You know.. it would be nice to not make mistakes. It's even better when your mistakes aren't distributed to multiple schools for scrutiny. I had the wonderful opportunity of writing three assessments for moderation groups all at the same time, two tests (one for 3A MAT and one for 2C MAT) and an EPW (for 2C). Tests did not exist that could be pulled off the shelf and I didn't want to use a Curriculum Council EPW as they have been widely leaked (yes I'm looking at you Curtin University!).
Anyhow, the 2C paper had an error (three circle Venn diagrams aren't part of the curriculum) and it was one of my complex questions along with another question that I changed at a teacher request to set notation. Unfortunately by doing so it also reduced them to non complex questions. The test (although broadly covering key concepts) did not have the required complexity.
Once marked the curve for my class was badly skewed. It's a bit embarrassing as it's the first time I've taught 2C and really wanted to do the right thing by my moderation group. The test had an error in it and I had to re-issue the marking key as well as the original one had mistakes in it too.
Hopefully the 3A paper is ok (it's harder than the 2C paper and I think my students are going to get a little wake up call) and I must say - the amount of work required to write a 2C EPW should not be underestimated. If you're interested in an original 2C Finance EPW based on spreadsheets leave a comment with your DET email address and I'll forward it to you (Your email address is safe, - I moderate all comments before release and I'll delete the comment before it goes online so that the email address is not made public).
I've been flat out trying to get it all done (and interim reports) and bed down my classes. Hopefully now it will settle as all of my NCOS assessments for term one have been done and I can start enjoying myself again working on the lower school courses. Ten year 9/10 students approached me today to run an afterschool extension class again. They're fun but a lot of work when you and the kids are hot and tired.
We'll see how it goes. Bring on the long weekend!
Friday, March 4, 2011
Wednesday, March 2, 2011
Solving Venn diagrams where the intersection is unknown
n=40
Today in 2C MAT we came across that old chestnut, the Venn diagram with the missing value in the intersection with a number in A, B and the outside region.
In many cases the easiest way is to use a guess and check approach and a lot of the time the answer will fall out by substituting into the intersection and revising your result based on the values
A union B + the outside region = n.
n=40
Another approach is to name the segments and solve a series of equations:
a = 20-b
c = 30-b
a + b + c + 5 = 40
By substitution (20-b) + b + (30 - b) + 5 = 40
Therefore b=15
Once the intersection(b) is known, finding "A only"(a) and "B only"(b) is trivial.
I was asked the question "why teach this technique?" and my response was that it was not formally taught, it was a logical answer for a question given. We have some unknowns, we have some equations, why not solve for them? This sort of problem solving "setting up of equations" technique is common in optimisation and linear programming - why not use it in a probability setting?
I remember a particular student that was renowned for having solutions of this nature where his answers always deviated from the answer key and he had the right answer (or was on the right track) more often than not. We still call intuitive answers like this after "that" student as they forced the marker to find the underlying logic rather than application of a given method (if that student is reading this - get offline and study for your uni courses, scallywag!)
Anyhow, a third and more common approach is to rearrange the property:
A U B = A + B - A intersection B
By rearranging the equation
A intersection B = A + B - AUB
Since we know that:
AUB = U - (the outside region)
to find AUB is fairly simple:
AUB = 40-5
= 35
Therefore:
A intersection B = 20 + 30 - 35
= 15 (as before)
This approach does have the advantage that you can talk about the intersection being counted twice when the union is calculated by adding A + B where A and B aren't mutually exclusive.
I can't really see how this problem could be classed complex given the second method exists. Perhaps, if combined with a wordy explanation, a question of this sort could be made complex but to my mind that would defeat the purpose of the syllabus points in defining complexity. After all, why should something be classed a "complex question" if the only reason was that the question was worded to be understood by students with strong English comprehension?
Today in 2C MAT we came across that old chestnut, the Venn diagram with the missing value in the intersection with a number in A, B and the outside region.
