tag:blogger.com,1999:blog-15081246429678572942014-10-02T06:46:08.530+01:00Some Sort of Mathematics...A blog about mathematics, teaching and all that lies betwixt.ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.comBlogger32125tag:blogger.com,1999:blog-1508124642967857294.post-80107517094446884352012-02-07T08:06:00.002+00:002012-02-07T08:07:30.573+00:00Mathematics Counts: The Cockcroft Report (1982)I've just found that the entire text of this report can be found <a href="http://www.educationengland.org.uk/documents/cockcroft/index.html">here</a>; it may well be 30 years old but it seems as fresh and relevant as ever. I thought I would reproduce paragraph 243 below to make it even easier to find it:<br /><br />"243 Mathematics teaching at all levels should include opportunities for <br /><br /><br />exposition by the teacher; <br /><br />discussion between teacher and pupils and between pupils themselves; <br /><br />appropriate practical work; <br /><br />consolidation and practice of fundamental skills and routines; <br /><br />problem solving, including the application of mathematics to everyday situations; <br /><br />investigational work. "<br /><br />More to come.ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com1tag:blogger.com,1999:blog-1508124642967857294.post-36339531949493877782011-09-29T08:28:00.000+01:002011-09-29T08:30:43.527+01:00Concepts of Area; or: how to link parallelograms and rectangles<br /><div style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img height="203" id="il_fi" src="http://www.intigral.com/images/s_parallelogram450.png" style="padding-bottom: 8px; padding-right: 8px; padding-top: 8px;" width="320" /></div><br />The year 7 teachers have this week been doing some work on the links between the area of a parallelogram and the area of a rectangle with the same dimensions. This has caused issues for a number of learners, most notably when trying to calculate the height of the shape. Spending time drawing and cutting up rectangles and parallelograms has proved fruitful but has taken longer than a lot of the teachers expected. We also found that the phrasing of the question also made a big difference to learners' attempts: "Does the rectangle fit inside the parallelogram?" led to a much different exercise than "Who can do it in the fewest cuts?".<br /><br />Whilst observing a class who had been given a sheet of one 8cm x 4cm rectangle and three 8cm x 4cm parallelograms, I was struck by the necessity of there being some <em>forcing of awareness</em> to make sure that connections between <em>concrete and abstract</em> forms were made. Learners set about measuring the sides of both the rectangles and the parallelograms getting 32 square cms for the rectangle and other answers for the parallelograms. Learners then set about cutting up the rectangle to fit it over one of the parallelograms; some very proficiently. However, at this point I noticed that conversation had changed to who had done it in the fewest cuts and the implications of the rectangle fitting perfectly were not being appreciated.<br /><br />It occured to me that some stages need to be moved through in order for there to be a deep understanding:<br /><br />-Establish clearly the area of a rectangle (what it means and how to find it)<br />-Make predictions of what you think the areas of the parallelograms will be<br />-Use concrete forms to rearrange one or the other area<br />-establish what the implications from the concrete forms are<br />-think about whether the original predictions need to be altered in the light of the implications.ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com1tag:blogger.com,1999:blog-1508124642967857294.post-43459293389876165422011-09-19T21:47:00.000+01:002011-09-19T21:47:41.351+01:00Discussion and Follow up in the Mathematics Lesson<div class="separator" style="clear: both; text-align: center;"><a href="http://www.atm.org.uk/shop/productpix/act080.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://www.atm.org.uk/shop/productpix/act080.jpg" /></a></div>Some of the difficulties that teachers have reported at the beginning of using <a href="http://www.atm.org.uk/shop/products/act080.html">Big Ideas</a> has been what to do after a class has had a discussion, what to do to ensure that a discussion has been a fruitful activity. Whilst believing that the act of taking part and engaging in a discussion in a mathematics lesson is positive and important in itself, I shared their concerns.<br /><br /><span class="fullpost"></span><br />For those of you who have been similarly unsure I thought it would be helpful to share some of the ideas that I suggested with you on the blog:<br /><br /><span class="fullpost"></span><br />1)Ask the learners to do some writing based upon their experience of the discussion: what have they learnt that they did not know before? What did they think before? What did other people think? <br /><br />2) Ask learners to summarise the debate for someone who wasn't there (and then pick and share some of them). Reading which points were important to different learners can highlight a lot for the teacher.<br /><br />3) Be sure to leave enough time at the end of the lesson for there to be a shift in mode of working, i.e. you might be expecting the learners to go from sitting in a circle discussing as a whole group to sitting and writing reflectively on their own. This require both a physical and mental shift and requires some effort.<br /><br /><span class="fullpost"></span><br />4) Be prepared to answer questions about what form the responses should take but give other learners the opportunity to make suggestions as well.<br /><br />How do you frame discussion in your lessons? What do learners do after discussing? What do you think of these suggestions? Leave a message...<br /><br /><span class="fullpost"></span><br /><br /><span class="fullpost"></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-38245865481433109682011-09-11T11:21:00.000+01:002012-02-07T08:08:40.075+00:00Big Ideas Released<br /><div class="separator" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"></div> My first publication has just been published with the <a href="http://www.atm.org.uk/">ATM</a> and is now shipping. Apologies about the delay of the CD but if you've already received the book you should receive the CD this week.