Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Saturday, December 8, 2012

Importance of self esteem

There are always groups of students that are difficult to reach.  Students that do not directly benefit from mathematics in the short term can lack motivation to attempt work, leading slowly to disengagement.  Success with these students often relies on making a personal connection with them, sharing part of your life indicating that students that do not go directly to university are still successful in life.

My approach for this centres around experiences when I was not much older than my students.  I am lucky that during my formative adult years I had little support and passed through many jobs; nightfill, fast food, labouring, kitchen hand, reception,  data entry, accounts, furniture removal.  These jobs were not  the high flying roles that I had later in my twenties, but they enabled me, I had skills and could recognise opportunities that those closeted in university may not have had access to.

It's an important message for kids not destined for direct entry university.  Many lack any vision of the future and don't have an understanding of hope - they're simply living for the now. The simple message that "if you're willing to work harder than anyone else, you'll start to get ahead" is an eye opener for them.  I couple this with some basic finance, setting a budget, learning about credit card debt, saving half your income, basic investment strategies and interest calculations to show them that the jobs they may be already in, can provide them with financial security with a small amount of planning.

A favourite lesson is valuing a dollar saved.  Most (if not all) kids do not recognise that a dollar spent is worth more than a dollar earned.  To spend a dollar we must have already paid taxes, the bills and all the costs of living.  An dollar saved may require three or four dollars to be earned first.  If students can get this ratio down to 1:1 they are on their way to financial independence.  When a third factor is introduced (investment) and they can cover expenses through investment dollars they can increase the time to enjoy life and enable retirement.

Many are destined for jobs they will not enjoy.  If working has a clear purpose, it will make for better employees that value their employment. I also tell them that a bit of life experience can help them understand the importance of education.  I didn't finish my degree until my thirties!

Another message is to give them is a multi-generational viewpoint.  All say they want their kids to go to better schools, they want houses, weddings, fast cars, plasma TVs.  If they understand the costs incurred during later life and can aim from the beginning to help their kids during their lives, it will promote a budgeting outlook rather than hand-to-mouth accounting.

I try and invoke the principle that taking pleasure in "giving" is the simplest path to happiness.  There are many occupations where the pleasure of working becomes a part of the attraction to the work.  You won't become rich but you will have a life of rich experiences and make fruitful contributions to society.  Teaching and nursing are two that spring to mind, and we do have a disproportionate number of students seeking math teaching and nursing each year.

These things, together with providing rich mathematical programmes (and not falling into the trap of assuming these kids need an impoverished curriculum purely because of low assessed results), can turn around students that are disengaging.

I think that seeing future pathway is a path to positive self image that can improve their self esteem.  Self worth of some of these kids is at rock bottom but it can take very little to get them excited again about their futures.  Lessons like these are part of a broader picture to get our kids thinking ahead.

I don't think I'm explaining myself well here, but I think the gist is present.  After six years of teaching here, the formula for delivering lower performing students (or students with a disrupted educations) is getting quite complex but some general strategies are emerging.

Tuesday, May 25, 2010

April Fools Joke - Not!!!

Today I had a look at the expected standards "C grade descriptors". This document outlines the requirements of a C grade for a student yrs 8-10.

"The descriptors have been informed by population testing data [NAPLAN], draft national curriculum materials and the professional knowledge of experienced teachers. During the consultation process teachers strongly supported the production of concise descriptors and welcomed the inclusion of examples." Department of Education April 2010.

The issue is that the end product was written devoid of common sense.

Here is an outline of the expectation for year 9 students from the C grade descriptor document.

"By the end of Year 9, students use number and algebra to solve routine and non-routine problems involving pattern, finance, rate and measurement including the calculation of area of triangles, circles, quadrilaterals and the surface area and volume of prisms, pyramids, cones and cylinders. They solve problems using Pythagoras’ Theorem and proportional understanding (similar triangles and the tangent ratio). They have a sound understanding of linear functions and are developing fluency with using quadratic and simple non-linear functions, such as with patterns involving doubling. They have a sound understanding of index laws pertaining to positive integral powers."

