Math-epol338 Course Notes - Mathematics Curriculum Study

EPOL338 Course NotesOnline Collaboration Team for VUW Diploma of Education Edited by Blair M. Smith July 18, 2010

1 Copyright c 2010, Blair M. Smith Please copy, modify and redistribute under the terms of the GNU Free Document Licence (GPL FDL) here: http://www.gnu.org/licenses/fdl-1.3-standalone.html

ContentsPreface Introduction and Warm-up Activities Reections on School Mathematics . . . . . . . . . . . . . . . . . . . . . . Calculations in Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Module 1An Introduction to Mathematics Teaching 6 7 7 9 13

Thoughts on the Way Mathematics Could be Taught . . . . . . . . . . . . 13 Interviews with AdultsThoughts on School Mathematics . . . . . . . . . 14 Task 1.1Attributes of a Quality Mathematics Programme . . . . . . . . 14 Task 1.2.1: Curriculum Achievement Objectives . . . . . . . . . . . . . . . 17 Task 1.2.2: Frog Hopping Problem . . . . . . . . . . . . . . . . . . . . . . 20 Mathematical Facts and Ideas . . . . . . . . . . . . . . . . . . . . . . . . . 25 Glossary of Some Terms Used in school Mathematics . . . . . . . . . . . . 28 Resources and Tools Familiarization . . . . . . . . . . . . . . . . . . . . . . 30 NCEA Achievement Standards . . . . . . . . . . . . . . . . . . . . . 30

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Module 2Introduction to Mathematics Lesson Planning 32

Module 2.1Planning for Mathematics Teaching . . . . . . . . . . . . . . 32 Comments on Developing Mathematics Understanding . . . . . . . 32

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Comments on Designing Rich Mathematical Experiences . . . . . . 33 Comments on Planning to Teach a Mathematics Lesson . . . . . . 36 A Checklist Mathematics for Lesson Plans . . . . . . . . . . . . . . . 37 Module 2.3Exploring Mathematics Teaching Resources . . . . . . . . . . 40 Comments on Using Assessment for Eective Learning . . . . . . . 40 3 Module 3The Numeracy Development Project 42

Module 3.1Overview of the Numeracy Development Project . . . . . . . 42 4 Module 4Number and Algebra 43

Module 4.1Teaching Number . . . . . . . . . . . . . . . . . . . . . . . . 43 4.0.1 Tips for Teaching Number & Algebra . . . . . . . . . . . . . . 43

Dealing with Misconceptions . . . . . . . . . . . . . . . . . . . . . . . 44 Module 4.2Algebra and Generalization . . . . . . . . . . . . . . . . . . . 47 Mathematics Needs for ESOL Students . . . . . . . . . . . . . . . . . . . . 48 5 Module 51 Algebra 53

Teaching Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Teaching Algebra53 Charting Algebra in the NZ Curriculum . . . . . . . . . . . . . . . . 54 Common Student Diculties with Algebra . . . . . . . . . . . . . . . 55 Teaching Strategies for Helping Students with Algebra . . . . . . . . 57 Helping Students with Algebra Word Problems . . . . . . . . . . . . 58

Forum Reections of Teaching Algebra . . . . . . . . . . . . . . . . . . . . 61 Variation and Invariance in Mathematics Pedagogy . . . . . . . . . . . . . 64 Spreadsheets in Mathematics Pedagogy . . . . . . . . . . . . . . . . . . . . 65 Visual Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Some Tips for Mathemagics . . . . . . . . . . . . . . . . . . . . . . . 68

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Using Technology to Teach Algebra . . . . . . . . . . . . . . . . . . . . . . 70 GeoGebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 CLUCalc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 A Final Word on Teaching Arithmetic & Algebra . . . . . . . . . . . . . . 71 6 Pedagogical Content Knowledge StudyComplex Numbers 6.1 72

Ideas for Teaching Complex Numbers . . . . . . . . . . . . . . . . . . 72 6.1.1 C Possible Teaching Sequences . . . . . . . . . . . . . . . . . . 72

C Possible Teaching Sequences . . . . . . . . . . . . . . . . . . . . . . 72 6.1.2 Dealing with C Misconceptions . . . . . . . . . . . . . . . . . 73

Dealing with C Misconceptions . . . . . . . . . . . . . . . . . . . . . 73 6.2 Some Tools for Teaching Complex Numbers . . . . . . . . . . . . . . 74 6.2.1 Resource of Good C Inquiry Projects and Questions . . . . . . 75

Resource of Good C Inquiry Projects and Questions . . . . . . . . . . 75 6.2.2 Advanced Examples . . . . . . . . . . . . . . . . . . . . . . . 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 81

Advanced Examples Reection Journal

Reections for Week One of Epol-338 . . . . . . . . . . . . . . . . . . . . 81 Reection on Features of a Quality Mathematics Programme . . . . . 81 Reections on School Mathematics Experiences . . . . . . . . . . . . 82 Reections on Week 1 Readings . . . . . . . . . . . . . . . . . . . . . 84 Why do we teach mathematics? . . . . . . . . . . . . . . . . . . . . . 89 Why do we need to teach numeracy? . . . . . . . . . . . . . . . . . 89 Module 1 ReectionMathematics Then and Now . . . . . . . . . . 91 Reections for Week Two of Epol-338 . . . . . . . . . . . . . . . . . . . . 92 The Art of Starter Activities and Interventions . . . . . . . . . . . . . 92 Interpretation of Teaching Areas of Triangles . . . . . . . . . . . . 93

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Reections for Week Two of Epol-338 . . . . . . . . . . . . . . . . . . . . 94 Reections for Week Two of Epol-338 . . . . . . . . . . . . . . . . . . . . 94 Reections for Week Two of Epol-338 . . . . . . . . . . . . . . . . . . . . 95 6.2.3 Reections on Polyas How to Solve It . . . . . . . . . . . . . 95

Reections on Formative Assessment . . . . . . . . . . . . . . . . . . . . . 97 Reections on My Personal Teaching Style . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

PrefaceThese are free collaborative collective course note for the 2010 online students enrolled in the VUW Epol338 course. Please copy and redistribute as you please, respecting the GPL-FDL copyright. We have included a lot of quotes from the online discussion forums, which has added to the length of this book somewhat. The suggestion is to not read this book serially, but to instead scan the topics and delve into the quoted paragraphs as your interest guides youthat way the book will hopefully not seem too daunting to read. Also, these course notes are not intended as substitutes for the course Module notes, textbook and readings. The idea is that this book will serve as a reference and memory jog for all of our future work in education, and not so much as an exam preparation guide for the course.

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Introduction and Warm-up ActivitiesThe portions of these notes written in the rst person voice can be considered the views of the editor. Other sections are collaborative. Margin symbols like are used to highlight what I think are essential tips for good teaching.

Reections on School MathematicsBelow under each question are my answers, followed by those of two adults whom I surveyed. Describe a typical mathematics lesson from secondary school. My answer. Teacher would turn up a few minutes after the bell. Possibly a quick review of the previous lesson. An introduction to the next topic, or problems put up on the board if continuing a topic. Sometimes the teacher asks the class a question while demonstrating a method or problem model solution or derivation. Students work basically alone to solve problems but discussion amongst friends at same group of desks allowed. Describe a mathematics lesson you have enjoyed. My answer. Fourth form mathematics teacher discusses knots. We go outside to test the strength of various slips and knots. Very little theoretical knowledge was transfered, but the topic was engaging and stimulated interest. The same teacher taught us how to analyze and solve problems involving logic gates, AND, OR, XOR and NOT gates. We may have spent a week on this among doing other things. Again, it was interesting and dierent from the usual lessons, so we were excited.

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What were some of the key features of that lesson? My answer. The aforementioned excitement and novelty of something that did not seem to appear in any textbooks. It seemed cool in way to be thinking of doing things that took computers nanoseconds (at that time probably fractions of microseconds) to accomplish and yet seeing that the mental processes involved were not entirely all that trivial. No matter how practiced we became there was a limit to how fast we could solve any given network of gates to work out the signals at some nodes given sucient data from others. Describe a mathematics lesson that you did not enjoyed. My answer. I vaguely recall a seventh form lesson introducing either summations of innite series or proof by induction. I remember thinking, this will be easy, I studied this last year, but the lesson was minimalist, lacked context and did not draw out my prior knowledge, so I felt lost for a long time and recall trying some very mechanical seeming thinking and strategy to try and grasp how to do the example problems. I was scared that the teacher would ask me for a solution, since I usually knew an answer or how to start one, but this time I did not. After the lesson I was not sure I could solve similar problems, so I had to go away and teach myself the skill, after which it became clear and simple. I was bemused by why the lesson seemed so dicult. That class had a lot of similar bemusing lessons and most students unked the course so-to-speak, in fact all but two of us. Why do secondary school students study mathematics? My answer. For many reasons. Also, many take mathematics classes and yet do not really study mathematics! Because they have to, when mathematics is not optional, and they desire a decent grade. They love the subject. They are good at the subject. They need to know a bit of mathematics to do well in other subjects or their planned future careers. They want to impress someone with their mathematical ability.

