Index laws and the lower ability group

It always surprises me what will work with a low ability group and what will not. Generally you have to hit the ability level spot on for the whole group for them to be able to grasp a concept (even if only momentarily).

Take indices. On one day grasping 3x3 = 3 squared = 9 was impossible, not to mention any attempts at 3^2 x 3^3 = 3^5. We went through a number of examples and by the end of the lesson I had 10 bored students and had lost half of my hair.

The next day I took a different investigative approach. This might be obvious to an experienced teacher but was fairly radical to me.

Sequence (imagine that ^3 is written as 3 superscripted):
a) Discuss nomenclature with notes (base, index, indicies, power, factor and power form)
b) Use calculator to evaluate single term powers - eg. 3^3 = ...
c) Add multiplying powers to the board (with positive index) - eg. 3^3 x 3^5 = ...
e) Look for a pattern in the numbers - supply the base after a few minutes.
f) Explain how multiplying powers works and supply notes including general forms
g) Rub off the answers, write as index addition - eg. 3^3 x 3^5 = 3^...+... = 3^8 = 6561
h) Add dividing powers to the board where answers are positive > 0 - eg. 3^5 ÷ 3^2 = ...
i) Look for a pattern in the numbers - supply the base after a few minutes.
j) Explain dividing powers, supply notes including general form (as ÷ and fraction)
k) Rub off the answers, write as index subtraction - eg. 3^5 ÷ 3^3 = 3^...-... = 3^2 = 9
l) Supply mixed problems

About 40 mins. I don't think I could have done this investigating factored form with these students as regrouping and cancelling bored them silly the previous day (I will revisit it later though). Using calculators to do the sum and examine the sum backwards worked far better. Special note was made from e) onwards about checking for same base, superscripting properly, neatness, identifying operator used in original sum and always referring back to general form to make sure the correct index operation is being done. By the end of class all 5 students were engaged and had grasped the concepts involved. yay!

Now some may ask 'why do index laws with a low ability group in yr 10?'. I suppose it is a philosophy problem put in place at uni. Students shouldn't have impoverished courses 'entertainment based/childcare oriented' purely because they are in a low ability group. If they could master simple algebra earlier in the year and ratios later in the year, I consider index laws and other more 'pure' maths well within their grasp even with behaviour difficulties. These students too should have the pleasure of mastering something that looks quite cool on paper, harder than they believe possible to learn and not feel inferior to peers when they walk into an upper school maths class.

Longitude and Latitude problems

We revisited longitude and latitude today just prior to a quick quiz on Monday of elevation/depression, longitude/latitude and bearings problems.

Firstly the section that we did on angles(complementary/supplementary) and bearings seems to have resolved many of the angle addition/subtraction problems students encounter in longitude and latitude (this makes sense). The second was that teaching how to solve latitude problems before longitude problems is far easier than vice-versa (this was not expected).

Sequence used:
a) nomenclature & uses - great circles:longitude+equator, small circles:latitude
b) how to draw latitude problems - uses of front vs side views
c) find the radius of small circles - r(small circle) = r(great circle) cos (longitude)
d) how to find the circumference of small circles - C(small circle) = 2 x pi x r(small circle)
e) how to identify the angle travelled - reading diagrams carefully & common errors
f) how to calculate the distance travelled - angle / 360 x C(small circle)

The reasoning for the second finding is that students were happy to learn new skills (eg. solve latitude problems)...

... students were even happier to be told that longitude problems were easier than latitude problems (eg. remove step c), draw the diagram a little differently - front view edge marked)

.. rather than teach longitude first and say that latitude problems were more difficult (as you need to add step c) and have to think harder about the diagram).

All in all, it can be taught in a lesson to a good group in year 10, maybe a bit longer if proving step c), with practice for a couple of sessions. I never would have identified the second finding if I hadn't chosen the wrong question as an example. Nothing like a random event to improve your teaching and give an insight into student thinking.

Different approaches to supporting teachers

It is interesting to view different teaching styles and different ways of supporting teachers. Support staff such as team leaders and HODs are important parts of school machinery. I've been to a couple of schools and have made a few observations about how these support staff operate.

Type 1: Student focused (Soft and mushy).
The soft and mushy students support person always has the student in the forefront of their mind. They are 'friends' of the student, listen to their grievances and reason through their issues. Issues with this person relate to support of students based on student reporting of incidents especially as students often only relate parts that they remember and conveniently forget negative portions. Positives of this type of support staff is that many issues can become non-issues without further teacher intervention and they can negotiate middle-grounds where teachers and students both feel POV is acknowledged and resolution has been achieved. They also have great rapports with parents. Generally supporters of BMIS management techniques.