My issue is that this is a C grade description. The number of students with "sound" understandings of linear functions (by my definition of sound) during year 9 is minimal and students that have a conceptual notion of quadratic and other functions at this stage are the "A" students that have been extended - not the C students. In fact given this description it would be difficult to give many C's or even a single A in many state schools.

If students entered high school have some algebraic knowledge, they may have some chance at reaching this standard. At present I would suggest that this is highly unlikely. As students delivered by national curriculum are 5 years away - starting assessment now at a national curriculum level is ludicrous.

It is obvious the scope and sequence has been modified to include national curriculum requirements (look for the * in the scope and sequence). You can see that linear functions was the main focus of year 9 and then quadratics and 'other functions' were dumped into the sequence with little consideration given as to how time will be found to implement the new curriculum especially as it was hard to fit in the old curriculum (I sat and wrote a lesson by lesson plan for year 9 based on the old scope and sequence and challenge anyone to do the same on the new scope and sequence given the current entry point of students in year 8).

I have no problem with lifting the bar for students, but it requires time to re-instill work ethic at a younger age and subject specialists having access to these students.

To grade students that have not been adequately prepared for national curriculum assessment is grossly unfair. How anyone could propose this for semester one grading 2010 indicates a lack of understanding of the change management required. Either schools will need to fudge grades (easy to spot when comparing NAPLAN to school grade) or masses of students will not get grades higher than a D or E.

When teaching students in low literacy settings, handing out D & E grades to students trying their utmost to succeed is tantamount to child abuse. It is demoralising, unfair and sets up an expectation of failure. I can't say this in stronger words. Someone needs to have a good think about what is being done to our children.

Link to national curriculum media release (Julia Gillard)
Link to expected standards (Department of Education)
Link to mathematics scope and sequence (Department of Education)

Sunday, May 23, 2010

Teaching of mathematical literacy

I have been teaching a low literacy class this year and it has taught me the perils of relying on the immersion technique to teach mathematical literacy. After seeing the positive effects of direct instruction of low literacy students, I see immersion as a lazy teaching technique for low ability students - immersion is slow, ineffective and generally detrimental to these students, especially in a large heterogeneous class.

Let me explain. When I teach area, I generally teach students to write a story that I can read. They draw a diagram, label the sides, write a general equation, substitute the values, evaluate the solution and check their answer. Once this technique has been learned I can then easily teach other concepts such as Pythagoras' theorem, trigonometric ratios, surface area and the like.

The explicit teaching of mathematical literacy (requiring specific layout and explicitly explaining the meaning and need for each component of the layout) is the key component in this exercise. By year ten, most can answer the area of a rectangle and write the answer, but cannot abstract the method to a circle. I attribute this to a lack of mathematical literacy and a failure to appreciate the true need for mathematical literacy.

Although modelling has a place in teaching mathematics (a key tool in immersion), we must be mindful that we need to teach literacy explicitly and not assume students will just pick up major concepts by observing a question being completed. The difference between a student answering a question correctly (after being given an example on the board) and being able to identify how to answer a question correctly from a range of tools (without prompting) is considerable.

Mathematics has grammar just like English. If students understand the grammar of mathematics, the meta-language of mathematics and the algebraic/arithmetic/visual representations/tools of mathematics, then their ability to solve problems increases exponentially.

And as mathematics teachers we can appreciate the benefits of exponential growth.

Sunday, March 28, 2010

Project scheduling "99" rule

The actual project scheduling rule goes like this:

"The first 90 percent of the task takes 10 percent of the time. The last 10 percent takes the other 90 percent."

The way we always said it was:

"The first 90 percent of the task takes 90 percent of the time. The last 10 percent takes the other 90 percent."

This is true of students. Often we can get them 90% of the way there but we don't have the time to get them the remaining 10%. That remaining 10% is found through practice and investigating the concept in a variety of contexts. I'm gaining an appreciation of the well written investigation that goes some of the way to assisting students gain this understanding.

