tag:blogger.com,1999:blog-91492433880874662342018-06-19T09:15:06.975+01:00Digital technologies for learning mathematicsTom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.comBlogger27125tag: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;"></div><div style="margin-left: 1em; margin-right: 1em;"></div><br />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 /><br /><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 /><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-lVA-pk2GvdM/Ww_DG0Cs9PI/AAAAAAAAG-s/mm2Mq_9zspY-nSW6jd-ZITTdcS7zvq5fwCLcBGAs/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://4.bp.blogspot.com/-lVA-pk2GvdM/Ww_DG0Cs9PI/AAAAAAAAG-s/mm2Mq_9zspY-nSW6jd-ZITTdcS7zvq5fwCLcBGAs/s320/26-pic-1b.png" width="320" /></a></div><div class="" style="clear: both; text-align: left;"><br /></div><div class="" style="clear: both; text-align: left;">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><div class="separator" style="clear: both; text-align: center;"><a href="https://www.blogger.com/u/1/blogger.g?blogID=9149243388087466234" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/u/1/blogger.g?blogID=9149243388087466234" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://3.bp.blogspot.com/-mNDzlxqw4Dw/Ww_C_BgRh6I/AAAAAAAAG-o/wLefrX7w4FcPeHbw2dCTp_bN2D4vBGPtwCLcBGAs/s1600/26-pic-2.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://3.bp.blogspot.com/-mNDzlxqw4Dw/Ww_C_BgRh6I/AAAAAAAAG-o/wLefrX7w4FcPeHbw2dCTp_bN2D4vBGPtwCLcBGAs/s320/26-pic-2.png" width="320" /></a></div><div class="" style="clear: both; text-align: center;"> </div><div class="" style="clear: both; text-align: left;">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 /><br />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 /><br />3<i>x</i> + <i>y</i> = 11<br /><i>x</i> + 5<i>y</i> = 13<br /><br />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 /><br /><div style="text-align: center;"><img alt="" src="data:image/png;base64,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" /></div><div style="text-align: left;"><br /></div><div style="text-align: left;">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><br /><div style="text-align: center;"><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 /><br /><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.com0tag: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 /><br />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 /><br /><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.com0tag: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 /><br /><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 /><br /><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> <o:OfficeDocumentSettings> <o:AllowPNG/> </o:OfficeDocumentSettings></xml><![endif]--></li></ul><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-k3AcLTCtL4w/WmsxkEotr7I/AAAAAAAAFiw/_R-kZ0YVD2od9LVi4nWnNI2KSiQxHEz8wCLcBGAs/s1600/24-pd-video-1-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="269" data-original-width="312" src="https://2.bp.blogspot.com/-k3AcLTCtL4w/WmsxkEotr7I/AAAAAAAAFiw/_R-kZ0YVD2od9LVi4nWnNI2KSiQxHEz8wCLcBGAs/s1600/24-pd-video-1-1.png" /></a></div><br />Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag: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.com3tag: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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-0zPlqReHyuM/WIDji2QVd7I/AAAAAAAAAlE/-Lonc4JKKPYCsnVYMMOJVzl-Ax_HoeLRQCLcB/s1600/23-graph.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-0zPlqReHyuM/WIDji2QVd7I/AAAAAAAAAlE/-Lonc4JKKPYCsnVYMMOJVzl-Ax_HoeLRQCLcB/s1600/23-graph.png" /></a></div><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.com1tag: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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-2_eKXlLw8Q4/Vud37trl5GI/AAAAAAAAAdg/nsYHMbCYNSY75Q5cyhr6YJFXeLrdsXAdw/s1600/21-myscript.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/-2_eKXlLw8Q4/Vud37trl5GI/AAAAAAAAAdg/nsYHMbCYNSY75Q5cyhr6YJFXeLrdsXAdw/s320/21-myscript.png" width="186" /></a></div><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 /><br /><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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-gj7_TgItXm0/Vud37pk20wI/AAAAAAAAAdk/H9GYoO6_LU4XPXbi_eddhrRZ67vPc6Ikg/s1600/21-desmos.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-gj7_TgItXm0/Vud37pk20wI/AAAAAAAAAdk/H9GYoO6_LU4XPXbi_eddhrRZ67vPc6Ikg/s320/21-desmos.png" width="186" /></a></div><br /><br />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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-pfadkkesXTw/Vud37iDhhgI/AAAAAAAAAdc/c5yjZZPoYMoHoQxCbkivJBIzmepyYOQwA/s1600/21-geogebra.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-pfadkkesXTw/Vud37iDhhgI/AAAAAAAAAdc/c5yjZZPoYMoHoQxCbkivJBIzmepyYOQwA/s320/21-geogebra.png" width="187" /></a></div><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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-mjLk-1ZuXEs/Vud35i5BYcI/AAAAAAAAAdU/tEUm8vBOrXom-1cgDrRiJ2V7ClwGlCoQw/s1600/21-sumaze.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-mjLk-1ZuXEs/Vud35i5BYcI/AAAAAAAAAdU/tEUm8vBOrXom-1cgDrRiJ2V7ClwGlCoQw/s320/21-sumaze.png" width="180" /></a></div><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 /><br /><h3>An example: gradient of the tangent to a curve at a point</h3><h3> </h3><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-vvoVch1YaY0/VpeHnsA05kI/AAAAAAAAAcc/26ApCAejWjw/s1600/20-ggb-android.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://4.bp.blogspot.com/-vvoVch1YaY0/VpeHnsA05kI/AAAAAAAAAcc/26ApCAejWjw/s400/20-ggb-android.png" width="225" /> </a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br />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.com1tag: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 /><br /><h2>Designing GeoGebra Tasks for Visualization and Reasoning</h2><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-y3J61az85GM/VbJuzyx41HI/AAAAAAAAAPk/Q8mmcUN-iwg/s1600/IMG_20150716_104522.