Mathematics Pathways 2009

On paper the new courses next year look to be improvements on the existing 2008 MIPS/Foundations/Intro Calculus/G&T combinations. In 2008 MIPS was for weaker students, Foundations for mid tier students and Intro Calculus/G&T was for the capable students (think Maths 2/3 if you're from my era). In Year 12 MIPS lead to Modelling, Foundations lead to Discrete, Intro Calculus/G&T lead to Calculus/Applicable mathematics.

There were definite issues with the old structure. First MIPS was very language oriented which caused serious difficulties for low literacy students. The Foundations course was more difficult than the year 12 Discrete Mathematics course that it lead to. Many capable students opted to take the easier option (Foundations/Discrete) rather than Intro Calculus/G&T students as the scaling was never quite right for(although Intro Calculus/G&T students did get the benefit of satisfying many pre-requisites in university and avoided bridging courses)

The new courses for 2009 on paper better cater to a range of students. These courses are semesterised and labelled 1B-3D (eg. 2C in semester 1 and 2D in semester 2 year 11, and 3A in semester 1, 3B in semester 2 year 12). Each year 11 course (if necessary) can be sat again in year 12 (eg. if a student failed 2c/2D in year 11 and repeated 2C/2D in year 12). There are lower courses aimed at students with learning difficulties (PA/PB/1A).

Weaker students have a more traditional year 9/10 type course in 1B/1C/1D/1E or 1D/1E/2A/2B (replacing MIPS/Modelling)
Weak mid tier students now have 2A/2B/2C/2D (replacing Foundations/Discrete)
Strong mid tier type students now have 2C/2D/3A/3B (replacing Foundations/Discrete)
Capable students have 3A/3B/3C/3D & 3A/3B/3C/3D specialist (replacing IC/G&T/Calc/Applic)

The strong/capable students are typically university bound, weak mid tier students may use their score for low requirement university courses or TAFE. Level 1 courses are generally for vocational students.

The real benefits are for mid tier and capable students that now have real options in selecting 2A-3A as a starting point in year 11. Students that sit higher end mathematics are now promised a more equitable scaling factor than was in place under the old system. The general opinion is also that the new courses are easier (in general) than the courses they replace. Only time will tell.

Class load next year

I don't know if I like or dislike my proposed timetable for 2009.

It looks like I will have 3 dot periods in a row next year (on a Thursday) which will make life a little difficult. The saving grace is that most of my class sizes are around 15 with taking the level three year 11 classes, modelling in year 12 and some of the alternate eduation kids. Only my year 10 class bodes to be larger than 17.

This time only 3 of my five classes are first timers rather than all five (I've taught year 10's and modelling before). The level 3 classes and the alternate education kids have the potential to draw a lot of time.

So far I'm one period down, which will make me a lightening rod for reliefs too. At least I'm not teaching out of area!

Upper 10's and exam preparation

My upper tens have begun revision for their exam. I've pushed them over the last week, getting them to complete twice as much work as my normal expectation per class as a prelude to expectations in year 11. Next I wanted them to experience the expectation of exams and model how to study in year 11. I set aside four days for them to revise and they did the following:

Day 1: Identification of material that could be in the exam
Day 2: Finding of resources (where in texts, portfolios) and identification of areas of weakness
Day 3: Preparation of notes / judging detail required via marks allocated
Day 4: Review and identification of material to study over weekend

Key findings
Students tend not to use contents pages in texts very well. Even when given words to find (solve, substitute, trigonometry) they can't find where in the text to examine. Students also need to consider that higher order questions are more likely to be in the exam therefore they should make sure they can answer questions at the end of exercises.

It was interesting to note levels of anxiety from the beginning (a couple of students with high levels) and the anxiety decreasing as they realised that they had mastered most of the content already. I look forward to next year when I can show them the progress made from year 9 to 10 by showing them their own year 9 exams and comparing it to their year 10 exam.

Students can now produce notes quite well and after a year of reinforcement they can see how to both use and benefit from them.

Students are starting to see that exam hints can be used to improve marks and how necessary it is to follow up and fix any areas of weakness.

Probability & statistics

I have running battle with my lower tens getting them started but found that a game of craypots generally helped. I've bought some foam cubes from Clark rubber (2 for $3.50) 125cm x 125cm x 125cm.

I've labelled the sides storm, storm, last, fine, fine, fine in red marker so that the students can see what the weather is and decide how to play their craypots to maximise the chance of profit.

It's been great to see students find the maximum possible value, the quickest way to get out of the game (eg. the riskiest route), see their mental mathematics in action and their ability to follow a process. It's a game that's easy to set up (it needs a worksheet and a teacher with dice) and one where the class highscore can be kept to be aimed for. Here's a copy of the online version. When I remember I'll bring home my worksheet for upload.

A fair few other dice games can be found here.

It's also a bit of fun to throw the foam cubes around the room near students not paying attention.