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Because their friends are studying mathematics. Because someone else they like is studying mathematics. Because their parents force them to. Because they know no better and could not think of other courses they would like better. What did you like about mathematics at school? My answer. It was challenging yet easy to master. Objectives were reasonably clear and a sense of accomplishment was easy to measure. Many topics were darned interesting. I knew I needed to be good at mathematics for studying other sciences. The increasing abstraction of mathematics, its structure and patterns and relationships between various topics and branches of mathematics, was all inherently fascinating. The less raw numbers became involved the more interesting the lessons became. It was also great to be thought of as brainy even thought I knew I was not a genius, merely because I was better at doing mathematics than most other students. There were few brilliant students who were really of genius level, and I seemed to get classied along with them as something of a bright spark, just by association through similar mathematical ability! That was quite fun for a while, until I was exposed as just an ordinary mind.. What did you dislike about mathematics at school? My answer. Getting incorrect answers to problems. Making silly mistakes like losing plus or minus signs, or misreading questions. Also, I hated unclear lessons that left me wondering what the heck the teacher was teaching. Failing to understand a proof or technique was horrible. Most of all, I hated the pressure of tests, yet was strangely exhilarated when the tests turned out to be fairly easy. Facing exams was a bit like a competition, such as a sprint race. Lots of adrenaline surges, some anxiety and nerves, followed by relief and exuberance when I felt I had done well.

Calculations in ContextFour questions are given below. The activity is to document the strategy you use to solve each problem, also try to jot down notes of possible alternative strategies. No calculators at rst.

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Sample questions.3 1. It is 11:40 am. You have 1 4 more hours of work and then a 25 minute drive to get home. What time will you get home?

2. The marked price of a shirt is $85. Everything is on sale with a 30% discount. How much will the shirt cost in sale? 3. Give an estimate of the cost of 320 booklets at $5.70 each. 4. After a meal in a restaurant your group decide to split the bill. What will you have to pay if you are paying for 3 of the 8 people in the group and the total bill is $175?3 Notes on question 1. Immediately I gure I will have to add 1 4 of time to the start time of 11:40, and then add another 25 minutes. Breaking this up I rst add one hour to get to 12:40pm (guring the change from morning to afternoon), then I add 3 of an hour which I work out (or just know) to be 45 mins (since a full hour 4 is 60 min) which brings the projected time rst to 1:00pm plus the dierence of 45 mins less the 20 min to get to 1:00pm, which brings me to 1:25pm. Finally add the 25 mins drive bringing the overall projected ETA to 1:50pm.

An alternative approach would be to add up the total remaining work+drive time 3 rst, giving 1 4 h +25m or 2 hours plus the dierence of 25 min less the 15 min to get to 2 hours, or 2 hr and 10 min. Then add this to the start time of 11:40am, which would bring one to 13:40 + 0:10 or 13:50 hr or 1:50pm as before. A nice graphical method would be to just tick o the time on a clock-face, it could be a fake clock sketched on paper. This might even be my preferred method performed with an internal mental clock face perhaps.30 Notes on question 2. I rst recall that 30% of something is a fraction 100 or 3 of the thing. Since Im not condent of doing this in my head straight o by 10 3 some sort of memorized times table, I need to multiply $85 by 10 . I can do this in a way to be error-free by multiplying $85 by 3 and then divide the result by 10. Even simpler I break this down using distributivity as (80 3 + 5 3)/10 so I get rstly 3 80 or (3 8) 10 using simplest factors of 80 and associativity to get 24 10 = 240. Then I still have to add 3 5 = 15, to get the intermediate result 240 + 15 = 255 (which I mentally view as the simple sum 40 + 15). Nearly there, I still have to divide by 10, but thats a simple shift of decimal place to the left, so the answer is 25.5 or in monetary units $25.50.

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An alternative method might be to divide 85 by 10 rst, then multiply by 3. Or if my head could easily hold multiplication tables for fractions I might do 10 (8 0.3) + 5 0.3, otherwise break it up as 10 (8 3)/10 + 5 3/10 which is nice because I cancel a factor of 10 to get 8 3 + 15/10 = 24 + 1.5 = 25.5 as before. Notes on question 3. For this estimation problem my strategy rst o the bat is to use rounding. One up one down for balance. So taking nearest nice numbers I gure a good estimate will be 300$ 6 = (3 6) 100 or $1800. I can also estimate this should be o by only about the twice (2 stdev) the product of the errors from rounding or 2 20 0.3 which is about 2 7 = 14, or conservatively I expect and error of between 10 to 20. I could not think of a more intuitive alternative algebraic method. Rounding 5.7 down to 5 and 320 up to 400 would be acceptable I suppose, and maybe easier on the mental multiplication times table in my head, for an estimate of $2000. Notes of question 4. First I say to myself, there are 8 people and the total bill is $175. So I should divide by 8 to get the cost per person. Then I will just multiply by 3 to arrive at the amount to pay. I wonder if rst multiplying 175 by 3 will give an easier division, but I discount that for the moment. Whichever way I choose, it is probably not a division my brain has stored. So for accuracy I need a good division strategy. I know 2 8 = 16, and with a factor of 10 that gets me closest to 175. so I start with a factor of 20. That leaves 175 160 = 15 remainder. 8 goes into this only once, so I have a dollar amount of 20+1=21 with remainder. The remainder is now 15 8 = 7. I have to go into fractions (cents), so appending a zero I divide 7.0 by 8 (or mentally 70/8) which from memory has 8 8 = 64 the closest I can get, so the running result is now 21.8 with remainder 7.0 6.4 which is 0.6. Again I need another decimal place so appending a zero I think about 60/8 which has 7 as the closest factor getting me to 56 with 4 remainder, or taking the decimal places into account this would be 0.7 with remainder 0.4. Now I immediately click and note the 8 5=40. No remainder. So Im home. I add the 5 cents and the total I pay comes to 21.874 or rounding to $21.90. I still need to multiply by 3. Using strategies noted above (decimating it up into parts) I will eventually arrive at 65.625 or NZ$65.65 to pay. The other way, 3 175 is not too hard to calculate as 525. But dividing this by 8 is another chore. 8 6 = 48 is a good start, with the rst chop being 60 with 480 525 = 45 remainder. 8 ves are 40 so I get to 65 remainder 5. Then use 8 sixes =48 again to get 65.6 remainder 0.2. Then 8 twos are 16 to give 65.62 with remainder 0.04. Then six 8s are 40 and Im home with 65.625 an zero remainder. So on reection maybe this way of associating the tasks was a bit simpler overall. I

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wasnt to know. It makes me wonder if multiplication by a rational is more often easier by starting with the multiplication by the numerator? I doubt this could be general, the division is always the mentally harder task I think, so the nicer association of operations depends on the divisor.

1. Module 1An Introduction to Mathematics TeachingThoughts on the Way Mathematics Could be TaughtIf you have never wondered before how mathematics might be best taught, a good read is the article (Begg, 2009) in the Epol-338 course textbook. This chapter is also available in audio from our course TWiki. The main idea is that traditional school teaching is about transmitting what we (as educators) think children should know about mathematics (the usual curriculum subjects, arithmetic, algebra, geometry, calculus, probability and statistics), and that this view of mathematics education needs challenging, and increasingly has been challenged and shown to be less than optimal for educating bright and inquisitive mathematicians. The new teaching pedagogies championed by Begg and researchers he cites are the methods of constructivism, discovery learning, rich mathematical experience and complexity. Most of these methodologies are discussed in the literature and will be familiar to you from the EPSY courses. The complexity pedagogy might be the least familiar. In a few words it could be described as an eclectic bunch of approaches to learning with the common theme of exposing students to the full glory of mathematics and science, allowing them to learn from their own investigations and questioning and research. The complexity of this approach refers variously to the deliberate avoidance of over-simplication of topics and the rich veins of connections that should be (ideally) formed within the minds of students as they go about their learning. It is an extension and modication of many constructivist pedagogies. Begg also writes about the knowing, doing and thinking, skills in mathematics as being interrelated and not separate skills. The complexity pedagogies are therefore holistic in naturethey try to avoid teaching mathematics in a way that

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Module 1An Introduction to Mathematics Teaching

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separates sub-topics and disciplines. They even attempt to relate mathematics to other curriculum subjects like art, history, sport, literature, and so on. The article becomes truly illuminative when discussing the personal and social domains of education, asking us to think about what our aims for our students are in mathematics classes, particularly in these two domains. Do we want students just to master the subject? Or are we aiming instead to develop the aective and intrinsic motivations and interest for mathematics in our students? These are not incompatible goals, but dierent teachers will place dierent emphasis on these areas of education. In the Reection Journal section of these course notes some further thoughts on these questions are given. The focus is on the start of the year and preparing students for learning.

