence in using technology in front of a class of students and I would recommend this to other teachers starting out using technology. <br />
<br />
There are still cases where I use prepared files now. This is mainly when I want to emphasise a certain concept and the construction would take too long, and be an unnecessary distraction, to do it live.<br />
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<h3>
What questioning strategies are effective when using graphing software?</h3>
One of the best questions to ask, especially when an object is moved in dynamic software, is "Describe how this object is moving and explain why is is moving in this way". The connections between the representations are often best expressed in terms of how a change in one representation will affect a change in another one. In this example here I used the change in the gradient of the tangent to the curve as the point moved along <i>y</i>=<i>x</i><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">² and </span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><i>y</i>=<i>x</i><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"></span>³. For the latter of these it helped the students see that the gradient would always be positive (or 0) and that this was consistent with an algebraic version of the derivative d<i>y</i>/d<i>x</i>=3<i>x</i></span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">².</span><br />
<br />
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">As a teacher it's a common to hear the observation that you understand something better if you have to explain it yourself. </span>"Describe how this object is moving and explain why is is moving in this way" is an opportunity for the students to engage in the activity of explain a mathematical relationship which can improve their understanding. This question works especially well because an object is moving - it feels very natural when observing the motion of an object to want to explain why it is moving in a certain way.<br />
<br />
An alternative way to ask a similar question is to tell student that you are going to change something and ask them to describe how the objects will move before you do so. You can then check the answer. Here's one to try:<br />
<ul>
<li>Plot <i>y</i>=<i>x</i><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">² and </span><i>y</i>=<i>bx</i>+1. </li>
<li>Describe how the midpoint of the points of intersection of the two curves will move as <i>b</i> is varied.<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"></span><!--[if gte mso 9]><xml>
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</ul>
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<br />Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com1tag:blogger.com,1999:blog-9149243388087466234.post-29197067985739364672017-06-16T15:10:00.000+01:002017-06-16T15:10:59.529+01:00What can the mathematics education community do to increase the use of digital resources by KS5 teachersI was recently asked the following two questions in an email. <br />
<ol>
<li>Teachers: What can the mathematics education community do to increase the use of digital resources by KS5 teachers? </li>
<li>Students: How can time be found in the KS5 teaching programme to bring up the students’ skills in digital resources?</li>
</ol>
I've reposted my response here:<br />
<h2>
1. Teachers: What can the mathematics education community do to increase the use of digital resources by KS5 teachers? </h2>
I have a model for this of 10:80:10 for teachers who are really interested in technology/will use technology if it helps/will rarely use technology. I think the most effective strategy is to concentrate on the middle 80%. For these the single most important criterion to judge any resource on, including a digital one, is whether it is the best tool to help students understand the concept. Many of these teachers don’t have the time to learn complicated software, such as Mathematica, but would use a graph-plotter in front of students if it’s user-friendly and demonstrates the mathematics better/more efficiently. The important role for the technology enthusiasts in the maths education community is to communicate to teachers how some very easy to use digital tools will have a positive impact on their students’ understanding.<br /><br />This needs to be done on an almost topic-by-topic basis. I’ll flesh it out with a couple of examples – the details aren't important - it’s more to demonstrate that for me there occasions where a graphing tool is simple, and adds something significant, and others where it isn’t:<br />
<ul>
<li>I would definitely use a graphing tool early on in students’ learning of differentiation. The concept that the gradient of the tangent to a curve varies and that the numerical value can be expressed as a relationship to the point on the curve is something that it significantly quicker to show in a graphing tool. Additionally, the dynamic element of seeing the tangent moves gives a very physical representation of why we are considering the relationship to where we are on the curve. Pencil and paper or purely algebraic approaches here are markedly inferior (I’ve tried them with students in the past!).</li>
<li>I probably wouldn’t use a graphing tool for the initial learning of binomial expansions. I’m aware I could do some expanding using technology (such as CAS) and/or graphing but I this is a bit fiddly and it’s not going to convince many of the teachers in the middle 80%.</li>
</ul>
So our role is to demonstrate these ideas for appropriate cases and show teachers how easy various tools are to use. We need to be cognisant of the fact that many teachers are directed to be assessment driven and consequently tie this way of teaching into how it better prepares students for the assessments. One thing I’ve been doing recently is drawing teachers attention to questions such as qn 3 on paper 1 of the Edexcel sample A level: <a href="https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-mathematics-sams.pdf">https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-mathematics-sams.pdf</a><br />
