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.
Sunday, February 27, 2011
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.
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.
Catering for gifted students
Catering for gifted students is one of the hardest parts of the job. These kids have been haphazardly accelerated in various topics resulting in them blitzing through some topics and requiring high levels of assistance at other times ahead of students in the normal programme.
It is near on impossible to cater for these students in a true heterogenous classroom as a beginning teacher. There is no possible way that a starting teacher has the skills to run multiple programmes in a room and diagnose issues for these students in a just-in-time manner. An experienced teacher can do it (with difficulty) but a beginning teacher cannot.
An analogy is the best possible way of explaining what I have come across.
Each child in the room has the combined computing power of every computer in the world today combined (there was a great article on this found via /. the other day). I would not expect a just graduated four year programmer to produce a programme that would optimise throughput via every computer in the world.
Yet we regularly ask 1st year out teachers to create optimised programmes (and IEPS)that cater for thirty such brains with 30 times our current worldwide computing capacity. Let's face facts.. the only reason teaching works is that over the last 2000 years we have stumbled across some methods that make the world more understandable for these underdeveloped intelligences.
And here we are again not giving baseline programmes to these graduate teachers. The national curriculum has failed to deliver something easily usable and assessible in the classroom (are we in education forever destined to repeat mistakes - maybe it was the lack of History in classrooms over an extended period??). I was very critical of the lack of production by the maths TDC's but at least at the end they tried to produce something for the classroom that could be modified to suit a learning environment.
As teachers in the system for some time, we need to be constantly aware of new teachers that will need our help and guidance - hopefully willingly, and sometimes reluctantly. Those 2000 years of education have some parts baby that shouldn't be thrown out with the bathwater.
We place our gifted students at risk every time they enter a classroom of where we do not cater to their needs. Without the need to strive, they coast, get lazy or find a private school that will cater to their needs (check to see if your school has a year nine exodus and then ask what is being done about it). We need to be careful that good teachers that need support are given it, students are optimally taught and environments are created that promote the benefits of learning.
I'm currently pointing the finger at middle schools over catering to pastoral needs and the national curriculum intent to remove the ability to provide developmentally appropriate classes in WA senior schools.
It is near on impossible to cater for these students in a true heterogenous classroom as a beginning teacher. There is no possible way that a starting teacher has the skills to run multiple programmes in a room and diagnose issues for these students in a just-in-time manner. An experienced teacher can do it (with difficulty) but a beginning teacher cannot.
An analogy is the best possible way of explaining what I have come across.
Each child in the room has the combined computing power of every computer in the world today combined (there was a great article on this found via /. the other day). I would not expect a just graduated four year programmer to produce a programme that would optimise throughput via every computer in the world.
Yet we regularly ask 1st year out teachers to create optimised programmes (and IEPS)that cater for thirty such brains with 30 times our current worldwide computing capacity. Let's face facts.. the only reason teaching works is that over the last 2000 years we have stumbled across some methods that make the world more understandable for these underdeveloped intelligences.
And here we are again not giving baseline programmes to these graduate teachers. The national curriculum has failed to deliver something easily usable and assessible in the classroom (are we in education forever destined to repeat mistakes - maybe it was the lack of History in classrooms over an extended period??). I was very critical of the lack of production by the maths TDC's but at least at the end they tried to produce something for the classroom that could be modified to suit a learning environment.
As teachers in the system for some time, we need to be constantly aware of new teachers that will need our help and guidance - hopefully willingly, and sometimes reluctantly. Those 2000 years of education have some parts baby that shouldn't be thrown out with the bathwater.
We place our gifted students at risk every time they enter a classroom of where we do not cater to their needs. Without the need to strive, they coast, get lazy or find a private school that will cater to their needs (check to see if your school has a year nine exodus and then ask what is being done about it). We need to be careful that good teachers that need support are given it, students are optimally taught and environments are created that promote the benefits of learning.
I'm currently pointing the finger at middle schools over catering to pastoral needs and the national curriculum intent to remove the ability to provide developmentally appropriate classes in WA senior schools.
Thursday, February 17, 2011
Fractions
My emphasis for the last week has been on establishing an idea of "one" with my year 9 academic class. We examined how our idea of one influences how we deal with fractions and algebra.
Firstly we looked at common denominator problems and examined in more detail the method for adding fractions with different denominators.