In many cases the easiest way is to use a guess and check approach and a lot of the time the answer will fall out by substituting into the intersection and revising your result based on the values
A union B + the outside region = n.
n=40
Another approach is to name the segments and solve a series of equations:
a = 20-b
c = 30-b
a + b + c + 5 = 40
By substitution (20-b) + b + (30 - b) + 5 = 40
Therefore b=15
Once the intersection(b) is known, finding "A only"(a) and "B only"(b) is trivial.
I was asked the question "why teach this technique?" and my response was that it was not formally taught, it was a logical answer for a question given. We have some unknowns, we have some equations, why not solve for them? This sort of problem solving "setting up of equations" technique is common in optimisation and linear programming - why not use it in a probability setting?
I remember a particular student that was renowned for having solutions of this nature where his answers always deviated from the answer key and he had the right answer (or was on the right track) more often than not. We still call intuitive answers like this after "that" student as they forced the marker to find the underlying logic rather than application of a given method (if that student is reading this - get offline and study for your uni courses, scallywag!)
Anyhow, a third and more common approach is to rearrange the property:
A U B = A + B - A intersection B
By rearranging the equation
A intersection B = A + B - AUB
Since we know that:
AUB = U - (the outside region)
to find AUB is fairly simple:
AUB = 40-5
= 35
Therefore:
A intersection B = 20 + 30 - 35
= 15 (as before)
This approach does have the advantage that you can talk about the intersection being counted twice when the union is calculated by adding A + B where A and B aren't mutually exclusive.
I can't really see how this problem could be classed complex given the second method exists. Perhaps, if combined with a wordy explanation, a question of this sort could be made complex but to my mind that would defeat the purpose of the syllabus points in defining complexity. After all, why should something be classed a "complex question" if the only reason was that the question was worded to be understood by students with strong English comprehension?
Labels:
teaching
Further exploring the properties of one
To find an equivalent fraction of a decimals, one way to explain it is to take the decimal part of the original number and place it over the lowest place value. Leave any whole numbers in front. (This only works for non-recurring decimals)
eg 0.123
The lowest place value is thousandths, the decimal part is 123.
therefore:
0.123 = 123/1000
An alternative way to explain it is using properties of one. The idea is that
a) numerators of fractions should be whole numbers and;
b) the fraction should be equivalent to the decimal.
We can ensure the fraction is equivalent if we only multiply or divide by 1 or more importantly a fraction that is equivalent to 1.
To satisfy part a)
To make 0.123 a whole number we have to multiply it by a power of 10 - 1000 (10^3). This was a concept we had investigated earlier.
..but if we multiply by 1000 we will change the original number from 0.123 to 123 - it will no longer be equivalent.
So to satisfy part b)
We multiply by 1000/1000 (or 1!)
Thus:
.0123 = .123/1 x 1000/1000
= 123 / 1000
I like this because it continues to explore how fractions are constructed, the connection between decimals and fractions and why decimal conversion works. I wouldn't try it in classes with low ability due to the possibility for high levels of confusion if understandings of multiplication and commutative properties are not properly understood.
An earlier article exploring one and fractions can be found here.
Viola.
eg 0.123
The lowest place value is thousandths, the decimal part is 123.
therefore:
0.123 = 123/1000
An alternative way to explain it is using properties of one. The idea is that
a) numerators of fractions should be whole numbers and;
b) the fraction should be equivalent to the decimal.
We can ensure the fraction is equivalent if we only multiply or divide by 1 or more importantly a fraction that is equivalent to 1.
To satisfy part a)
To make 0.123 a whole number we have to multiply it by a power of 10 - 1000 (10^3). This was a concept we had investigated earlier.
..but if we multiply by 1000 we will change the original number from 0.123 to 123 - it will no longer be equivalent.
So to satisfy part b)
We multiply by 1000/1000 (or 1!)
Thus:
.0123 = .123/1 x 1000/1000
= 123 / 1000
I like this because it continues to explore how fractions are constructed, the connection between decimals and fractions and why decimal conversion works. I wouldn't try it in classes with low ability due to the possibility for high levels of confusion if understandings of multiplication and commutative properties are not properly understood.
