The big day has arrived and we're getting ready to go to the hospital. Fingers crossed all goes well. We have no idea what we're supposed to do today except wait for contractions to be 10 mins apart... They're about 12 now. Very nervous.
but... Yay!
3D trig, bearings and angle of elevation have proven themselves a stumbling block again. Once a diagram was constructed from a worded problem, students were able to complete problems. Walking students through constructing answers seems to have raised their confidence. Next year I will suggest adding a full lesson to just constructing (not solving) 3D, bearing and angle of elevation problems.
Oblivion has me hooked again. That stupid game is a weekend wrecker.
Monday, February 23, 2009
Saturday, February 21, 2009
A generational approach vs the "me" generations
Baby boomers and generation y have a lot in common. Both have come in boom times and both are self centred. It is no wonder why they have such difficulty relating to their parents and their social responsibility.
I look forward to the slow shift where we look to setting up future generations and not just where we have enough loan capacity available to get our next plasma TV and leaving debt to future generations.
How can you expect to have loving relationships when your personal philosophy centres around the pleasure principle? The decline in birth rate and the fact that couples are deferring having children gives me hope as it indicates that current relationships are starting to understand the responsibility required for a happy family, and seek to raise children in an environment free of the financial and emotional strains seen in their divorced parents. I for one am one of those - paranoid about debt levels and very cautious before embarking on child raising.
For me it was a case of taking a holistic approach to our future, considering all aspects and trying to balance them - career, marriage, security, need for purpose, even spiritual requirements. Teaching was great as it was something I enjoy doing, is relatively stable, has a greater purpose and allows me time with my family +1 - to provide them with support. It did take ten years of work in other industries to be able to afford to enter teaching.. but it was time well spent as I can use this experience to help my students. Hopefully as my child grows up, our well established relationship (now in its 13th year) and my new educational knowledge will help raise a well adjusted and capable child. In time, all that I have will be theirs and hopefully they will continue and mark their mark on the world (I can hear you laughing - but I can try!).
You can't plan for everything, but to not plan is to never fully appreciate success or to learn effectively from failure. Only a few days to go before the baby arrives... Yay!
I look forward to the slow shift where we look to setting up future generations and not just where we have enough loan capacity available to get our next plasma TV and leaving debt to future generations.
How can you expect to have loving relationships when your personal philosophy centres around the pleasure principle? The decline in birth rate and the fact that couples are deferring having children gives me hope as it indicates that current relationships are starting to understand the responsibility required for a happy family, and seek to raise children in an environment free of the financial and emotional strains seen in their divorced parents. I for one am one of those - paranoid about debt levels and very cautious before embarking on child raising.
For me it was a case of taking a holistic approach to our future, considering all aspects and trying to balance them - career, marriage, security, need for purpose, even spiritual requirements. Teaching was great as it was something I enjoy doing, is relatively stable, has a greater purpose and allows me time with my family +1 - to provide them with support. It did take ten years of work in other industries to be able to afford to enter teaching.. but it was time well spent as I can use this experience to help my students. Hopefully as my child grows up, our well established relationship (now in its 13th year) and my new educational knowledge will help raise a well adjusted and capable child. In time, all that I have will be theirs and hopefully they will continue and mark their mark on the world (I can hear you laughing - but I can try!).
You can't plan for everything, but to not plan is to never fully appreciate success or to learn effectively from failure. Only a few days to go before the baby arrives... Yay!
Friday, February 20, 2009
Evidence Based Education
Evidence based education is the new buzzword in education. It shares an acronym similar to OBE but is remarkably different in its approach to teaching. We need to be careful in engaging in more edubabble.
At face value it sounds much like common sense. From what I have read it can be summed up as following: "Systemic change in education should only occur with valid, statistically supported evidence." Today, where information is so easily shared there is no excuse for implementing blanket change without scientifically verified support (note that I don't say pseudo-scientific).
You can imagine some resistance from teachers as this sounds that it might take any chance of introducing new ideas/methods into the classroom. To my mind it is hard to argue against a process that ensures that the majority of what teachers do is tried and tested (if it fails, it is easier to detect why) and new ideas are limited to a small portion of the curriculum/pedagogy.
The prescribed evidence based methods would be imbued at teacher training programmes. This ensures that the majority of teachers perform at least at a minimum standard by being educated in tried and true methods. Higher risk alternative strategies or optimisation strategies would only be used as an adjunct to the tried methods.