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://2.bp.blogspot.com/-y3J61az85GM/VbJuzyx41HI/AAAAAAAAAPk/Q8mmcUN-iwg/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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-I49Wykkj7CE/VbJuL0xO88I/AAAAAAAAAPc/5ZZwk9hBhXE/s1600/19-parallel.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="243" src="http://2.bp.blogspot.com/-I49Wykkj7CE/VbJuL0xO88I/AAAAAAAAAPc/5ZZwk9hBhXE/s320/19-parallel.JPG" width="320" /></a></div><br /><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 /><br /><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 /><br /><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 /><br /><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 /><br /><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 /><br /><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 /><br />The only mathematics worth doing is mathematics that helps us understand the relationship between numbers/shapes or helps us solve a realistic problem.<br /><br /><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 /><br />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 /><br /><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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-9-loNSvJZI4/VGzip969rRI/AAAAAAAAAL0/L6tFq0EymrE/s1600/18-calc.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-9-loNSvJZI4/VGzip969rRI/AAAAAAAAAL0/L6tFq0EymrE/s1600/18-calc.jpg" height="182" width="320" /></a></div><br /><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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-ZscYneUK-Fc/VGzip2fFzFI/AAAAAAAAAMI/cp7dX-LHpro/s1600/18-1-product.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-ZscYneUK-Fc/VGzip2fFzFI/AAAAAAAAAMI/cp7dX-LHpro/s1600/18-1-product.png" /></a><a href="http://4.bp.blogspot.com/-o1wbPXyUrrk/VGzip-UF0cI/AAAAAAAAAME/HnH1dR2g9tI/s1600/18-2-product.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-o1wbPXyUrrk/VGzip-UF0cI/AAAAAAAAAME/HnH1dR2g9tI/s1600/18-2-product.png" /></a></div><br /><br /><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 /><br /><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 /><br /><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 /><br />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 /><br /><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 /><br />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 /><br /><h3>How to Break Excel</h3><br />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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-hPnVQOpF4os/VF0Ar5JI9fI/AAAAAAAAALY/FtjRCqaVz7o/s1600/16-excel.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-hPnVQOpF4os/VF0Ar5JI9fI/AAAAAAAAALY/FtjRCqaVz7o/s1600/16-excel.jpg" height="320" width="160" /></a></div><br /><br />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 /><br />Similarly 0.9 in base 10 is stored in binary as<br />0.11100110011001100110011001100110011001100110011001101<br /><br />This means that every time you use 0.1 in base 10 Excel is actually using:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-1ZB3tujyPh8/VF0Arz_hXdI/AAAAAAAAALg/001LSYTmVP8/s1600/16-onetenth.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-1ZB3tujyPh8/VF0Arz_hXdI/AAAAAAAAALg/001LSYTmVP8/s1600/16-onetenth.gif" /></a></div><br />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 /><br />I think this is happening because it is rounding 2/3 in binary to:<br />0.10101010101010101010101010101010101010101010101010101<br /><br />As a fraction this is:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-gWOSjQTE3-Q/VF0Ar21brDI/AAAAAAAAALU/QyEA0cth7lI/s1600/16-twothirds.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-gWOSjQTE3-Q/VF0Ar21brDI/AAAAAAAAALU/QyEA0cth7lI/s1600/16-twothirds.gif" /></a></div><br />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 /><br />But, if you enter <b>=(-8)^(1/3)</b> into Excel it does give the result <b>-2</b>.<br /><br />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 /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-JUuB2wj69P0/VEqHuU7qVZI/AAAAAAAAAK0/l-hvLBuGpuI/s1600/15-photomath.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-JUuB2wj69P0/VEqHuU7qVZI/AAAAAAAAAK0/l-hvLBuGpuI/s1600/15-photomath.gif" height="194" width="320" /></a></div><br /><br />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 /><br /><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 /><br />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 /><br />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 /><br />Start with a square of variable side with one vertex on the origin and one on the positive x-axis:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-mTCKa-18KEI/Uk72G1qi28I/AAAAAAAAAGo/QDwy4cNLQoU/s1600/14-square-geogebra.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-mTCKa-18KEI/Uk72G1qi28I/AAAAAAAAAGo/QDwy4cNLQoU/s1600/14-square-geogebra.png" /></a></div><br />The problem is to create a rectangle with the same area as the square whose sides are in the ratio 2:1.<br /><br /><h3>Creating the initial square in GeoGebra</h3><br /><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><br /><h3>Creating the initial square in TI-Nspire</h3><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-fTCdTWzUAnM/Uk72H9NmXdI/AAAAAAAAAGw/FJ7Vm_vCsg4/s1600/14-square-nspire.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="http://1.bp.blogspot.com/-fTCdTWzUAnM/Uk72H9NmXdI/AAAAAAAAAGw/FJ7Vm_vCsg4/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><br /><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 /><div class="MsoNormal">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><div class="MsoNormal"><br /></div><div class="MsoNormal">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><div class="MsoNormal"><br /></div><div class="MsoNormal">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><div class="MsoNormal">Interactive force diagrams can be created where the vectors for the forces acting on an object are represented dynamically. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">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><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-yln2rx4lKs8/UR05YCwU08I/AAAAAAAAAEQ/bbw6BAXPnpU/s1600/dynamic_force.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="214" src="http://1.bp.blogspot.com/-yln2rx4lKs8/UR05YCwU08I/AAAAAAAAAEQ/bbw6BAXPnpU/s320/dynamic_force.png" width="320" /></a></div><div class="MsoNormal"><br /></div><h2>Animations of position<o:p></o:p></h2><div class="MsoNormal">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><div class="MsoNormal"><br /></div><div class="MsoNormal">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><div class="MsoNormal"><br /></div><div class="MsoNormal">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><div class="MsoNormal"><br /></div><div class="MsoNormal">The example below shows the position, velocity and speed for a particle where these are functions of time.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-UTzG0h-Bx2Y/UR05pBWT87I/AAAAAAAAAEY/R7oB1Uxq-lA/s1600/dynamic_position2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="315" src="http://1.bp.blogspot.com/-UTzG0h-Bx2Y/UR05pBWT87I/AAAAAAAAAEY/R7oB1Uxq-lA/s320/dynamic_position2.png" width="320" /></a></div><div class="MsoNormal"><br /></div><h2>Further examples<o:p></o:p></h2><div class="MsoNormal">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.com2tag:blogger.com,1999:blog-9149243388087466234.post-22339077645691085652013-01-22T19:05:00.000+00:002013-01-22T19:05:25.999+00:00Could the drive for using CAS come from students?<br />In a previous post I wrote about how Computer Algebra Systems (CAS) are really useful tools for teaching when writing questions: <a href="http://digitalmathematics.blogspot.co.uk/2011/08/using-cas-for-writing-questions.html">http://digitalmathematics.blogspot.co.uk/2011/08/using-cas-for-writing-questions.html</a> This focussed on teachers using them for preparation but not necessarily exploiting them in the classroom.<br /><br />Many teachers are reluctant to use them in lessons when they aren’t allowed in examinations. There is also the view that the purpose of mathematics lessons is for students to learn processes and therefore a tool that can perform those processes will hamper students’ progress. <br /><h3>CAS is out there</h3>There are free tools available online that have CAS built into them such as websites like <a href="http://www.wolframalpha.com/">Wolfram Alpha</a>, and software such as <a href="http://www.geogebra.org/">Geogebra</a> and <a href="http://www.microsoft.com/en-gb/download/details.aspx?id=15702">Microsoft Mathematics</a>. If students find these they can use them when set basic algebraic tasks. A series of questions on factorising quadratic equations can be performed very quickly on Wolfram Alpha!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Wth_98uKaCM/UP7iHGRrwKI/AAAAAAAAADw/wMecYr51pF8/s1600/12_cas_wolfram.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="162" src="http://2.bp.blogspot.com/-Wth_98uKaCM/UP7iHGRrwKI/AAAAAAAAADw/wMecYr51pF8/s320/12_cas_wolfram.jpg" width="320" /></a></div><br />Once students are aware of these then the standard tasks set to them may appear pointless. They may also question why the school subject of mathematics is not using a widely available tool that everyone has access to. This could be the driver that leads teachers to think about how CAS can be used effectively in the teaching of mathematics, as opposed to ignoring it or banning it.<br /><br /><h3>When to use CAS</h3><div><br /></div>CAS is very powerful and used appropriately can give students access to results very quickly that they would not necessarily be able to find accurately without it, at least not without taking a lot of time. This gives the scope for relationships to be investigated without the limiting factor of always need to perform processes by hand.<br /><br />A parallel can be drawn with four-function calculators and arithmetic. If a problem required students to divide two numbers I wouldn’t always want them to demonstrate that they could divide these numbers with a pencil and paper method, especially if the numbers were large or given to a lot of decimal places. I would often be looking for them to know that division was need and that they could give the answer to a suitable degree of accuracy when found using a calculator. That is not to say there is or there isn’t a place for students learning a pencil and paper method for division but that is a debate for another time.<br /><br />If the same logic is applied to students who have access to a CAS calculator this would mean that problems could be set where I would be interested in whether students could select the correct process and give a suitable answer. For example, in a question that required a derivative I would want the students to know that a derivative was needed and to be able to work with the expression for the derivative obtained using CAS but not demonstrate that they could find this by pen-and-paper (such as using the quotient rule). Again, this is not to say that there isn’t a place for learning the quotient rule.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-H-iMj2I65x4/UP7iJB6nlsI/AAAAAAAAAD4/255Invei6II/s1600/12_cas_geogebra.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-H-iMj2I65x4/UP7iJB6nlsI/AAAAAAAAAD4/255Invei6II/s1600/12_cas_geogebra.jpg" /></a></div><br />A lot of the decision as to whether to use CAS comes down to an individual’s perception of what mathematics is. For those who perceive maths as a series of processes where the mechanics of them are important then CAS can appear like cheating; for those who see it as the relationship between objects then it can be a useful tool for freeing-up students to investigate these relationships.