<div style="text-align: center;">
"A circle C has equation <i>x</i><span class="st">²</span>+<i>y</i><span class="st">²</span>–4<i>x</i>+10<i>y</i>=<i>k</i> … State the range of possible values for <i>k</i>."</div>
This is one where seeing it vary dynamically gives a really good way in to the question so teachers are tempted to use something dynamic in the classroom. I also think we need to meet the challenge head-on of teachers saying that they won’t have a computer with a graph-plotter in the exam by pointing out that they do lots of things in the classroom that aren’t allowed in the exam: asking the teacher a question, discussing with other students, checking the answers in the back of the book …<br /><br />All of this is an attempt to answer the question: “Why should KS5 teachers use digital resources?”. If we have a clear answer to this then it will be easier to suggest what we should do. <br /><br />
<h2>
2. Students: How can time be found in the KS5 teaching programme to bring up the students’ skills in digital resources?</h2>
This is a trickier question to answer. I’m personally convinced that students’ understanding of mathematics can be enhanced by using digital technologies for mathematics. I’ve observed in my own learning, as well as students’ learning, that the process of constructing mathematical objects in digital technologies very closely models the internal process of constructing the understanding of mathematical relationships. It requires rigour, the ability to understand how objects relate to each other (such as through dependencies) and an understanding of the nature of different mathematical objects. There are many times where I have found constructing something in mathematical software has given me an insight that has allowed me to solve a problem. There is a democratising element to digital tools – they remove many routine aspects allowing time to focus on the bigger picture. This is evident with the way that numerical calculations are not a barrier to understanding when one has a calculator (and also in music technology too where it’s possible to create tunes without much formal musical training!).<br /><br />So this question again comes down to the “Why?”. If we present this as just something that’s needed because the curriculum "says so" then the sensible response would be to do the bare minimum. Again, as with question 1, I think it is our role to present opportunities where students will learn more efficiently and in a deeper way, by using digital technologies themselves. If we can do this then the time question becomes irrelevant. One thing I’ve seen recently that makes me think this is possible is the improvement in smartphone maths apps in the last year or two. I’ve observed students using Desmos on their phones in class and ease of use, the mathematical structure of the software and the fact it was on a device that was personal to the students meant that they engaged with it and were able to very quickly explore ideas that deepened their understanding.<br /><br />Within MEI we’ve been attempting to map the use of technology to the new A level curriculum including lots of examples of tasks focussing on student use of technology: <a href="http://mei.org.uk/integrating-technology">http://mei.org.uk/integrating-technology</a> <br /><br />Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com8tag:blogger.com,1999:blog-9149243388087466234.post-90732711944567885852017-01-19T16:07:00.001+00:002017-01-19T16:07:16.203+00:00The new Maths A level: Graphing families of curvesLast week the MEI new A level for Maths was accredited – the first full A level to be accredited of all the specifications. The specification includes advice on using technology and the sample assessment materials have questions which lend themselves to the use of technology when teaching the topic.<br />
<h2>
Use of graphing tools for families of curves</h2>
The MEI specification includes guidance for activities that should be carried out during the course. The first, and possibly most important, of these is:<br /><br />"Graphing tools: Learners should use graphing software to investigate the<br />relationships between graphical and algebraic representations, e.g. understanding the effect of changing the parameter <i><span style="font-family: Times,"Times New Roman",serif;">k</span></i> in the graphs of <span style="font-family: Times,"Times New Roman",serif;"><i>y</i> = 1/<i>x</i> + <i>k</i></span> or <span style="font-family: Times,"Times New Roman",serif;"><i>y</i> = <i>x</i>² <span style="font-size: 11pt; line-height: 115%;">– </span><i>kx "</i></span><br /><br />The ability to plot a family of curves, and observe the effect on the graphs of dynamically changing a parameter, is an incredibly powerful tool in helping students understand mathematical relationships. Understanding how a mathematical object changes is greatly enhanced by considering how its graphical representation moves.<br />
<h2>
An example from the sample assessment materials</h2>
The sample assessment materials include the following question:<br /><br />Determine the values of k for which part of the graph of <span style="font-family: Times,"Times New Roman",serif;"><i>y</i> = <i>x</i></span><span style="font-family: Times,"Times New Roman",serif;">² <span style="font-family: "Calibri","sans-serif"; font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-GB; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;">–</span> <i>kx</i> + 2<i>k</i></span> appears below the <span style="font-family: Times,"Times New Roman",serif;"><i>x</i></span>-axis.<br /><br />Plotting this in GeoGebra gives the option to vary <span style="font-family: Times,"Times New Roman",serif;"><i>k</i></span> to see how the curve changes. <br />