As a parent you would not want your child subject to scientifically unsound/unproven educational methods. But.. with proof comes a necessity for time to find proof and judge what level of proof is required. This is not conducive to the immediate political needs for success in education. The issue it brings to mind is, "can the need for evidence march at the pace required for results?" ..and will the need for results and commercial lobbying compromise any findings made to the point of irrelevancy. I suggest this irrelevancy and bastardisation of results is exactly what will happen.
Change of this type would take at least 15 years to develop content, rotate in new teachers, start new students on their 12 year journey and develop change management processes to effectively monitor progress. Can you imagine a minister standing up and saying we will start the process now and if a subsequent government doesn't mess with it, we will have a good educational results in 2024? The long term nature of education is a good reason to divorce it from political imperatives and place it into a vehicle responsible to government via regulated requirements. I imagine this was first intended with the Curriculum council or SEA.
OBE was an opportunity for teaching to become a true profession again with each teacher becoming a curriculum expert for a given range of students. For those trained in the concept and with the necessary horsepower and support to use it, OBE is a wonderful curriculum development tool. Unfortunately the poor implementation of OBE, the tangling of cooperative learning with OBE and issues with assessment/levelling made it an unmitigated disaster in WA.
Despite this it still would be a mistake to throw the baby out with the bathwater.
EBE is the pendulum swinging and brings a new range of opportunities with the pressure of performance on teachers falling back to professional curriculum writers and systemic decision makers. It too needs to be taken with care until any evidence of success justifies the hype.
At face value it sounds much like common sense. From what I have read it can be summed up as following: "Systemic change in education should only occur with valid, statistically supported evidence." Today, where information is so easily shared there is no excuse for implementing blanket change without scientifically verified support (note that I don't say pseudo-scientific).
You can imagine some resistance from teachers as this sounds that it might take any chance of introducing new ideas/methods into the classroom. To my mind it is hard to argue against a process that ensures that the majority of what teachers do is tried and tested (if it fails, it is easier to detect why) and new ideas are limited to a small portion of the curriculum/pedagogy.
The prescribed evidence based methods would be imbued at teacher training programmes. This ensures that the majority of teachers perform at least at a minimum standard by being educated in tried and true methods. Higher risk alternative strategies or optimisation strategies would only be used as an adjunct to the tried methods.
As a parent you would not want your child subject to scientifically unsound/unproven educational methods. But.. with proof comes a necessity for time to find proof and judge what level of proof is required. This is not conducive to the immediate political needs for success in education. The issue it brings to mind is, "can the need for evidence march at the pace required for results?" ..and will the need for results and commercial lobbying compromise any findings made to the point of irrelevancy. I suggest this irrelevancy and bastardisation of results is exactly what will happen.
Change of this type would take at least 15 years to develop content, rotate in new teachers, start new students on their 12 year journey and develop change management processes to effectively monitor progress. Can you imagine a minister standing up and saying we will start the process now and if a subsequent government doesn't mess with it, we will have a good educational results in 2024? The long term nature of education is a good reason to divorce it from political imperatives and place it into a vehicle responsible to government via regulated requirements. I imagine this was first intended with the Curriculum council or SEA.
OBE was an opportunity for teaching to become a true profession again with each teacher becoming a curriculum expert for a given range of students. For those trained in the concept and with the necessary horsepower and support to use it, OBE is a wonderful curriculum development tool. Unfortunately the poor implementation of OBE, the tangling of cooperative learning with OBE and issues with assessment/levelling made it an unmitigated disaster in WA.
Despite this it still would be a mistake to throw the baby out with the bathwater.
EBE is the pendulum swinging and brings a new range of opportunities with the pressure of performance on teachers falling back to professional curriculum writers and systemic decision makers. It too needs to be taken with care until any evidence of success justifies the hype.
Thursday, February 19, 2009
Driving school curriculum
The connection between middle school (yr 8-9) and senior school (yr 10-12) in the absence of a prescribed syllabus is cause for concern. When a teacher is limited by timetabling to only teach middle school classes they can lack the perspective gained by teaching senior school classes.
Similarly teachers only teaching in senior school that never teach lower school classes can lack an understanding of the difficulties in teaching younger students.