<br /><br />I am really excited to hear from those of you who have picked up the book and have started using some of the ideas. Please leave a comment below to let myself and others here about any successful lessons or possible pitfalls that you have had.<br /><br />Looking forward to working with you all!<br /><br />ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-67845973428411192742011-07-19T14:56:00.004+01:002011-07-19T15:07:25.663+01:00Dan Meyer: Math Class needs a makeover<!--copy and paste--><object width="446" height="326"><param name="movie" value="http://video.ted.com/assets/player/swf/EmbedPlayer.swf"></param><param name="allowFullScreen" value="true" /><param name="allowScriptAccess" value="always"/><param name="wmode" value="transparent"></param><param name="bgColor" value="#ffffff"></param> <param name="flashvars" value="vu=http://video.ted.com/talks/dynamic/DanMeyer_2010X-medium.flv&su=http://images.ted.com/images/ted/tedindex/embed-posters/DanMeyer-2010X.embed_thumbnail.jpg&vw=432&vh=240&ap=0&ti=855&lang=eng&introDuration=15330&adDuration=4000&postAdDuration=830&adKeys=talk=dan_meyer_math_curriculum_makeover;year=2010;theme=unconventional_explanations;theme=media_that_matters;theme=design_like_you_give_a_damn;theme=a_taste_of_tedx;theme=how_we_learn;event=TEDxNYED;tag=children;tag=education;tag=math;tag=student;&preAdTag=tconf.ted/embed;tile=1;sz=512x288;" /><embed src="http://video.ted.com/assets/player/swf/EmbedPlayer.swf" pluginspace="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" wmode="transparent" bgColor="#ffffff" width="446" height="326" allowFullScreen="true" allowScriptAccess="always" flashvars="vu=http://video.ted.com/talks/dynamic/DanMeyer_2010X-medium.flv&su=http://images.ted.com/images/ted/tedindex/embed-posters/DanMeyer-2010X.embed_thumbnail.jpg&vw=432&vh=240&ap=0&ti=855&lang=eng&introDuration=15330&adDuration=4000&postAdDuration=830&adKeys=talk=dan_meyer_math_curriculum_makeover;year=2010;theme=unconventional_explanations;theme=media_that_matters;theme=design_like_you_give_a_damn;theme=a_taste_of_tedx;theme=how_we_learn;event=TEDxNYED;tag=children;tag=education;tag=math;tag=student;"></embed></object><br /><br />A really exciting talk about the damage we do in 'helping' our students; and the changes we can make to encourage our learners to become self-motivated and independent.<br /><br />The 'tweaking' of mathematical tasks is a stand-out topic in this talk and lead me to reflect on the tasks that I give to my learners. Think of a task you have offered a class recently, to what extent did it encourage learners to act on their own initiative?ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-3006862420747249362011-03-30T09:19:00.002+01:002011-03-30T09:27:25.212+01:00An Open Way of WorkingI was recently asked for some tasks to use with KS3/4 that would help teachers work in a more 'open way'. I decided that it is not so much the task but the way of working itself that is vital. The tasks themselves are what comes after the way of working has been established. Have a look at what I wrote and let me know what you think.<br /><br /><span class="fullpost">In terms of working in 'this way', I have found that there are a number of intermediary steps required before specific tasks will be of any use. For a start it is important that you are constantly working with your class on 'becoming mathematicians' this can be explicitly stated to students and perhaps discussed. How would a mathematician approach this task? What would a mathematician do that is different from a non-mathematician? These questions, if used consistently and given enough time, can change learners perspectives about what it means to 'do mathematics'. <br /><br />Other important shifts in practice would include how resources and tasks are used. A task that has the potential to be open and exploratory and interesting for learners can easily be undermined should the teacher fall into the trap of 'telling'. As soon as we are arresting initial considerations, discussions, attempts and ideas and replacing them with 'the way to do it' we might as well not have bothered finding the task. Unfortunately this will mean quite a few lessons where learners are frustrated: if they are used to the teacher being the person who tells them what to do they are going to be annoyed and upset. Some structure towards this way of working is thus needed. <br /><br />I have found that the easiest way to work like this with a class is to do it from day one. Obviously you don't currently have this luxury (but you soon will!) so this might mean with your current classes building in 10-15 minutes at the start of each task a time where learners are going to try and make sense of the task themselves. You might refuse to answer any questions until then. After this time you could ask the class for any observations: what do they think about the task? What approaches are they considering? What vocabulary do they not understand? This discussion in itself could last another 10-15 minutes. Eventually you might get to the point where the period of initial consideration could last most of the lesson with very little input from you. <br /><br />Beyond this it is important to find opportunities to empower the learners. Instead of giving them 10 questions all the same, why not give them two and ask them to write their method and then invent 3 more of their own...that way they can answer them and set them for other learners. They could even share them on the board. You will be surprised: learners will often think of far more challenging examples than you would have set them. I really believe that Nrich has more than enough starting points for you to explore with your class but I would say before you use them try them yourself...they are often quite tricky and require some getting used to. Support during this period is not an issue: in fact it is probably vital: what is important is how, over time, this support fades away. <br /><br />Two final thoughts: <br /><br />1. There is no such thing as a bad resource, just a bad use of a resource. <br /><br />2. Don't think with questioning it is as simple as open questions - good, closed questions - bad. The task 'write me a sum with the answer 12' will be a lot less interesting than 'write me a sum which includes at least one negative and at least one decimal and has the answer 12'.