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<br /><br />This immediately gives some insights into what is happening:<br />
<ul>
<li>How the graph changes for different values of <i><span style="font-family: Times,"Times New Roman",serif;">k</span></i></li>
<li>The graph is sometimes above the <i><span style="font-family: Times,"Times New Roman",serif;">y</span></i>-axis for all values of <i><span style="font-family: Times,"Times New Roman",serif;">x</span></i></li>
<li>For many values of <i><span style="font-family: Times,"Times New Roman",serif;">k</span></i> the graph will be below the <i><span style="font-family: Times,"Times New Roman",serif;">x</span></i>-axis between two values of <i>x<span style="font-family: Times,"Times New Roman",serif;"></span></i></li>
</ul>
<br />Using the software does not answer the question for the students but it does give them a picture which they can use to understand what the question in asking them. I think this is Ofqual's intention with their statement that the use of graphing tools should permeate the study of A level Maths - these tools are easily accessible for all students and can have a massive positive impact on the way they build their understanding of mathematical objects.Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com3tag:blogger.com,1999:blog-9149243388087466234.post-65138349750414498312016-04-13T13:46:00.002+01:002016-04-13T13:46:47.594+01:00Use of Technology in the new A level Mathematics qualificationsLast Friday (8th April) the DfE published the GCE subject-level guidance for mathematics. This guidance is for awarding bodies to help them in designing their specifications and assessments. The full document can be found at: <a href="https://www.gov.uk/government/publications/gce-subject-level-guidance-for-mathematics">https://www.gov.uk/government/publications/gce-subject-level-guidance-for-mathematics</a><br /><br />
<h3>
Requirement for awarding bodies to explain how use of technology will permeate the study of mathematics</h3>
<br />
In the Overarching themes and use of technology section:<br /><br />“<i>Paragraph 8 of the Content Document states that –<br /><br />8. The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A level mathematics.<br /><br />This statement should be interpreted primarily as indicating the desired approach to teaching GCE Qualifications in Mathematics. <br /><br />However, this statement also has implications for assessments. Consequently, in respect of each GCE Qualification in Mathematics which it makes available, or proposes to make available, we expect an awarding organisation to explain and justify in its assessment strategy for that qualification how this statement has been reflected in the qualification’s design.</i>”<br /><br />I think this is very good news in terms of the design brief given to the awarding bodies and, if it applied in the way it is intended, should result in greater and more effective use of technology in the A level mathematics classroom. I look forward with interest to seeing how the awarding bodies justify that their assessment strategies are ensuring that technology permeates the study.<br /><br />
<h3>
Strategies I would like to see</h3>
<br />
There are two main strategies that I would like to see employed: an explicit and an implicit one.<br /><br />I expect to see questions that explicitly refer to the use of technology. This could be through means of a statistical test that a candidate would perform on their calculators or by referencing spreadsheets in the questions. It will be clear to teachers that in order to prepare candidates for the assessment they should be using technology in the teaching and learning.<br /><br />In addition to this I would like to see questions where, although there is no requirement for the candidates to use technology in answering them, they will be better prepared for them if they have using technology in their studies. For example a question asking a candidate to explain the impact of the parameter <i>b</i> on the graph of <i>y</i>=<i>x</i>²+<i>bx</i>+4 is likely to be answered better by students who’ve been using graphing tools to explore curves in their study. This is an implicit strategy but can still be very powerful in encouraging use in the classroom.<br /><br />I look forward to seeing the specifications and sample assessments when they are produced!<br />Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-50625643159265819282016-03-15T02:52:00.000+00:002016-03-15T02:52:13.514+00:00Maths on a SmartphoneI recently gave a talk about doing Maths on a smartphone. I chose four of my favourite apps. I like these apps because in all of them there are opportunities to think and work mathematically, not just passively observe prepared material.<br />
<h2>
MyScript Calculator</h2>
MyScript is really easy to use – you just write the calculations with your finger.<br />
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<br />Android: <a href="https://play.google.com/store/apps/details?id=com.visionobjects.calculator&hl=en_GB">https://play.google.com/store/apps/details?id=com.visionobjects.calculator&hl=en_GB</a> <br />iOS: <a href="https://itunes.apple.com/gb/app/myscript-calculator-handwriting/id578979413?mt=8">https://itunes.apple.com/gb/app/myscript-calculator-handwriting/id578979413?mt=8</a> <br /><br />Problem to try: What’s the maximum product of a set of positive numbers that sum to 19?<br />
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<h2>
Desmos</h2>
<br />Desmos is a very user-friendly graphing calculator. In my experience most people find the interface intuitive and are able to work with it very quickly.<br />