Where there are middle schooling practices, we have to be ever so careful to ensure that the "developmental" curriculum in middle school is dovetailed into the prescribed and effectively "streamed" curriculum of senior school... or else we end up with cream puffs with little resilience and disparate understanding/skills.. that when faced with pressure of performance they sadly crumble.
To not have cooperation between the middle and senior school is a real recipe for disaster. By not pushing hard enough in middle school, students find it difficult to adjust to the rigor of senior school. By pushing too hard, students become disengaged and arrive at senior school unable to enter effectively into more demanding subjects.
I believe that the only way forward is with strong leadership and it has to come from the staff that have taught and have an understanding of the needs of all five years. As staff with courses of study experience only exist in the senior school, under normal circumstances, this is where you would normally find that person. Typically they are well respected and driven individuals, although recently they may be feeling disempowered and ill used.
Once that person is identified they need to be empowered with the ability to make change such that the transition of students through middle to senior school promotes the highest outcomes from students. Being part of performance management and hiring processes for learning area staff may be a good idea.
Secondly they need to identify areas of intervention that will improve results across all five years at school, in assessment practices, student motivation, ICT implementation, curriculum development, statistical analysis of results and pedagogy ideas.
Thirdly they need to be a conduit to other learning areas to measure the amount of transference of information, examining opportunities for application of mathematical concepts effectively.
Lastly these leaders need to be recognised for their achievements and clear measurables need to be defined to show them any progress that is made. This needs to be a positive process with focus on success and examining failures for better ways to achieve desired outcomes.
With the dogs breakfast that currently exists with the lack of a current syllabus and the imminent failure of the development of a national curriculum, the opportunity exists to enhance state schools through effective identification of issues and the subsequent change management. State schools that face this challenge head on will avoid the catastrophe that is approaching post the disaster of the OBE implementation.
Similarly teachers only teaching in senior school that never teach lower school classes can lack an understanding of the difficulties in teaching younger students.
Where there are middle schooling practices, we have to be ever so careful to ensure that the "developmental" curriculum in middle school is dovetailed into the prescribed and effectively "streamed" curriculum of senior school... or else we end up with cream puffs with little resilience and disparate understanding/skills.. that when faced with pressure of performance they sadly crumble.
To not have cooperation between the middle and senior school is a real recipe for disaster. By not pushing hard enough in middle school, students find it difficult to adjust to the rigor of senior school. By pushing too hard, students become disengaged and arrive at senior school unable to enter effectively into more demanding subjects.
I believe that the only way forward is with strong leadership and it has to come from the staff that have taught and have an understanding of the needs of all five years. As staff with courses of study experience only exist in the senior school, under normal circumstances, this is where you would normally find that person. Typically they are well respected and driven individuals, although recently they may be feeling disempowered and ill used.
Once that person is identified they need to be empowered with the ability to make change such that the transition of students through middle to senior school promotes the highest outcomes from students. Being part of performance management and hiring processes for learning area staff may be a good idea.
Secondly they need to identify areas of intervention that will improve results across all five years at school, in assessment practices, student motivation, ICT implementation, curriculum development, statistical analysis of results and pedagogy ideas.
Thirdly they need to be a conduit to other learning areas to measure the amount of transference of information, examining opportunities for application of mathematical concepts effectively.
Lastly these leaders need to be recognised for their achievements and clear measurables need to be defined to show them any progress that is made. This needs to be a positive process with focus on success and examining failures for better ways to achieve desired outcomes.
With the dogs breakfast that currently exists with the lack of a current syllabus and the imminent failure of the development of a national curriculum, the opportunity exists to enhance state schools through effective identification of issues and the subsequent change management. State schools that face this challenge head on will avoid the catastrophe that is approaching post the disaster of the OBE implementation.
Sunday, February 15, 2009
Baby Week
We're all prepped and ready for the arrival of our first new family member..
nursery finished... tick
house tidy... tick
bag packed ... tick
classes prepped for next week... tick
baby?? late of course.. just like it's mother.
My wife had a dream that she was having a Labrador. We checked on the ultrasound and it definitely is a baby.
I'll take a day off when it's due and then another four days when they come home. It's all very exciting and we're both looking forward to meeting our new family member. I've been telling my students that we'll name the baby Eunice or Eugene to make sure they grow up a maths dork, study hard, don't get boyfriends/girlfriends until they are at least 40 and make lots of money.