<br /><div><br /></div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com1tag:blogger.com,1999:blog-9149243388087466234.post-86987335509219952782012-03-18T13:29:00.000+00:002012-03-18T13:29:05.132+00:00Student-centred use of technology<div class="MsoNormal"><span style="font-family: inherit;">A lot of the use of technology in mathematics can be very teacher-centred.</span><span style="font-family: inherit;"> </span><span style="font-family: inherit;">Tools such as Interactive Whiteboards can the unintended consequence of reducing student-centred use. Student use of technology can be much more powerful in terms of increasing students’ understanding.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;"><b>Using technology for problem solving</b><o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;">A few years ago I wanted to construct a dynamic file that would display a cubic based on knowing the position of two turning points. The task was a lot less trivial than I had initially thought but in solving it I think I learnt more about cubic functions in that hour than I had learnt in all the years previous to that! My thoughts on completing it were that if this was such a powerful tool for me to learn mathematics through investigation and discovery then I should be giving my students the opportunity to experience this. This is unlikely to happen through watching me operate software at the front of the class.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">As an aside, this is a really nice task. Using some geometry/graphing software, such as Geogebra or TI-Nspire, add a pair of points to the screen then construct the dynamic cubic that will have these two points as the turning points.<o:p></o:p></span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-HTVE3BEvbLE/T2Xi-lTUcCI/AAAAAAAAADE/sCjtNRUKwaI/s1600/student_centred.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="http://4.bp.blogspot.com/-HTVE3BEvbLE/T2Xi-lTUcCI/AAAAAAAAADE/sCjtNRUKwaI/s1600/student_centred.jpg" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;"><b>Encouraging student-centred use of technology</b><o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;">One of the issues is giving students access to the technology. One of the solutions to this is set problems for students to complete outside the classroom. The use of free tools, such as Geogebra, makes this realistic.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">An alternative is to use handheld devices, such as TI-Nspire. These don’t require access to a computer lab and students are often receptive to their use as they can use many of these in their exams.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;"><b>Tasks</b><o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;">As well as giving access to the technology it is important to have effective tasks that encourage learning. Some examples of tasks (for A level Maths) are available at <a href="http://www.mei.org.uk/?section=resources&page=ict">http://www.mei.org.uk/?section=resources&page=ict</a></span> <o:p></o:p></div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com1tag:blogger.com,1999:blog-9149243388087466234.post-65358214083276025122011-11-18T18:17:00.001+00:002011-11-18T18:51:46.598+00:00Programming for learning mathematics: Project Euler<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-ARehBEEyDsQ/Tsagqi977RI/AAAAAAAAACc/heSU9wtjNUU/s1600/prog_nspire.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></div><div style="font-family: inherit;">Last week I went to two events where the subject of programming came-up, and specifically how useful a tool it is for the learning of mathematics. Essentially to be able to program a computer to perform a process you need to understand it first and also the process of thinking about how you would construct a program can help you to understand an idea.<br /><br /><b>Project Euler</b><br />At both events Project Euler was mentioned as a great resource/community to encourage people to learn maths through programming. Michael Borcherds (<a href="http://twitter.com/mike_geogebra">twitter.com/mike_geogebra</a>) suggested that I might be interested in it at the Computer Based Math summit: <a href="http://www.computerbasedmath.org/">www.computerbasedmath.org/</a>. Then a couple of days later Matt Parker (<a href="http://twitter.com/standupmaths">twitter.com/standupmaths</a>) promoted it at MathsJam: <a href="http://mathsjam.com/">mathsjam.com/</a>.<br /><br />Project Euler (<a href="http://projecteuler.net/">projecteuler.net/</a>) is a series of mathematical/computer programming problems that require some mathematical insight and a little bit of programming knowledge to solve. However, an understanding of a FOR … NEXT loop and an IF … THEN statement should be enough to get going. The website is structured so that you aren’t restricted to any particular programming language – you just enter your numerical answer generated by your program and it checks the answer. It also features a list of the problems you’ve solved and (once you’ve solved a problem) lets you see the forum for that problem.<br /><br /><b>Programming on the TI-Nspire</b><br />I tried the first problem on the TI-Nspire. The Nspire has TI Basic built into it and is pretty easy to get going on.</div><div style="font-family: inherit;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-ARehBEEyDsQ/Tsagqi977RI/AAAAAAAAACc/heSU9wtjNUU/s1600/prog_nspire.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="http://4.bp.blogspot.com/-ARehBEEyDsQ/Tsagqi977RI/AAAAAAAAACc/heSU9wtjNUU/s320/prog_nspire.jpg" width="320" /></a></div><div style="font-family: inherit;"><br />One additional advantage is that it has all the mathematical functions built-in and easy to access which is especially useful if you’re using the CAS version.