Interviews with AdultsThoughts on School MathematicsOne female noted that even though their father was an engineer who used a lot of mathematics, they nevertheless felt uninspired by their parents job. they spoke of the use of a more self-centred role model. The interviewee spoke about how it wouldve been useful to be exposed to successful female role models who uses mathematics in their careers and who had aspirational careers. So their jobs seemed interesting. Have them come into class and talk to students and provide mentoring opportunities. In short, giving students, particularly females, greater aspirational motivators would have been more inspirational, for this interviewee, than just attempting to make mathematics seem like fun.

Task 1.1Attributes of a Quality Mathematics ProgrammeThe rst forum task for this module is to answer the following question. What are the features of a quality school mathematics programme? Contribute four or ve aspects that you would consider important in a quality school mathematics programme. Examples could include opportunities for practical work or work set at appropriate levels for dierent ability groups in the class.


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In addition to the suggested two qualities here are some others. Involving the students in mathematical tasks that have interesting real world applications. Stimulating student interest in the aesthetic pleasures of mathematics and its intrinsic beauty. Providing opportunities for mathematical discovery and original thought. Gradually increasing student mathematical literacy to a high level, starting from basic number skills up to high level abstract algebra and analysis. Making sure students get time to be aware of what they are learning when solving practical problems! That is, always linking practical stu back to mathematical abstractions and generalizations. Exposing students to the accessible cutting edge of mathematics (this might be decades old developments, such is the extreme nature of modern mathematics). Providing numerous research projects of a suitable level and diculty for each student. Allowing regular discussion sessions where student ideas and their possible misconceptions can be aired. Achieving all the legal and curriculum aims purely as an incidental side-eect of having fun and exploring mathematics (with the inevitable struggles and frustrations that seem only fun after the fact). Recreate mathematical historysee if students can rediscover already proven theorems and conjectures. Dont just teach the historical facts. Prime them, but tell the students about the full history afterwards! (Why should Gauss et al have had all the fun inventing complex numbers?) Contributed by Kate: New ideas/concepts continue on from prior learning so students can build on knowledge previously acquired. Students are given the opportunity and encouragement to think for themselves. Creative thinking is promotedStudents have a chance to adapt and apply appropriate strategies to solve problems.


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There is continued assessment of knowledge that aligns with curriculum and learning objectives. Learning is connected to students lives and cultures to assist with engagement and meaningfulness. Contributed by Rewa: Clearly establishing levels of prior knowledge and plan lessons to build on this. Get to know the students. Recognizing and catering to a varying level of abilities and rates of progression. Setting challenging, realistic learning goals. When possible include students in this goal setting. Establishing relevancy for the learnerswhy should they learn this or more importantly why should they want to learn this. Allow adequate practice time for internalization of concepts. Accessible support and regular, positive feedback time, honest and encouraging evaluation. Contributed by Michael: Establish the relevance for the learner, i.e., if the learner can see how this is important they are more likely to try and master the area. Give everyday examples that the learner is already familiar with e.g., sale price less 30%, how much will I save etc.,. . . For theories, give the historical signicance. What was Pythagoras trying do when he developed his theory? This may enable the learners to link this with their own situations. Look for practical applications in the learners own lives and use these as examples in the teaching. Where memorization or practice is important, try and make it fun. For example, use games that aid in learning; say scoring a game of cribbage (helps with rapid addition). Contributed by Dave: Recognize what is the knowledge that the students come with (prior knowledge and what level standard they are really at).


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Dierentiate learning to meet individual needs (as students within the same class have dierent levels of knowledge). Encouraging multicultural interactions in the class when teaching some topics like for example, area ( ask the student about the area of their country that they came from),etc.,. . . Practical tasks should be included in the program, as many as possible (including games) so that the students will have the chance to accomplish these task in peers or groups and this will encourage the students (a) To interact with each other more eectively whereas most students feel safe in group environment to express themselves and be more relaxed to learn from each other without any embarrassment (specially if there is an EAL learners in the class ); (b) To have their autonomy will make the learning experience enjoyable and meaningful. On going assessment to inform teaching and learning. Regular and meaningful feed back to encourage students engagement towards accomplishing required tasks throughout the whole maths program. As much as it is possible relate maths teaching to every day situations like discounts, sales, and percentages, to show the students the importance of studying maths (value) as they will relate maths learning to their life experience.

Task 1.2.1: Curriculum Achievement ObjectivesHere are some collected examples from online colleagues. The task recall was to select an Achievement Objective (AO) from the NZC 2009 Mathematics & Statistics curriculum, state the chosen Strand and Level, and write a description of a meaningful context and a brief indication of how you would use the context to support students to understand the concept. AO, Number and Algebra: Understand operations on fractions, decimals, percentages and integers. Number and Algebra Strand, Number strategies and knowledge. Level 5. (Contributed by Rewa.) A basic lesson could be centred around the eect of increasing GST from 12.5% to 15%. The lesson could start with news article/s on the subject and a discussion of how it would eect the students/families. You could also expand


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the idea to include the eect on the overall household income and/or the raised revenue the government will receive. OR Start with a selection of advertising leaets/newspapers which promote sales centred around so many percent o, some even give you more % o the more you buy etc. Make sure to include goods/suppliers of interest to students within a range of goods/suppliers. An exercise could be to nd the best buy (require a bit of devising but could be set to varying levels). OR Another common sales pitch these days is the no deposit/interest free for so many months. Again start with the relevant ads and discuss what do the students actually think a purchaser would end up paying under such purchase agreements. The next example is a bit of a stretch for the curriculum. But what the heck, you only live once. AO, Statistics: Use simple fractions and percentages to describe probabilities. Level 3 or 4. (Contributed by Blair.) The context is sport. Specically a game of darts. Some ground rules would need to be set since it is a dangerous lesson. Alternatively a Velcro dart board could be used. The student maturity level dictates the safety precautions. The target should be square and also have a circle inscribed. Most students will have a characteristic distribution of scatter for their dart throws. We want to set up the board so that they will roughly evenly hit a square area with almost uniform probability. So adjust the target or studentto-target distance to achieve this. Then they can start having fun throwing darts. If they miss entirely it doesnt matter. The more attempts the better. Awesome! Get the students to help each other tally the hits in the square and in the inscribed circle region. Noting the distribution pattern as well. What can they then say about the probability of hitting the square versus that of the circle? What chance is there of hitting the square and missing the circle or vice versa?


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By Level 5 I reckon the students should be able to reason about the theoretical probabilities assuming their distributions of scatter were about uniform. Its just a ratio of areas. This can be visually checked of course. Some brighter students might note that the ratio of the hits in the square to those in the circle aught to be roughly /4. If not the teacher can guide them towards this result and then they can check if it is numerically about the same with their data. AO, Geometry and Measurement: Level 5. Measurement. Find the perimeters and areas of circles and composite shapes and the volumes of prisms including cylinders. (Contributed by Michael.) Background: Students are from a rural area with a low decile rating. The natural environment features strongly in their recreational lives. They swim and sh in the local rivers and kaimoana is important to them. Proposed Unit Plan Duration: 5 lessons We will look at what is involved in creating tanks to hold live eels for a tangi. The holding density will be set at 30 kg of eels per cubic metre of water. The class will decide on the number of people attending the tangi, how much eel the average person will eat, the meat recovery from a live to processed eel (this will be a weekend home work task; they can catch some eels and weigh them pre and post processing) and from this to how many eels that will be needed to be caught and held. Using the 30 kg m3 they will calculate the required tank volume and they will need to think about area above the water line so the eels dont jump out of the tanks. From here they will calculate tank sizes. To aid in this work I may have some drawings on the wall of various shapes and volumes to give them some perspective of volume. Say a photo of the local swimming pool with water volume, a bath tub, water storage tank (many come from areas where rain water is collected) etc. AO, Geometry and Measurement: Level 6. Shape. Use trigonometry ratios and Pythagoras theorem in two and three dimensions. (Contributed by Kate.) Lessons for this AO could be based around production depending on the interests and background of the class, for example building a house. Can use hands on activities and even head outdoors to look at buildings.


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Could start the lesson by questioning what skills and information we need to say build a roof on a house, moving into ways we could gather the information and introducing scenarios of what if we dont have that information available. Here we can introduce Pythagoras for the building requirements, for example: to nd how long our piece of wood will need to be for the roof if we only know two measurements. Then in subsequent lessons advance to trigonometry when the class is ready, using what if we only know this information and draw up several examples sticking to the theme of building a house and eve just focusing on the roof. This theme could be used for many other AO within this classroom.