I'm finding the 3A MAT Trig test difficult to construct as my first attempt was too easy. I'll have to beef it up again later.
:-)
nursery finished... tick
house tidy... tick
bag packed ... tick
classes prepped for next week... tick
baby?? late of course.. just like it's mother.
My wife had a dream that she was having a Labrador. We checked on the ultrasound and it definitely is a baby.
I'll take a day off when it's due and then another four days when they come home. It's all very exciting and we're both looking forward to meeting our new family member. I've been telling my students that we'll name the baby Eunice or Eugene to make sure they grow up a maths dork, study hard, don't get boyfriends/girlfriends until they are at least 40 and make lots of money.
I'm finding the 3A MAT Trig test difficult to construct as my first attempt was too easy. I'll have to beef it up again later.
:-)
Tuesday, February 10, 2009
Escalating issues with students.
Some teachers are really good at identifying a potential behaviour issue with a student and bopping it on the head. Occassionally a student just gets a bee in their bonnet and won't let it go. What starts as a shoosh directed at a student when I start my lesson, ends with the student on in-school suspension for multiple misdemeaners during the lesson and ongoing issues for months thereafter.
There's a knack for diffusing students and when I'm concentrating usually I can pick the student and prevent them from doing stupid things. My favourite list of things to prevent these events is as follows:
Sleep well: If I'm tired I'm bound to miss the signs of a student ready to blow and probably respond with less patience than I normally would.
Maintain firm class rules: Respect, responsibility, doing your best.
Look for storm clouds: Student body language on entering the room can give an indication to their mood.
Use of humour: Sometimes a simple laugh can turn a student around.
Check their understanding of the topic: If a student feels hopeless they may compensate with poor behaviour to hide the issue.
Low key responses: Have a range of responses that don't draw attention to the student (eg. hand signals, proximity, diversion, interacting with nearby students, sending on an errand.)
Backup responses: Moving students, talking to them out of class, preventing students sitting near disruptive influences, extra homework, class detention.
If all these fail (and the student continues to disrupt the class) or a critical incident occurs (abuse of teacher/student, uncontrollable anger, damage to school equipment, visibly upset/crying) then upline referral is required - probably resulting in suspension. This of course causes further issues (my estimate) is that it takes 4 days to catch up for every day missed in senior school. The quality of the upline support will dictate how easy it is to re-introduce the student to class and resolve the issue.
When suspension occurs I see this as my failure - albeit sometimes I wish I knew some of the 'confidential' information within the school so that I could modify my responses to errant behaviour accordingly.
There's a knack for diffusing students and when I'm concentrating usually I can pick the student and prevent them from doing stupid things. My favourite list of things to prevent these events is as follows:
Sleep well: If I'm tired I'm bound to miss the signs of a student ready to blow and probably respond with less patience than I normally would.
Maintain firm class rules: Respect, responsibility, doing your best.
Look for storm clouds: Student body language on entering the room can give an indication to their mood.
Use of humour: Sometimes a simple laugh can turn a student around.
Check their understanding of the topic: If a student feels hopeless they may compensate with poor behaviour to hide the issue.
Low key responses: Have a range of responses that don't draw attention to the student (eg. hand signals, proximity, diversion, interacting with nearby students, sending on an errand.)
Backup responses: Moving students, talking to them out of class, preventing students sitting near disruptive influences, extra homework, class detention.
If all these fail (and the student continues to disrupt the class) or a critical incident occurs (abuse of teacher/student, uncontrollable anger, damage to school equipment, visibly upset/crying) then upline referral is required - probably resulting in suspension. This of course causes further issues (my estimate) is that it takes 4 days to catch up for every day missed in senior school. The quality of the upline support will dictate how easy it is to re-introduce the student to class and resolve the issue.
When suspension occurs I see this as my failure - albeit sometimes I wish I knew some of the 'confidential' information within the school so that I could modify my responses to errant behaviour accordingly.
Friday, February 6, 2009
Rewarding effort of teachers
Today I received a certificate from the Principal in recognition of the summer school run for our 3A maths students and a letter of appreciation from the school. It acknowledged the engagement of students in the summer school programme and recognised the planning required for such an undertaking. Our other two mathematics teachers in the senior school received similar commendations. This is really positive performance management (albeit in normal business it would be accompanied with pay rises or condition benefits). One really negative thing about teaching is that career progression requires entry into management roles and lateral "teaching students" advancement is not really catered to (level 3CT is the one exception and IMHO I haven't met a worthy recipient).