For instance, we recently completed an investigation on radians. By the end of the investigation, students had worked through a variety of uses of radians, applied formula in a variety of ways and had to think about what the formula was comprised of and how it was derived.

This though is rarely done in lower classes - and is a real flaw in use of texts. If questions are presented in method-> practice exercise form, students rarely have to think about what is required to solve a problem. This causes a lack of retention and poor examination results (if examinations are done at all).

I've been thinking that a revision week may have their place in the programme, once a term where students are forced to reconsider earlier work with an element of training how to revise for exams.

I will think on this further.

Friday, March 5, 2010

Absolute value

I spent a fair bit of time thinking about absolute value problems in the form |x+a| - |x-b| = c. Many students were struggling with visualising what these functions actually look like. What was happening when we try and solve them?

For example:
|x+5| - |x - 2| = 6
How could I display this equation graphically to give students an understanding of the underlying algebra to solve it?

I tried graphing y = |x+5| - |x - 2| and y=6 to find the intersection but was unsatisfied with the result as y = |x+5| - |x - 2| is not something easily tied to the absolute value concept or 'v' shaped absolute value graphs.
I was eventually satisfied with graphing y= |x+5| and y = |x-2| and then examining each part of the graph until I found a section of the graph that was 6 units apart.

For those wondering how to put it into a graphics calculator while exploring the concept

Go Menu -> Graph & Tab
Edit -> Clear All -> ok
at "Y1:" ->Softkeyboard->mth tab->select 'x'->type "x+5)" (it will change from abs(x+5) to |x+5|)
at "Y2:"->select 'x'->type "x-2)"
ensure that the boxes next to "Y1" and "Y2" are ticked

Now the temptation is to assume the answer is the intersection point.

but if we look at the equation |x+5| - |x - 2| = 6, it is asking "for what value of x is the value of |x+5| (the dotted line) subtract the value of |x-2| (the solid line) equal to 6". When is the gap between the two functions +6.

We can ignore values of x<= -5 as y=|x+5| is below y=|x-2| and the subtraction will only give negative values (we are looking for a gap of +6 which is a positive value).

We can also ignore values up to the intersection point as this also will only result in negative values.
The next place I looked is at x>=2 as the gap is constant and positive after this point (both functions have the same gradient).
at x=2, |x+5| is equal to 7 and |x-2| is equal to 0. |2+5| - |2-2| = 7. We can ignore values where x>=2 as the answer is not +6.

In fact the only possible solution has to lie between the intersection point (x~-1.5) and 2 and is probably closer to 2.
For y=|x+5| all values are positive between -1.5 < x < 2
For y=|x-2| all values are negative between -1.5 < x < 2
To ensure positive values for x-2 in the range -1.5< x < 2 we need to take the negative of (x-2) when solving the equation |x+5| - |x - 2| = 6.

x+5 - (-(x-2)) = 6
2x+3 = 6

Check answer:
|x+5| - |x - 2| = 6
Let x=1.5
|1.5+5| - |1.5-2| = 6
LHS =| 6.5| -| -.5|
= 6.5 - 0.5
= 6.0


It would also be interesting to explore |x+a| + |x-b| = c,  |x+a| = |x-b| and -|x+a| = c in a similar way.

Here is a link to other CAS calculator posts.

Thursday, February 25, 2010

Manual Subtraction

An interesting question was posed to me today.

How do I subtract two number manually when the answer is negative??

For instance, 3896 - 4321 (to which the answer is -425).

I originally set up the problem in vertical columns


and tried to subtract..


which obviously does not work.

So I thought about it.. the only obvious solution was to say, when subtracting always put the larger number on top.


As this answer is positive, it is still incorrect. It requires an additional rule, that when the order is changed, the sign of the answer is negative. Thus the answer is -425.

I'm sure everyone knows this (and it's just one of those odd cases I haven't come across before), but it could be an interesting short investigation for upper primary or lower secondary doing directed number exercises.