Task 1.2.2: Frog Hopping ProblemHere is one think aloud exposition of the frog problem. If we ignore the sliding, then the minimum number of leaps required can be deduced by sequentially shifting frogs by sequential hops. Without too much eort it is clear that if the number of, say green, frogs on the left is n and the number of, say red, frogs on the right is n, then each of the n red frogs must be hopped over all m green frogs, which requires n m jumps. To check you can imagine instead jumping the green frogs, youd expect the answer to be the same: which is just jumping all m green frogs over all n red frogs, and the answer is still m n = n m, as expected. Now including the sliding moves. To jump a red frog to the right over a green frog we need one slide at rst to put a green frog immediately adjacent to the nearest red frog. This slide is repeated for the remaining m 1 green frogs to shift the leftmost red frog to the left of all the green frogs. So that is a total of m slide moves for the rst red frog. Then it is apparent that all the frogs will be bunched with no gaps between them, so to make a space for the next jump+slide sequence we have to slide all n 2 red frogs on the right, then we can jump the rightmost green frog over the 2nd red frog. Then we are in the same pattern as before but with just n 1 red frogs to move. But at this point we have another option, we could move the rst red frog to the left and then move m 1 green frogs leftwards to make a gap for the 2nd red frog to jump the mth green frog. At this point I start to worry about other strategies that might be more ecient.


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Then I realize that there should be extra restrictions that were not mentioned in the Module 1.1 notes, but are given in Frankcoms article. The red frogs can only move leftwards and the green frogs can only move rightwards. A somewhat articial constraint, but it makes solving the problem clearer, there are fewer strategies now. The critical insight is that we must avoid a pattern whereby two or more green frogs block two or more red frog on the reds left, and similarly we get stuck if two or more red frogs are blocking two or more green frog to the greens right. In either of these congurations there is no move to remove the blockage. The solution proceeds more or less uniquely. This is very interesting! adding the proper constraints has led me to the solution that is optimal, because now I realize that the critical thing for a minimal move solution is to achieve a nal conguration that has all frogs minimally displaced from their original posiitons. So the minimal move solution would have initial conguration, 1 2 . . . m m + 1 m + 2 . . . n + m (slot numbers) g1 g2 . . . gm r1 . . . rn initial positions and a nal conguration, t 1 2 . . . n 1 n n + 1 n + 2 n + 3 . . . n + m (slot numbers) r1 r2 . . . rn g1 g2 g3 . . . gm nal positions. Assuming m n, we can say that we cannot possibly complete the rearrangement in less than n(m + 1) moves for the reds and n moves for the frog g1 and (m 1)n moves for the frogs g2 . . . gm . Noting that there are mn jumps which move 2 positions at a time, we need to subtract mn/2 from each of these two tallies, making, n(m + 1) and n + mn n + m mn mn = + m moves for the greens 2 2 mn mn = + n moves for the reds 2 2


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and so the lower bound on the minimum number of moves is, Nmin mn mn +n+ + m = mn + m + n. 2 2

So if we can nd a specic strategy that yields this number of moves then we have the solution, otherwise we have to keep looking. After some playing around and trial and error I realized there is indeed an algorithm guaranteed to achieve this minimal move solution, it goes as follows (there is more than one way to describe it): 1. Move gm left. 2. Jump as many red to the left as possible (rst time you can only jump one, r1 ), without doubling up any adjacent red frogs. Keep the position marked at the last place you move to, call this p. 3. Slide one r frog left or slide the next g frog right if there is no r frog on the right of the current place, p to move left. 4. Jump as many g frogs to the right as possible without doubling up any g frogs. Note the new position p. 5. Slide one g right or slide the next r frog left if there is no green frog to the left of p. 6. Repeat from step 2 until all red frogs are on the left of all the green frogs. 7. The nal move is just to slide the g1 frog into position n + 2 to get all of them adjacent. An alternative way of describing the same algorithm implicitly is to say: start with any valid move, proceed by jumping and moving without allowing two or more of the same colour frogs to become adjacent unless (a) they are already adjacent, or (b) the new adjacent block has no opposite coloured frogs in front of them (as dened by their colours uni-direction of motion). Do not make any move that puts a frog beyond the positions 1 to n + m + 1, that is, use the constraint that there are only m + n + 1 boxes or spaces. It should be clear, after experimenting a bit, that this description gives a unique solution, the same as the formal solution, which is what I call the 1D Chinese checkers solution, because it proceeds by creating spaces between like coloured frogs allowing the other colours to hop along a chain. I was quite amazed that the enforcing the constraint actually led me to the general minimal move solution. Without it I was starting down the wrong algorithm road. After realizing the constraints suddenly the optimal solution was clear, with or without the constraint.


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VariationBackwards shifts are allowed, nal conguration unimportant. What if we allow backwards slides and jumps? Probably that wont help narrow down the minimal moves since going backwards costs extra moves. But what if we also dont stipulate that the nal frogs need to be adjacent? We also allow boxes to be positioned to hold frogs to the left and right of the initial places. Paradoxically this freedom initially confounded me. I rst had a strategy that was less ecient (in terms of total number of moves required) than the more constrained problem when m and n are not both in the set {1, 2, 3}. However, for these lower values of n and m my strategy was quicker than the no-backwards moves constrained solution but only when either n or m was =1. The challenge of worrying about multiple ways of accomplishing the end goal is intrinsically intriguing, so that is a good attribute to have for a class activity. There are now innitely many ways to move the frogs, most of which are total lunacy, but at least two broad strategies seem apparent. One is to use loose sequences of shift+jumps to move the red frogs past the green frogs using slide moves for the fewer red frogs and jump moves for the green frogs. The second broad strategy is to think more about spacing the frogs out so that jumps can be coordinated like a game of Chinese checkers. Intuitively the rst Loose strategy seems to be likely to be more ecientbut except for small n and m cases it isnt, as we will show below. Maybe there is a variation or combination of the two strategies that is optimal for all n and m? Heres the Loose strategy illustrated with m = 3 and n = 2. green frogs labelled g1 , g2 , g3 , red frogs r1 , r2 Minimum number of moves is 11. To emphasise the asymmetry the extra initially empty slot on the RHS is shown from the start. g1 g1 g1 g1 g1 g1 g2 g3 g2 g3 g2 g 2 r1 r1 r1 r1 g 1 r1 g 1 r1 g 1 r1 g 1 r1 g 1 r1 r1 r1 g2 g2 g2 g2 g2 r2 r2 r1 r2 r2 g 3 r2 g 3 r2 g 3 r2 g 3 r2 g 3 r2 r2 r2 r2 g 2 g2 g1 g2 initial state slide jump slide jump slide jump jump slide jump slide jump, and nish.

g3 g3 g3 g3 g3

Clearly this is less ecient than the Chinese checkers strategy when n and m are reasonably large, in fact merely greater than 2.


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Another cute thing about the problem is that the tight strategy is asymmetric. If we have m > n then it is almost always quicker to work by jumping the m green frogs over the sliding red frogs, provided neither m nor n is =1. If n or m equals 1 and the other number is less than 4 (i.e., say m = 1 and n {1, 2, 3} then it turns out to be optimal to slide the frogs of greater number, that is slide the red frogs if m < n and jump the green frogs, and vice versa if m > n. If either n or m is greater than 3 then it is always optimal, using this strategy, to slide the frogs of fewer number and jump the frogs of greater number. The general formula for the number of moves can be derived for this strategy, N = 2 + 2n(m 2) + n(n + 3)/2 Plotting N versus the pairs (m, n) will then reveal the asymmetry. This constructive argument does not prove that it results in the minimum number of moves. Recall that the optimum for the highly constrained leap frog problem, where the nal conguration has to have all red frogs adjacent with one gap then all the green frogs adjacent was, mn mn Nmin +n+ + m = mn + m + n. 2 2 We can get the best result using this Chinese checkers strategy for the unconstrained problem by simply subtracting the last move (the move the puts the frog g1 adjacent to its neighbour g2 ), so, Nmin = mn + m + n 1

N indicating this is the number of moves for satisfying the new unordered arrangement problem. The question is, is their a faster strategy? I could not nd a better method when n and m are not both in {1, 2, 3} with one of them equal to 1. For all other cases I looked at it seemed evident, and logical, that I could not do any better than the Chinese checkers solution. indeed, there are only two edge cases where a faster asymmetric strategy exists, these are (m = 2, n = 1) and (m = 1, n = 3). The dierence is just one move. Why does this asymmetric strategy work better for these two edge cases? The gist of it is in that just for these two special cases one can use the extra jump to get a frog moved two places in one move. The Chinese checkers solution algorithm ignores this possibility because of its constraint to the original place holdings. Heres the illustration that we (m = 2, n = 1) g1 g2 g1 g 1 r1 r1 g 1 can do one move fewer for these two cases. First r1 initial state r1 nish, in 3 moves,

g2 g2 g2


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whereas Chinese checkers uses 4 moves for this case. And for (m = 1, n = 3) Chinese checkers uses 6 moves for this case, compared to g1 r1 g 1 r1 g 1 r1 r1 g 1 r1 r1 r2 r3 r2 r3 r2 r2 r2 g 1 r2 initial state r3 r3 r3 r3 g1 nish, in 5 moves.