Diagnostics

Sometimes teaching year 10's you get so caught up in your lesson you miss the bloody obvious.

Today we were looking at distance from origin, Perth to Meekatharra, Perth to Newman and so forth using trip counters and odometers. My students were finding it hard to find the distance between two locations (I forget the real distances so please excuse my made up ones)..

They could understand that the 259km and the 1600km indicated distances from Perth but could not understand how to determine the distance between Meekatharra and Newman. So I tried something similar but with smaller numbers.

When asked the distance between Mary's house and the shops, this time students were able to say 1km. I asked them how they did that and they gave me the answer 9+1=10. I asked if they could think of another way. A sea of blank looks from the new kids in the class. I think to myself OMG. No wonder these kids couldn't do the previous problem. I showed them that subtraction could work and they were able to complete the exercise finding distances between various destinations. By the end it was obvious that they could learn the pattern for solving the problems but did not have an understanding of the underlying maths - thus had forgotten the method by the next problem.

So I went a bit further and used my favourite diagnostic tool. I wrote the following on their page.

10
99
109
119
1109
1229

And asked them to add one to each number. Thankfully only one student couldn't do this. Then I write 10526 and 52679 at the end of the list and moved to each asking them to say the numbers quietly to me. This was not as successful. This was the trigger to split the class (again) and start these kids on a different programme of work.

The integration of low ability or underschooled students into the mainstream and removal of dedicated remedial classes has meant that many students are falling through the cracks. It is no wonder that some of these kids can be disruptive, disheartened and are having confidence issues. Students that cannot read numbers above 100, can't identify simple operations or understand place value have little chance in mainstream classes. It's enough to make you physically upset and reminds me why I originally intended teaching year 7 rather than senior secondary. The impact of intervention is greater at an earlier age than now.

I have many of these kids next year and have identified areas of focus, we're designing entry tests for new students and aim to improve numeracy with low performing students. It's frustrating the amount of time that it takes to find these issues and detect the underlying factors behind avoidant behaviour. Time will only tell how much success can be made of it.

Reflecting on streaming

We streamed the year ten mathematics classes this year. To our mind it was a success. Other learning areas had heterogeneous classes and had a lot of difficulties with behaviour management issues that streaming helped us avoid. We have our 3A, 2C, 2A and 1B classes for 2009 where many doubted whether they would occur.

The success that we have had has put some pressure on the lower school to stream classes to better cater to our more capable students. There has been some regrouping in lower school classes and teachers have reported improvements in the ability to teach mathematics topics.

The start of the Maths/English streaming debate started this week. Should we stream on mathematics results or English results? Being a mathematics teacher, to my mind it requires little consideration. English teachers on the whole don't want to stream - after all English is a subject that lends itself to the heterogeneous approach - an essay can be assessed on many levels. Mathematics on the other hand tends to be hierarchical with a concept impossible to learn without the building blocks before it. Therefore stream on mathematics.

It doesn't need to be that black and white either. Some clever timetabling was done for us and now Maths, English, SOSE, Science can use a good compromise. We have grouped all yr 10 students into two bands, an upper ability band (class A & B) and a lower ability band (class C&D). A&B's are timetabled at the same time and C&D's are timetabled at the same time.

So for the situation above, in the first example maths students in period 1 are streamed into four classes (movement of students between A&B or C&D can be done freely for each learning area). In period 2 for English, the upper ability group is mixed into two classes and the students from C&D are mixed.

The main issue occurs when students in the C are not streamed correctly (eg. maturity raises their output, students are not assessed correctly etc.) and need to be promoted to the upper ability block. This involves the changing of many classes. All four areas have to be flexible in the promotion of students and the consideration of who can move to the upper band. We try to avoid movement by setting entrance tests before the start of the year and re-examining students after four weeks at the start of term 1. New students are to sit the tests before entering an ability block. The main advantage is the reduction of the level of teaching diversity required - there is less gap between the top and bottom student in each class.

It is not perfect as students may not settle into classes as well as they constantly encounter different student configurations (as typically happens with options classes).

Furthermore, it would be interesting to know if our success would have been the same if students had been streamed in all classes. Maybe the novelty of the streaming process is a factor in the success itself.

Practical application of effective leadership

Today we wrote the test to help us stream the year 9's into year 10 classes for next year effectively. We decided on two one hour exams testing number facts & space/Measurement then Algebra and problem solving.

If I was to write the two exams, I'd have to sit with the outcomes, read half a dozen books to find questions that adequately test the outcomes, do the test, record how long it would take me, estimate how long it will take students, re-evaluate the order of the questions, write a marking key, ensure that the test adequately covers the material originally intended and then two days has passed with little sleep.

or..

Our curriculum leader wrote the exam on a bit of paper off the top of his head, completed the answer key, allocated marks, I typed it up and it was done in two hours. Looking at it, it is far better than what I could have done alone. I worry that the difficulty level is a little high but I am happy to see what happens. If the students underachieve it will be easier to show their progress by the end of year 10 next year.