Type 2: Authoritative (Firm and stern)
For me, the easiest support person to deal with as many issues are black and white. They will investigate, report and resolve issues. In most cases will lean to the side of the teacher. Negatives is that some students will feel that their POV is not given enough weighting. Positive is that student POV is not given weighting over teacher and holistic classroom position. Middle ground resolution tends to focus on end results and measurable future performance. Generally supporters of the "teacher is boss" type mentality.

Type 3: Administrative (Paper pusher)
This is the worst type to my mind as nothing gets resolved but all paperwork is in order. Chaos tends to follow a support person of this type and responsibility is delegated to anyone but the support person for ultimate resolution of an issue. For specific issues they are often unavailable as they are pursuing "pet" projects. They tend to be active self promoters. The main negative is the sheer amount of unresolved issues pushed back onto teachers - students are not sent to support staff unless an unresolvable issue within the classroom has been encountered and pushing unresolvable situations back to teachers does not help maintain programme momentum. The only positive is that major incidents have well recorded backgrounds. Generally supporters of give anyone the responsibility as long as it is not me.

As I think of more I will add them to the list.

Curriculum Council Moderation

Here are words that inspire fear in the most confident graduate teacher - "Moderation". We had ours and the one thing it showed was the need for experienced teachers to guide the less experienced teacher. Our most experienced teacher took the work of students, repackaged it to make it easy to be viewed and removed much of the stress.

I'm not saying that moderation was not stressful - the idea of redressing issues 3 weeks before the end of a course kept me up at night for a number of days. When it happened though, it was great to have recognised that the grading given was correct and also be given ideas about how to improve the course.

In my non-TEE course I now understand that they like to see context heavily developed into the course - coursework specially developed for the cohort. If this can be demonstrated in assignment pieces this can be a good thing.

Another insight was that moderators look at the intake as well as the results especially in TEE courses - to check if the students selected for courses has been done appropriately. It was interesting to hear of restrictions placed on student subject selection in other schools to (I imagine) protect school TEE rankings and student self esteem. As schools reduce the number of subjects on offer and students have reduced options, I wonder if we will see more issues in this area.

Our moderator liked to see cognitive type assessment. I shall seek out some more of this sort of thing (to be honest I have no idea what they are (isn't an investigation typically cognitive?) - but will ask around).

It shouldn't be underestimated the time it takes to prepare for moderation and the disruption it causes to other classes. I strongly suggest for portfolios of all assessment to be gathered and kept for all students in a class by the teacher. If you don't have these pre-done, prepare a couple of each for moderation or you may have your D turn into a C and have to find another portfolio at the last minute to fill the gap (...I wonder how I know this??).

Keeping portfolios from students is detrimental to test preparation (as you have the portfolios rather than students for study) but is far preferable than trying to gather materials in the lead up to moderation dates (especially as for us they fell just before the start of TEE mock exam preparation). After moderation dates portfolios can also be a great study tool for exams that careless students wouldn't have at the end of the year.

A quick hello!

Hello to those reading!! I quick thank you for the tripling of volume this month to the blog (300 unique visitors or so this month), the public and private comments I have received and the 25% of people that have been interested enough to have a good read and spend a few minutes with my ramblings.

Reflection is something that we all find hard to do regularly. Having it here helps keep up the motivation, I would heartily recommend anyone interested in blogging to have a go and start your own!

Rates and Ratios

Rates and ratios is one of those topics that I always have troubles teaching - especially to weak classes with poor fractions.

It is a large topic and covers many areas:

• parts of a whole (eg. 2 boys: 3 girls)
• parts of the whole (2 boys: 5 students total)
• scaling factors (1:100)
• equivalent ratios (2:4 as 1:2)
• conversion of fractions to ratios (2 blue:3 green = 2/5 blue)
• comparing unit values (1.25L of Coke @ $1.15 & 2L of Coke @ 2.35)
• unit conversions (dilute 1mL : 2L -> how much for 5L)

I don't think teaching this topic to 7's or 10's that I've established a sequence that enables students to grasp the concept easily. A few successful strategies have been corners and calculating ratios of m&m's and smarties. Mapping activities have been good (find ten cities in ten different countries and find the actual distance travelled) for large unit conversion activities. The grid activity where students grid a picture and then construct an enlarged grid and redraw the picture inside also works well.

I think next year when the topic is introduced I'll define ratios as a way of 'describing a groups of things' and then launch into proportions.

This is definately one that needs more attention on the programme and the sourcing of more resources for next year.