After all this work we still have no exact proof of the optimal strategy for the general unconstrained box position variation of leap frogs. The constrained Chinese checkers solution must surely be near optimal, even with its constraint on the nal adjacencywe know this simply because by design it results in minimal unidirectional shifts of frogs away from their initial positions. But I cannot be 100% sure if it is truly optimal for the more general unconstrained conditions. We know the Loose strategy is sub-optimal, but for (m = 1, n = 2) and (m = 1, n = 3) it is optimal, which is simply proven by brute force enumeration of all possibilities. At this stage time constraints meant I had to put aside these investigations, but this should at least provide a lot of food for thought when preparing this lesson activity for high school classes. VariationCost incurred by jumping then nd lowest cost solution. Suppose the frogs have to collectively achieve their end with minimum energy expenditure, and sliding takes half as much energy to perform as jumping. What then is the best solution? It may involve more slides and a total greater number of moves, but we know that there must still be at least mn jumps, that is unavoidable! So the Chinese checkers solution with the no-backwards move constraint is probably still the best for most m and n not both in the smaller set {1, 2, 3}. Further reections on this problem are noted in the Journal section page 88.

Mathematical Facts and Ideas[Editor.] This (Neyland, 2009), the third reading for Module 1.2, was interesting albeit a bit nutty. I like nutty mathematics. First the weird stu. I thought the whole discussion by Neyland on the theme of mathematics as only existing when humans invent it is just wrong. So this is a philosophical dierence of opinion, and one that cannot be resolved since it depends upon personal belief, and the nature


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of mathematical ontology is not a decidable thing. Consider Neylands discussion of 1+1=2: The ideas behind so-called facts can often be revealed by inquiring about their origins in time.. . . When did one plus one rst equal two? Was this true when the dinosaurs were roaming the Earth, long before humans existed?. . . the answer to the last question, according to some mathematicians such as Hersch, is no. . . . So the statement One brontosaurus and another brontosaurus make two brontosauruses is not a statement about numbers; it is a statement about brontosauruses. . . . The noun two is abstracted from the multitude of pairs of real physical objects encountered in human experience. . . . The critical point is that it took human beings to abstract two-ness from these pairs. So One and one make two is a statement that does not predate human kind. It is time dependent. OK, so the problem is that Neyland takes a materialist philosophical stand. But that is a prejudice. He gets into linguistic analysis over a matter of no importance whether 1+1=2 is a fact or an idea. You can take it either as an axiom or prove it from other axioms, thats all there is in mathematics. So the whole idea versus fact dilemma is really irrelevant, unless it is something we need to disabuse our students ofin which case it is an interesting debate, but we do not need the social constructivist philosophical point of view to correct students. We only need a good understanding of mathematical consistency, logic and axiomatic schema. Moreover, how do we know some alien civilization on another planet did not come up with this statement millions of years before humans. So Neylands view of the idea of 1+1=2 is not only time-dependent, it is place and contingency dependent. Thats a poor foundation for mathematics IMHO. An alternative view is that humans discover mathematical ideas. Mathematical ideas are physically pre-existent because they do not depend upon anything physical. If there was a sentient mind before humans, or even before what we know of as the physical universe, then it is at least logically possible that that mind or minds may have formulated mathematics. There may be other inequivalent consistent and incomplete formal systems like Zermelo-Frankel set theory or Peano arithmetic, but there nothing physically dependent about these formal systems. Humans bring these formal systems to life in physical representation (in textbooks, our neural excitation patterns, scribbles in sand, television programmes, computer codes, and so forth). So the physical manifestations of these eternal ideas is time-dependent, naturally. That does not mean the consistent ideas themselves are time-dependent.


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This contrary view to Neylands (called neo-Platonism, if you need a name for it) also does not place mathematical ideas on the pedestal of facts, they are still ideas. In isolation they have no meaning, but in relation to other ideas they form consistent tautologies, which are also not facts. If they are not consistent then they are by denition not part of the same mathematical system. So there is something there for people like Neyland, who seem to think mathematics is socially constructed, to hold onto. If they dont want facts then they dont need to have them. Also, as an aside, humans have no way of proving that physically represented mathematics, at least as powerful as Peano arithmetic, is consistent. We do however have a notion of what consistency means, namely that it should not be possible to formally prove a statement and its contradiction. This notion is a necessary dening feature of any mathematical system. A last point on the philosophy before moving on. Neyland seems to think that Pierpont came to an appalling yet inescapable conclusion that mathematics must by dened by logic and consistency, in other words by the arithmetic rather than by the ideas of humans. I nd this to be a false dichotomy. Mathematics is still an art. It is also a logical science based on consistency and axiomatic method. Whats the problem here? Humans do mathematics using creativity and artful play. The result is, at best, a physical and mental representation of logical consistency. There is no dilemma here for Pierpont to worry about. Mathematics is a humanist activity (except for when it is not practised by humans). The result is that mathematicians (human or otherwise) discover Platonic truths, if they are lucky, and they may even discover, as we have, that they cannot know the absolute truth about all theorems of mathematics. Theres the link between the beauty and aesthetics of mathematics and its cold hard arithmeticization. I do not understand why this worries Pierpont an Neyland, or why it is seen as appalling. Perhaps I do not understand their argument. Most modern mathematicians probably side with Neylands view, if they have a strong view on these philosophical things. So Im just putting up an alternative perspective for the readers interest, which I might add, is a view I share with any brilliant mathematical logicians, Gdel included1 . o Anyway, this is a philosophical dierence of opinion and is not all that relevant to pedagogy, so with all this said lets look at the positive things in Neylands article. It really is a good article! Probably the main thing to take away from this reading is that it is pretty destructive to present students with mathematical ideas as if they were facts castThe late William Hatcher is another. I had the pleasure of conversing with Hatcher on a few occasions before he died.1


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in stone. A simple way to avoid this is to construct some toy formal systems for the students to play with (see Hofstadter, 1979, for some examples). Then guide them towards the realization that some statements can be treated like facts, and we call them axioms, but that there is still nothing eternally true about them. They simply dene the mathematical system that we are dealing with. This should prime students with the important sensibility of having a questioning and curious attitude towards mathematical knowledge. [EDITOR: comments are welcome. Please add any.]

Glossary of Some Terms Used in school MathematicsHigh school mathematics covers very broad range of topics. So it is not surprising if many professional mathematicians are unfamiliar with at least one or two terms used in school classes. Here is a short list of some of the more unusual terms, plus a few familiar terms that one might forget about from time to time. Addend any one of a set of numbers to be added. Assignable causes (of variability) causes of variability in data collected over timewhich could not have been predicted ahead of time. Hence, usually associated with an abrupt change or drift in a process. Clinometer instrument for measuring vertical angles. [OK, but I still do not know what this looks like?] Concurrent lines lines that all pass through one common point. Congruent geometric objects have the same size and shape, indicated by the relation symbol =. Decomposition a method for subtraction whereby the larger numbers are rewritten as a sum to simplify the mental process of subtraction: e.g., 74 28 = (60 + 14) (20 + 8). Dividend in 12 3 = 4, the number 12 is the dividend (the number being divided up) Dot plot represents tallies as dots on a scale.


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Eulers relation V + F E = 2, where V = number of vertices (nodes, F = number of faces, and E = number of edges of a polyhedron or network. Figurate numbers numbers that can be represented by dots arranged in a gure (on a grid?) Function a set of ordered pairs where the rst element of each pair occurs only once in the entire function list, e.g, {(a, 3), (b, 2), (c, 3), (d, 9)}, all letters are dierent numbers. Golden section the ratio a/b with the property a/b = (a + b)/a. Hectare 10, 000 m3 Icosahedron polygon with 20 faces. Kite quadrilateral with two pairs of congruent adjacent sides.

Strip graph Represents frequencies as a proportion of a rectangular strip.

I added the Mori kupu terms to my computers Mori-English translator. a a [TODO: add a few more terms in the L to Z range.]


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Resources and Tools FamiliarizationNCEA Achievement StandardsAssume you have browsed these, starting a search here http://www.nzqa.govt.nz/ncea/index.html. No need to comment on them here.