A common idea is to find common multiples or factors of the denominator and then multiply both the numerator and denominator of the fractions until common denominators are found.
eg. 1/2 + 1/3 -> common denominator of 6 (LCM of 2 and 3)
We then need to find equivalent fractions with denominators of six.
eg 1/2 x 3/3 = 3/6
1/3 x 2/2 = 2/6
Now we have common denominators we can add the fractions..
eg 2/6 + 3/6 = 5/6
But.. why does multiplying by 2/2 and 3/3 work??? Understanding "One" is the answer!!!
1/2 x 1 = 1/2
3/3 = 1
Therefore by substitution 1/2 x 3/3 is just multiplying 1/2 by one. Any number multiplied by one is equal to the original value thus any resulting fraction must be equal to 1/2!
This illustrates two different ideas related to one.. "Multiplying by One" and "Dividing a number by itself".
We also looked at cancelling and why it works..
2m / 3m, we commonly use the skill cancel the m's and 2/3 is what is left.
By re-examining how multiplication works with fractions we find that we can rewrite
2m/3m
as
2/3 x m/m
..but we know that anything divided by itself is 1 (other than zero of course!)
Therefore we can simplify to
2/3 x 1
and we know that anything multiplied by one is equal to the original value.... thus we can see why cancelling works..
Quite a fun little lesson.
Russ.
Firstly we looked at common denominator problems and examined in more detail the method for adding fractions with different denominators.
A common idea is to find common multiples or factors of the denominator and then multiply both the numerator and denominator of the fractions until common denominators are found.
eg. 1/2 + 1/3 -> common denominator of 6 (LCM of 2 and 3)
We then need to find equivalent fractions with denominators of six.
eg 1/2 x 3/3 = 3/6
1/3 x 2/2 = 2/6
Now we have common denominators we can add the fractions..
eg 2/6 + 3/6 = 5/6
But.. why does multiplying by 2/2 and 3/3 work??? Understanding "One" is the answer!!!
1/2 x 1 = 1/2
3/3 = 1
Therefore by substitution 1/2 x 3/3 is just multiplying 1/2 by one. Any number multiplied by one is equal to the original value thus any resulting fraction must be equal to 1/2!
This illustrates two different ideas related to one.. "Multiplying by One" and "Dividing a number by itself".
We also looked at cancelling and why it works..
2m / 3m, we commonly use the skill cancel the m's and 2/3 is what is left.
By re-examining how multiplication works with fractions we find that we can rewrite
2m/3m
as
2/3 x m/m
..but we know that anything divided by itself is 1 (other than zero of course!)
Therefore we can simplify to
2/3 x 1
and we know that anything multiplied by one is equal to the original value.... thus we can see why cancelling works..
Quite a fun little lesson.
Russ.
Saturday, February 12, 2011
Moderation - advice for new players.
Moderation is the local equivalent of peer assessment at a teacher level. If your class is small (less than 12), it is assumed that it is too difficult to give fair grades thus you need to find other small schools to check your grades against. If you are having trouble locating a group tell your HoD/TiC then contact the curriculum council.
Moderation sounds like a pain (and it is) but there is one major advantage. Generally, not always, when you do this you share assessment. This means that you may only need to write half (or a third/less depending on the number of schools involved in your group) of the assessment for the course. If your group has teachers that are organised it can create some great discussion and access to course materials that are often hard to find (such as investigations). Sometimes teachers are not organised, are difficult by nature or have a different opinion to you as to the content and difficulty level of assessment. When they are a combination of these you end up with conflict. Especially if assessment is given late and other participants do not have time to check the difficulty level and breadth of assessment. This is reasonably rare and you can always decline letting them into your next small group. It's in nobody's interest to have a slacker in your group. If you are the slacker for a good reason (such as sickness at home or an unrealistic load at school) then make sure you nurture a good relationship with the rest of the group. Don't let the resentment fester.
If you are terrible at investigations (I own up to this one, I rarely get the difficulty level right), then ask for a later investigation in the year and start now, using your mentor teacher as a guide for where to go with the project. Hunt around for one that hasn't been done for a few years at your school. There are some fantastic investigations being dreamed up at the moment as teachers are finally finding that they have more time with courses bedding down.