Experience always tells. There is no doubt, when writing the exam, in the instant between brain and hand, he had done all the things that I would have had to do; and even if I had written the exam knowing that our curriculum leader would look over it and make suggestions is a relief and takes away some of the pressure. Having someone you respect looking over things can make the difference between a new idea being accepted or rejected out of hand. That level of support and challenge is so necessary in your early years.

It is more than just experience though.

We have come up with a heap of hair brained schemes that you could see he doubted would be effective, but rather than dismissing them out of hand, he let us try. Sure enough, some of them had limited effect, but others have helped us understand the students better (like morning classes), others helped organise classes more efficiently (using more common assessment tasks and assessment schedules), others aim to assist students next year (like regrouping the 10's into their COS classes in term 4), helping us by bailing us out of duty when we have over committed (and need extra time to spend with students) and support our school wide initiatives (like a marks book for all classes or detailed programmes for junior school).

Earlier in the year we had moderation and intervention was needed to make sure the material we were presenting was demonstrative of our performance. The material I had prepared was inadequate and he suggested a number of things we had to do with presenting assessment prior to moderation, fixed the issues and our course was judged spot on.

Experienced staff commonly know who to talk to, what procedure is required, how long something will take for approval and what shortcuts are possible. They can save embarrassment from suggesting an idea that has been tried and failed or an idea that is unsuitable for a number of unthought of reasons.

Experienced staff tend to know what resources work and can lay their hands on them - in maths this is especially true with logic puzzle/investigation/problem solving activities that are hard to source.

Our curriculum leader likes to come into my room and takes great pleasure in finding mistakes with my board work (not one of my favourite habits) - but... I'd rather someone that had a clue than someone that didn't care enough to check that the senior school teachers are doing the right thing.

So to sum it up.. good leaders have superior content sequencing & resource knowledge (expert power), be willing to intervene where required and advocate to senior management for junior staff (management expertise), have respect of fellow staff (be charismatic), understand process (administrative expertise) and encourage risk taking (be entrepreneurial).

Developing problem solving, reading and comprehension skills

My little challenging group of year 10's can be quite difficult to engage at times. A real issue with their maths is getting them to read the question effectively. Completely out of character, they have loved exercises in the book Logic Mysteries by Jane Molnar. Although it misleadingly states grade 3-5 on the cover, the year 10 students have loved the idea of reading these problems and solving them. When I first introduced it, I abandoned the rest of my normal planned lesson as I had not seen these students this enthused and engaged since the algebra topic.

Each mystery has a story and is solved by eradicating options that do not exist. A grid is set up to record the findings as they go through the mystery.

Many great mathematical concepts can be investigated. For instance complementary events become obvious, if she has a bird - all the options in the bird column that are boys can be eradicated. Inequalities can be investigated through clues like Jane's age is less than Mary's. Sets can be investigated through concepts like Mary's item fits in a school bag... and so on..

The main thing is that it requires the students to read the clues that are not necessarily in order, requiring reading and re-reading until they are all done.

A similar book Quizzles or More Quizzles by Wayne Williams has proven very successful with my upper class of year tens. These logic puzzles are multi dimensional and can be quite difficult so be warned!

For these to be successful I invited students to attempt them themselves for 5 minutes then modelled how to complete a problem. Then the following day I gave another problem at the start of class.

Either way, improving comprehension and reading ability is more and more important in mathematics (the temptation to enter into a diatribe as to why we need to teach English in maths here is near on irresistible - I shall try though!). These three books have been some of the more enjoyable methods of developing literacy skills thus far.

Ethical reasoning and streaming

Streaming is a difficult topic as it raises a number of questions regarding student capture, teacher judgement, assessment, and social justice.

Student capture for me is the most critical aspect of a classroom. Capturing a student's interest is a perpetual task, a combination of selling your subject and moving fast enough to keep their interest, yet slowly enough to allow them to fully grasp a subject. For some it can be done through connections with the teacher's personality, others through mathematics success, others through contribution to the class and others by connections with peers. If you can capture a student and get them to consistently have a positive attitude towards your subject then this is the first criteria met for a student to be placed into a difficult mathematics class. Streaming captured students into a class can greatly assist in improving possibility for success.

Teacher judgement is the next criteria. Does the student have the intellectual horsepower to complete the work? No amount of mathematics tutoring will assist a student that has extreme difficulty in reading a question, has too many holes in their skill base or takes too long to understand a new skill. A teacher needs to be able to identify that bit of extra practice that will move the student from being able to use a skill when directed, to be able to apply a skill undirected, to be able to identify the right skill from a range of available skills. It is possible that having to continuously assist a student on a continuous basis will destroy the flow of a class and disadvantage all within it especially in upper range classes.