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Android: <a href="https://play.google.com/store/apps/details?id=com.desmos.calculator&hl=en_GB">https://play.google.com/store/apps/details?id=com.desmos.calculator&hl=en_GB</a><br />iOS: <a href="https://itunes.apple.com/gb/app/desmos-graphing-calculator/id653517540?mt=8">https://itunes.apple.com/gb/app/desmos-graphing-calculator/id653517540?mt=8</a> <br /><br />Problem to try: What’s the effect of varying a in the graph of <i>y</i>=<i>x</i>^3+ax+1 ?<br /><br />
<h2>
GeoGebra</h2>
GeoGebra is a very powerful mathematical package that I’ve discussed many times on here. Currently the app is available for Android but not iOS. You can see my thoughts on the app at <a href="http://digitalmathematics.blogspot.co.uk/2016/01/geogebra-app-for-android-phones.html">http://digitalmathematics.blogspot.co.uk/2016/01/geogebra-app-for-android-phones.html</a> <br /><br />In addition to the app you can also open files from the extensive set of materials at <a href="http://www.geogebra.org/materials/">http://www.geogebra.org/materials/</a> using a browser. At the time of writing there are over 360,000 materials on there.<br />
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<br /><br />Android: <a href="https://play.google.com/store/apps/details?id=org.geogebra.android&hl=en_GB">https://play.google.com/store/apps/details?id=org.geogebra.android&hl=en_GB</a> <br />iOS: not currently available for iOS phones<br /><br />Problem to try: Add the points A and B on the x-axis and C on the y-axis. Find the equation of the quadratic that will always go through A, B and C wherever they are moved to.<br /><br />
<h2>
Sumaze!</h2>
<br />Sumaze! is a mathematical puzzle app that requires you to move a block around a maze with various routes involving operations or restrictions on the value of your block. It’s a great puzzle and features lots of maths including arithmetic, inequalities, the modulus function, indices, logarithms and primes.<br />
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<br /><br />Android: <a href="https://play.google.com/store/apps/details?id=com.mei.sumaze&hl=en_GB">https://play.google.com/store/apps/details?id=com.mei.sumaze&hl=en_GB</a> <br />iOS: <a href="https://itunes.apple.com/gb/app/sumaze!/id1045060091?mt=8">https://itunes.apple.com/gb/app/sumaze!/id1045060091?mt=8</a><br /><br />Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-56903799334560537532016-01-14T11:38:00.002+00:002016-01-14T11:38:57.788+00:00GeoGebra App for Android phonesGeoGebra have recently released a version for Android phones. Having played around with it it seems very responsive. The ability to select/drag objects and the speed that it updates appears to be really good - much better than when viewing GeoGebra worksheets via a browser.<br />
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<h3>
An example: gradient of the tangent to a curve at a point</h3>
<h3>
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This is an example that shows how the gradient of the tangent to a curve at a point varies with the point. The app is so quick and easy to use that this took me 17 seconds to create (I timed myself!).<br /><br />
<h3>
Use of smartphones in classrooms</h3>
This app presents a fantastic opportunity to put dynamic maths software into the hands of students. As I've commented on before, I think the real benefits of technology come when students are using it. In addition to this there are significant advantages when this is on a device that students have an attachment to and feel ownership of. Most people feel their own phone is a device that is very personal to them and this means students are more likely to be well-disposed to software on it.<br /><br />Having GeoGebra on their phones means that students can harness the power of the software wherever they are: at home, on the bus, ... However, many teachers have reservations about students using phones in class. There are concerns that this presents a major classroom management issue. This is an ongoing debate and there is some evidence that banning phones in schools has a positive impact on achievement: <a href="http://www.theguardian.com/education/2015/may/16/schools-mobile-phones-academic-results">http://www.theguardian.com/education/2015/may/16/schools-mobile-phones-academic-results</a>. A lot of these arguments focus on general mobile phone use in class and it would be interesting to see some experiences based on students using subject-specific apps such as GeoGebra with effective tasks designed to improve their understanding using the software.<br /><br />
<h3>
Downloading the the Android app</h3>
The Android app can be downloaded from: <a href="https://play.google.com/store/apps/details?id=org.geogebra.android">https://play.google.com/store/apps/details?id=org.geogebra.android</a>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com2tag:blogger.com,1999:blog-9149243388087466234.post-73939379806737355162015-07-24T18:00:00.000+01:002015-07-24T18:04:21.518+01:00GeoGebra Global Gathering 2015Last week I attended the GeoGebra Global Gathering in Linz, Austria. There were lots of fantastic ideas being demonstrated but these are a few of my highlights that I think I’ll be making use of …<br />
<br />
<h2>
Students creating animations</h2>
It was great to see some ideas from Fabian Vitabar from Uruguay about student tasks that involve them creating animations in GeoGebra. There were two different suggestions for doing this – one was to get the students to create an animation such as a bouncing ball by animating points appropriately and then adding images to make the animation look like a real scene. The other was to create some more pure maths based animations such as moving points around a polygon. You can see a GeoGebraBook of his talk and some examples at:<a href="https://tube.geogebra.org/b/1408699#material/1408705"> https://tube.geogebra.org/b/1408699#material/1408705</a>. I think it could be very motivating to students to create animations like these and there is a lot of maths that they’ll need to sort out for themselves to get them to work correctly.<br />