Last but not least are the technical issues. Sort out whether you are running concurrent or sequential. Ensure that you know what the weightings are for each assessment and where the marks are coming from (take home and/or validation). Check if notes or calculators are allowed in each assessment. Send your marks to all members of your group and check where your students lie - this will change your approach during semester. Agree on grade cutoffs for semester 1 well before the end of term 3.
Have Fun.
Russ.
Moderation sounds like a pain (and it is) but there is one major advantage. Generally, not always, when you do this you share assessment. This means that you may only need to write half (or a third/less depending on the number of schools involved in your group) of the assessment for the course. If your group has teachers that are organised it can create some great discussion and access to course materials that are often hard to find (such as investigations). Sometimes teachers are not organised, are difficult by nature or have a different opinion to you as to the content and difficulty level of assessment. When they are a combination of these you end up with conflict. Especially if assessment is given late and other participants do not have time to check the difficulty level and breadth of assessment. This is reasonably rare and you can always decline letting them into your next small group. It's in nobody's interest to have a slacker in your group. If you are the slacker for a good reason (such as sickness at home or an unrealistic load at school) then make sure you nurture a good relationship with the rest of the group. Don't let the resentment fester.
If you are terrible at investigations (I own up to this one, I rarely get the difficulty level right), then ask for a later investigation in the year and start now, using your mentor teacher as a guide for where to go with the project. Hunt around for one that hasn't been done for a few years at your school. There are some fantastic investigations being dreamed up at the moment as teachers are finally finding that they have more time with courses bedding down.
Last but not least are the technical issues. Sort out whether you are running concurrent or sequential. Ensure that you know what the weightings are for each assessment and where the marks are coming from (take home and/or validation). Check if notes or calculators are allowed in each assessment. Send your marks to all members of your group and check where your students lie - this will change your approach during semester. Agree on grade cutoffs for semester 1 well before the end of term 3.
Have Fun.
Russ.
Tuesday, February 8, 2011
Bullying
In a school with strong personalities, bullying can be a real problem. Typically physical bullying with the boys and psychological bullying with the girls. Bullying can and does break good students. A success story of our school is the lack of bullying despite public perception.
It is one area of the school where the middle school and the counselling group excels. The kids that come through to the senior school typically aren't bullies; those that try get counselled to death and the source of their bullying painfully exposed. I can't imagine being told "you are a bully and you need to have a look at yourself" is a wonderful experience.
There is always room for improvement. Especially with new kids. Assimilation can be tenuous at time especially in well settled groups. Each teacher needs to be conscious of isolates within a class and subtly discourage them. Each teacher needs to be conscious of niggles that rise during the year. Each year an issue defines a group: race issues, bitchiness, physical agression, complacency, lack of work ethic, teacher conflicts, lower than expected performance. How we deal with those issues makes or breaks a year group.
A nice thing is that regardless smart students at our school are looked up to - there are safe areas in the school for them, for the weird kids, for the popular kids, for the sporty kids. Inside a class anyone can answer a question without fear of a smartness stigma. Amongst all the "over" worldliness of our kids is an innocence that comes with a lack of funds and a questionable future. There are few students that have a future guaranteed by a parent's bank account. Education is one pathway out of the poverty trap. It's a source of pure hope.
It's a real responsibility to find a pathway for this hope through education into the workforce for each of our kids, whether VET or TEE and we all have a part to play.
It is one area of the school where the middle school and the counselling group excels. The kids that come through to the senior school typically aren't bullies; those that try get counselled to death and the source of their bullying painfully exposed. I can't imagine being told "you are a bully and you need to have a look at yourself" is a wonderful experience.
There is always room for improvement. Especially with new kids. Assimilation can be tenuous at time especially in well settled groups. Each teacher needs to be conscious of isolates within a class and subtly discourage them. Each teacher needs to be conscious of niggles that rise during the year. Each year an issue defines a group: race issues, bitchiness, physical agression, complacency, lack of work ethic, teacher conflicts, lower than expected performance. How we deal with those issues makes or breaks a year group.
A nice thing is that regardless smart students at our school are looked up to - there are safe areas in the school for them, for the weird kids, for the popular kids, for the sporty kids. Inside a class anyone can answer a question without fear of a smartness stigma. Amongst all the "over" worldliness of our kids is an innocence that comes with a lack of funds and a questionable future. There are few students that have a future guaranteed by a parent's bank account. Education is one pathway out of the poverty trap. It's a source of pure hope.