</span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com2tag:blogger.com,1999:blog-1508124642967857294.post-29192698875635690972010-10-08T11:18:00.002+01:002010-10-08T11:32:48.711+01:00John Taylor Gatto, Dumbing Us Down and Enforced Schooling<a href="http://emergent-culture.com/wp-content/uploads/2009/04/dumbing-us-down-by-john-taylor-gatto.jpg"><img style="FLOAT: left; MARGIN: 0px 10px 10px 0px; WIDTH: 264px; CURSOR: hand; HEIGHT: 395px" alt="" src="http://emergent-culture.com/wp-content/uploads/2009/04/dumbing-us-down-by-john-taylor-gatto.jpg" border="0" /></a><br /><div><div>I read Dumbing Us Down for the first time this week (Flu and Chest Infection made this possible) and was shocked by how much the ideas within spoke to me. </div><br /><div>Gatto's concern is that through forcing children to go to school, to shuffle around to the sound of a bell, to follow instructions, to accept our judgements of them, we are disabling them. We are making students dependent.</div><br /><div>Gatto's solutions are radical: claiming that the systems as they are cannot be fixed and need to be destroyed to stop this. I don't know what I think of this being a teacher myself. It's scary.</div><br /><div>What I do know however is that every time I reach over to snatch the pen out of a student's hand to complete a task for them, or go to finish their sentence, or presume to second-guess them or begin to interrupt them I will think about the possible effects that is having.</div></div>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-17056547545072719772009-10-24T22:20:00.004+01:002009-10-24T22:31:24.883+01:00Thinking about working Pt.1<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://heyugly.org/images/QuestionMark.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 311px; height: 310px;" src="http://heyugly.org/images/QuestionMark.jpg" alt="" border="0" /></a>There are a number of ways of working on mathematics and a similar number of ways of encouraging others to work on mathematics...but is there much consensus on what is the most effective way...or a way that works consistently? And how do we judge that something has worked?<br /><br /><span class="fullpost">Things can be difficult for teachers, often being required to make assessments of things which are quite difficult to assess: what grade will this learner get in two years time? What does this learner know? Why did this learner behave like this?<br /><br />I feel that if assessments are going to be made about learners these assessments need to be used while they are still relevant i.e. in the moment that they are formulated. What do these learners appear to know right now and how can I use this to inform my teaching? It is this way of working that has affected my practice recently, allowing learners to inform and direct where learning will go. What are yor views?<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-6288526337490718372009-04-12T14:49:00.004+01:002009-04-12T15:04:46.579+01:00Dance Squared<object width="320" height="265"><param name="movie" value="http://www.youtube.com/v/yXL4DP_3dJI&hl=en&fs=1&rel=0"><param name="allowFullScreen" value="true"><param name="allowscriptaccess" value="always"><embed src="http://www.youtube.com/v/yXL4DP_3dJI&hl=en&fs=1&rel=0" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="320" height="265"></embed></object><br />A fantastic video showing a multitude of transformations and symmetries of the square. Marvel as the square bisects, reflects and dances to the music. A link for the equally amazing Notes on a Triangle follows.<br /><span class="fullpost"><br /><a href="http://www3.nfb.ca/includes/player/player_full.php?_onfplr_sel=viewfull&film=id=10581&formats=default&speeds=default&use_cc=no&use_dv=no&f=flash&t=normal&s=hv&pm=rtmp%3A%2F%2Fflash.onf.ca%2Fcollection%2Ffilms%2F300_10659.flv&w=640&h=512&c=http%3A%2F%2Fwww3.nfb.ca%2Fincludes%2Fplayer%2Fnfb_global_player.css&pp=http%3A%2F%2Fwww3.nfb.ca%2Fincludes%2Fplayer%2Fonf_U_plr_v1_full.swf&cn=objan&ct=2500000&ttl=_full&url=http%3A%2F%2Fwww3.nfb.ca%2Fincludes%2Fplayer%2F&lg=en&ss=Films%20%E0%20voir&pmvroot=%2Fvar%2Fnfb%2Fapache%2Fhtdocs%2Fphplib%2F&pmvurl=%2Fstats%2F&pmvsid=20&pmvpage=Visionnements%2FNotes+on+a+Triangle&pmvglob=0">Notes on a Triangle</a> is a similar <span style="font-style: italic;">geometric ballet</span> to Dance Squared only this time with a triangle. How are such transformations possible with the triangle? What other geometrical shapes would contain such symmetry? Why not try it with different quadrilaterals?<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-63274923584262708022009-01-17T21:46:00.007+00:002009-01-17T22:29:35.984+00:00Communication in Mathematics and Mathematics Teaching<span style="font-style: italic;font-family:verdana;" ><span style="font-weight: bold;">L: </span>hey guess what</span><br /><span style="font-style: italic;font-family:verdana;" ><span style="font-weight: bold;">me:</span> what?</span><br /><span style="font-style: italic;font-family:verdana;" ><span style="font-weight: bold;">L:</span> your lesson on algebra with your year 8 (?) inspired me the following day</span><br /><span style="font-style: italic;font-family:verdana;" >I scrapped the syllabus and we just did a lesson on proof of (n-2) *180</span><br /><span style="font-style: italic;font-family:verdana;" ><span style="font-weight: bold;">me:</span> :)</span><br /><span style="font-style: italic;font-family:verdana;" ><span style="font-weight: bold;">L:</span> they spent the whole lesson doing it</span><br /><span style="font-style: italic;font-family:verdana;" ><span style="font-weight: bold;">me:</span> that's fantastic, how did they get on?</span><br /><span style="font-style: italic;font-family:verdana;" ><span style="font-weight: bold;">L:</span> </span><span style="font-style: italic;font-family:verdana;" id=":eb" >yeah, really well, at the end two of them presented their proofs and the others asked questions when they'd finished.