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<h2>
Designing GeoGebra Tasks for Visualization and Reasoning</h2>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS-AjDS7765vxQmqS4cTVsA4a132_qX5qItXP0CrwSqgOY2CzuueRVktyf4sEXixXvPYdS4zt9o1mfFBWnabLxcAhTxk0bBevdrxdLlY-89LEzEe9cBYIBiu5djv1wtkh0gnsXQjvA15g/s1600/IMG_20150716_104522.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS-AjDS7765vxQmqS4cTVsA4a132_qX5qItXP0CrwSqgOY2CzuueRVktyf4sEXixXvPYdS4zt9o1mfFBWnabLxcAhTxk0bBevdrxdLlY-89LEzEe9cBYIBiu5djv1wtkh0gnsXQjvA15g/s320/IMG_20150716_104522.jpg" width="320" /></a></div>
<br />
Anthony Or from Hong Kong gave a fantastic talk on Designing GeoGebra Tasks for Visualization and Reasoning. You can see a GeoGebraBook of his talk at <a href="https://tube.geogebra.org/b/1405633">https://tube.geogebra.org/b/1405633</a>. There’s much more in it than I’ll be able to do justice to here but the main theme was linking the process of students constructing objects in GeoGebra to enhancing their mathematical reasoning skills. One useful idea was the contrast between robust and soft constructions: a robust construction maintains properties when objects are dragged and a soft one doesn’t but can be used for investigating. I’d not really thought about emphasising soft constructions as a way in to robust constructions before but I think it could be a useful technique. A final highlight from this session was the challenges section at the end. Constructing an equilateral triangle on a set of three parallel lines is a particular favourite – I managed to solve it on the train journey back from Linz to Vienna!<br />
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<h2>
Problems that Challenge Intuition</h2>
Diego Lieban from Brazil had some examples of very nice problems that challenge intuition: <a href="https://tube.geogebra.org/b/971211">https://tube.geogebra.org/b/971211</a>. The first one about the shape of a net of a tube when sliced diagonally is particularly tricky as it’s hard to make the cut on a real cardboard tube without creasing it.<br />
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<h2>
New features on their way – Phone apps, Badges and Groups </h2>
Some upcoming features were presented that are very exciting. The work on the phone apps is developing and there is a beta version of the Android one available for testing now. I’ve tried this and I’m really impressed – it’s very responsive when objects are dragged and, contrary to what I was worried about, it seems to work well on such a small screen when selecting and dragging objects. Badges are coming soon for GeoGebraTube accounts – these will automatically display when users have created specific objects or used specific tools/views. I can see these as being very effective in encourage people to continually develop their skills. The last feature that I’m excited about seeing is being able to create groups of GeoGebra users. This will be especially useful for Professional Development workshops – it will mean that teachers from these can form a network and continue to support each other afterwards.<br />
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<h2>
Smart Board software</h2>
I’d been aware for a while that GeoGebra was built-in to Smart Board software but hadn’t seen it in action. The ability to hand write an equation of a curve and then drag this into a GeoGebra widget looks like a handy tool and one that I plan to investigate further. There’s more instructions online about how to do this at: <br />
<a href="http://onlinehelp.smarttech.com/english/mac/help/notebook/14_3_0/Content/Product-User/InsertingContent/InsertingContentFromGeoGebra.htm">http://onlinehelp.smarttech.com/english/mac/help/notebook/14_3_0/Content/Product-User/InsertingContent/InsertingContentFromGeoGebra.htm </a><br />
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<h2>
Other sessions</h2>
There were lots of other great sessions and plenaries at the gathering. A full list of GeoGebraBooks for the talks is at: <a href="https://tube.geogebra.org/student/bDgPocAYy#material/1375207">https://tube.geogebra.org/student/bDgPocAYy#material/1375207</a>.<br />
<br />
Ben Sparks and I gave two sessions too:<br />
Professional Development for practising teachers including live online sessions - <a href="https://tube.geogebra.org/material/show/id/1367449">https://tube.geogebra.org/material/show/id/1367449</a> <br />
Creating effective teaching resources and using GeoGebra in examinations? - <a href="https://tube.geogebra.org/b/1367455">https://tube.geogebra.org/b/1367455</a> <br />
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<br />
<br />Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com3tag:blogger.com,1999:blog-9149243388087466234.post-49776155414013751122014-11-19T18:36:00.000+00:002014-11-19T18:36:47.688+00:00To use or not to use calculators: a false dichotomyThe issue of whether to use calculators when teaching students maths seems to worry a lot of people. The argument for is often presented as the use of calculators is part of the modern world and, by having access to answers to calculations rapidly, students are more likely to be able to understand relationships. The argument against is that they reduce students' calculation skills which can impact negatively on understanding and also that they over-rely on them and are unable to spot errors as they believe the number on calculator must always be the correct answer.<br />
<br />
I think that this is a false dichotomy that is caused by a misunderstanding of what mathematics is and the low quality of many of the questions that students are asked to do in mathematics classes.<br />