Education revolution - Bah humbug!

When someone uses revolution in a sentence, I usually put the following talk in the same category as others that include inclusive, flexible, robust and my all time favourite.. well... I try not to be cynical but election campaigns are always full of sensational headlines and little else when you dig a little deeper.

Here comes the usual garbage through my letterbox from state government politicians talking about greening the planet whilst cutting trees putting junk mail into my letterbox that goes direct to the bin. The latest junk being performance tables for schools.

I am not an advocate for school league tables. A better indicator for prospective parents is to speak to local parents about how students were treated by the school over the five year period. A school's reputation is its life blood. Conversely collating data and independent analysis privately reviewed is a great idea. Acting on these issues and seeking remedies over time in the public's best interest would be fantastic.

The only reason to release league tables is to accept that the government cannot act in the best interest of the public and the public needs access to the information to decide for itself. To accept this is to accept that the public has lost the faith in government. This is a direct indictment on the quality of our political leaders and public servants. One only has to look at how elections are won and lost - often on the actions of individuals rather than on their ability to govern. If parents had faith in government, the decision of parents would be geographical or financial as it has always been. Devolution of responsibility to schools is open acceptance that government cannot accept the responsibility/risk associated with governing schools. They are not capable or willing to remedy issues and are relying on market forces to do it for them. This is the heart of the safety net path I discussed in an earlier post (we'll take them if no one else will).

There are many reasons for poor results and to release the data with no analysis of why it has happened is not fair on a school. A school has poor results for many reasons - poor teaching, demographic change, a change in leadership, weak leadership, a lack of experienced staff, a large number of inexperienced staff, behavioural issues with specific students/classes, resourcing, a weak cohort, changing curriculum, socioeconomic reasons, issues with feeder schools. The list is unending.

Will a poor school be able to attract better students to raise its status after negative reports? Will poor reports relegate a school to a slippery slide of not attracting better students or teachers for fear of the school closing?

And what makes a good school? Is it the results of the top ten percent? The number of students without criminal records after five years? University entries? Students that gain incomes over $100,000? Students that don't end up divorced? Students that do well/better on standardised tests? Schools where parents are happy? Low teacher turnover?

How will statistics provide a fair and equitable benchmark for measurement of performance when high and low performance of students, teachers and administration is nigh on impossible to define and measure? People get very clever at analysing how scores can be manipulated.

I must say that I think these questions raise more questions than answers and that politically it seems it is just a smokescreen aimed to reduce inflationary pressures caused by police/teacher/nurse/public service wage claims and take away the focus from economic forces.

Impost of Mathematics NCOS in our schools

Let's look at the impost on schools and students for the introduction of NCOS in mathematics.

Firstly handmedowns instantly become useless to families. Old textbooks cannot be passed on as new courses are semesterised and the curriculum has changed. Students are at a significant disadvantage if using hand-me-down graphics calculators next year and old calculators cannot be resold relegating $147 calculators to $20 adding machines.

Next comes the impost of new texts. Each course (two per year) requires a book at $22 each (thank goodness Saddler books are reasonably priced) - $88 over two years or $8800. In year 11, students need to purchase a new CAS calculator at $187 - adding to another $18,700. In a school that supports 100 senior school maths students, this means that parents of senior school students as a group need to find $27,500 to cover costs just for mathematics.

Now knowing how hard it is to get $6 out of students for an excursion in difficult times, we should not discount the impost this is causing families. Suggesting to students that working and buying calculators rather than MP3's and PSP's may be the only way that some of these kids will be able to do mathematics. Perhaps this needs to be the christmas present for a number of students.

That it has come to students picking other subjects because they are cheaper says poor things about our schooling system. I know that some families have to make great sacrifices to find that $187 on top of requirements for other subjects, stationery and uniforms.

Then we look at the cost of implementation - PD, adjusting to new texts and calculators, course redesign, lack of available TEE materials, adjustments in subject selection, timetabling issues, preparation of new tests/ assignments/ worksheets/ homework /investigations, adjustment to non-calculator sections, the unpreparedness of 10 new courses all implemented at once. The time available for senior teachers to mentor junior teachers over the next couple of years may be severely limited. There will be very few knowledgable in the intricacies of these courses for some time. For confirmation of the limits of knowledge ask a mathematics teacher how the level 2/3 courses are treated in calculating TER scores. Good luck understanding the answer (it's here).

Course change is not something to be taken lightly- nor something that should be done in a rush.

I hate to think what will happen as books and calculators go missing or courses go awry next year - there are going to be some very sad parents, teachers and students.