.. and I have a stupid cold, my nose is dripping like a tap and I can't hug my daughter. It's made my day!

Thursday, January 28, 2010


I sat through another round of someone extolling the benefits of rubrics/analytic marking keys/explicit marking keys. There was no doubt a lot of effort went into constructing them, but the usual issues were there amongst the generic template.

Assessment is supposed to be Fair, Explicit, Comprehensive, Educative, Valid

Rubrics vary between too vague to be of benefit (fails the explicit test - makes marking easy but cannot be easily connected to assignment without 'dejargoning') or so explicit that most students can get an A if they put some effort in (fails the comprehensive/valid test - can a student do it without the rubric??).

The position put forward was that marking should be quick. I'm afraid I can't see how this is true. The only comments students read, are ones in red pen. If you circle where students lie in a marking key, they normally just skip to where the final grade is. Students will read every line written in red pen and ask for clarification of it.

This is where investigations today fall down a little. Typically we guide students through the investigation (so that it becomes more like self teaching than investigating) - but the other side of the coin is that students can't be expected to rediscover what mathematicians took millenia on their own. We need to find a middleground.

We have collected a wide range of investigations, categorised and standardised them. I must admit I have struggled with selecting, generating, marking and guiding students with regard to investigations and marking keys. It needs more work and thought.

Wednesday, December 9, 2009

Interesting facts about Perth Teachers

WACOT released the following figures about Perth teachers in their latest publication InClass, found here.

38,125 Registered Teachers
• 7,749 Provisionally Registered Teachers
• 363 Limited Authority to Teach
• 26 Associate Members

34,256 female teachers
12,007 male teachers.

The imbalance between female and male teachers is astonishing. Of the 46,263 teachers of varying registrations, only 26% are male. That's a real lack of male role models in our workforce. I wonder if it was the reverse (eg. more males than females) if we would be having a recruiting drive and financial incentives for females to enter the industry?

At our school, I would hazard that the male percentage is much higher than that. In low socio-economic schools, where single parent percentages are normally higher, I would also suggest that this is a good thing.

Another interesting statistic is that 17% of the workforce is in training/probation/being actively mentored (on provisional registration). Last year only 2.5% moved from provisional registration to full registration (another 2.5% re-registered as provisional registration not meeting the criteria for full registration).

Thursday, November 26, 2009

NCOS and consolidation of knowledge.

One criticism of senior school and mathematics in general is the lack of consolidation of topics - especially when the course is prescribed as is the case with NCOS. Funnily enough, the NCOS has brought about an opportunity for consolidation that did not exist under the old courses.

The new courses allow for repeating of yr 11 subjects - which makes sense under an outcomes approach where learning speed is not being measured, just knowledge and skills gained (this is an issue in itself that needs investigating if TEE scores are to remain a predictor of university success).

Students that cannot withstand the pace of the course in year 11 have in year 12 the option of consolidating (by repeating the course), remediating (by completing a lower course) or advancing to the next course. This approach allows teachers to make more aggressive subject selection recommendations in year 11 that promotes striving for excellence without fear of being locked into advancing and failing the yr 12 course. The recent trend of conservative subject selection could be broken!

For example, a student doing yr 11 3A MAT has the option in year 12 of doing 2C (remediating) 3A (repeating) or 3C (advancing).

I doubt this was the original intent (in other subjects teachers must teach another context - but only one context really exists in maths/science courses).

I fail to see the issue in repeating or remediating although I know some humanities teachers think it unfair - students that repeat will have the option to gain a deeper understanding at some level and a further opportunity to apply their skills - having a second bite at the cherry.

It will be interesting to see if the old adage that 'repeaters don't succeed' will bear true next year. For the lazy student - repeating/remediating will not work, but for those that have good work ethic but need more time logic says they should succeed (more time better results!).