<br /><br /><b>Python</b><br />I tried the second problem using Python. Python is an open-source programming language that is popular because it is also easy to read.</div><div style="font-family: inherit;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-uIASVHfxzWI/TsagrDFzhiI/AAAAAAAAACg/ZginSvaBoRA/s1600/prog_python.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-uIASVHfxzWI/TsagrDFzhiI/AAAAAAAAACg/ZginSvaBoRA/s320/prog_python.jpg" width="256" /></a></div><div style="font-family: inherit;"><br />I downloaded Python from <a href="http://python.org/">python.org/</a> and then installed the Ninja IDE front-end from <a href="http://ninja-ide.org/">ninja-ide.org/</a> (and then pointed it at where I’d installed Python). Ninja is a lot easier to use than the command-line version of Python. <br /><br /><b>Programming for learning maths</b><br />As generalisation is often the aim in mathematics programming is an excellent tool for learning the subject. When programming is viewed as explaining a generalisation to a computer it is easy to see why it is so powerful.</div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com3tag:blogger.com,1999:blog-9149243388087466234.post-72002438653116470782011-11-03T18:15:00.001+00:002011-11-03T18:21:50.626+00:00Using an Interactive Whiteboard (IWB) effectively for teaching Maths<div style="font-family: inherit;">Many mathematics classrooms are installed with interactive whiteboards (IWBs); but I often get asked by teachers how they can use them to their full potential. There is a lot more that can be done with an IWB other than displaying a static demonstration. The Ofsted report of 2008 highlighted that the full potential of IWBs is not being capitalised on in Maths lessons: “... too often teachers used (IWBs) simply for PowerPoint presentations with no interaction by the pupils.” (see <a href="http://www.ofsted.gov.uk/node/2255">http://www.ofsted.gov.uk/node/2255</a>). I discussed some of the problems with PowerPoint in my previous post: “<a href="http://digitalmathematics.blogspot.com/2011/10/problem-with-powerpoint-in-mathematics.html">What’s wrong with PowerPoint for Teaching Maths</a>”. In spite of this there are some features of IWBs that make the particularly useful for teaching maths relating to the immediacy of dynamic software.<br /><br /><b>The immediacy of dynamic software </b><br />When using an IWB the projector is (almost) always on therefore it is easy to use a piece of mathematical software for a small component of a larger lesson. This is especially useful when there is an object that can be dragged where it is on the screen (as opposed dragging it using a mouse which his hidden from the students’ views).<br /><br />In comparison in the days before IWBs, when there were only a couple of projectors in the college I was teaching in, I found that if I signed the projector out for a lesson I had a tendency to overuse it – what could have been a very effective 5 minute explanation using some dynamic software turned into a longer and less-effective part of the lesson. <br /><br />There are many pieces of dynamic software that can be used with an IWB but two that I find particularly effective are Autograph and TI-Nspire.<br /><br /><b>Autograph</b></div>Autograph has an IWB mode that can be enabled from the top menu (shown in the red square in the image below). This implements a few very useful features:<br /><ul><li>On-screen keyboard (this is a very useful keyboard with built-in mathematical characters that works in other applications)</li><li>Multiple select without shift</li><li>The scribble tool: this allows you to make annotations that remain when the software is changed</li><li>Thicker lines</li></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-kWoKeGVFRg0/TrLZoB8hGEI/AAAAAAAAACM/GwmermtDwe0/s1600/iwb_auto.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="257" src="http://3.bp.blogspot.com/-kWoKeGVFRg0/TrLZoB8hGEI/AAAAAAAAACM/GwmermtDwe0/s320/iwb_auto.jpg" width="320" /></a></div><br /><div style="font-family: inherit;">A single side of A4 with useful IWB tools can be found at: <a href="http://mei.org.uk/files/ict/autograph_tools.pdf">http://mei.org.uk/files/ict/autograph_tools.pdf</a>. This can be printed and stuck to the wall next to your IWB.<br /><br /><b>TI-Nspire</b><br />Nspire is a fantastic bit of software that I’ve also blogged about previously: “<a href="http://digitalmathematics.blogspot.com/2011/04/ti-nspire-30.html">TI-Nspire 3.0</a>”. One particularly useful feature when using it on a IWB is the Keypad in the Documents Toolbox panel. This displays an Nspire handheld keypad on the screen which can be used to fully operate the software. This means that the software can be used without leaving the front of the class and, perhaps more importantly, the students can follow the steps on handhelds. This is a very easy way to encourage student use of ICT.</div><div style="font-family: inherit;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-wEycWFjp1Hs/TrLZouQeGGI/AAAAAAAAACQ/7bB8XxZsUTs/s1600/iwb_nspire.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="211" src="http://4.bp.blogspot.com/-wEycWFjp1Hs/TrLZouQeGGI/AAAAAAAAACQ/7bB8XxZsUTs/s320/iwb_nspire.jpg" width="320" /></a></div><div style="font-family: inherit;"><br /><b>Developing your IWB use</b><br />Subject-specific IWB professional development is not always easy to find so most teachers will develop their skills through using one. The process of learning to use an IWB is an important one and it is likely that in the first instance teachers are going to want to use it in the same was as a traditional whiteboard. This is not something to feel guilty about; however, it is useful when do this to consider how skills can be developed so that over time it is being used to its full potential.