It is nice to put another letter/certificate in my Portfolio!
Now for planning next year's summer school with involvement from schools and students in adjacent areas. It would be great to be able to get some publicity/media attention and for participating students to be able to choose lecture/tutorials based on student needs and interests.
On another front, in the senior school we have a rolling class strategy in year 10 math, where each of the three senior school math teachers take turns following a top year ten class through to year 12. The middle school maths programme (developed last year by the senior school) has made a difference already, with the current year 10 coordinator already noticing that our top students have made solid gains in algebra compared to last year. Our middle school teacher has done well by these kids implementing the programme - a real achievement for the school.
The year 11 3A course is moving along swimmingly thus far. Yay!
It is nice to put another letter/certificate in my Portfolio!
Now for planning next year's summer school with involvement from schools and students in adjacent areas. It would be great to be able to get some publicity/media attention and for participating students to be able to choose lecture/tutorials based on student needs and interests.
On another front, in the senior school we have a rolling class strategy in year 10 math, where each of the three senior school math teachers take turns following a top year ten class through to year 12. The middle school maths programme (developed last year by the senior school) has made a difference already, with the current year 10 coordinator already noticing that our top students have made solid gains in algebra compared to last year. Our middle school teacher has done well by these kids implementing the programme - a real achievement for the school.
The year 11 3A course is moving along swimmingly thus far. Yay!
Thursday, February 5, 2009
Absolute value and the 3A MAT course
Ok. Absolute value - easy enough, to take the absolute value of a number, make it positive if it wasn't already. easy peasy...
I started by displaying the graph on the board using the overhead gadget for the Classpad.
I selected Graph&Tab and entered Ix-3I for y1 and I2x+4I for y2. For some reason the graph workpane doesn't allow you to use the 2d tab absolute value option - so use the abs() function under the cat tab in the soft keyboard. When you hit enter it will restore the absolute value notation.
Using the intersect function under analysis in the menu bar we know that the two lines intersect at -1/3 and -7 therefore the interval is x<=-7, x>=-1/3.
Ix-3I<=I2x+4I
I then asked students to draw a number line with the intervals marked and substitute the values back into the original inequality. We numbered the three intervals. The first interval represented x<=-7, the second -7<=x<=-1/3 and the last x>=-1/3.
Here is a link to other CAS calculator posts.
... until you start to look at IyI=IxI and ask students to graph it..
... then ask them to find the intersection of Ix-3I=I2x+4I algebraically
... then ask them to find Ix-3I<=I2x+4I
... then ask them to find Ix-3I<=I2x+4I
Students really bogged down when they reached inequalities. The approach I used was similar to that by Sadler in his 3A book for MAS. The problem was that I really wasn't sure they understood what they were doing.. they could follow the algorithm but understanding was eluding them.
I started by looking at absolute numbers and explained models for solving using number lines, graphing and algebraically. Then I used a composite approach to assist students visualise what it was they were doing with problems like:
Ix-3I<=I2x+4I
I started by displaying the graph on the board using the overhead gadget for the Classpad.
I entered Graph&Tab from the menu workpane (using the menu icon at the base of the workpane).
I selected Graph&Tab and entered Ix-3I for y1 and I2x+4I for y2. For some reason the graph workpane doesn't allow you to use the 2d tab absolute value option - so use the abs() function under the cat tab in the soft keyboard. When you hit enter it will restore the absolute value notation.
Make sure both y1 and y2 are ticked (if they are not place the cursor on the line using your stylus and hit exe). Hit the graph button in the toolbar (the first icon with the top formula pane selected). The following graph should appear:
We then looked at the original inequality again and I asked what did it really mean?
Ix-3I<=I2x+4I
One way of thinking about it was, "when is the graph y=Ix-3I less than or equal to the graph of y=I2x+4I?"
We looked at the graph and found that the part marked red on the line y=Ix-3I satisfied the inequality.
Using the intersect function under analysis in the menu bar we know that the two lines intersect at -1/3 and -7 therefore the interval is x<=-7, x>=-1/3.
We had discussed that we could also do this algebraically by using the property if IxI=IyI then x=y or x=-y to find the points of intersection.