</span><br /><br />Communication is a very important factor to comprehension in mathematics, this fact has lead to a distinct shift towards activities that utilise communication in mathematics learning. It is with this in mind that I think about pedagogy. Is it just mathematics comprehension that is aided by communication? Or is it something that can aid learners of all disciplines? The dialogue above is from a conversation between a fellow maths teacher and myself. <span class="fullpost"><br /><br />On wednesday we had a conversation about teaching and I happened to mention a way of working that I had been trying to foster in my year 8 class, one in which noting observations, asking questions, hypothesising and testing were the norm. It has been difficult and at times frustrating but it seemed to be particularly successful of late. This in turn caused L to reflect and to try something else (and as you can see it went well!). This to me seems telling. If a short conversation prompted a fellow maths teacher of mine to think about her upcoming lessons and to try something she hadn't necessarily planned to, then what difference could be made if this were standard in all departments across the country? If it were standard for colleagues in maths departments to ask each other about what they have done recently and in what way, about the maths and the unique way in which this subject can be (re)presented.<br /><br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-23470315891954668052009-01-09T13:43:00.006+00:002009-01-09T13:57:42.987+00:00Why Am I so Lazy?<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Trigonometry_intro_circ_triangle.svg/361px-Trigonometry_intro_circ_triangle.svg.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 361px; height: 365px;" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Trigonometry_intro_circ_triangle.svg/361px-Trigonometry_intro_circ_triangle.svg.png" alt="" border="0" /></a>Hey Guys, I'm sorry that I haven't posted for so long - I have no excuse. I guess I'm just hoping to let it slide and that - like my wii fit trainer - you guys will pretend like nothing has happened. <span class="fullpost"><br /><br />There's been a lot of stuff that I have been thinking of in the realm of Maths Education of late with particular reference to a couple of books that I have just read that were fantastic. So in the next couple of days/weeks I'm going to be posting some thoughts and some ideas based around them.<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-25857563221179058452008-08-06T12:34:00.004+01:002008-08-06T12:42:25.073+01:00Recipes and Algorithms<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://christopher.j.martin.googlepages.com/SANY0072.JPG/SANY0072-medium;init:.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://christopher.j.martin.googlepages.com/SANY0072.JPG/SANY0072-medium;init:.jpg" alt="" border="0" /></a>That's a picture of some blueberry muffins I made on Monday - taste good, look good I was very pleased. Pleased until I started thinking: what am I pleased about? That they came out well I guess, but what was my input? I followed a recipe which is, by its very nature, an algorithm so would I have known what the problem was if something had gone wrong? Probably not.<br /><br /><span class="fullpost">Which brings me back to the question: what am I pleased about? It seems to me that I'm pleased that the muffins 'worked' after I blindly followed some instructions - sure I recognised some of the ingredients but did I understand what was going on? Not really. This got me thinking about mathematics teaching. How often do our students blindly follow recipes to be pleased when the right thing appears? How often are they able to see what the problem is in their working? How often can they tell when a solution doesn't fit with the problem? These are all things which we should be considering when we are teaching mathematics - is the understanding there?<br /><br />I'm hungry...off to get a muffin!<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com4tag:blogger.com,1999:blog-1508124642967857294.post-10038903841734742412008-08-04T18:10:00.003+01:002008-08-04T18:30:06.078+01:00The Sound of Silence<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://www.massimochiodi.com/illustrations/caro_silence.gif"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://www.massimochiodi.com/illustrations/caro_silence.gif" alt="" border="0" /></a>I guess the post title seems relevant right now. I'm sorry OK, to those of you who have noticed...and to those who haven't well I haven't posted for a while. The post title goes beyond the emptiness of the blog however, it connects to some thoughts I've been having about teaching. <span class="fullpost"><br /><br /><a href="http://www.teaching.iub.edu/finder/wrapper.php?inc_id=s2_1_lect_04_quest.shtml">Studies</a> have shown that the average wait time between a teacher asking a question and expecting a response is as small as 0.9 seconds. That's not much time. It has raised some thoughts in my mind: about what it is that we expect of our pupils. If I'm ever puzzled by a question I often require a sizable gap in which to think, do we not want our students to be puzzled? To think?<br /><br />This leads me on to a method that I came across in a variety of sources (my <a href="http://www.edstud.ox.ac.uk/courses/pgce/index.php">PGCE course</a>, <a href="http://www.allbookstores.com/book/9780199190096/Colin_Stephen_Banwell/Starting_Points_For_Teaching_Mathematics_In_Middle_And_Secondary_Schools.html">Starting Points</a>, and fortunately for me at school) a numerical relationship is presented on the board in front of the class in complete silence. Pairs of numbers are drawn up, and that is it. Students are then invited (silently) to suggest other pairs that may fit the relationship. Simple as it sounds this method can have a real power. Students are often captivated by the silence, and are able to sit and try to work out what is going on. Why don't you try it?<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-33043917842712588692008-05-18T21:43:00.005+01:002008-05-18T22:13:15.013+01:00Mathematics and Art (Part 3)<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://bp2.blogger.com/_E5IClm-J5Mw/SDCVMnNqz9I/AAAAAAAAAGk/o-F_Qxm46po/s1600-h/statue.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://bp2.blogger.com/_E5IClm-J5Mw/SDCVMnNqz9I/AAAAAAAAAGk/o-F_Qxm46po/s320/statue.jpg" alt="" id="BLOGGER_PHOTO_ID_5201821613530206162" border="0" /></a>Myself and <a href="http://myenemieslist.blogspot.com/">Ollie</a> had an art lesson with Justin on Friday. The theme was <a href="http://mathsphotos.blogspot.com/2008/05/value-study.html">Value</a>, and how we could represent this using only discrete lines.<br /><br />In seeps Mathematics. As we discuss iterations and binary, series and value I recognise the truth in the statement that <a href="http://www.atm.org.uk/about/">"the ability to cooperate mathematically is an aspect of human functioning which is as universal as language itself"</a>. The value studies above represent a four-stage discrete approximation of the continuous scale from darkness to lightness.