Assessment is the next criteria. Assessment supports teacher judgement not the other way around. To stream purely on assessment is a recipe for disaster. This is especially true for students riding the end of their ability curve and coasting or loafing. These students, when they hit the wall and finally need to study can be hurt, confused and looking for those to blame for their lack of performance. If these students have not been properly coached before the 'big drop' in results, they can drop morale in a class at a rapid rate. Sometimes (especially in this case or the case where students are having external difficulties) it is best to ignore assessment and use the first two criteria to stream students.

Social justice is the final criteria and it has to be very carefully applied. An injudicious use of social justice to students when streaming will produce weak streams and deprecate the benefits of streaming. Just because a student has a legitimate reason for underperforming does not mean that in time a student will perform. Some say that streaming a class is a social justice issue in itself but watching students being unable to complete work that the rest of the class is working on and suffering self esteem issues or dumbing a class down to the lowest common denominator is not a solution to my mind.

The hardest part of establishing a stream is that it is not an exact science. A student performing at an optimum level with one teacher may not perform at all with another (this is especially true with boys). When creating a stream (especially in small class sizes) team dynamics play a large part - if you can create a team of peers and the teacher anything is possible. It's why I think traditionally the upper classes have been sought after - despite requiring the most skill to make work - they are the ones where there has been most flexibility in construction.

Changing role of senior school

Senior school, years 10,11,12 have traditionally been the home of the most experienced teachers. These teachers generally have a vast amount of experience that is tapped from time to time by other teachers when need arises, either in behaviour management, content knowledge and generally are aware of how things work, what has been tried before and how to get things done. They have the experience to guide our students through to TEE, university entry or into VET pathways where necessary.

Now I say this as an observer (as I am neither experienced, nor the most capable in senior school). I have no ambitions for a HoD role and actively promote the idea that the HoD should teach the most capable class and other senior school teachers should do an apprenticeship of sorts with mid range classes to hone technique and pedagogy first. ... and I enjoy classroom teaching too much to get involved with the admin required to do the job properly.

Somewhere along the line I think we have lost track of what senior school teachers bring to the school. We have lost our heads of department in Mathematics/English/SoSE/Science to other areas such as literacy experts and careers guidance, L3 adminstrative roles. Responsibility now for the performance of learning areas has fallen to those incapable of measuring success or failure as they may not have ever taught the subject.

An issue that is currently rising is the lack of time to complete yr 12 COS in time for the TEE exams. With 1 term lost to the exam process, it leaves only 16 weeks per semester to complete the course. A possible way to increase the amount of teaching time for COS is to use term 4 year 10 to start the COS process and to start the year 12 course a term early.

Staffing of this is a real issue because if a unit starts in term four, few teachers are willing to take on an overloaded teaching schedule to make this happen. At this time of year the temptation arises to utilise senior school staff to fulfil this role as in many cases they will be teaching these students in the following years anyway.

I think we need to resist this happening especially for our HoD's. If our best and most capable are not given unallocated time to identify and remedy issues within learning areas it is only likely that over time things will get worse. The time that they put into improving staff ability and student output is clearly underestimated and is not being adequately nurtured. It would be good to see the complete opposite occur and HoD's given the time, recognition, responsibility and pay to make things happen.

Class size & the concept of 'Intervention Time'

I have heard many times that reduced class size is not a factor in learning or that it has minimal effect. Reduced class sizes is not the panacea to improved student learning but it is a handy tool when used correctly. To have an early intervention strategy there must be adequate class time for intervention.

If you have a high performing class of motivated students (with 3 levels between the top and bottom performing students), class sizes of around thirty in year ten can be managed. You would need to use a fair amount of skill to keep them motivated as after instruction and settling time (say 30 mins per class, two blocks of instruction, h/w and pack up) it would be hard to get to every student every class to identify issues, correct them (say 1 minute of intervention time per student per class) and maintain their learning inertia. You would more reliant on picking up issues during the homework, quiz, revision, assessment and corrections teaching cycle and complete more marking out of class.

In a mid performing class (with four levels between the top and bottom performing students) with around 20 students and 20 minutes of instruction and settling time you could get to each student twice (eg. average of 2 mins of intervention time per lesson). This seems feasible.

If you have a low performing student group in mathematics (with five levels between the top and bottom performing students), I would say that a class of 30 is lunacy (there are usually valid and disparate reasons why students are this far behind) and would send the best teachers barmy. Under normal circumstances in these types of classes there are not enough corners in the room to separate disruptive students. Each student in a class of that type requires constant attention to fully enjoy and appreciate mathematics. For example in my class, one student required behavioural attention once every 3 minutes (I timed him), each time requiring further attention to settle him. In a class of thirty that would make teaching nigh near impossible. For a class of this type it is preferable to have intervention time around 3-4 minutes per student, limiting class sizes to 13-16 students. This size of class would also promote more collaborative work, especially if other teachers are willing to assist during their DOTT or if a T/A is available.