</div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com2tag:blogger.com,1999:blog-9149243388087466234.post-49633794551553293722011-10-11T19:38:00.002+01:002014-11-14T15:02:17.023+00:00The problem with Powerpoint for teaching maths<div style="font-family: inherit;">There are a lot places on the internet where the are negative opinions about Powerpoint. I don't want to repeat these arguments here, instead I wish highlight particular problems relating to the use of Powerpoint in the teaching and learning of mathematics. These are that:</div><ul style="font-family: inherit;"><li>It reinforces a view of mathematics that it is a series of algorithms to be rote-learned;</li><li>It can reduce the amount of student-centred use of ICT in learning mathematics;</li><li>It is usually a static form of mathematics and there are many easy tools for creating equivalent dynamic forms of mathematics.</li></ul><div style="font-family: inherit;"><span style="font-size: small;"><b>Reinforcing a limited perception of mathematics</b></span><br />Many Powerpoint presentations for mathematics feature a question with the stages of a solution presented. This can have a negative impact of students' perceptions of mathematics. There is a link between students' perception of mathematics and how successful they are. Students who perceive maths a series of unrelated recipes for solving problems are less successful than those who see it as series of related ideas. Displaying a single, predetermined method to solve a problem can reinforce the perception that mathematics is about learning the method for each type of question which reinforces the perception of it being about unrelated recipes that need to be rote learned.<br /><br />Similarly it doesn't allow space for students to ask "what if...?" type questions or to suggest alternative methods for solving a problem or opportunities for linking with other areas of mathematics. For example if the question is "solve x² + 5x + 6 = 0" and the solution presented is to factorise the students may perceive there is no value to sketching the curve, completing the square or applying the quadratic formula, or possibly, and even worse, that they if they'd tried to solve it using one of these methods that they are "wrong".<br /><br /><b>Reducing the student-centred use of ICT</b><br />The potential that digital technologies, or ICT, can have in the mathematics classroom is widely acknowledged. However, there is a danger that using Powerpoint as a presentational tool can be seen as fulfilling this requirement and consequently additional, more powerful uses of ICT, such as student-centred use of ICT, may be overlooked. This is missing a huge potential given the impact student-centred use can have on learners' understanding when compared to passively watching a presentation. <br /><br /><b>It is easier to produce a dynamic version of mathematics</b><br />The mathematics presented in powerpoint is static: if there is a function this cannot be altered easily in the presentation. By contrast if the function is created in mathematical software it will be easy to alter. For example, in teaching the relationship between the roots of a quadratic equations and the factorised form, it is straightforward to graph a quadratic function and observe the relationship between the equation and the intersections with the axis. Not only does this provide a more generalisable demonstration of the relationship, it is easier to do than producing a powerpoint. </div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com1tag:blogger.com,1999:blog-9149243388087466234.post-62962750598791008462011-09-10T14:03:00.000+01:002011-09-10T14:03:09.165+01:00Using the Guardian Data Store for teaching statistics<div style="font-family: inherit;">There are many sites on the internet with data that can be used for teaching statistics but one of the best, and most topical is the Guardian Data Store. The Guardian Data Store can be found at <a href="http://www.guardian.co.uk/data">http://www.guardian.co.uk/data</a></div><div style="font-family: inherit;"><br /><b>Raw data on many topical news stories</b><br /><br />The data store contains the data associated with stories that are in the news including such varied items as the riots and deprivation, the attendance of MPs and the full data on all the Doctor Who villains. For many of the items there is a Google Docs spreadsheet of the raw data to download.</div><div style="font-family: inherit;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-NBHcjczKyHA/Tmtf6IWKgMI/AAAAAAAAABk/q0q1MZAkCQU/s1600/07_data.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-NBHcjczKyHA/Tmtf6IWKgMI/AAAAAAAAABk/q0q1MZAkCQU/s320/07_data.jpg" width="302" /></a></div><br /><div style="font-family: inherit;"><b>Importing the data into software</b><br /><br />Whilst it is possible to analyse that data within a Google Docs spreadsheet you can do a lot more by importing it into a statistics package. Two of the easiest to use are TI-Nspire and Autograph. With both of these it is very quick to just to copy the data in the spreadsheet and paste it into a list. All the analysis and the diagrams built-in to these packages can then be applied to the data.</div><div style="font-family: inherit;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-kWPmqEx0vP0/Tmtf51YGPMI/AAAAAAAAABg/MbJpc5-4pvU/s1600/07_boxwhisker.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-kWPmqEx0vP0/Tmtf51YGPMI/AAAAAAAAABg/MbJpc5-4pvU/s1600/07_boxwhisker.jpg" /></a></div><div style="font-family: inherit;"><br /></div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-89098428538184763242011-08-04T16:41:00.000+01:002011-08-04T16:41:44.464+01:00Using CAS for writing questionsComputer Algebra Systems (CAS) are very powerful tools for mathematics but they are underused in the classroom probably because they aren't allowed in examinations (at least in the English school system). This lack of use by students means that teachers often overlook how useful they could be for themselves.