<br /><br /><span class="fullpost">And here is the rest of it. The lesson was followed by a day trip to the <a href="http://www.vam.ac.uk/">V&A museum</a> in Kensington. We applied some of the ideas that we had practised on some human form. Discussions on representation were had and conclusions crystallised: if what we were doing was an imperfect approximation of the real thing anyway, why not take license?<br /><br />Which brings me finally to my point. If looked at closely a number of geometric forms can be discerned within the picture to the left. This is of course license taken by me but taken consciously. Are these forms not there to be seen if one sets oneself to attend to them? Even if they are formed primarily by negative space is it not still that they are there?<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-82853201428395323482008-05-12T20:18:00.002+01:002008-05-12T20:38:42.108+01:00On Noticing<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://aaalab.stanford.edu/child_development/dev_fractions.html"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://aaalab.stanford.edu/images/project_images/child_development/fractions/child.png" alt="" border="0" /></a>I have written about noticing before, but it is something that occurs and reoccurs in thinking and reflecting on my practice. What is noticing? What do you notice? What form does your noticing take? How do you attend to that which you notice?<br /><br /><span class="fullpost">And here is the rest of it. Have you ever been surprised by the multiple occurrences of something which you have only just spoken about? Does this happen very often? Is this just a co-incidence do you think? Perhaps there's something else? There is certainly something about this feeling: the fact that it happens to people so often (well to me anyway). I feel that it has something to do with what it is we <a href="http://mcs.open.ac.uk/jhm3/">"set ourselves to notice"</a>.<br /><br />I noticed recently that a lot passes me by and that I notice very little, in the order of things anyway. It is so easy for someone observing to have seen something that we 'in the fray' didn't see at all. This is where reflection is very important: by setting ourselves to notice specific things, be they ways in which we react, patterns of behaviour in students, the intonation of our voice; we are much more likely to be able to attend to those things we want to change or at least notice.<br /><br />The idea of being more in control of my own classroom behaviour seem to me to be empowering.<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-21946723429471849612008-05-08T21:04:00.012+01:002008-05-08T21:41:21.205+01:00Mathematics and Art (part 2)<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://paintingsdrawingsarithmetic.blogspot.com/"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://bp1.blogger.com/_E5IClm-J5Mw/SCNk0u_uO4I/AAAAAAAAAGM/P25LnbcqqQQ/s400/brown%2Bcourdoroys.jpg" alt="" id="BLOGGER_PHOTO_ID_5198109252046961538" border="0" /></a>What is <a href="http://en.wikipedia.org/wiki/Recursion">recursion</a>? Is it only Mathematical? Where can it be found? If you look closely at the picture on the left you'll see a fascinating exemplification of recursion. Note how the picture occurs as an element of itself.<br /><div style="text-align: justify;"><br />This is in much the same way as a recursive function will feature within its own definition.<br /><br />For example <a href="http://mathworld.wolfram.com/RecursiveFunction.html">this</a> is a recursive function; where S(x) = x + 1 is the successor function. It defines the function for multiplying x by y.<br /><br />See if you can think of an example of a recursive function. The one above is a bit tricky; don't be scared to think simply.<br /><br /></div>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com2tag:blogger.com,1999:blog-1508124642967857294.post-69692675700329895412008-05-04T09:19:00.009+01:002008-05-04T10:15:35.502+01:00Mathematics and Art (part 1)<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://mathsphotos.blogspot.com/"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://bp3.blogger.com/_E5IClm-J5Mw/SBzqkFMNEYI/AAAAAAAAAFs/ncSMJZqmUfI/s400/cont.jpg" alt="" border="0" /></a>I started drawing recently (yesterday in fact) with <a href="http://paintingsdrawingsarithmetic.blogspot.com/">Justin</a>. It is a skill I have never had much confidence in doing, there is very little in the way of drawing that I can say that I have been pleased with. This is going to stop though. I think I need to be more patient and understand that it can take time. I have come to the conclusion that my impatience is in part to blame.<br /><br />You may be asking what place this has on a blog about maths, and you'd be right to ask. The reason that I am posting about this is that whilst drawing I found myself thinking about mathematics and how it could be used to describe the processes and the image.<br /><br /><span class="fullpost">We started by discussing the nature of parallel and perpendicular and the effect that they have upon form. Gradient and juxtaposition. Two concepts which also came out in thinking about shadow and light.<br /><br />It seems to me that I perceive that which is different, but I also perceive in context. I see a parallelogram when the black lines are juxtaposed with the white paper, in an isometric context however I may perceive a rectangle.<br /><br />As I sketched the container I attended to the curvature particularly. The way in which I thought of the top of the base intersected closely with graphs. "It stops at this point" I thought, "this is a stationary point". I noted this as it was a moment of unforced mathematical behaviour. I was not trying to think about how mathematics related to what I was doing, it happened because it <span style="font-style: italic;">was</span> related.