My prediction is that (when counselled and supported correctly) repeaters and remediators will do far better than advancers and scaling will be applied to these students (compared to advancing students) in future years. It will be interesting to see if the scaling factor of 10% between 3AB and 3CD will be enough to compensate (I can't see how having two years to master a course can't cause better than a 10% increase in low/mid performing students between the two groups). The scaling may already be heavier for repeaters - but I'm not aware of it.

Monday, November 23, 2009

Don't forget to vote in the new poll!

There's a poll on the right hand side asking how we found the new courses - are they better than the ones they replaced?

For parents who are interested:

The harder parts of Intro Calc / G&T / Applic / Calculus (for strong science/maths/engineering bound uni students) was replaced by 3ABCD MAS (with some changes)
The easier parts of Intro Calc / G&T / Applic / Calculus (for capable science/maths/engineering bound uni students) was replaced by 3ABCD MAT (with some changes)
Foundations / Discrete (for capable Uni bound students) is now 2CD 3AB MAT
Foundations / Discrete (for weak Uni bound students) is now 2ABCD MAT
MIPS /Modelling (for students needing some maths - TAFE/Uni bound) is now 1DE2AB MAT
MIPS /Modelling (for remedial maths students - work or TAFE bound) is now 1BCDE MAT
No real maths course under old system (for Ed support or struggling maths students) is now PA PB 1A MAT

Now don't forget to vote on the left!

Sunday, July 12, 2009

Arkanoid Clone

Here's a basic outline of an Arkanoid clone, made by modifying the space invaders code of last week (it was done in a day!).

Click here to play.

Click here for source.

To complete this I had to use polar coordinates, bearings, reflection, complementary/supplementary angles, recursion and a lot of boolean logic.

The paddle reflection is not perfect and the game could use more levels. There is an issue with ball speed that I need to resolve. It doesn't work in other than 1024x768.

A link here to the space invader clone of last week.

As always, the games are written in Java using Netbeans. All of the game logic written by yours truly. Feel free to do use the code as you see fit.


Wednesday, July 8, 2009

Space Invaders

Here's a little space invaders program that I wrote on the weekend. If it doesn't work the first time, maximise it and try again.. I haven't optimised it for anything other than 1024x768 yet.. Maybe tomorrow.

It was written in Java using Netbeans and some free sound and graphics from the web. With exception of some of the graphics, database and sound routines, the game logic was all written by yours truly.

It has a high score table and multiple levels. It should auto load using Java Web start (fingers crossed).

I've been writing these games with the hope of starting a programming group of students. The basic ideas gained have direct application in mathematics, especially in algebra, coordinate geometry, functions, recursive algorithms and trigonometry.

Click here to play game

Click here for source code

Best of all.. all of the software used to make it was free!

Pretty cool huh?

Saturday, June 20, 2009

Lockharts Lament

Lockhart's Lament was recently /.'d and is a really good read about one person's thoughts of where mathematics has diverged from being an art to a science (yes you read that correctly, was an art form now a science). It was interesting to read how a classroom could be transformed from a fact finding mission to a place focusing on the development of ideas, more akin to a history and art class than as a science. I've read a few different variations on the same theme, but this is one of the better ones.

It would be funny (thinking of pure maths as an art form) if there wasn't some truth in it. True creativity and inspiration is at the heart of any discovery. On the other hand, if I was hiring people to build a bridge or a skyscraper I'd want a person doing it that had been drilled in maths and understood how to apply it rather than some introspective, dreamy, philosophy driven hippy. The approach suggested infers putting even more language in mathematics, running the risk of removing maths even further from those students that find it a refuge from humanities based subjects.

There's nothing stopping us implementing or re-introducing some of the ideas in the article into the curriculum. I'm always looking for ways to reinvigorate my classes and this may be one! I would though be wary of any approach that took more maths out of mathematics. After all, we don't all have the genius to discover maths the way of Gallileo, Pythagoras, Archimedes, Newton and Leibniz... but like the writer I can appreciate and revere the simplicity and elegance of their findings.