Problem SolvingHave a look at sample activities here: http://www.nzmaths.co.nz/node/449. Choose one and work through it, then act out teaching it with a class. Note any diculties or interesting thoughts that occur during or after doing this little exercise. Below are three samples. Hineas Watch Hands Problem. How many times in a day do the hour and minute hands of an analog watch coincide? http://www.nzmaths.co.nz/resource/hineas-watchs-hands Comments: This is a nice little problem. The intuitive answer is 24, since the miniute hand sweep out 24 revolutions in a day. When solved algrbraically, using say the relative speed of the hands, i.e., m = 12h and solving thetah m = n2, we get h = n2/11 with solutions limited to the range h (0, 4). This formally has 23 solutions not including 23:59:60. Why not 24? Because the n =24th solution has h = [24/11]mod2 6.854 > 2, so at this time the day has past, and we are into the next day. But if we also exclude 00:00:00 (i.e., excluding n = 0), then we get only 22 solutions! I would force the algebraic approach by asking students to nd the exact hh:mm:ss at which, say the fth, coincidence occurs. The guiding questions suggested in the lesson plan are I think necessary, most Level 5 students will probably not be able to gure out the algebraic approach without guidance. But I would give them plenty of time to ponder and struggle with it and jog them to get to it. After a bit of playing around with actual analogue clock faces I would get the students to obtain some numbers and maybe look for patterns if they cannot immediately jump to the abstract algebraic formulation. * * *


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Hineas Other Watch Problem. How many times in a day do the hour and minute hands become perpendicular? Comments: This is a nice follow-up because it uses the same principles, and hence both tests and cements previous understanding, as well as having the surprise of a completely dierent numerical answer. Here we have h = (2n + 1)/2, with constraint 0 h 4 as before. Its gentle also, because the surprise is not too shocking, and most students will see why in short time. * * *

Laras Equiprobable Dice. This was not a closed problem but rather a series of similar problems. http://www.nzmaths.co.nz/resource/probability-trees. We dont need to copy it here, it is quite a long plan. After following the suggested exercise to act out teaching this activity I had the following comments. Comments: Combinatorics was and is one of my weaknesses in mathematics. It always takes me longer than I expect to get solutions out and accurate. assuming my students may be similarly challenged (at least some of them) the use of combinatorics in probability theory is probably a very worthy classroom exercise. This problem set is a nice introduction, and there is opportunity for gifted students to derive the formulae for permutations and combinations that arise. At this stage I cannot be sure what year or level of class such activities would be most suitable, but there is probably something here to be gained for any secondary school level.

2. Module 2Introduction to Mathematics Lesson PlanningModule 2.1, 2.2Planning for Mathematics TeachingFour readings, (Goos, Stillman, & Vale, 2007; Neyland, 1994; Chambers, 2008; Goulding, 2004) and a Journal Reection task.

Comments on Developing Mathematics UnderstandingThe reading is (Goos et al., 2007). While not tremendously informative, I enjoyed this article. Some highlights: Student responses to question: how do you know when you understand something in mathematics? Not surprisingly the most frequent answer was when I can do the exercises and get the correct answer. What wed prefer to see in students and in ourselves) is the response, I know I understand when I can explain it to others. So why do we not try to breed this metacognitive understanding? This appreciation that true understanding is deep and is not just being able to apply technique, but the highest test is to be able to explain it and teach it to others. So mathematics, or any school subject, teachers should try to inculcate this belief and appreciation in students and in their approach to teaching. Group activities are great for this, both for guring out one does not really understand a subject and for practising explaining things to colleagues. The Pirie-Kieren levels of understanding. While a bit academic they are perhaps useful to keep in mind as levels of understanding that we look for in 32

Module 2Introduction to Mathematics Lesson Planning

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students progress. They are, 1. Primitive knowing. 2. Image making. 3. Image mental construction (having an image). 4. Property noticing. 5. Formalising. 6. Observing. 7. Structuring. 8. Inventising. The solving a simultaneous equation anecdote. This was cute and probably familiar to many teachers and tutors. The lesson for us is what it implies for improving teachingdo not be prescriptive, or you risk students copying your demonstrations verbatim, and missing the uidity of mathematics. Better to teach simultaneous equations with a few solved examples (concrete experience) but before letting students loose on problem solving give them the deep understanding of what we are really doing in trying to eliminate variables. The dierences between the two schoolsPhoenix Park (progressive, constructivist oriented) and Amber Hill (traditional, direct instruction). Its not surprising but still interesting and valuable to know that the more progressive school does indeed win on all countsbetter deep understanding and students do better in the standardized exams than the highly coached school, despite not being taught to the tests.

Comments on Designing Rich Mathematical ExperiencesThe reading is (Neyland, 1994). This is a terric article1 , even though it dates back to 1994. It is quite prescient. The wierd thing is that perhaps Neylands warnings and suggestions have not really been heeded in many schools. One gets the feeling the problem might be not so much that todays mathematics teachers do not appreciate the importance of providing a rich mathematical experience, but rather that todays teachers do not quite know how to provide a rich mathematical1 I may have been a bit rude about Jims humanist construction philosophy of mathematics earlier in these notes, but he redeems himself here in my view [Editor].


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experience given the constraints imposed on them by curriculum, NCEA and school administration. So what is a rich mathematical experience ? Neyland denes it as at a minimum providing the following ten aspects , Activities that are accessible. Allow further challenge and be extensible. Invite students to make decisions. Involve students in speculation, conjecturing, testing, proving, explaining, reecting and interpreting. Should not restrict pupils from searching in non-standard directions. Promotes discussion and communication. Encourages originality and invention. Encourages what if and what if not questioning. Has an element of surprise. Is enjoyable. Not much mention of mathematics and numbers in that list! So it is nice and general and could be a good list for richness in any teaching subject. Note also the fourth aspect is really about seven in one. The art of a good starter. This is another theme in common with all teaching. Capturing the students attention at the start of a lesson is almost a necessity for a successful lesson. One can recover a lesson without a great starter, and one can botch a lesson even with a great starter, but it sure helps to start with a big bang. As with a lot of art, the choice and design of starters for school lessons is partly obvious and dictated by the subject matter and learning intention but also very dicult to master and deliver with aplomb. Also, one does not really want to take up the whole lesson with the starterunless the entire lesson is built up smoothly from the starter activity. Sometimes a lesson plan will be so awesome that it is the starter, core, and end, all in one.


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The interventiondiscouragement divide. We have to thank Jim Neyland for broaching this topic. Most teachers do (I suspect) intervene too early, but then others who want to promote student self-discovery are probably guilty, if not at risk, of allowing too much discouragement and de-motivation to develop. My answer is, Do not feel guilty! So what if a student gets discouraged by lack of success? This is what will happen in the mathematical research world. Sure, its important to oer encouragement and advice, especially when asked, but we can, I think, be highly successful mathematics teachers without supplying any answers to students for the problems they are assigned to solve, that is, not until they have made a full attempt. So have no guilt. But dont leave students entirely to their own devices. Be true to your teaching philosophy or change it if you get evidence that it is simply not working. The art of asking good intervention questions. Intervening to help students need not be just giving them the answer or showing them directly a solution to an analogous problem. Neyland suggests some good question strategies. Can you explain what you were trying to do? Can you make a guess and test it? Do you have any questions about what you are doing? Have you tried another strategy? Have you tried solving a similar problem, maybe an easier one for starters? Have you tred breaking the problem down into smaller simpler steps? Why not try working backwards? Can you illustrate or sketch the problem out? In time, a newbie teacher will acquire the skill to almost automatically ask an appropriate prompting question, without it being too leading. NOTE TO SELF: Its a good idea to refer back to this article by Neyland for his nice little examples of paired starter and intervention questionings. TODO: Make starter and intervention questions part of the detail of all my lesson plans. Even if they are not used or if they are attached to a separate page of the lesson plan, have some handy.


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Planning for rich mathematical activities. The advice is to do what I just jotted down! Have good starter and intervention questions on hand, appropriate for the topic, and designed to laser-like focus student attention on the learning intentions. Then assess, evaluate and improve the lesson plan afterwards. You can always make improvements. Also, do the activity yourself. Rehearse by videoing yourself delivering the lesson and watch it. Try your best to anticipate all the varied ways a student might be tempted to think about the problem as you have introduced it in your rehearsal.

Comments on Planning to Teach a Mathematics LessonThe reading is (Chambers, 2008). Some highlights and comments: Textbooks are not your lesson plans. Indeed, textbooks are really for students, not so much for teachers, though they can be a useful source of interesting problems. Elements of good lesson plans. These can be held in the teachers mind, but novice teachers are strongly advised to write it all down for reference. Ive begun a long checklist of elements of a lesson plan that Ive gleaned are important. It is currently in my Epsy302 Course Notes and excerpted to the course TWiki. Interpreting a plan developed by someone else. The main focus of our study of this reading is to follow the Module notes instructions: On page 67 of the Chambers chapter, there is a lesson plan for nding areas of triangles. Assume that you were asked to teach the Main Activity from 9:15 till 9:50 and that you have the text book exercises available, try to visualise how you would go about teaching this section of the lesson. Draw a picture showing what you would record on the whiteboard, and how you would organise it. For some of the time you would be working at the whiteboard. How would you involve students during this presentation time? The plan states After 1520 minutes stop the class and cover an example of a nonright angled triangle, use prepared resource to demonstrate that the formula used is the same. Make a drawing of what you would use for the prepared resource. What would you say as you demonstrated this idea? Make up 3 or 4 exercises that could be in EX 10F and Ex 10G. What would you be doing when you were not working at the board? Act out this part of the lessonpreferably with at least one person in the role of student.