<br /><br />One time-saving use for teachers is writing questions with certain properties. For example if you know you want a cubic with a repeated root you could multiply out appropriate brackets by hand, or you could use CAS to do it:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-3kBT-ZJ8R8A/Tjq84wdrb1I/AAAAAAAAABc/5Rn7pUXlFtI/s1600/CAS_nspire.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-3kBT-ZJ8R8A/Tjq84wdrb1I/AAAAAAAAABc/5Rn7pUXlFtI/s1600/CAS_nspire.jpg" /></a></div><br />Similarly you may want a quadratic with specific complex roots:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-58pDxw0zJ10/Tjq84nnHczI/AAAAAAAAABY/G4gwfpujanQ/s1600/CAS_microsoft.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-58pDxw0zJ10/Tjq84nnHczI/AAAAAAAAABY/G4gwfpujanQ/s1600/CAS_microsoft.jpg" /></a></div><br /><br />Most CAS engines also feature calculus tools too so you could use the integration function to find a function with a specific derivative:<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-F6sLy4_KuNA/Tjq84H4RKiI/AAAAAAAAABU/03PAZZaTN-E/s1600/CAS_maxima.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-F6sLy4_KuNA/Tjq84H4RKiI/AAAAAAAAABU/03PAZZaTN-E/s1600/CAS_maxima.jpg" /></a></div><br />In addition to using CAS to write questions teachers can also use to check students' answers.<br /><br />There are many CAS tools available. The three that I have used here are:<br /><ul><li>TI-nspire <a href="http://education.ti.com/html/nspire_uk/">http://education.ti.com/html/nspire_uk/</a> (this is available as a CAS and non-CAS version - students will likely want to buy the non-CAS version so they can use it in exams but teachers are advised to buy the CAS version)</li><li>Microsoft Mathematics <a href="http://www.microsoft.com/education/products/student/math/">http://www.microsoft.com/education/products/student/math/</a> (free download)</li><li>Maxima <a href="http://maxima.sourceforge.net/">http://maxima.sourceforge.net/</a> (open-source, free download)</li></ul>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-90535878471047223872011-05-25T09:00:00.000+01:002011-05-25T09:00:20.102+01:00Spreadsheet Algebra<div style="font-family: inherit;">Spreadsheet algebra is a very powerful tool that can be used to teach algebra to students of all abilities and all ages. The major reasons for this are: </div><ul><li>It is a genuine example of how algebra is used in the outside the mathematics classroom</li><li>It reinforces the concept of a variable</li><li>It reduces the likelihood of numerical errors obscuring the underlying mathematics </li><li>It emphasises the importance of correct syntax</li></ul><div style="font-family: inherit;">Many students do not see the ‘point’ of algebra. However, they are aware that they may need to use a spreadsheet after they’ve left school and entered employment and so may be more willing to learn mathematics in format that they perceive as more relevant. <br /><br />Many students do not fully understand the concept of a variable. The use of x as the unknown is alien to many students and can produce misunderstandings. This is not helped by the fact that many students’ first experience of algebra is to solve equations. The result is that they see x as an unknown quantity whose value should be found as opposed to a variable which can be used to define a relationship. A formula in a spreadsheet changes when the variable(s) are changed: this allows students to observe how the output of a function varies as the input varies. An additional advantage is that instead of typing in the cell-reference when entering a formula you can just click on the cell you want: this makes algebra a physical activity, where you ‘point’ at the variable you want.</div><div style="font-family: inherit;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-BqqhM_xp4yA/Tdy11qEnRUI/AAAAAAAAAA0/-HAvbfL22hk/s1600/5_spread_geo.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-BqqhM_xp4yA/Tdy11qEnRUI/AAAAAAAAAA0/-HAvbfL22hk/s1600/5_spread_geo.jpg" /></a></div><div style="font-family: inherit;"><br /></div><div style="font-family: inherit;">Students can often miss the point when investigating mathematical ideas because a numerical error is giving a false result which is obscuring the mathematics. When using a spreadsheet as a tool to investigate mathematics students can rely on the numerical values of calculations and therefore focus their attention on trying to identify and understand any relationships.</div><div style="font-family: inherit;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-eFyt6XOIy9U/Tdy12Pnz-YI/AAAAAAAAAA4/kGUt_mfZ8vA/s1600/5_spread_nsp.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="187" src="http://3.bp.blogspot.com/-eFyt6XOIy9U/Tdy12Pnz-YI/AAAAAAAAAA4/kGUt_mfZ8vA/s320/5_spread_nsp.jpg" width="320" /></a></div><div style="font-family: inherit;"><br />The correct syntax for written algebra can be confusing: e.g. you don’t write a multiplication sign; 2 + 3x means multiply by 3 first; etc. Spreadsheet algebra has a slightly different syntax (though helpfully often still uses BIDMAS). This is analogous to learning a foreign language: it will have different grammatical rules, but learning these will improve your understanding of grammar in both of the languages and emphasise why it is important. Learning spreadsheet algebra will improve students’ understanding of the syntax of written algebra.</div><div style="font-family: inherit;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-zOCusWH5JYQ/Tdy12ZycAmI/AAAAAAAAAA8/xjavW4D7O-U/s1600/5_spread_xl.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="98" src="http://1.bp.blogspot.com/-zOCusWH5JYQ/Tdy12ZycAmI/AAAAAAAAAA8/xjavW4D7O-U/s320/5_spread_xl.jpg" width="320" /></a></div><br /><b>Examples of tasks</b> <br /><ul><li>Setting up an order form that will calculate total cost when different quantities of products are ordered.</li><li>Solving equations by trial and improvement.