<br /><br />When we began considering the container Justin began by asking me to sketch it 10 times for a minute each time. After this I sketched it twice, each time for 10 minutes. I finally sketched it for 30 minutes. At each stage I noticed more detail, more shadow, more light. I am currently reading <a href="http://www.amazon.co.uk/Researching-Your-Own-Practice-Discipline/dp/0415248620/ref=sr_1_1?ie=UTF8&s=books&qid=1209891638&sr=8-1"><span style="font-style: italic;">Researching Your Own Practice: The Discipline of Noticing</span></a> by John Mason and the drawing seemed to serve as a fitting analogy to thinking about teaching. We attend to that which we set ourselves to notice, and it is through reflection and consideration that the finer details reveal themselves.<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com2tag:blogger.com,1999:blog-1508124642967857294.post-20346659836769119262008-04-29T20:50:00.005+01:002008-04-29T21:04:28.949+01:00More of a puzzle than a 'pinion...<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://www.atm.org.uk/mt/archive/mt167.html"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://bp3.blogger.com/_E5IClm-J5Mw/SBd7fVMNEVI/AAAAAAAAAFU/Nd_9F8UFhs0/s400/SANY0041.JPG" alt="" border="0" /></a><br />I don't know if you've seen this sequence before but it was on the front of MT167. It has plagued me for a long time now and I have no idea how it was generated. <a href="http://bp3.blogger.com/_E5IClm-J5Mw/SBd7fVMNEVI/AAAAAAAAAFU/Nd_9F8UFhs0/s1600-h/SANY0041.JPG">Have a look</a>...if you can't read it in the pic the sequence is posted after the jump.<br /><br /><span class="fullpost">The sequence goes as follows:<br />1, 4, 7, 12, 16, 20, 23, 28, 33, 37, 40, 46, 52, 60, 68, 75, 82, 91, 99, 107, 113, 122, 131, 142, 152, 162, 171, 182, 193, 203, 209, 218, 227, 238, 248, 258, 267, 278, 289, 299, 304, 312, 320, 330, 339, 348, 356, 366, 376, 385.<br /><br />I have thrown a lot at it and got close to nowhere. Please drop me a comment if you have any clues!<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com2tag:blogger.com,1999:blog-1508124642967857294.post-53769809577310245602008-04-26T16:14:00.008+01:002008-04-27T20:36:56.820+01:00In response to connections, fluency, coherence<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Trigonometry_intro_circ_triangle.svg/300px-Trigonometry_intro_circ_triangle.svg.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Trigonometry_intro_circ_triangle.svg/300px-Trigonometry_intro_circ_triangle.svg.png" alt="" border="0" /></a>With the last post in place I started the term with my year 10s in mind. With trigonometry in mind. With understanding in mind. I felt a panic/rush/excitement/worry about this and about problems that I may have caused with previous lessons.<span class="fullpost"><br /><br />We started the lesson with a discussion of what trigonometry evoked in the students: I asked them to think and discuss for a few minutes. Students suggested triangles, angles, labelling sides, lengths, inverses, right angles... they also suggested SOH CAH TOA... "But what does that mean" a lot of them asked. A good question. What does that mean?</span><span class="fullpost"><br /><br />The students were then asked to think of questions in which they would use trigonometry (they have had some experience of it before). Lots of triangles were drawn (right-angle on the right). "OK... what can you do now? Try something!" Most students' examples gave the lengths of 2 sides (e.g. opposite and hypotenuse) and an angle to which they were trying to apply sine. "But what do I have to work out?" This gave a good discussion point...why use sine here when you have the angle, the opposite and the hypotenuse? What do you want to work out? Why would you want to include an unknown?</span><span class="fullpost"> How would that help?<br /><br />Rearrangement of the quotients led to discussion. "Why have you done it like that?" "I looked at what I did before and I swapped them" It seemed that thinking of sin 46 as a number proved challenging. Thinking of sin as a function was also challenging: "Can you divide sin x by sin to get x?"</span><span class="fullpost"><br /><br />There was a lot of positive discussion and a lot of challenge. The class were at such different points in this topic that using this method of teaching gave the class an opportunity to work on aspects which they themselves found challenging. Whilst I took this opportunity to clarify some methods and why we did them, some students plotted sine as a graph and experienced it as a function.</span><span class="fullpost"><br /><br />The richness of looking at a small group of functions for an extended period of time excites me and helps me to realise connections and coherence. I also see points at which other 'topics' can be assessed as they come into use throughout. I can see myself doing Trigonometry for a while!<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-5017959768891358772008-04-19T16:20:00.016+01:002008-04-19T17:30:00.439+01:00That Which is Always there... (part 2)<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://en.wikipedia.org/wiki/Unit_circle"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 400px;" src="http://christopher.j.martin.googlepages.com/SineAnim.gif" alt="" border="0" /></a>The past couple of days I have been thinking a lot about trigonometry. This has been spurred in part by the fact that I will be doing some lessons on it soon and in part by <a href="http://www.edstud.ox.ac.uk/uploaded/ATM%20MA%20NANAMIC%20AMET.ppt">Anne Watson's excellent opening address</a> at the recent ATM conference. The address forced me to think about how joined-up maths really is, but also about how disconnected it is often taught. Starting with this post I am going to work on ways of encouraging <a href="http://academic.sun.ac.za/mathed/174/Skemp.pdf">'Relational Understanding'</a> in my students, an ability to see these connections for themselves.<br /><br />This post looks at the links between linear sequences, gradients, algebra and trigonometry.<br /><br /><span class="fullpost">I was struck yesterday by the image of someone using a right angle triangle to calculate the gradient of a line. "Hold on" I thought, "are they not calculating the <a href="http://en.wikipedia.org/wiki/Trigonometric_functions">Tangent</a> of some angle? And more to the point isn't that the angle that defines the steepness of the line? And more to the point do we not call this the gradient?" This was quite striking to me. Stop me if this is painstakingly obvious, but here are two processes completed to varying degrees of success by thousands of students around Britain that I have not once seen taught as interconnected.