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Assumming weve all done this. The thing to do now is write some reection journal notes on the exercise. A sample is given in the Journal section of these notes on page 93.

A Checklist for Mathematics Lesson PlansI plan to be totally unsatised unless all of the following boxes can be ticked for any nal lesson plan I devise [Ed.]. Lesson topic/name (make it catchy and memorable). Why should the students be learning this? How should they learn this? Cross check with all the strategy checks. Lesson year and level. Estimated and preferably the rehearsed time. Clear statement of lesson objectivewhat should the students learn? Relevance to National curriculum achievement objectives clearly stated. Clear statement of lesson intentionslinked nicely to achievement objectives. Key words and key concepts. Clear statement of success criterialinked to learning intentions. Homeworkif necessary (not just a catch-up) Background of relevant student knowledge and context. Background of previous lessons of relevance. Resources neededall of: physical, human, and ICT. Notes on preparation of resources and any clean-up or follow up afterwards. Details for arranging possible additional teacher aids, guests, support sta if required. Extra safety arrangements if needed. Seating arrangement needed if necessary. Starter activities, if not part of the main lesson. Does the starter activity get students immediately thinking?


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Are the starter and plenary activities adequate to check and assess learning? Strategy for dierentiation, to cater for student diversity. Strategy for ensuring safe and supportive learning environment. Timed sequence of main proposed teacher and student activitieslearning experiences. Notes on possible anticipated departures from the timed sequence. Is the strategy appropriate for achievement of the learning intentions? Does the strategy make good use of student knowledge, suggestions and examples? If you have a group activity, is it well-planned and have you selected a group structure that suits the lesson? Is this going to be an enjoyable lesson, one that you would enjoy? Key questions to ask: starters and intervention questions. Notes on likely student misconceptions, errors, thinking traps, and plan how to respond. Outlets for potentially disruptive students that will keep them focused in some way on the learning intentions. Lesson closing activity, summary, extension suggestions and review. Back-up plan in case a main or critical path component of the lesson fails. In general some sort of out or escape in case of some unanticipated break down in the lesson or teacher brain freeze. (This has to be something that will still lead to valuable learning.) Assessment criteriawill you be able to judge whether learning has been achieved or not? Details on how assessment will be performed or recorded. Where administration and other notes will be recorded (non-assessment stu). Evaluation goals for reviewing the lesson and improving the plan. Have you rehearsed the lesson? Have you consulted your reection journal for tips and reviews, as well as teacher guides, past exam papers and examiner reports? If not, then re-check this


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entire check list after doing so. Are your entering and exiting routines clear and consistent with previous standards for the classroom? If there enough time to complete the lesson, or will you need a few short-cuts and/or extensions up your sleeve? Look over the lesson plan again, and ask, will the students immediately get a clear vision of where the lesson is coming from and heading towards in the big picture of the years grand plan and goals? If you havent ticked the previous box, is it because this lesson is a special one-o topic? If not, then re-do the plan and re-check this check list. Mathematics Specic checks. Some topics in mathematics and sciences completely fail as lessons if the background knowledge is not understood by students. This is more of a weakness in mathematics and sciences than in other subjects where teachers can more easily ad lib. Rigorous check on prior understanding required? Plan on what to do if lack of required understanding is found. Prepared questions to use to gauge student understanding. Is the warm-up or starter simple enough to draw students in rather than discourage them? What will you do for students who clearly struggle with the concepts? How are you going to involve students in the development of ideas and conjectures? Is your jargon level minimal? Have you consulted relevant past examination papers and examiners reports, and learned from them? Have you checked pertinent teacher guides for potential tips and snags? Have you worked through the proposed exercises yourself? Note any peculiarities you may like to raise in class. What will you do when students are working alone or in small groups? Have you checked computer equipment, experimental apparatus, and OHP machines are all in good working order?


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? TIP:Try sometimes using newspaper clippings or other news stories as starters when available and appropriate.

Module 2.3Exploring Mathematics Teaching ResourcesThe tasks for this module were to explore some mathematics resource websites and to practice dong and grading an NCEA exam paper. I found these activities quite useful. There are too many websites to review here. I started listing a few on the course wiki Epol-338 resources.

Comments on Using Assessment for Eective LearningThe reading is (Lee, 2001). This short article has real mathematics teaching gems, but they are useful for general teaching. The main ideas that arose from various reported problems include, Guiding and advising students is more benecial than marking their work. Cooperative work is better than competition. Teachers should discuss their methods with each other, and with students. Quality is more important than quantity in education. Assessment is useless if not analysed and used to improve teaching. The main solution ideas were, 1. Start lessons often with, have you got any burning questions? 2. Ask questions worth thinking about, collect answers from all students

invite comments on insightful answers

a good answer is thought provoking, not necessarily correct or wrong


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ask groups to devise solutions to complex problems and invite them to present their work to the whole class. 3. Explore complex questions that last more than one lesson. 4. Get students to research topics and write their own questions. 5. Use problem and misconception focus questionsstudents mark their own work on sets of questions designed to reveal misconceptions, then let the teacher know what they think they need to focus on. De-emphasise marks, emphasise problem areas and misunderstandings. 6. Use complex homework questions and peer-evaluation using teacher-supplied fully detailed answer sheets, and tell students to accept solutions that are convincing and seem to work and discuss the dierences with the prepared solutions. 7. Get students to work on complex homework tasks and before sending them away get them to think about their burning questions, assign themselves how am I going to complete this task? and go over these in class. 8. Use the last ve minutes of a period to have students reect on and write down what they have learned (what they know now that they did know know before). We can identify two strong themes in all the ideas of this article. Design lessons to make students reect, think, and reveal their thinking openly to their peers and the teacher. Give pupils a clear voice, freely and often. The other theme I thought particularly important, and in agreement with my own philosophy, was to avoid giving the students simple dull exercises. Always use complex questions, every lesson, and make them questions worth asking and answering. This was a common tactic reported in the article that helped engage students, get them thinking and maximised the usefulness of the teacher as a guide. Doing lots of simple drill problems seems less important, even if the school focus is on exam preparation. Using complex questions, as well as tasking students with more realistic problems, can easily extend a class over more than one lesson period, which is a good thing in a way since it allows deeper consolidation of knowledge and skills.

3. Module 3The Numeracy Development ProjectModule 3.1Overview of the Numeracy Development ProjectThe reader is referred to the Numeracy Development Project (NDP) Book series published by the NZ Ministry of Education Te Thuhu o te Mtauranga. To write a a anything more about this project here would be entirely redundant. The Numeracy Development Project is typically useful only for school children below year 9, so it has less relevance to secondary school. However, the general pedagogical principles used by the NDP is broadly applicable and useful for teaching at any year level, provided the content is accordingly adapted. So it is thoroughly worth studying. There are also a couple of audio lectures recorded from the Numeracy Development Project Books available here on our course TWiki.

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4. Module 4Number and AlgebraModule 4.1Teaching NumberFor convenience I have recorded most of this module on audio. See the Epol-338 TWiki.

4.0.1

Tips for Teaching Number & Algebra

The Epol-338 forum asked us to research ideas useful for teaching number and algebra concepts. Here are some contributions. Contributed by Blair: I quite liked the look of the resource on estimating for accuracy: http://www.nzmaths.co.nz/resource/estimating-accuracy Here are some comments: 1. Its a very handy skill to be able to estimate arithmetical operations for business, or engineering, science, or virtually any other application of numbers. 2. Having a feel for the size and likely result of a computation is a great way to easily spot potential mistakes, especially when the use of calculators is in play and one can too easily just rely upon buttonpushing. Ive already seen many students put their trust in a calculator and get wrong answers by making button push errors, and not even question their results. 3. The estimation for accuracy resource can be tailored for use at many ability levels and for a wide variety of number topics. Even at advanced levels students can be engaged in estimation problems (e.g., 43

Module 4Number and Algebra

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how may atoms are there in the universe?) Contributed by Michael: The students in the year 9 class that I taught really struggled with fractions. This was a common thread I found in all my interviews. The following game, as a starter, is Easy Fractions Game. Found at http://www.nzmaths.co.nz/resource/easy-fraction-game This is suitable for students at level 4. It can be played as a whole class starter or used with small learning groups. Nice fun game using dice and paper. It reinforces value of fractions and the adding of fractions. Contributed by Rewa: I like the Shopping for Saving lesson found at http://www.nzmaths.co.nz/resource/shopping-savings . This is a lesson designed around comparing prices and savings on grocery bills. What I like about it is relative to real life incorporation of fractions, decimals and percentages use of stations would allow for diering abilities focus on reasonableness of answer

you can build in your own exibility re pace of lesson(s), complexity of work,use of groups or whole class etc It would be great to link it to a school camp or similar e.g., family celebration/hangi

Module 4.2Dealing with MisconceptionsMy main comment for this section is that misconceptions and errors are great! How else can we say we are making progress teaching students unless we are correcting errors? If students are not making errors then they are arguably not being challenged optimally. The course readings by Swan (2000) and Tanner & Jones (2000) are excellent places to start learning about how to deal with student misconceptions. So I have recorded them on audio for convenience and posted them on my Epol-338 TWiki. A sweet summary of how to deal with misconceptions might be a simple three step process as follows. Students with dierent misconceptions need:


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examples chosen to highlight their errors, emphasis on the ideas they do not understand, good basic instruction stressing the underlying principles, in common with everyone else.