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<h2>
A question</h2>
Here's a fairly typical question: what is 36 × 9? <br />
<br />
Do you think students should do this with or without a calculator?<br />
<br />
My answer is (to both questions) - I don't care!<br />
<br />
This is an absolutely pointless question. What does it matter what 36 times 9 is? In the absence of any realistic or mathematical context it is meaningless and there is no reason to calculate this product.<br />
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The only mathematics worth doing is mathematics that helps us understand the relationship between numbers/shapes or helps us solve a realistic problem.<br />
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<h3>
Better questions:</h3>
<br />
How much would a worker earning £9 per hour earn for a 36 hour week?<br />
<br />
A rectangle has sides length 36 units and 9 units. What is the area of the rectangle and will it be a square number?<br />
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Both of these questions require the product of 36 and 9; however, in both cases the important skill here is to be able to identify that product is required, once a student has identified that the answer will require a product I would hope that they had a range of strategies available to them and that they can use the most appropriate one.<br />
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<h2>
Strategies</h2>
<h3>
Strategy 1: using a calculator.</h3>
<br />
In the 21st century most people (at least in the UK) have a powerful calculator to hand most of the time (in the guise of a mobile phone). It should be part of students' mathematical education to learn this. <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1TJodxzfVVxNXvtcObVt2CXLatuDtwWDXZe-nuyeDjnulz5TmBb35O8pHDnHy14zszPWGaIxc-ENfSm3VbEYMkam86r4ZJtSmi7IVtBpfLilCw4SzQzUvjyAia-GuU0V3tYnV_MY162c/s1600/18-calc.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1TJodxzfVVxNXvtcObVt2CXLatuDtwWDXZe-nuyeDjnulz5TmBb35O8pHDnHy14zszPWGaIxc-ENfSm3VbEYMkam86r4ZJtSmi7IVtBpfLilCw4SzQzUvjyAia-GuU0V3tYnV_MY162c/s1600/18-calc.jpg" height="182" width="320" /></a></div>
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<h3>
Strategy 2: pencil and paper methods</h3>
<br />
These are often useful; however, I would argue for developing alternative strategies for processes. I also think it is important to emphasise methods that aid relational understanding over those that are merely efficient. For example, of the two presented here I think that box method gives a clearer representation of the 2-dimensional nature of multiplication.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhffPV4B44DZJmx56VPlTR_dvtaFhLdoy8EmVgDbXuuy7ejMqg9tMK2g9lLqFQoBJAiGrT-VZfrfq0lNdyr547kVgf-MuOsrUJrHG-si26rRfpa-QHHFRERSt5BFK9nR9pHKDEJP8X0wrQ/s1600/18-1-product.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhffPV4B44DZJmx56VPlTR_dvtaFhLdoy8EmVgDbXuuy7ejMqg9tMK2g9lLqFQoBJAiGrT-VZfrfq0lNdyr547kVgf-MuOsrUJrHG-si26rRfpa-QHHFRERSt5BFK9nR9pHKDEJP8X0wrQ/s1600/18-1-product.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzoMj_AbGswyiTMNuR6wnOHdJV_C_n2mtaeK6-sm6M5wJISf3tOPeGT3VO6w3ZUCq8VeFHyBXypgnkGrVYCfKktuOB3_5j-YIuabcIJ1g3JKVyNiwx8bl_Q_rKWXxp6RAiRqSTFLP6vIQ/s1600/18-2-product.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzoMj_AbGswyiTMNuR6wnOHdJV_C_n2mtaeK6-sm6M5wJISf3tOPeGT3VO6w3ZUCq8VeFHyBXypgnkGrVYCfKktuOB3_5j-YIuabcIJ1g3JKVyNiwx8bl_Q_rKWXxp6RAiRqSTFLP6vIQ/s1600/18-2-product.png" /></a></div>
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<h3>
Strategy 3: in your head</h3>
<br />
If I wanted to multiply 36 by 9 I would probably do 36 times 10 to get 360 and then subtract 36 to get 324. There are many other strategies that can be used for this but I think it is important to develop a playful nature with numbers through these kind of mental calculations.<br />
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<h2>
Which is the best strategy?</h2>
<br />
The three different strategies presented here will be useful in different situations - it would depend on the context. Once a student has correctly identified that a product is needed I would be happy with them using any of these three but I think students should be encouraged to develop their skills in all of them.<br />
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<h2>
Developing skills in using these strategies</h2>
<br />
For all of these strategies it is important that students develop their skills in using them and, with any skill, the best way of doing this is through practice. This practice can either be in the form of drills or play. The problem with much of the mathematical activity that students are asked to do is that the practice of particular skills, such as multiplying a two digit number by a one digit number, becomes an end in itself as opposed to a useful tool to have in one's mathematical toolkit to solve problems. Over-emphasis on practising performing mathematical operations is boring whether they are on a calculator, pencil and paper or in your head.<br />
<br />
If the process of multiplying two numbers by hand is seen as an end in itself then obviously using technology to do this could be considered "cheating". I would suggest instead that the goal of the mathematics taught to students should always be explicitly about solving realistic problems or understanding mathematical relationships. If this is the case then where there is a need to develop and practise strategies for performing processes it will be natural to consider both calculator and non-calculator strategies.<br />