</span><span class="fullpost"><br /><br />Now let me re-assure those of you who are reading this, mouth agape, ready to fire off an email telling me that you've been teaching this for years, I am a mere NQT my experience is sorely lacking. However this was an important realisation for me. The fact that I had students using the same skill in several areas but who were unaware of it was an exciting prospect!</span><span class="fullpost"><br /><br />My mind then moved from gradients and tangents to linear sequences. How could this be connected to both trigonometry and gradients? Think about the connection between similar triangles and trigonometry... any thoughts? Leave a comment.</span><span class="fullpost"><br />Now when introducing tangent to my class I presented a set of right-angled triangles all with a 21ยบ angle and let them work out opposite over adjacent...surprise!!! They all give the same thing! I'm starting to wonder now whether surprise was the best emotion to connect with this fact however. Based upon the experience that they already have (linear sequences, similar triangles) could they not have been scaffolded towards an understanding that things cannot be otherwise? That the tangent of an angle is necessarily unique? I suppose that this is something to think about.</span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com2tag:blogger.com,1999:blog-1508124642967857294.post-5124726937123374842008-04-18T16:00:00.007+01:002008-04-22T08:00:09.220+01:00Counting Counters (with Mathematical Journeys)<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://en.wikipedia.org/wiki/Tiddlywinks"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://bp0.blogger.com/_E5IClm-J5Mw/SAyjTd71vaI/AAAAAAAAAEo/no8FuLHvAis/s200/04G.jpg" alt="" id="BLOGGER_PHOTO_ID_5191704025299795362" border="0" /></a>I was reading through the ATM's excellent <a href="http://atm.org.uk/buyonline/products/act065.html">Mathematical Journeys</a> when I came across this excellent puzzle. But before I go into that a little history...Mathematical Journeys is a compilation of some of the puzzles from the excellent <a href="http://atm.org.uk/buyonline/products/dnl003.html">Departure Points</a> series. This set of four books (plus one for primary) were the result of an extended ATM conference session at the 1977 conference. They serve as excellent starting points that let learners (of all ages!) discover and investigate mathematics in all its aspects.<br /><br /><span class="fullpost">The puzzle to investigate is this:<br /><br /><span style="font-weight: bold;">Leila has a jar of counters, when she counts them into piles of four, she has two left over. When she counts them into piles of five she has one left over. How many counters could she have had?<br /><br /></span>This is wholesale taken from the Mathematical Journeys I make no denial, but I really like how it encourages working with modulo numbers...when is x (mod 4) = 2 and x (mod 5) = 1? The possibilities to investigate are massive, introducing another count into piles of 6? Changing the two numbers? Giving bounds (more than 10 less than 100?)...a fantastic problem.<br /><br />Thanks to Greg for spotting the mistake!<br /></span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-13041590284036611162008-04-16T13:15:00.011+01:002008-04-16T13:59:30.597+01:00On the Conference and Why the ATM is the Pro Org to Join<div style="text-align: justify;"><div style="text-align: left;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://www.atm.org.uk/conferences/conference2008.html"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://www.atm.org.uk/conferences/conf2008stuff/atm-ma-conferece-2008-vert.jpg" alt="" border="0" /></a>April 2nd to April 5th 2008 was my very first ATM conference. This year it was a joint conference with the <a href="http://www.m-a.org.uk/">MA</a> and so the theme - very fittingly - was <a href="http://www.atm.org.uk/conferences/conference2008.html">Joined Up Mathematics</a>.<br /><br />In this post I will give some more detail on the conference itself and explain why every teacher of mathematics should be a member of the <a href="http://www.atm.org.uk/">ATM</a>.<br /></div><br /><span class="fullpost"><br />The conference consisted of a number of sessions focused on some element of mathematics or <a href="http://www.answers.com/pedagogy&r=67">pedagogy</a> (or both). People at each session then worked together, discussed or presented ideas that at the very least forced you to rethink some aspect of your notions of mathematics and teaching. At the very best they had you <a href="http://www.atm.org.uk/conferences/conf2008stuff/david-cain.jpg">jumping up and down in your chair</a> and then realising you were counting in binary, but I guess you had to be there. The great thing about these sessions was that everybody sat down with each other without having met and so you got to work with people from all areas of maths education.<br /><br />The conference is very sociable, each day you speak to new people with new ideas and and new experiences: this is what really makes the conference so worthwhile. I came away feeling completely re-energised, feeling very very positive and feeling that the best thing about this job is the massive wealth of opportunity that we have to make our classes and above all ourselves think. This was seen especially in the workshop, a room that was open all day every day and was there to play, to think and to explore (the resources available were immense!).<br /><br />The <a href="http://www.atm.org.uk/">ATM</a> is the driving force behind these conferences but along with these they also produce a wealth of <a href="http://www.atm.org.uk/buyonline/browse-alphabetical.html">resources, books and software</a> to support your teaching. These resources are <a href="http://www.atm.org.uk/buyonline/products/act057.html">very</a>, <a href="http://www.atm.org.uk/buyonline/products/dis002.html">very</a> <a href="http://www.atm.org.uk/buyonline/products/act067.html">thoughtful</a> and not only that, compared to the price of some of the junk that is available to buy it is incredibly cheap...on top of this if you <a href="http://www.atm.org.uk/join/">join</a> you get 25% off! On the subject of joining we haven't yet mentioned one of the biggest benefits of all and that is the <a href="http://www.atm.org.uk/mt/">journal: MT</a>.