It's a real responsibility to find a pathway for this hope through education into the workforce for each of our kids, whether VET or TEE and we all have a part to play.
Monday, February 7, 2011
NAPLAN preparation
There are lots of times you are surprised as a teacher. Today I did some NAPLAN revision of decimal numbers with my year 9 class. It really surprised me how difficult students find the concept of decimal numbers.
Here's something to try with your child.
Draw a number line and place 4.5 at one end and 4.6 at the other.
Place a marker in the middle and ask your child what number would go there.
The answer is 4.55 and many students may get this right, but many would not be 100% sure.
Split the number line again so that it is now in four equal sections. Ask your student to label the new sections.
You may get a wide variety of answers and weird looks.
The answer is 4.5, 4.525, 4.55, 4.575 and 4.6
If your child cannot do this they are not alone. Try again using whole numbers and break it into ten equal sections. Try asking for points between intervals.
Errors like these indicate an issue with both division and place value. It can easily be remedied with some place value exercises (to check if they understand that 4.6 is bigger than 4.59), some estimation exercises (to check if their answers are feasible/reasonable), determining how to find the width of set intervals (using division), learning how to add on intervals and how to find midpoints of intervals.
Here's something to try with your child.
Draw a number line and place 4.5 at one end and 4.6 at the other.
Place a marker in the middle and ask your child what number would go there.
The answer is 4.55 and many students may get this right, but many would not be 100% sure.
Split the number line again so that it is now in four equal sections. Ask your student to label the new sections.
You may get a wide variety of answers and weird looks.
The answer is 4.5, 4.525, 4.55, 4.575 and 4.6
If your child cannot do this they are not alone. Try again using whole numbers and break it into ten equal sections. Try asking for points between intervals.
Errors like these indicate an issue with both division and place value. It can easily be remedied with some place value exercises (to check if they understand that 4.6 is bigger than 4.59), some estimation exercises (to check if their answers are feasible/reasonable), determining how to find the width of set intervals (using division), learning how to add on intervals and how to find midpoints of intervals.
Friday, February 4, 2011
Fractions and year 10
We're reviewing fractions and my academic 10's sheepishly owned up to not being confident at fractions. The issue was traced back to poor tables (without it students get hopelessly stuck with LCD methods).
PARENTS NOTE: TEACH YOUR CHILDREN TABLES.
I'm shouting because it's seemingly not PC to rote learn anything. It is hard to get this message heard. People are too busy to do the little things. Curriculum is too full to teach tables in lower school (nonsense), parents are working multiple jobs and don't have time (you can't afford to not find the time), students are too lazy (they have always been too lazy, this hasn't changed), students have little discipline. We are setting students up to fail if we don't take minimum effort to assist them learn key content.
Anyhow, the second element of students not knowing fractions is a lack of actual teaching of what fractions are and how they work. After 60 mins of learning time they could add subtract and multiply fractions and there were a lot of happier students in the room. Here's the method I used.
I started by drawing two objects, one in halves, one with two quarters (colouring in the selected parts) and described fractions as a way of describing the proportion of an object selected. Both objects were the same size and were split into equal parts. I wrote 1/2 and 2/4 (vertically) next to the objects and discussed numerators were the parts selected and denominators were the number of equal parts in each object
I then asked students what would happen if I added the two objects. Students responded that I would have a whole of an object. This was good as it indicated that they had some understanding of a fraction. We discussed how we would expect 2/2 and 4/4 for a whole.
I then added the numerators and denominators and students could see that this was wrong (3/6). I drew what 3/6 would look like.
I then split the 1/2 into quarters and relabelled the 1/2 object 2/4. We talked about equivalent fractions and lowest common multiples at some length.
I then added the numerator and denominators again. This time we had 4/6. I drew this. It was still wrong. Students pointed out not to add the denominators. We noted that adding denominators made no sense as the denominator described the number of parts. Good! We now had 4/4.
We then talked about multiplication. They were happy to accept that to multiply fractions, multiply the numerators and multiply the denominators.
Now we discussed the effect of multiplying by one, how 2/2, 3/3, 4/4 was really one; and used this fact and multiplication to construct equivalent fractions. I pointed out that without tables it was difficult to find lowest common multiples or factors (for denominators) and that simplifying large fractions was a poor alternative for knowing multiples and factors. We then looked back at the cross multiplication method that many had been taught and how that aligned with what we were doing.