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In short - if you are asking whether students should be using a calculator or not it's not that you're asking the wrong question it's that you're asking the wrong questions.Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-5040046946831565862014-11-14T07:34:00.001+00:002014-11-14T07:34:11.404+00:00Video: Integrating technology into the teaching and learning of mathematicsThis a video featuring Charlie Stripp and me talking about integrating technology into the teaching and learning of mathematics, recorded at this year's MEI Conference.<br />
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<iframe allowfullscreen="" frameborder="0" height="293" src="//www.youtube.com/embed/nsF5jA4-utM" width="480"></iframe>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-9782424707788790312014-11-07T17:35:00.000+00:002014-11-07T17:35:29.046+00:00How to Break ExcelLast weekend was the annual MathsJam Conference. If you haven’t been (or even if you have) I highly recommend it. For more details see: <a href="http://www.mathsjam.com/conference/">http://www.mathsjam.com/conference/</a><br />
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The format is talks of maximum length 5 minutes on anything you find interesting. This year I gave a talk on “How to Break Excel”.<br />
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<h3>
How to Break Excel</h3>
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A commonly occurring “error” in Excel happens when you type <b>1</b> into cell A1, <b>=A1-0.1</b> into cell A2 and then drag this down to cell A11.<br />
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This problems occurs due to the way Excel stores numbers: it uses floating point arithmetic with 1 bit for the sign, 53 bits for the mantissa and 10 for the exponent. This means that the number we think of as 0.1 in base 10 is actually stored in binary as:<br />
0.0001100110011001100110011001100110011001100110011001101<br />
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Similarly 0.9 in base 10 is stored in binary as<br />
0.11100110011001100110011001100110011001100110011001101<br />
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This means that every time you use 0.1 in base 10 Excel is actually using:<br />
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At each stage in the subtraction Excel rounds to 53 significant binary figures. There are two places where this introduces an error: when subtracting 0.1 from 0.8 there’s an error in the 53<sup>rd</sup> binary place and when subtracting 0.1 from 0.4 there’s an error in the 55<sup>th</sup> binary place. This results in a total error of 2<sup>-53</sup> + 2<sup>-55</sup> = 1.38778×10<sup>-16</sup><br />
<sup><br /></sup>
<h3>
Fractional Powers of Negative Numbers</h3>
<br />
Another error occurs when you enter <b>=(-8)^(2/3)</b> into Excel . It gives the result <b>#NUM</b>.<br />
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I think this is happening because it is rounding 2/3 in binary to:<br />
0.10101010101010101010101010101010101010101010101010101<br />
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As a fraction this is:<br />
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The denominator of this fraction is 2<sup>53</sup>, which is even. Consequently this requires finding an even root of a negative number which isn't real!<br />
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But, if you enter <b>=(-8)^(1/3)</b> into Excel it does give the result <b>-2</b>.<br />
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I’m not sure why, but I think that a negative number to the power one over an odd number has been hard-coded in as a special case but a negative number to the power of any other number over an odd number hasn't.Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-53757087674552948122014-10-24T18:10:00.001+01:002014-10-24T18:11:17.422+01:00“Any teacher that can be replaced by a computer should be”There’s been a lot of talk on Twitter this week about an app called PhotoMath. It scans mathematics live and then gives an “answer” to the question. This is a screenshot of it:<br />
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Some comments on Twitter seem to think it’s cheating whilst others think it’s the future (although to be honest most seem to mirror my experience of struggling to get it to read anything sufficiently accurately to evaluate or solve it).<br />
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<h3>
Any question that can be answered by a computer should be</h3>
Arthur C Clarke once said “Any teacher that can be replaced by a computer should be”. It’s a great quote and I’d like to offer a variation on this:<br />
<i>Any question that can be answered by a computer should be.</i><br />
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The whole point of mathematics is to investigate the relationships between mathematical objects or solve real problems (I’ll skirt round the definition of “real” here!). There are many processes that are useful in doing this but where they can be reduced to algorithms (such as solving linear equations, square rooting or dividing) there is no inherent value in performing these algorithms manually as opposed to outsourcing them to technology. <br />
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Whilst this app might not be perfect there are many (free) CAS tools that will solve equations and evaluate expressions that are typed in to them. It doesn't worry me that a computer can solve a linear equation and that a student might “cheat” on their homework. What does worry me is the excessive focus of many maths questions on getting students to learn algorithmic processes – technology presents an opportunity to refocus the curriculum on the important skills of making connections within the subject and solving problems.Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-59140707835413716262013-10-04T18:13:00.000+01:002013-10-04T18:14:56.542+01:00My favourite technology-based maths problemThis is my favourite technology-based maths problem, and possibly my favourite maths problem. It needs to be attempted in software that has linked geometrical and algebraic views (and preferably some others too). GeoGebra or TI-Nspire are ideal for this.<br />