Eg.
Ix-3I<=I2x+4I
x-3 = 2x+4
-x = 7
x=-7
x-3 = -(2x+4)
x-3=-2x-4
3x=-1
x=-1/3
I then asked students to draw a number line with the intervals marked and substitute the values back into the original inequality. We numbered the three intervals. The first interval represented x<=-7, the second -7<=x<=-1/3 and the last x>=-1/3.
We then selected a value within each of the intervals and substituted them into the inequality. If they were true then this indicated values of x that satisfied the inequality.
Ix-3I<=I2x+4I
Interval 1 (x=-8)
I-8-3I<=I2(-8)+4I
I-11I<=I-12I
11<=-12 (true)
Therefore x<=-7 is a valid interval.
Interval 2 (x=-5)
I-5-3I<=I2(-5)+4I
I-8I<=I-6I
8<=-6 (false)
Therefore -7<=x<=-1/3 is not a valid interval.
Interval 3 (x=0)
I-0-3I<=I2(-0)+4I
I-0-3I<=I2(-0)+4I
I-3I<=I6I
3<=6 (true)
Therefore x>=-1/3 is a valid interval.
The inequality Ix-3I<=I2x+4I is valid over x<=-7, x>=-1/3
Drawing students attention from the graph and back to the algebraic representation released the tension in the room, the screwed up faces and suddenly lights went back on.
Thank goodness!
Here is a link to other CAS calculator posts.
Wednesday, February 4, 2009
Eureka.. one problem solved!
Teaching students to solve equations with the balancing method can be difficult as many different skills are required. Collecting like terms, fractions, multiplying pronumerals, dealing with coefficients and the like. When adding all the complexities together students can really struggle.
Surfing around yesterday I found the following link using a classpad calculator to assist students check their understanding of how to use the balancing method. It has worked fabulously well and yr10 students that typically hate algebra (and maths) are all smiles...
Here's the sequence of lessons up to this point (first week of term one)..
I then introduced the balancing method saying we could get to the same result by making x the subject of the equation by examining the LHS and thinking what operation could we do to isolate the x value.
A student suggested that we subtract 5 and I said great.
Then I said to students that the crux of the balancing method was that anything we did on the LHS of the equation has to be done to the RHS. I wrote on the board
x + 5 = 7
x + 5 -5 = 7 - 5
x = 2
and asked how did that compare with our original answer. We then did the following example:
5x + 5 = 20
A student offered the following step:
5x + 5 - 5 = 20 -5
5x = 15
Typically students get stuck at this stage as 5÷5 =1 is not an intuitive step. For once I told them that I would divide by five and showed them how it works.
5x ÷ 5 = 15 ÷ 5 (please excuse the division symbol, I actually used fractions but it is too hard in html)
x = 3
And here's the real magic.. I then took out CAS calculators borrowed from the senior school and they did a number of examples with them. For the following example:
2x - 2 = 15
Their brains started making connections and they actually were using the calculators to check that their logic was correct rather than to give them just answers.
You can see from the example that each step in the calculator mimics the steps to answer the problem on paper. It is easy to see how after each operation (+2, ÷2) x becomes the subject of the equation and ultimately becomes solved with x=8.5
Common errors become obvious earlier. Students decide what operation needs to be done and see what that operation would do. Take this common case:
The student has multiplied by 2 before they have subtracted. They can instantly see their mistake (the LHS of the equation looks more complex rather than simpler so the student starts again. The second attempt subtracting 5 gets them closer to making x the subject of the equation.
For many of us, this is how we learnt to transpose equations - a little trial and error. Lots of practice. Lots of heartache. Lots of looking at the back of the book.
The students found using the calculator fun... and the calculator only gave them guidance - not just solving the answer. It was a mix between the old inverse operations method (change the sign/change the sign that causes all sorts of difficulties when fractional terms/multiple terms are introduced) and the balancing method. To be honest, I've never found the 'scales' explanation that typically accompanies the balancing method useful - but the CAS introduction way I think may have real promise.
The other great thing is that they were recording their answers really well on paper.
For the above example I would see (with equals signs aligned):
2x-2=15
2x-2+2=15+2
2x=17
2x÷2=17÷2
x=8.5
To see mid tier students lay out work like this rather than
1) 8.5
was fantastic.
Here is a link to other CAS calculator posts.