Students completed 60 questions of increasing difficulty. All completed working and checked their own answers. Note that there was no "fractions" specific method (such as cross multiplication and lowest common denominator) used here. It simply flowed from their own mathematical understandings.
Finally we discussed that order was important with subtraction. Division was left for another lesson. Formal notes were then given. 60 mins. Happy faces. Job done. Tick.
I'm not saying that this would work with students that have no understanding of fractions. I am saying that proper consolidation of teaching done in upper primary and lower secondary is not difficult with average students and this topic.
The trick will be to consolidate this in algebra, indices and trigonometry topics so that key concepts are not lost in future.
Russ.
PARENTS NOTE: TEACH YOUR CHILDREN TABLES.
I'm shouting because it's seemingly not PC to rote learn anything. It is hard to get this message heard. People are too busy to do the little things. Curriculum is too full to teach tables in lower school (nonsense), parents are working multiple jobs and don't have time (you can't afford to not find the time), students are too lazy (they have always been too lazy, this hasn't changed), students have little discipline. We are setting students up to fail if we don't take minimum effort to assist them learn key content.
Anyhow, the second element of students not knowing fractions is a lack of actual teaching of what fractions are and how they work. After 60 mins of learning time they could add subtract and multiply fractions and there were a lot of happier students in the room. Here's the method I used.
I started by drawing two objects, one in halves, one with two quarters (colouring in the selected parts) and described fractions as a way of describing the proportion of an object selected. Both objects were the same size and were split into equal parts. I wrote 1/2 and 2/4 (vertically) next to the objects and discussed numerators were the parts selected and denominators were the number of equal parts in each object
I then asked students what would happen if I added the two objects. Students responded that I would have a whole of an object. This was good as it indicated that they had some understanding of a fraction. We discussed how we would expect 2/2 and 4/4 for a whole.
I then added the numerators and denominators and students could see that this was wrong (3/6). I drew what 3/6 would look like.
I then split the 1/2 into quarters and relabelled the 1/2 object 2/4. We talked about equivalent fractions and lowest common multiples at some length.
I then added the numerator and denominators again. This time we had 4/6. I drew this. It was still wrong. Students pointed out not to add the denominators. We noted that adding denominators made no sense as the denominator described the number of parts. Good! We now had 4/4.
We then talked about multiplication. They were happy to accept that to multiply fractions, multiply the numerators and multiply the denominators.
Now we discussed the effect of multiplying by one, how 2/2, 3/3, 4/4 was really one; and used this fact and multiplication to construct equivalent fractions. I pointed out that without tables it was difficult to find lowest common multiples or factors (for denominators) and that simplifying large fractions was a poor alternative for knowing multiples and factors. We then looked back at the cross multiplication method that many had been taught and how that aligned with what we were doing.
Students completed 60 questions of increasing difficulty. All completed working and checked their own answers. Note that there was no "fractions" specific method (such as cross multiplication and lowest common denominator) used here. It simply flowed from their own mathematical understandings.
Finally we discussed that order was important with subtraction. Division was left for another lesson. Formal notes were then given. 60 mins. Happy faces. Job done. Tick.
I'm not saying that this would work with students that have no understanding of fractions. I am saying that proper consolidation of teaching done in upper primary and lower secondary is not difficult with average students and this topic.
The trick will be to consolidate this in algebra, indices and trigonometry topics so that key concepts are not lost in future.
Russ.
Attacking a subject
I always tell my students to attack a subject and it worries me when I get a class of passive students - especially in stage three courses.
Students that are attacking a course:
a) come in bright eyed and bushy tailed
b) are on time
c) have all of their resources (books, calculators, pens ...) ready on day 1
d) attend regularly
e) have pre-read the chapters
f) get stuck into their coursework and are not afraid to have a go
g) natter about their current question with other students
Students that wait to be prompted and expect to be spoonfed, wait for you to find that they are stuck and look like deers in headlights make me concerned. Students that seek personal information from the teacher, natter during instruction, dawdle in late, are disrupting the whole class with nonsense annoy me. They make me think "Is this student in the right place?". This is after all senior school, the pointy end of education.