<br /><br />Throughout my PGCE year I found that having these brightly coloured and exciting journals turn up every 2 months really motivated me at times where I felt stressed or pressured. They gave me new ideas, amusing stories and above all lots to think about. I still rely on them this year (I'm an NQT) to do exactly the same thing and they still work as well as ever.<br /><br />So there you go a whole-hearted and unashamed plug for the <a href="http://www.atm.org.uk/">ATM</a> and its <a href="http://www.atm.org.uk/mt/">journal</a>.<br /><br /><a href="http://www.atm.org.uk/join/">Join NOW!</a><br /></span></div>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com2tag:blogger.com,1999:blog-1508124642967857294.post-86412831915015416982008-04-15T14:00:00.009+01:002008-04-16T13:00:05.369+01:00Mistakes that they make that they make...<div style="text-align: justify;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://www.robeastaway.com/"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://www.robeastaway.com/buses_cover.jpg" alt="" border="0" /></a>I have been reading the excellent <span style="font-style: italic;">Why do buses come in threes?</span> by Rob Eastaway and Jeremy Wyndham and it has brought a number of interesting mistakes to my attention. Well mistake is a strong word, I think surprises is a better one and in this post I intend to bring one of them to your attention.<br /><br /><span class="fullpost"><br />Take for example the Problem of the cricket pitch. Each year a Lord holds a cricket match on his grounds. To make it a more authentic experience he sets up a boundary with a picket fence. This provides the pitch with a 50m radius i.e. one needs to hit the ball 50m to score a 4 or 6. This year however, six metres of fence has been lost...how much shorter does that make the radius of the pitch? </span><span class="fullpost">(Guess before you read on)</span><br /><span class="fullpost"><br />It turns out that it is one metre. Nothing too surprising there. However the story continues. At the very same time somewhere far far away another Alien Race is also playing a form of cricket. Their radius however, is billions of metres wide. By some strange coincidence they have also lost six metres of fence. How much smaller is their radius? (Guess before you read on)<br /><br />Now the strange thing, at least at first glance, is that their pitch is also smaller by one metre...but why?<br /></span></div>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-62377429896564090712008-04-12T12:06:00.024+01:002008-04-16T14:00:09.173+01:00That Which is Always there... (part 1)<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://en.wikipedia.org/wiki/Infinity"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://bp0.blogger.com/_E5IClm-J5Mw/SAIRrXVH_JI/AAAAAAAAADo/IMvOEqemt1M/s200/Infinity_s.jpg" alt="" id="BLOGGER_PHOTO_ID_5188729157378899090" border="0" /></a><div style="text-align: justify;">Thinking about mathematics leads inevitably to generalities and it is these generalities that imbue maths with the power that allows us to talk of ideas "<a href="http://en.wikipedia.org/wiki/Caleb_Gattegno">shot through with infinity</a>".<br /><span class="fullpost"><br /><br />The title of the post above was intended to refer to algebra but the more I write the more I feel the presence of infinity within and throughout my thinking. Could it be that part of our use of algebra is to allow us to deal with the infinite? What does this mean to you?</span><br /><span class="fullpost"><br />The reliance upon some sort of algebra in all aspects of our lives is quite phenomenal but it takes a lot to become conscious of when exactly we are doing it. Try and think of a time when you may have applied algebra (it will be a lot more meaningful than any example from me) were you aware of applying algebra? Could you generalise based on that application? Could you deal with infinity? Would that even be meaningful in this context?<br /><br /></span><span class="fullpost">What is it about algebra that makes for it being such a stumbling block? That makes it a point of which so many adults look back on as some alien language? Try to recall your experience of algebra at school and compare it to a time you feel that you have used it informally in your life. What is different?</span></div>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0tag:blogger.com,1999:blog-1508124642967857294.post-71553629127565007492008-04-11T00:54:00.009+01:002008-04-11T01:21:10.164+01:00How many primes?<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://en.wikipedia.org/wiki/Eisenstein_primes"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 200px;" src="http://upload.wikimedia.org/wikipedia/commons/3/3d/Eisenstein_primes.png" alt="" border="0" /></a><a href="http://en.wikipedia.org/wiki/Prime_numbers">Prime numbers</a>...just how many are there? And how do we know? If we look at the first few we get the following sequence:<br /><div style="text-align: center;"><a href="http://www.research.att.com/%7Enjas/sequences/A000040">2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...</a><br /><div style="text-align: justify;"><br />Seemingly no real rhyme or reason, but what can we say about them?<span class="fullpost"> Well to start with we have the <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic">Fundamental Theorem of Arithmetic</a>...sounds grand no? It's actually pretty simple (we use it every time we do factor trees!) and it's just that every number can be written as a unique product of primes...<br /></span></div></div><br /><div style="text-align: center;"><span class="fullpost">e.g. 180 = 2 x 2 x 3 x 3 x 5<br /><br /></span></div><span class="fullpost">We can use this to help calculate the number of primes in total...<br /><br /></span><span class="fullpost">Start by assuming that there are only a finite number of primes who knows how many? 100? 1000? 1000000000? We're not sure so let's just say n...</span><br /><span class="fullpost"><br />Multiply these n primes together and then add 1...now what is the unique product of primes that make this number? It doesn't divide by any of our n primes (as we have + 1 on the end) and so we either have a new prime or there is a prime not in our list that divides into our number...<br /><br /></span><span class="fullpost">Both of these outcomes give us an extra prime not on our list...so how many primes are there?</span>ATM Memberhttp://www.blogger.com/profile/07330705273424394556noreply@blogger.com0