After the issues with the 2C test it was nice to have a good day. My 10's were responsive and worked well whilst our Principal was in the room for a whole hour doing his impromptu visits. It's good that he does them, but it can be a bit harrowing. We investigated how to use our CAS calculators to build spreadsheets and will now start looking at the results to investigate compound interest further.
There were lots of things I would do differently with the lesson itself but I can't fault the kids in that they followed instruction, were able to use formulas and solve a compound interest problem using technology by the end of the lesson. After replacing most of the batteries in the morning, only two failed during the lesson which was ok.
I checked my 9's homework and that was a different story. I used some old fashioned "I will do my homework when my teacher asks me to otherwise I will have to write this." x 100 to ensure that students had some encouragement to do their homework in future. Those that did their homework enjoyed it if nothing else.
My 1B's are going ok, they finished the exercise but are not fully understanding cumulative frequency, so we will need to redo that lesson. I must remember tomorrow morning to hunt out a worksheet that will reinforce the connection between cf and median (and xf and mean).
A nice change from Friday.
Russ.
Tuesday, March 8, 2011
Friday, March 4, 2011
Making mistakes
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!
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!
Labels:
teaching
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
Wednesday, February 23, 2011
Drawing the first derivative
Teaching students how to visualise the first derivative in 3B MAT has been problematic over the last two years. This morning I had a bit of a breakthrough in that students weren't looking at me as if I was speaking Alien.
The major difference was that I didn't use the arrow approach. Here's what I did.
I drew a positive cubic on the board and identified the turning points. I identified clearly the x axis and the y axis and identified the coordinates for each TP. I drew their attention to (x,y)
Then I drew a second pair coordinate plane directly underneath and identified/labelled the x axis. I then deliberately (as in made a big song and dance) labelled the other axis y' asking students to think what this might mean.
I then went to the first turning point on the x,y plane and asked students what the gradient was at this point. They said zero straight away.
I then went to the second axis and said coordinates on this plane were (x,y'). Given that the TP we were examining was at (0.25) and y'(0.25) = 0, the coordinate(x,y') that we needed was at (0.25,0). We repeated this for the other turning point.
I then drew vertical dotted lines through both coordinate planes. We then looked at the slope to the left of the TP. Being a cubic (with a positive coefficient of x cubed) the slope was +ve. On the second plane I wrote +ve above the x axis to the left of the TP above the x axis. We then examined the second area and noted the slope was negative (making special note of where the point of inflection was - it wasn't mandated by the course but made sense in the context). I labelled the graph -ve underneath the x axis to the right of the TP. I then wrote +ve in the third area above the x axis.
<- It looked like this.
Once the areas were labelled it was trivial to join the dots starting where y' was positive (y' at +ve infinity), leading to where y' was negative and then changing direction midway between the x intercepts on y', back towards to the x axis until y' was +ve again (again until y' at +ve infinity). It was also a good time to discuss the type of function produced (eg a concave up quadratic) if you differentiate a cubic with a +ve coefficient of the cubed term and how that related to our y' graph.
We then repeated the process for a quartic.
yay!
The major difference was that I didn't use the arrow approach. Here's what I did.
I drew a positive cubic on the board and identified the turning points. I identified clearly the x axis and the y axis and identified the coordinates for each TP. I drew their attention to (x,y)
Then I drew a second pair coordinate plane directly underneath and identified/labelled the x axis. I then deliberately (as in made a big song and dance) labelled the other axis y' asking students to think what this might mean.
I then went to the first turning point on the x,y plane and asked students what the gradient was at this point. They said zero straight away.
I then went to the second axis and said coordinates on this plane were (x,y'). Given that the TP we were examining was at (0.25) and y'(0.25) = 0, the coordinate(x,y') that we needed was at (0.25,0). We repeated this for the other turning point.
I then drew vertical dotted lines through both coordinate planes. We then looked at the slope to the left of the TP. Being a cubic (with a positive coefficient of x cubed) the slope was +ve. On the second plane I wrote +ve above the x axis to the left of the TP above the x axis. We then examined the second area and noted the slope was negative (making special note of where the point of inflection was - it wasn't mandated by the course but made sense in the context). I labelled the graph -ve underneath the x axis to the right of the TP. I then wrote +ve in the third area above the x axis.
<- It looked like this.
Once the areas were labelled it was trivial to join the dots starting where y' was positive (y' at +ve infinity), leading to where y' was negative and then changing direction midway between the x intercepts on y', back towards to the x axis until y' was +ve again (again until y' at +ve infinity). It was also a good time to discuss the type of function produced (eg a concave up quadratic) if you differentiate a cubic with a +ve coefficient of the cubed term and how that related to our y' graph.
We then repeated the process for a quartic.
yay!
Sunday, February 20, 2011
School Fights
Many teachers feel intimidated when a fight occurs in the playground. Fights are things that are skirted around by teaching institutions and rarely spoken of in PD other than in strict legalistic terms.
I'm of reasonably slight build and am considerably smaller than many of the year 11 and 12 students. I'm bigger than many of the female staff also on duty.
So what happens when a fight occurs? How do you, as a teacher, alter an out of control situation when you are physically incapable of stopping students from injuring others and yourself.