In practice each student does not need (or get) an individual minute of your time and is normally able to do their work without individual intervention through the teacher identifying classwide issues and modifying instructional techniques (eg. more modelling), by using peer assistance, having effective instructional notes, by increasing participation in after class discussion or by bringing groups of students back to the board. What the intervention time model does is provide a benchmark of performance and can help identify structural issues vs teaching issues with classes that are clearly not working.

Using a model of this nature we could measure the learning capacity of student groups (by creating class sizes and monitoring teaching/intervention/disruption time) and the approximate class sizes required to teach them optimally. This has the potential to greatly assist in designing and justifying appropriate class sizes for our students.

Measuring Teacher Performance

I stumbled upon this article from 1999 stating clearly issues raised by teachers regarding student performance in primary and secondary schools. It is just as relevant today as it was then. This shows a number of areas of difficulty measuring teacher performance. I have highlighted some of the areas of student performance impacted by teachers. Many carry through to high school from primary.

I grouped the results into behavioural (primarily learned behaviours brought to the classroom), genetics, environmental (factors with limited control by the teacher), structural (constraints imposed on a classroom) and societal factors to isolate factors solely controlled by teachers within the classroom. Pedagogy(teaching methods), content knowledge are the two major factors teachers contributing to teaching students.

Primary

• students who are not doing well tend to give up, refuse to try, and this makes the problem worse - this behaviour gets worse as they get older and they start to compare their work with those of other students (behavioural)
• high achieving students can taunt low achievers and this makes the problems worse
  students with psychological problems (eg, trauma experienced in the home) have trouble learning (behavioural)
• sometimes teachers can’t work out why students can’t learn - it can be the problem of the teacher who hasn’t worked out how to engage students (getting inside the walnut) (pedagogy)
• parents refuse to have their children placed in classes for students who have intellectual disabilities (structural)
• students lack academic ability (genetics/environmental)
• teachers don’t explain concepts clearly (pedagogy/content knowledge)
• parents indulge their children so they won’t pay attention in class (societal)
• parents don’t take an interest in children’s school work (societal)
• students are transient and so miss a lot of school (societal)
• it’s more difficult these days to get students placed in classes for students with intellectual disabilities there are children with attention deficit disorder who have difficulty concentrating in class (structural)

Secondary

• students haven’t been well taught in earlier years at school (historical)
• students don’t value school work (behavioural/societal)
• parents don’t value their children’s school work (societal)
• students lack ability (genetics/environmental)
• the system allows students to progress through grades without passing subjects (structural)
• maturational level - students mature at different rates - they may not be able to grasp concepts now but they could in a couple of years’ time (genetic/environmental)
• poor teaching (pedagogy/content knowledge)
• teachers blame the students for poor performance when it’s the teachers’ fault (pedagogy)
• students have psychological problems because of unhappy home lives (environmental)
• teachers don’t have a good mathematics background (pedagogy/content knowledge/structural)
• students’ poor behaviour in class means they don’t pay attention to the work - discipline problems in schools are on the rise - it’s part of wider societal problems (behavioural/structural/societal)
• students lack self discipline - they’re not prepared to work (behavioural)

It is clear to see that student performance is a poor measure of teaching ability as many other factors exist to influence this criteria. To blame teachers for poor performance of students based purely on teacher pedagogy (teaching methods) or lack of knowledge of content ignores a host of other possible reasons.

Passion, student behaviour and being fiery

One of the issues in classes today that stems from the home is that students have trouble accepting that a teacher has authority in the classroom. At home they argue with parents in a very democratic fashion. Students believe (wholeheartedly) that they have a right of reply to any misconception that they face.

I must admit this gets me fired up especially in my 'A' class. Any student willing to take responsibility for the care, nurture, learning needs and welfare of thirty students, get a degree as a minimum requirement for teaching can have my job if they can prove they would do it better. Until they do this, if I ask a student to be quiet or stand in the hall, see the team leader, copy off the board or attempt a question they may believe they can't do, I expect them to attempt to follow my expectation.

They will fail sometimes, and this is ok. This does not give them a right to argue and waste teaching time. It should prompt some introspection as to why they didn't understand how to do it and hopefully seek assistance from friends, pay more attention when solutions are put on the board or seek assistance at an opportune moment during class or after class. Maybe it would be a good idea to get them to journal why they have had such trouble understanding a concept and identify ways they could better understand a topic. Bringing the correct materials to class (eg. CAS calculators, pens, paper, texts), paying attention during instruction, fostering friendships with those that do understand, reading their notes (and keeping them in a place they can be used) - attending school regularly (my favourite) and catching up after sickness may be a good start.

These students do not have a right to insist on help at a time that suits them. To use a claim for help to justify poor or avoidant behaviour is not acceptable. I would love to be able to provide just-in-time intervention to every student all of the time. In a class of thirty it just is not possible. The belief that getting instant help is a right is infuriating and I don't know where it is being fostered. Maybe I should enquire into how many are only children (and thus do not have to compete for attention) and also examine my own methods of helping during practice time (maybe I am a contributer to the problem!).