Monday, June 15, 2009

Just in time intervention

Since starting teaching I've endeavoured to provide kids with just-in-time intervention. I'm not sure where the concept originated but I use the term as an in between to "early intervention", "delayed intervention" and "too bloody late intervention".

Just in time ("JIT") intervention is finding an area where the curriculum has failed (such as weak performance found after a test in an unexpected area) and plugging the gap by providing extra tuition or resources to fix the issue, immediately after the issue has been discovered (thus the "just in time"). Examples when JIT intervention would be be needed would be finding BIMDAS problems during the teaching of percentages or discovering negative number issues when expanding brackets in algebra topics. Students needing JIT intervention typically can master new topics but can't apply their new learning due to an associated issue - leading to poor retention of the new concept. Fix the associated issue and fix retention problems of new concepts.

JIT intervention is different to early intervention as early intervention is typically preventative and is sprayed around like a weed killer - "catch the issue before it occurs and hopefully we will stop what happened last year". It's different to delayed intervention as this can be seen as "the next teacher can try again with the same sort of material next year (only more difficult) and try to find success" and too bloody late intervention which occurs in senior school where students are finally streamed into classes where they can find success but have little time left in school to do so.

Until now I have focused on finding worksheets and doing lunch workshops for particular areas of the curriculum. I have avoided online resources as until now they have been overly focused on fun and are not focused enough on addressing requirements of students. As there is only one of me, workshops and worksheets have had limited success - in senior school if you scratch the surface it wells with underlying issues that require attention, more than any one person can address.

Our latest attempt to provide JIT intervention is to leverage some of the developed online tutorials that have shown some promise and direct kids to them. In an art imitates life experience (think 'the Simpsons'), the free McDonalds sponsored "MathsOnline" project is getting a guernsey at our school as the tutorials have found success with indigenous students - which we hope will extend to other struggling students needing help. We are setting up a maths lab that allows students access to the MathsOnline resources and will use them in conjunction with maths resources bought from the ESL budget for low literacy students. The mathsLab is adjacent to my room (connected with a concertina wall) and I aim to be able to monitor students as they attempt to rectify a range of issues and assist where possible.

I do like the maths online implementation as it is not "button mashing" or "timing based competition" but requires listening to a short tutorial, working out answers on paper and then checking them against an online marking key - similar in concept to the pizzazz or mathomat/mathmagic type worksheets (without the awful maths jokes (that I tend to laugh wayy too heartily at))... It also records the attempts of students so that I can investigate when the tutorial is insufficient.

We presented mathsonline today to the IT committee and hopefully they see some benefit in it. Our principal was positive with his praise of the initiative - now we have to find some success to warrant the praise.

Thursday, May 21, 2009

Content vs Process

Here is the first of (I imagine) many articles on the importance of teaching content in schools and the reemergence of the idea that developed conceptual understanding can only be achieved by having a baseline of subject knowledge.

It has always been to my mind counter-intuitive to request a student to "understand" a topic without having facts to scaffold that understanding upon. There is no use in giving students methods of learning information if time to learn the information is not given and valued. The constant devaluation of content knowledge vs developing process has lead to a flawed education system.

I have to agree with the writer that being a yr 11/12 subject teacher with a deep understanding of a course requires more ability than that of 6/7/8/9 or 10. These experts in their fields deserve to be paid more and gain recognition for the guiding of students at this critical point in their lives. It is high pressure work with success leading to recognition for the school and the making of careers for students. Failure can lead to pressure from parents, administration and (more damaging) self criticism and confidence depletion.

Having experienced now 7,8,9,10,11,12 there is no doubt in my mind that the pressure involved in getting students over the TEE line far outweighs anything in earlier years. I have utmost respect for those that do it successfully over long periods of time.

Wednesday, May 20, 2009

Mid Semester Exams

Yesterday I was asked why run exams in year 8 & 9.