Discussion of MisconceptionsThe Epol-338 forum asked us to discuss various aspects of handling misconceptions. Here were some entries: Contributed by Blair: I had a high ability year 9 class on TE, they had a few misconceptions but were generally smart enough to see their own errors relatively quickly. So here I found it more interesting to look at the reading by Swan (2000), page 111 of the EPOL338 book of readings. The suggested way of dealing with the misconceptions appealed to me. Namely, to use a discursive approach rather than telling the students rules for comparing fractions. It makes more sense to me to take the extra time and eort to allow students to explore fractions and compare them by converting to decimals or using other means (shading in blocks, etc). It may take them longer to resolve their cognitive conicts, but when they do it the way Swan suggests I would feel more condant that a deeper understanding has been gained by the students. I guess only some sort of formative assessment could determine this for sure, but at rst blush it seems to me that the discovery approach is superior in the long term. For the percentages misconceptions, I just love strategies like the one proered for correcting a sales assistant who suggests a 20% discount amounts to a 40% discount when selling two items: the remedy is to not bother correcting the sales-person directly but rather tell them youve changed your mind and would like to buy ve items! Awesome! It seems to me that when misconceptions arise in a school class there is always an opportunity to formulate such cognitive conicts. After all, once an incorrect answer is given by a student a skilled teacher can always twist it and use it to create an absurdity. Sometimes this trick is not warranted if the misconception is a trivial one, but when it seems like a deep misconception I just love the idea of being able to twist the students wrongness into a blatant nonsense.


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Then of course I like the idea of giving the student time to stew with the conicting nonsense. It does take up more class time and many teachers I suspect would be tempted to just give up and teach by rote. I nd that hard to stomach. Why waste a great learning opportunity simply for the sake of time? Of course, thats another issue: taking the time to teach when a school might be overly focused on getting through a curriculum. (Thats what I found sad about TE.) Contributed by Rewa: I had the pleasure of teaching fractions and decimals to my lower achieving year nines. Inevitably I encountered many misconceptions. Previous work in EPOL338 had made me aware that these two maths areas were a mineeld of misconceptions so in preparation I tried to t in as much research in this area as possible (along with other sites I found the decimal web site refered to in our module and the chapter in our textbook but didnt nd the readings in our book of readings - much to my disappointment; they would have been a great source of help). My associate teacher also gave my some clear steering with regard to using clear and consistent terminology throughout the lessons. For teaching fractions I (with the help of my nine year old and husband) made 11 fraction sets (colourful, laminated circles and fractions), these were worth all the eort. With some of the class it was actually seeing the conict in their reasoning that led to them restructuring it. Similarly with decimals I got hubby to make me one LABit certainly demonstrated how it would be a great resource - several students made comments, as it was demonstrated to them, about how they clicked that 0.1 was 1/10 and how 1/100 is smaller again. . . etc. Unfortunately I only had the one in a class of 16+ so the students didnt get the full benet of it. BUT the most frustrating thing about teaching these subjects, with all these existing mines, was the extremely limited time in which I was given to teach them! Contributed by Michael: During TE 1 I had the same year classes for both Math and Science. In science we were looking at living things and how they impact on their environment. To help the students link with their own experiences I built a pilot scale eel farm in the back of the science lab.


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I used this to help the students in visualization and calculations for volumes and capacity of cylinders. It also allowed us to work on area of circles, area of quadrilaterals [circular tanks made from welded plastic sheets] and area of triangles. The space between tanks. One problem that emerged was that students had diculty in using the correct unit of measure; cm2 for area or cubic cm for volume. To help reinforce the ideas I used materials. Stickit type blocks 1cm3 were used and the students made up various block congerations to cover certain areas and to ll small cardboard boxes. Using materials helps the students to visualize the concept.

Module 4.3Algebra and GeneralizationFor the next forum I wrote about the sieve of Eratosthenes. This is a simple activity to set up and invites plenty of explorative, discovery and pattern recognition elements. The cool thing would be to try to get students to articulate or in some other way explain how the algorithm works, and then ask them when it doesnt work (for primes greater than the rst few million!) Heres a good reference: http://primes.utm.edu/glossary/. . . SieveOfEratosthenes Its cool how one keeps forgetting some of the mathematics of antiquity and then rediscovers it every now and again! Its like always being a kid. . . something I would not experience if I had a better memory! So I gure, teach kids about the Greeks, if they ever forget the lessons then they will also hopefully enjoy the thrill of re-discovery! We never fully forget things, so theres always a bit of Prousts madeleine cake laying in wait for us in life, especially as teachers. Here is Mikes contribution to the forum: Generalisation is important because it allows students to see patterns. During TE 1, one teacher I observed would use generalisations to assist students make their own rules for calculating problems. E.g., in working out area of a quadrilateral students would use stickit blocks. Say a rectangle 5 high 4 wide of blocks. Students would stick them together and count up the total number of blocks. Sooner or later someone would recognise that you only have to multiply b h. The teacher would right on the white board Alphas rule: Area of a quadrilateral = bh


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AND: a Fraction Fraction in Book 8 page 24. [Ed. The idea here is that multiplying fractions (say f1 f2 ) follows easily from nding the areas of rectangles. One takes two copies of an original rectangle, use one to shade into the parts for the rst fraction, select the number of parts, e.g., split into ve parts and select three of the parts for say f1 = 3/5. From the shaded arts do the same to just the shaded protion for the second fraction. Comparing with the original copy we get a visualization of f1 f2 .] The use of materials to help the student visualise the problem is great and after a few trials someone in the class/ learning group is going to come up with the rule. This is a great example particularly in a rural area like up here on the East Coast, something the students would identify with. Rewas contribution to the forum: Generalisation is important in identifying generalisations from simple situations these can then help us solve more specic complex problems. From NDP Book 8 Id like to share two consecutive activities; Dividing Fractions page 21 which then leads into Harder Division of Fractions page 22. The rst activity can start with using materials/pictures to nd the answers; I found actually seeing a problem and the solution (particularly if you pretend it is cake) really helped my students. They also enjoy the doing involved in such activities. Then the stepped development of the concept through Imaging and Using number properties particularly through to the Harder Division of Fractions appears well set out and paced. Student guided in this way will develop the division of fraction generalisation by their own deductions and this should result in them feeling more ownership and understanding of the generalisation. They should be able to appreciate why it works.

Mathematics Needs for ESOL StudentsThe ChallengeWhile the language of mathematics may be universalthe language of mathematics instruction is not. The challenge is how to teach maths to students whose rst


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language is not English. Consider the following: The language skills needed for mathematics are two years ahead of the ocial system. This means that a year 8 maths problem often requires year 10 comprehension and reading skills. Mathematical discourse and syntactical structures include features that make it dicult for ESOL students to gauge meaning, as they often do not conform to the usual norms of language. Vocabulary in mathematics classrooms not only includes specialised maths terms, but also everyday terms that take on new meaning when used in a maths context, for example table, column, product. Also, there are the some tricky homophones such as sum and some, addition and audition, angle and ankle, factor and factory.

SolutionsStart with skills assessment Often, our bilingual students will have some ability in maths, but not be able to communicate that ability. While it may be convenient to simply assume that a less uent speaker of English is not good at maths, an assessment of current competence is an essential staring point. Obviously, it is vital that maths skills are appraised on the basis of cognitive ability across a range of areas, and not on the basis of the students prociency in English. Incorporate the teaching and communication methods outlined in the remainder of this article in your assessment design. Remember also, the tools you create for use with ESOL students can be used with the rest of the class. Work problem to solution, not vice-versa Historically, we teach maths skills, then apply them to problems. However ESOL students better understand by experiencing the problems rst, then developing the solutions. In application, this means that rather than starting with numbers, processes and

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Math-epol338 Course Notes - Mathematics Curriculum Study