When instructed on where their actions are errant I expect nothing less than silence especially with those talking during teaching time - this is done in the hall outside my room. Try my patience and half the school hears. It's fun watching them open their mouth and then hear them "but you won't let me talk to explain". If you talk during my teaching time and I have to stop the only thing I wish to hear is I'm sorry and then see an end to that behavior. Woebetide the student that interupts me again. My other students have the right to learn and it must be protected.

There must be a line between teacher expectation and student behaviour. There must be a consequence if this is crossed. A lecture, for many of my kids is enough to get the message. If they get the message, no further consequence. If it continues -they start the path to BMIS.

The argumentative nature of students at correct times needs to be fostered (we don't want meek students) - but it must be cultivated with manners and knowledge that there is a time and place to discuss the finer points of an issue. I always offer time after class for extra assistance and am happy to discuss any issues or problems from a class at this time. Funnily enough rarely is this offer taken up by these students during lunch or their own time.

Creating inspirational students

Students aren't born inspirational. They're born rather podgy blobs that whinge a lot... Some never change...

This week I spent a bit of time reminding my year 10's that they are inspirational. Lower school students look to them for cues on how to behave, on determining what is important and setting the tone within the school. If they want a happy school - be happy. If they want a school with a million rules - do stupid things. If they want a school based on success of students, show the lower years that our school can perform at a high level.

For this I think it is important that we create opportunities for them to be successful and protect those that foster these activities. It might be taking an interest in a student that is doing an afterschool ESL class, or not getting grumpy with the dance teacher that is taking students out of class for a recital, being supportive of the physical education staff and their events, supporting SOSE excursions by providing extra supervisor bodies or helping out with relief classes.

I think it also means looking for information that might help inspire kids. I recently found two books by the actress that played Winnie on the Wonder Years (Kevin's girlfriend for those of you ancient enough to remember). One is called 'Math doesn't suck' and the other is 'Kiss my Math'. The books themselves may be just the thing to get a student going and get them to believe that you care about how they think. The maths is a bit dodgy in places ('Highest common factor' becomes 'greatest crush factor') but it has a go at making maths pop culture ready and that's a good thing.

Another bit of success I've had is to let them into my life a little. Last class we created tally tables on the best baby name that we had selected. Next time I'll have a silent poll as it was a case of many just following the leader. Maybe this is a discussion in itself. We've also used my history to investigate stocks, examine salary ranges and evaluate priorities on what is important in life.

Another opportunity has been with my guitar. I am worse than hopeless, but the kids see that I am still learning well beyond school.

Lastly whenever a leadership event occurs I draw their attention to it and suggest that they pay heed to things done well or poorly as they will soon be in that position. If they can learn good leadership habits now, they will be in better stead going forward.

Casio Classpad 330, Finding the mean and missing values

I posed the following question to my year 10's in order to continue our learning of the new calculator. It is an example of solving a problem where the mean is known but a value in the sample is not.

"Q: A class had 5 students. Student results in the last test was {50,56,64,72,81}. Isabella joined the class and the new mean became 68. Did Isabella score higher than the old mean and what was her score?"

H: If the mean of {50,56,64,72,81} is less than 68 then Isabella has scored higher as a higher score by Isabella will raise the mean. Since we know the new mean (68) we can work out Isabella's score by working backwards.

Set up a working pane with a main application and a list editor. Title a column 'list1'. Add the 5 student results to the list editor.

Click in the main application and type mean(list1) using the soft keyboard. Hit the blue exe button. The answer is 64.6 .
A: The old mean 64.6 is less than 68 therefore Isabella has scored higher.

To find Isabella's score click in the list editor and tap the next empty cell in list1. Press the x button. Click in the main application pane on the line that says mean(list1). Press the blue exe button.

This will return a sum to work out the mean of the list for values of and value of x i.e. (x+323)/6.

As we know the new mean alter the first line to read mean(list1)=68. Highlight the solution sum and tap Edit in the menu bar and then Copy. Paste the sum on the next line in the main application pane. Highlight the sum, tap Interactive on the menu bar, then tap Advanced on the sub menu and then tap solve. Tap ok at the base of the dialog box. The answer is x=85.

A: Isabella's test score was 85.

Click here for other CAS calculator articles

Revisiting fractions

My 10D class has revisited fractions over the last week. For many fractions is like another language others have managed it in the past but have forgotten basic principles. The sequence I have used leading up to percentages of amounts is as follows

Drawing and identifying numerators and denominators
First exercise was identifying a variety of numerical fractions from pictorial form and then constructing pictorial fractions from numerical forms. We spent a lot of time looking at mixed numerals and converting between mixed numerals and improper fractions using pictorial means.
eg. for 3 2/3: draw 3 lots of 3 boxes with all boxes coloured and 1 lot of 3 boxes with two boxes coloured. When students counted the coloured boxes they had 11/3.