The school and how students view the school is a big part of this. I am lucky in that students at our school respect teachers and despite diffusing multiple fights in my career (with male students many times larger than myself and females that had little control over their actions) in all cases my status as a teacher has meant that I have not been at risk. Students seem to know a line that they cannot cross.
Yet I fear this may not always be the case. Students with disabilities are common in school grounds and anecdotal evidence suggest that mainstream students are becoming less able to control their actions.
Practical (not legal) training of staff is necessary before real injury becomes more common. My suggestions are based on practical observation.
1) When on duty stay in line of sight of another teacher on duty. Be prepared to render assistance at short notice. Know the parts of the duty area where you pass from line of sight from one teacher to another.
2) Survey who will take the primary role in diffusing a situation.
3) Issue a command(using full teacher voice) to stop to both parties and (if wise) get between the two students. Hopefully you can skip stage 4 if both students react appropriately. If you are taking the secondary role call for assistance (preferably from a deputy or someone that students are more likely to take seriously.) Seek out the amateur camera people and ensure that they are dealt with.
4) Have the secondary escort at least one of the parties to a safe area (such as the main office, tell the student where to go if you are the only one present and restraining the other student). Do not try to ascertain blame at this point. You may need to restrain the most out of control student for a short time to prevent a running fight towards the office if you, other students or the out of control student themselves are at risk of harm. Speak in a soothing tone to the student being restrained. As soon as the other student is in a safe zone release the student. Be prepared to restrain the student again if he has not regained control and is at risk of causing further bodily harm. Restraint is a last resort and usually indicates that intervention was too late. Holding a wrist is often sufficient. Usually they will seek somewhere quiet although be mindful of students seeking self harm at this point. Damage to property is repairable, staff and student injuries may not be.
5) Diffuse the audience and escort the remaining student to a team leader or deputy.
Students need you as teacher to be in control. Being calm is a key part of this. Don't do anything extra during a crisis time that is unnecessary to the safety of the students. If you are not able to fulfil your responsibilities in stage 4 then consider the legal ramifications of your actions and the risk of injury to other teachers and students.
I am not a lawyer and suggest this article only as a way to promote discussion within your school. I am not a principal - it is your school executive that will dictate what you may or may not do as a teacher on duty. This is an article purely of opinion and you as a teacher need to decide what you are willing to do in the course of being a teacher.
I'm of reasonably slight build and am considerably smaller than many of the year 11 and 12 students. I'm bigger than many of the female staff also on duty.
So what happens when a fight occurs? How do you, as a teacher, alter an out of control situation when you are physically incapable of stopping students from injuring others and yourself.
The school and how students view the school is a big part of this. I am lucky in that students at our school respect teachers and despite diffusing multiple fights in my career (with male students many times larger than myself and females that had little control over their actions) in all cases my status as a teacher has meant that I have not been at risk. Students seem to know a line that they cannot cross.
Yet I fear this may not always be the case. Students with disabilities are common in school grounds and anecdotal evidence suggest that mainstream students are becoming less able to control their actions.
Practical (not legal) training of staff is necessary before real injury becomes more common. My suggestions are based on practical observation.
1) When on duty stay in line of sight of another teacher on duty. Be prepared to render assistance at short notice. Know the parts of the duty area where you pass from line of sight from one teacher to another.
2) Survey who will take the primary role in diffusing a situation.
3) Issue a command(using full teacher voice) to stop to both parties and (if wise) get between the two students. Hopefully you can skip stage 4 if both students react appropriately. If you are taking the secondary role call for assistance (preferably from a deputy or someone that students are more likely to take seriously.) Seek out the amateur camera people and ensure that they are dealt with.
4) Have the secondary escort at least one of the parties to a safe area (such as the main office, tell the student where to go if you are the only one present and restraining the other student). Do not try to ascertain blame at this point. You may need to restrain the most out of control student for a short time to prevent a running fight towards the office if you, other students or the out of control student themselves are at risk of harm. Speak in a soothing tone to the student being restrained. As soon as the other student is in a safe zone release the student. Be prepared to restrain the student again if he has not regained control and is at risk of causing further bodily harm. Restraint is a last resort and usually indicates that intervention was too late. Holding a wrist is often sufficient. Usually they will seek somewhere quiet although be mindful of students seeking self harm at this point. Damage to property is repairable, staff and student injuries may not be.
5) Diffuse the audience and escort the remaining student to a team leader or deputy.
Students need you as teacher to be in control. Being calm is a key part of this. Don't do anything extra during a crisis time that is unnecessary to the safety of the students. If you are not able to fulfil your responsibilities in stage 4 then consider the legal ramifications of your actions and the risk of injury to other teachers and students.
I am not a lawyer and suggest this article only as a way to promote discussion within your school. I am not a principal - it is your school executive that will dictate what you may or may not do as a teacher on duty. This is an article purely of opinion and you as a teacher need to decide what you are willing to do in the course of being a teacher.
Labels:
discipline
Harry the goat
If anyone missed the Harry the Goat article on the 7.30 report go grab it off the web here.
It's what a 13 year old is capable of.
What a fantastic feel good story that shows the power of imagination.
It's what a 13 year old is capable of.
What a fantastic feel good story that shows the power of imagination.
Labels:
inspiration
Subscribe to:
Posts (Atom)