I could think of 8 reasons:

1. To reduce fear of exams (students use the idea of exams as a bugbear for not attempting level 2 subjects.)
2. It gives an anchor to the idea of study/revision.
3. It is good practice for upper school and identifies students bound for more difficult courses.
4. It provides feedback on what has been achieved by individual students during the semester.
5. It supports grades allocated by teachers put into reports.
6. It provides a benchmark of performance from year to year.
7. It is the backbone of academic rigour in a school, short of doing a personal project (which is impractical in most public schools).
8. Students gain confidence in doing exams by.. well.. doing exams.

Then I heard the excuses and heard what was really going on:

1. Such and such is just rewriting the NAPLAN test (fine if that is all you have taught in Sem 1!).
2. It's a lot of work (it's our job!) for little return (see 1-7 above).
3. I have to mark it (well.. yes.. but we teach math, compare that to issues in English & S&E, we have it easy!).
4. The kids can't do exams (some can, and they are severely disadvantaged compared to the rest of the state if the first time they see an exam is term 2 year 10(think league tables, think school numbers people! No results.. no school)).
5. I can't write an exam (huh?? ..nor can anyone else, we don't know what you have taught, nor do we know the level of your students! If you need help with formatting we have loads of support staff and teachers willing to help).

There is some argument that there is a level of over testing in year 9 due to NAPLAN but exams and NAPLAN have very different focus. NAPLAN looks at the student compared to the student cohort of the state. The exam should show a snapshot of the learning and retention of the most recent semester.

I can also understand the argument that some students should not sit an exam. If a student has a learning difficulty or is miles below the level of the exam (and a special exam has not been prepared for them) then it makes sense to exclude them.. these are our 1B kids.

Sunday, May 17, 2009

3A MAT Ex. 8B Annuities and Amortisation

Ok.. AP's and GP's are now a thing of the past (what?? huh?? when did that happen - in 8A of course!).. we're now onto applications of growth and decay.. nope.. (we did that in 8A too.. huh?? what??)..

MAT 8B We're now onto Annuities and Amortisation - growth and decay with payments.

The calculator handles this under the Financial, Sequences or Spreadsheet.

Starting with Financial:
Once in Financial, select Compound interest.

n - represents the number of installment periods
I% - is the interest p.a.
PV - is the present value (the initial investment)
PMT - is the payment per period
FV - is the future value (the investment at period N)
P/Y - is the number of installment periods per year (how often a payment is made)
C/Y - is the number of times interest is compounded

Let's look at a simple problem say 8B q.3 in 3A MAT. Kelvin invests $620,000 into an account giving 5.8% pa. interest compounded annually from which her withdraws $50,000 at the end of every year.

a) How much is left after 10 withdrawals (N=10, FV=?).

Leave the cursor on FV and press solve (at the bottom left hand corner of the window)

b) For how many years will Kelvin be able to withdraw 50000 per year

Find when the account is exhausted of funds (eg. N=? when FV=0)

Leave the cursor on N and press solve (at the bottom left hand corner of the window)

N=22.52 therefore for 22 years.

If anyone can explain why PV is negative I would be very appreciative. I know from last year's course that it is but have no idea why.

Now Sequence:
This could also have been done through the Sequence tool using recursion
a) Tn+1=Tn*1.058-50000; T0=620000. Find T10
b) Tn+1=Tn*1.058-50000; T0=620000. Find n Where Tn=0

I'll leave the spreadsheet method for another day.

Here is a link to other CAS calculator posts.

Wednesday, May 6, 2009

One for Ornithologists

Was caught today using one of my lessons from uni. It is about drawing histograms and the following data was used:

score. f
1...... 2
2...... 2
3...... 8
4...... 2
5...... 1

It was quite funny then but not really appropriate for a maths class.

Being a know-it-all

Sometimes I read past entries and think that perhaps I said that a bit strongly, and what is an opinion is stated a bit too much like fact.

Well.. today I was caught by my own enthusiasm and after a little assessment have proof that some of the coursework was above the level of the students as I see retention of information from first term way below the level expected (On a scale of little recalled to perfect recall it fell off the scale in the OMG category).