Investigating fractions of amounts
It seemed strange to do this here, but funnily enough it worked well as it established relevancy of the topic for many students. We started with a problem 3/4 of $24 is to be given to John and 1/4 to Mary.
I explained it as:
3/4 of 24 is: $6 per part (24/4)
I drew a box and split it into 4 equal parts (drawing attention to the denominator)
I put $6 in each box.
I coloured in three sections that represented John's portion
then counted $6 x 3 parts = $18 for John

I then repeated the same steps for Mary
1/4 of 24 is: $6 per part (24/4) then $6 x 1 part = $6 for Mary

We checked our answer to ensure all the money had been accounted for ($18+$6=$24). Students then completed a number of examples.

Investigating multiples and factors & Equivalent fractions
Next day we looked at multiples and factors. I explained this through examples, showing them examples of multiples and factors, then getting them to find the first five multiples for 2,3,4,7 and then the first five multiples for 2,3,5,7 over 100. After this they found factors of 10, 15, 24 and 42. We investigated patterns in factors (none greater than 1/2 the original valure other than itself, how it helped knowing your tables, factor pairs, 2 is always a factor for even numbers)

Students were then given a fraction wall and identified equivalent fractions in preparation for adding and subtracting fractions. The idea was put forward that fractions rely on parts to be equal otherwise the idea of equivalency would not be able to be used.

Adding and subtracting fractions
In the third lesson we looked at the problem of 1/3 + 1/2 using paper strips. The aim was to establish why equal parts is essential to an understanding of fractions. We used our fraction wall to look for equivalent fractions that allow us to add equal parts. After a few pictorial examples I started to show students how to use multiples and factors to assist in finding common denominators.

Next lesson we look at multiplying fractions...

Recharging students for success in mathematics

Being in a low socio-economic school sometimes is disheartening. The students don't believe that they are able to achieve academically. This is reinforced by parents, teachers and the school in subtle ways throughout the year.

A parent complains that the student is only doing lower maths and does not need a $175 calculator. The timetable allows many non-TEE subject to run, but only a few TEE subject selections are available. Portfolio entry is seen as a primary pathway to university rather than a backdoor entry for extreme cases. Lower school programmes lack the rigour of programmes in more academic schools. A single student or groups of students can disrupt classrooms for an entire year, but little coordinated effort can be made to limit the damage being caused. The idea of secondary graduation is diminished by the idea that 'anyone' can graduate. Cohorts of students are labelled challenging and good students lose opportunities as classes are aimed to manage the lower students and keep them engaged to detriment of academic achievement by top students.

Charging academic students for success is a mentality that must be driven - it doesn't just happen. Kids need to be told that they have the ability to succeed, shown possible outcomes, be given opportunity to try/fail/succeed and be mentored as they go along. Setting clear standards sets the groundwork for success.

Things that I consider serious issues in my A class:

• Not being quiet and ready to start work within 2 minutes of entering the room
• Being late for class and not entering the room quietly
• Complaining, whining and whinging before attempting work
• Not paying attention when instruction is given
• Relying on friends or personal attention of the teacher for instruction rather than some level of personal investigation
• Not attempting homework
• Failing a test or assignment ( lower than 1 standard deviation from mean)
• Not seeking assistance when required

Students that continuously fall into these issues risk demotion to BCD classes. For some, demotion is the right option, for others the motivation to be moved down is enough for them to alter negative behaviours. For a relative few, it identifies students with ability but are unlikely to succeed at TEE level. This year, boys in particular have been a real issue and a focus for the course next year (I think this is the most significant issue at our school).

Things that I do to promote positive attitudes towards mathematics and address issues:

• Look for opportunities to congratulate students on achievement
• Attempt to talk to each student each class
• Allow friendship groups to remain together only when learning is occurring
• Ensure that new topics include new material
• Promote the A class as being a privilege and a responsibility
• Reinforce that attitude is as important as aptitude
• Change the difficulty level regularly to allow for opportunities for success/failure and stretching of the mind.
• Question their own beliefs of their ability and remind them of progress made
• Use personal experiences to enhance class material
• Focus the basis of enjoyment in mathematics in achievement rather than entertainment by the teacher (though the converse may be more important in lower classes)
• Encourage students to self monitor behaviour and provide peer feedback
• Create opportunities for students to see the different rapport with yr 11/12 TEE students than with yr 10 students

Casio Classpad 330, Creating a Histogram

Today in class we looked at how to produce a Histogram using the list editor. A Histogram is used when data is continuous (there is no gap between intervals).

Class interval (Frequency)
0 <= x <>=80 (1)

Tap in the list editor. Tap Edit in the menu bar. Tap Clear All. and tap Ok in the dialog box. If a graph is open tap the StatGraph pane to select it. Tap the cross in the top corner of the window to remove the graph.

Name a column in the list editor ‘classmid’using the soft keyboard. Put the midpoint of each class into the classmid column. eg. {5,15, 25, .., 85} (make sure you name the column before putting the data in!).

Name a column in the list editor ‘freq’ using the soft keyboard. Add each corresponding frequency into the freq column. eg. {3,10,16,..,1}.