An earlier article exploring one and fractions can be found here.
Viola.
Labels:
teaching
Sunday, February 27, 2011
PD Days & Collegiality
One of the bugbears of PD days is the difficulty of engaging 60-70 university trained professionals of widely diverse interests, usually during times of high stress with timelines bearing down on you.
One idea is to use this time for learning area planning. This is usually unsuccessful and the planning time instead used for a wide variety of other tasks (general discussion, marking, personal planning). Why?
Some suggested reasons:
a) No deliverables are defined
b) Time frame for deliverables are unrealistic, ill defined or aspirational
c) Require sharing of resources that are thought of as proprietary (such as programmes developed in own time)
d) Require interaction between staff members that are oppositional
e) Processes are poorly lead and easily high jacked
f) Deliverables are not measured
g) No consequences for not meeting deliverables
Most of these are just indicators of poor school based management but many are problems that have arisen due to systemic ineptness. The lack of collegiality is a growing phenomenon that is occurring as competitiveness between teachers for promotional positions is rising and teaching moves from a vocational profession to an occupation. If schools do not actually manage the transfer of information and the information loss as teachers move between positions and schools, the school loses knowledge and effectiveness (especially cohort or area knowledge) with each transfer. Teachers tend to gain knowledge working in schools such as ours (on their path to effective teaching in low SES schools) rather than the other way around. Those entering these schools can encounter strong resistance to new ideas (especially if it is thought the ideas have been tried before), underestimate implementation issues or be unwilling to share until quid-pro-quo is found.
It should also be recognised that with the rapid changes in syllabus, the ability for a school to develop a working curriculum (that can be further developed over a number of years) has been made significantly harder. The weight of curriculum development has been placed on many occasions in the hands of the incompetent through no fault of their own (teaching out of area, beginning teachers, sole practitioners rather than team members, those lacking analytical skills but are fantastic teachers, administration staff that cannot measure effectiveness of a programme etc)
PD days are one opportunity to stop this information loss but it needs people that can define clearly a task to be done that would serve a real long term purpose and then measure the effectiveness of it. It is just another aspect of change management.
One idea is to use this time for learning area planning. This is usually unsuccessful and the planning time instead used for a wide variety of other tasks (general discussion, marking, personal planning). Why?
Some suggested reasons:
a) No deliverables are defined
b) Time frame for deliverables are unrealistic, ill defined or aspirational
c) Require sharing of resources that are thought of as proprietary (such as programmes developed in own time)
d) Require interaction between staff members that are oppositional
e) Processes are poorly lead and easily high jacked
f) Deliverables are not measured
g) No consequences for not meeting deliverables
Most of these are just indicators of poor school based management but many are problems that have arisen due to systemic ineptness. The lack of collegiality is a growing phenomenon that is occurring as competitiveness between teachers for promotional positions is rising and teaching moves from a vocational profession to an occupation. If schools do not actually manage the transfer of information and the information loss as teachers move between positions and schools, the school loses knowledge and effectiveness (especially cohort or area knowledge) with each transfer. Teachers tend to gain knowledge working in schools such as ours (on their path to effective teaching in low SES schools) rather than the other way around. Those entering these schools can encounter strong resistance to new ideas (especially if it is thought the ideas have been tried before), underestimate implementation issues or be unwilling to share until quid-pro-quo is found.
It should also be recognised that with the rapid changes in syllabus, the ability for a school to develop a working curriculum (that can be further developed over a number of years) has been made significantly harder. The weight of curriculum development has been placed on many occasions in the hands of the incompetent through no fault of their own (teaching out of area, beginning teachers, sole practitioners rather than team members, those lacking analytical skills but are fantastic teachers, administration staff that cannot measure effectiveness of a programme etc)
PD days are one opportunity to stop this information loss but it needs people that can define clearly a task to be done that would serve a real long term purpose and then measure the effectiveness of it. It is just another aspect of change management.
Labels:
curriculum
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