As a new teacher, pitching classes at the right level is a little hit and miss at times (sometimes coursework is too easy other times too hard).. but oh boy.. this one was a doozy. It's not that the situation is irretrievable or that any real harm has been done (later learning will be done faster through introduction of the topic now) but it does raise the point that pre-testing and having the experience to estimate ability accurately is a real bonus once out of your initial years of teaching.

Pitching a lesson series at the wrong level creates a raft of issues. Firstly it damages the confidence of students. Secondly it upsets the sequence of learning and lastly it can cause behavioural issues as students turn off and look for other activities to stimulate them.

Being absent whilst baby was born hasn't helped either, as I may have caught the error earlier.


Tuesday, May 5, 2009

Literacy and the need for developing capacity

A common catch cry in schools is that we need to improve literacy. Each year the same old rubbish is wheeled out in the guise of cross curricula scaffolding, pro-forma templates and a bunch of clever sounding words that achieve little.

I think if we actually looked at what each learning area is actually doing, literacy is a common component that does not need to be explicitly looked at as a 'literacy' issue. Let's take mathematics for example and the current rhetoric.

Literacy Statement:
Gone are the days where you can teach and test a skill. To adequately support literacy in a school we need to implement literacy in every learning area. Texts used need to support literacy initiatives.

Maths Reply:
Mathematics is typically a text dependent subject. A good mathematics text typically has three components. Each section starts with explanatory text, an area where a student has the content explained - such as a worked example. Following the explanatory text is usually some form of text bank that identifies key words within a section and their meaning. Each word in the text is identified by the teacher and used in context to assist students expand their mathematical vocabulary. Following each bank of words is a section of practice starting with straightforward examples and leading to word problems that require varying degrees of English comprehension and analysis. Mathematical comprehension is verified against answers supplied to questions.

Literacy Conclusion:
Over time, whilst immersed in examples of the mathematical form, the student gains contextual understanding, developing processes and strategies guided by cues for usage. Students are encouraged to reflect upon the level of their skills through answer keys and response items in assessment. Students develop independent learning strategies through investigative tasks to extend their growing understanding."

At this point some people (predominantly non teachers and skeptics like myself) will go "what a load of BS". This is not "literacy" rocket science but just old fashioned teaching (no surprises here.. the maths response was teaching from a text with some testing).

Unfortunately a lot of the literacy movement seems to be just hot air .. a lot of documentation that outlines what we already do, with no defined outcomes or outcomes so unmeasurable that they are worthless.

When parents ask for literacy improvement they usually mean can my student paragraph, write a coherent sentence, deconstruct a problem, understand a text. These tasks are typically issues addressed in English departments as specific skills taught over five to ten years. In the same way we teach supporting mathematics for SOSE and Science, we need English to teach grammar, comprehension and reading skills to assist us. This seems to have been the first positive outcome from NAPLAN testing and the national curriculum debate.

The main issue with the literacy debate and to a lesser degree "the whole of language approach" is that core skills in English (and to a lesser degree other subjects) have been given a backseat to experiential learning and by distributing responsibility for learning language based skills we have watered down the ability and accountability for learning areas to deliver their subject specific content (and undervalued the real skill of English teachers). The value of cross curricula learning has been overestimated, with few realising the amount of work it takes to establish a working cross curricula programme.

As someone that couldn't write a paragraph properly until year 10 (when my English teacher forced us to write an essay every Friday afternoon last period for a whole year) I recognise that this is not a new problem.. but we have had 15 years since I was in school to identify the issue and pinpoint better ways of solving it than the current mess. When responsibility for written skills is devolved to many, responsibility for success is also distributed to the point often that no-one is responsible. Written skills (although supported by all learning areas) need to be the responsibility of English departments in the same way that mathematics is guided in a school by a Mathematics department.

I think that strong, visible and active English and Mathematics departments in a school are clear indicators of a good school.

We need to consider that developing capable English and Mathematics departments is not optional in schools.. it is a necessity and priority for success.