</li><li>Investigating reverse percentages – what is the cost of an item without VAT?</li><li>Solving simultaneous equations by trial & improvement and elimination.</li><li>Setting up a spreadsheet that solves the quadratic equation ax² + bx + c = 0, when the values of a, b and c are entered in separate cells.</li><li>Setting up a spreadsheet that calculates mean (and standard deviation) from a frequency table.</li><li>Investigating sequences and series.</li><li>Investigating exponential growth and decay.</li><li>Multiplying and finding the inverse of a matrix.</li></ul><b>Software</b><br />In addition to Microsoft Excel spreadsheets are also available in other software such as Geogebra and TI-Nspire.<b> </b> <br /><div style="font-family: inherit;"><br /></div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-1980405058442508942011-05-07T13:02:00.000+01:002011-05-07T13:02:44.129+01:00Euclid – Geometric Constructions (iPhone app)<div style="font-family: inherit;">I’ve been playing with the Euclid iPhone app this week. It’s a fantastic app and very addictive. The basic idea is to turn the process of ruler and compass geometric constructions into a game. You start with some basic constructions, such as midpoint, and as you progress you get set more difficult ones, such as square roots.<br /><br />The progressive levels of difficulty, as you would expect with a game, works really well and contributes to a sense of achievement when levels are completed! A particularly nice feature is, as you would expect for something based on Euclid’s Elements, is that when you have completed some of the constructions, such as perpendicular bisector, this then gets added as a tool you can use.</div><div style="font-family: inherit;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-2AOAMHBh2v0/TcU0xfbRXeI/AAAAAAAAAAw/mvV3L8rqf8s/s1600/euclid.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="153" src="http://3.bp.blogspot.com/-2AOAMHBh2v0/TcU0xfbRXeI/AAAAAAAAAAw/mvV3L8rqf8s/s320/euclid.jpg" width="320" /></a></div><div style="font-family: inherit;"><br />It’s really pleasing to see maths envisaged in a puzzle game format in this way – I’ve certainly learned some geometry from playing it – and would be interesting to know if any teachers have used this with students. The app can be downloaded from: <a href="http://itunes.apple.com/gb/app/euclid-geometric-constructions/id432735893?mt=8%20">http://itunes.apple.com/gb/app/euclid-geometric-constructions/id432735893?mt=8 </a><br /><br /></div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0tag:blogger.com,1999:blog-9149243388087466234.post-62977617143426662382011-04-13T17:25:00.001+01:002011-06-07T09:29:22.957+01:00TI-Nspire 3.0<div style="font-family: inherit;"></div><div style="font-family: inherit;">Last Friday saw the launch of the latest version of the TI-Nspire software - version 3.0. This has a few new features in addition to the already excellent version 2:</div><div style="font-family: inherit;"><br /></div><div style="font-family: inherit;"><b>Adding images</b></div><div style="font-family: inherit;">You can now add images to Nspire pages, including as the background to a graphing or geometry page. This is a really powerful tool for relating mathematics to students' experiences outside the classroom. </div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-WuofR72T0hA/TaXOD2B9zRI/AAAAAAAAAAs/QEbUrGgBPOI/s1600/3_nspire_pic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="http://3.bp.blogspot.com/-WuofR72T0hA/TaXOD2B9zRI/AAAAAAAAAAs/QEbUrGgBPOI/s320/3_nspire_pic.jpg" width="320" /></a></div><div style="font-family: inherit;"></div><div style="font-family: inherit;"><br /></div><div style="font-family: inherit;"><b>Differential equations</b></div><div style="font-family: inherit;">You can also plot first order differential equations on version 3.0, where the derivative is a function of x and y. It plots a slope field indicating the shape of the general solution and particular solutions can be shown by entering initial conditions (as a single value or list).</div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-0nvW5WYcez0/TaXODvbQUNI/AAAAAAAAAAo/rOZeOb9IJYo/s1600/3_nspire_de.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="http://2.bp.blogspot.com/-0nvW5WYcez0/TaXODvbQUNI/AAAAAAAAAAo/rOZeOb9IJYo/s320/3_nspire_de.jpg" width="320" /></a></div><div style="font-family: inherit;"><br /></div><div style="font-family: inherit;"><b>3D Graphing</b></div><div style="font-family: inherit;">The 3D graphing will plot graphs of the form z=f(x,y). The graphs are displayed really nicely and the window is easy to move. I'm very hopeful that later iterations of version 3 will have the ability to add points and vectors to the 3D graphs so it could be used to for the vectors/3D geometry in A2 Core and A2 Further Pure.</div><div style="font-family: inherit;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Wp22xT3Nmts/TaXODQKrlBI/AAAAAAAAAAk/u4OOrSmmLNc/s1600/3_nspire_3d.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="218" src="http://4.bp.blogspot.com/-Wp22xT3Nmts/TaXODQKrlBI/AAAAAAAAAAk/u4OOrSmmLNc/s320/3_nspire_3d.jpg" width="320" /></a></div><div style="font-family: inherit;"><br /></div><div style="font-family: inherit;"><b>Publish to web and new handhelds</b></div><div style="font-family: inherit;">A feature that isn't enabled yet but will be really useful is the ability to publish TI-Nspire files on webpages which can then be viewed (and interacted with!) in any browser. I'm hoping to have some on here as soon as this feature is live. The other major advance is the new handhelds featuring high-resolution colour screens - gone are the days when a pixelated screen meant you couldn't tell the difference between an asymptote and a vertical line and this really brings the technology into the 21st century.</div>Tom Buttonhttp://www.blogger.com/profile/09032352216611295198noreply@blogger.com0