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Start with a square of variable side with one vertex on the origin and one on the positive x-axis:<br />
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The problem is to create a rectangle with the same area as the square whose sides are in the ratio 2:1.<br />
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<h3>
Creating the initial square in GeoGebra</h3>
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<ol>
<li>Add a point A at the origin.</li>
<li>Add a point B on the x-axis.</li>
<li>Use the Regular Polygon tool selecting vertices at A and B and set the number of vertices to 4.</li>
</ol>
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<h3>
Creating the initial square in TI-Nspire</h3>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1OeubLbOoUu3dk6XJlz4KiQNoD0XKyd_kaip8-pUffzcQMKsYkXf4bNcl99aSVwhhUjuCreDI0y4yPXWIumf-eW_17MXT-dp9-or2pJ4k7iLRmxf_kd7d0Kf-VTcOdDOwmMLIDP_3Ul8/s1600/14-square-nspire.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1OeubLbOoUu3dk6XJlz4KiQNoD0XKyd_kaip8-pUffzcQMKsYkXf4bNcl99aSVwhhUjuCreDI0y4yPXWIumf-eW_17MXT-dp9-or2pJ4k7iLRmxf_kd7d0Kf-VTcOdDOwmMLIDP_3Ul8/s320/14-square-nspire.jpg" width="320" /></a></div>
<ol>
<li>Add a new Graphs page.</li>
<li>Add a point at the origin by finding the Intersection Point of both axes .</li>
<li>Add a new Point On the x-axis.</li>
<li>Display the Coordinates of this point.</li>
<li>Use Measurement Transfer to transfer the value of the x-coordinate to the y-axis.</li>
<li>Add a line Perpendicular to the axis through the point on x-axis.</li>
<li>Add a line Perpendicular to the axis through the point on y-axis.</li>
<li>Find the Intersection Point of the two lines.</li>
<li>Add a Polygon through the four points. NB it is important to use the Polygon tool and not the Rectangle tool.</li>
<li>Measure the Area of the square.</li>
</ol>
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<h3>
Solutions</h3>
If you have a correct solution the area of the square and the rectangle should remain equal as you drag the point on the the positive x-axis. I know 8½ distinct methods of constructing a solution using GeoGebra and 6½ using Nspire. How many can you find?Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com1tag:blogger.com,1999:blog-9149243388087466234.post-76307309567473233822013-02-14T19:24:00.000+00:002013-02-14T19:24:51.642+00:00Modelling mechanics in software<br />
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Mechanics is more than just an application of algebra and
geometry, it is <i>the</i> <i>primary </i>application of geometry and
algebra. Newton’s development of
calculus was to answer problems in what we would now call mechanics and much of
his work was geometrical in nature.
However, students study less geometry than they use to and this, along
with difficulties in algebra, can result in issues when they start studying
mechanics.<o:p></o:p></div>
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As a remedy to this software that links geometry and algebra
is perfect tool in which students can practise modelling situations in
mechanics. This can complement their
studies in mechanics and enhance their skills.<o:p></o:p></div>
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In particular two main methods that lend themselves well to
being represented in software are interactive force diagrams and animations of
position.<o:p></o:p></div>
<h2>
Interactive force diagrams<o:p></o:p></h2>
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Interactive force diagrams can be created where the vectors
for the forces acting on an object are represented dynamically. <o:p></o:p></div>
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The diagram below shows the forces acting on a block when
there is a pulling force at some angle to the upwards vertical.<o:p></o:p></div>
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<h2>
Animations of position<o:p></o:p></h2>
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Many other situations in mechanics are about describing how
objects move and this movement is often defined as function as time. This may be something as straightforward as
the position being given as a linear function of the time (in the case of
constant velocity) or more completed examples where the velocity, acceleration
or force are given as functions of the time.<o:p></o:p></div>
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Most graph-plotters and dynamic geometry software allow for
a function, or the coordinates of a point, to be defined in terms of a
parameter, t. This means that if you
know the x and y-position of a particle as a function of the time these can be
entered to give the path of the particle.<o:p></o:p></div>
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To further enhance this a slider can be added for t and the both
the position of the particle and vectors for velocity (and acceleration) can be
shown. In Geogebra once a slider has
been added this can be animated. <o:p></o:p></div>
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The example below shows the position, velocity and speed for
a particle where these are functions of time.<o:p></o:p></div>
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<h2>
Further examples<o:p></o:p></h2>
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A full A level Mechanics 1 paper converted to dynamic Geogebra
files can be found at: <a href="http://www.mei.org.uk/?section=resources&page=ict#geogebra">http://www.mei.org.uk/?section=resources&page=ict#geogebra</a>
<o:p></o:p></div>
Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com2