My 9's, 10's and 2C course are going gangbusters. They demand notes on everything. They attempt questions that I haven't asked them to do as well as the ones I have. They are working on revision books. They are playing with their calculators. Good for them.
My 1B's and 3A courses are another story. Where's the ego? Where's the work ethic? Where is the focus? Hopefullly they're more awake next lesson.
Students that are attacking a course:
a) come in bright eyed and bushy tailed
b) are on time
c) have all of their resources (books, calculators, pens ...) ready on day 1
d) attend regularly
e) have pre-read the chapters
f) get stuck into their coursework and are not afraid to have a go
g) natter about their current question with other students
Students that wait to be prompted and expect to be spoonfed, wait for you to find that they are stuck and look like deers in headlights make me concerned. Students that seek personal information from the teacher, natter during instruction, dawdle in late, are disrupting the whole class with nonsense annoy me. They make me think "Is this student in the right place?". This is after all senior school, the pointy end of education.
My 9's, 10's and 2C course are going gangbusters. They demand notes on everything. They attempt questions that I haven't asked them to do as well as the ones I have. They are working on revision books. They are playing with their calculators. Good for them.
My 1B's and 3A courses are another story. Where's the ego? Where's the work ethic? Where is the focus? Hopefullly they're more awake next lesson.
Wednesday, February 2, 2011
Multiplying and dividing by powers of 10
I had my academic year nine class for the first time today and had a lot of fun. I had been warned about a few students, but they were arms deep in the trenches having a good go.
I took an experimental approach today with the class examining how to multiply and divide powers of 10. The idea was to create algorithms in student terms for solving simple equations.
We started with simple examples using whole numbers
5 x 10 = ...
"When multiplying by 10, 100, 1000... count the zeroes and put them on the end of the number you are multiplying."
25 ÷ 10 = ...
This time students considered the position of the decimal point:
"Count the zeroes in the number after the division symbol [divisor] and move the decimal point right of the other number [the dividend] that many digits"
We then looked at the case
2.5 x 10 =
and discovered our first algorithm didn't work as by our first algorithm 2.5 x 10 = 2.50
This lead us to a similar algorithm as for our division case:
"Count the zeroes in the multiplier and move the decimal point left of the other number [the factor] that many digits"
Using the whole number cases gave students an additional method of multiplying powers of ten than the messy loops moving decimal places method. The idea of this lesson was not to deny them mathematical language - but to give them an opportunity to explore a mathematical concept before formal language was introduced. It was a lot of fun for me and engaged them during the lesson.
We then looked at a few cases where the multiplier and divisor were not powers of 10. This exposed that students had difficulty with long division and long multiplication and were over dependent on calculators - which has the possibility of causing issues in non-calculator sections in upper school. We'll now go own to examining factor trees and ease into indices.
We also looked at 250 ÷ 10 where we converted the expression to a fraction and cancelled the zeroes - although we didn't consider why this works and will need to revisit it later.
I took an experimental approach today with the class examining how to multiply and divide powers of 10. The idea was to create algorithms in student terms for solving simple equations.
We started with simple examples using whole numbers
5 x 10 = ...
"When multiplying by 10, 100, 1000... count the zeroes and put them on the end of the number you are multiplying."
25 ÷ 10 = ...
This time students considered the position of the decimal point:
"Count the zeroes in the number after the division symbol [divisor] and move the decimal point right of the other number [the dividend] that many digits"
We then looked at the case
2.5 x 10 =
and discovered our first algorithm didn't work as by our first algorithm 2.5 x 10 = 2.50
This lead us to a similar algorithm as for our division case:
"Count the zeroes in the multiplier and move the decimal point left of the other number [the factor] that many digits"
Using the whole number cases gave students an additional method of multiplying powers of ten than the messy loops moving decimal places method. The idea of this lesson was not to deny them mathematical language - but to give them an opportunity to explore a mathematical concept before formal language was introduced. It was a lot of fun for me and engaged them during the lesson.
We then looked at a few cases where the multiplier and divisor were not powers of 10. This exposed that students had difficulty with long division and long multiplication and were over dependent on calculators - which has the possibility of causing issues in non-calculator sections in upper school. We'll now go own to examining factor trees and ease into indices.
We also looked at 250 ÷ 10 where we converted the expression to a fraction and cancelled the zeroes - although we didn't consider why this works and will need to revisit it later.
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