Tap SetGraph in the menu bar. Tap Setting. Select Histogram in the Type dropdown, select classmid in the XList dropdown and freq in the Freq dropdown. Make sure the Draw option is on. Tap Set at the base of the dialog box.

Tap the StatGraph icon in the icon bar to display the graph. Make HStart 5 (midpoint of first interval) and HStep 10(size of intervals).

A Histogram will appear. Tap the StatGraph pane and then tap Analysis in the menu bar. Tap Trace in the menu.

A flashing crosshair should appear above the first column of the graph. Use the blue cursor key to navigate column values in the graph. You can use these values to create your histogram on graph paper. The xc at the base of the graph are horizontal axis values and the Fc are your vertical axis values.

viola!

Other educationWA articles on CAS calculators
How to navigate through menus (what's a menu bar?) Click here
How to create a list (what's a list editor??) Click here

Here's a link to an index of other CAS calculator posts.

Casio Classpad, day 1 with students

As I play with the calculator things become a little more obvious. It was good fun with my year 10's showing them how to find the mean of

S:{10,12,13,14,15}

with the CAS calculator during p5 on a 35°C day and then set Maths for WA3 10C with 50 items in the sample. I was upfront in saying to my students that learning all the new content next year and learning how to use the calculator was a bad idea (lights went on... ahh, that's why I need to get one this year!!). For those students still unsure, I made them find the mean of a 50 item sample with their scientific calculators. They promised to buy a CAS calculator tomorrow.

Anyhow.. this is one way of finding the mean with the CAS calculator. There are many better ways but the idea was to learn how the calculator works (the picture is the end result).

Open a main application in the work pane.

1. The last icon in the tool bar should be a graph. Click the dropdown to the right of the graph. Tap the icon that looks like three columns in the sub menu. The list editor will open in the bottom pane below the main application.
2. We need to give our list a name. Tap the top of the first column. “list =” should appear at the base of the list editor.
3. Press the blue Keyboard button. The list editor will temporarily move to the top pane. The soft keyboard will appear in the bottom work pane.
4. There are four tabs in the soft keyboard. Tap the abc tab with the stylus. A qwerty keyboard should appear. Name the first column in the list editor ‘list1’ if it is not already. You may need to click again in the list editor list= textbox first.
5. Press blue Keyboard to get rid of the soft keyboard. The main application should reappear in the top pane and the list editor in the bottom pane
6. Use the stylus, tap the first cell in list1.
7. Using the number keys press 10 then exe (bottom right hand corner of the keypad). This should put the first number in the list. Not that the cursor has dropped to the next item in the list without having to use the stylus. Now enter 12 then exe. Your list should now have two entries. Add the remaining entries.
8. Click in the main application. Raise the soft keyboard with the blue Keyboard button. Open the abc tab and type list1 and press exe. {10,12,13,14,15} should appear.
9. Click Action in the menu bar and tap List-Calculation. Tap mean from the options provided. 'mean(' should appear in the main application.
10. Complete the action by typing ‘list1’ using the soft keyboard and the button ‘)’. You should now have ‘mean(list1)’ displayed. Press exe. The answer 64/5 will appear. To get a decimal representation, highlight ‘64/5’ with the stylus and click the first icon in the icon bar.

viola. You should be able to finish the tutorial by finding the median yourself. (An alternate way is to type list1, highlight it, tap the Interactive item in the menu bar, tap list calculation in the sub menu and then median and then select ok at the base of the dialog box.) You could also use statistics mode (tap Main on the icon bar, then tap Statistics.) The Statistics application is very similar in structure to the stats mode on the fx graphics calculator).

Here's a link to my last article on learning how to use a CAS calculator.
Here's a link to an index of other CAS calculator posts.

My Casio Classpad 330 Journey

Second weekend playing with the calculator.

When I was doing phone support often I could not see what the person on the other end was doing. I became quite adept at directing customers on quite difficult tasks blind. The most important thing to do was to adequately define things up front.

With the CAS calculator the windowing system can be quite confusing at first. It is important to name things in such a way that students can listen to your direction and follow it, rather than needing snapshots all of the time.



In the worksheets I have created, the calculator is divided into the screen and the buttons. The screen in divided into the menu bar, the tool bar, the work pane, the status bar and the icon panel. The buttons are blue, grey and black.

i.e.
The Edit, Action, Interactive text at the top is the menu bar
The icons underneath the menu bar is the tool bar
The area underneath is the work pane, it can be split into the top pane and the bottom pane. The work pane is currently filled with the main application.
The bit beneath the work pane (eg. Alg, Standard, Real, Deg, battery indicator) is the status bar.
The stylus and buttons are used to enter data and operations into the calculator.

Here's a link to the "How do I.. ???? on a Casio Classpad" book that I have been using.
Here's a link back to my first article on CAS calculators
Here's a link to an index of other CAS calculator posts.