Surfing around yesterday I found the following link using a classpad calculator to assist students check their understanding of how to use the balancing method. It has worked fabulously well and yr10 students that typically hate algebra (and maths) are all smiles...
Here's the sequence of lessons up to this point (first week of term one)..
- review of algebraic terminology
- review of collecting like terms
- review of multiplying algebraic terms
- solving simple equations
I then introduced the balancing method saying we could get to the same result by making x the subject of the equation by examining the LHS and thinking what operation could we do to isolate the x value.
A student suggested that we subtract 5 and I said great.
Then I said to students that the crux of the balancing method was that anything we did on the LHS of the equation has to be done to the RHS. I wrote on the board
x + 5 = 7
x + 5 -5 = 7 - 5
x = 2
and asked how did that compare with our original answer. We then did the following example:
5x + 5 = 20
A student offered the following step:
5x + 5 - 5 = 20 -5
5x = 15
Typically students get stuck at this stage as 5÷5 =1 is not an intuitive step. For once I told them that I would divide by five and showed them how it works.
5x ÷ 5 = 15 ÷ 5 (please excuse the division symbol, I actually used fractions but it is too hard in html)
x = 3
And here's the real magic.. I then took out CAS calculators borrowed from the senior school and they did a number of examples with them. For the following example:
2x - 2 = 15
Their brains started making connections and they actually were using the calculators to check that their logic was correct rather than to give them just answers.
You can see from the example that each step in the calculator mimics the steps to answer the problem on paper. It is easy to see how after each operation (+2, ÷2) x becomes the subject of the equation and ultimately becomes solved with x=8.5
Common errors become obvious earlier. Students decide what operation needs to be done and see what that operation would do. Take this common case:
The student has multiplied by 2 before they have subtracted. They can instantly see their mistake (the LHS of the equation looks more complex rather than simpler so the student starts again. The second attempt subtracting 5 gets them closer to making x the subject of the equation.
For many of us, this is how we learnt to transpose equations - a little trial and error. Lots of practice. Lots of heartache. Lots of looking at the back of the book.
The students found using the calculator fun... and the calculator only gave them guidance - not just solving the answer. It was a mix between the old inverse operations method (change the sign/change the sign that causes all sorts of difficulties when fractional terms/multiple terms are introduced) and the balancing method. To be honest, I've never found the 'scales' explanation that typically accompanies the balancing method useful - but the CAS introduction way I think may have real promise.
The other great thing is that they were recording their answers really well on paper.
For the above example I would see (with equals signs aligned):
2x-2=15
2x-2+2=15+2
2x=17
2x÷2=17÷2
x=8.5
To see mid tier students lay out work like this rather than
1) 8.5
was fantastic.
Here is a link to other CAS calculator posts.
Tuesday, February 3, 2009
Talented students and self confidence
If a talented student at the start of year 11 wants to leave your class what do you do?
It's a question I don't know the answer to and is a difficult one.
Paths I've taken in the past have included:
a) Discuss their choices and investigate their motivation for leaving
b) Direct them to school counsellors
c) Do nothing
Invariably before now I have been sucked into option a). This last time I've decided to do c). Whether it is peer pressure, lack of support from parents, lack of confidence in your abilities, interference from other learning areas to bolster numbers, laziness or poor work ethic; students feel compelled to make changes at the start of year 11. It will be interesting to see what they will do. I know though that I can't in all honesty tell them that they will pass if they think the grass is greener elsewhere. I'd much rather have those students that are enjoying themselves in the new courses.
The pall cast by students wishing to leave really dampens my enjoyment of classes as I was really looking forward to working positively with them. I suppose it's just the ups and downs of working with adolescents.
The main course affected seems to be the year 11 3ab MAS course. The introduction of new content seems to have spooked a few students. I am concerned that the MAT only students this year will struggle in 3C MAT next year without the additional practice provided by 3AB MAS. It is only guesswork at this stage.
It's a question I don't know the answer to and is a difficult one.
Paths I've taken in the past have included:
a) Discuss their choices and investigate their motivation for leaving
b) Direct them to school counsellors
c) Do nothing
Invariably before now I have been sucked into option a). This last time I've decided to do c). Whether it is peer pressure, lack of support from parents, lack of confidence in your abilities, interference from other learning areas to bolster numbers, laziness or poor work ethic; students feel compelled to make changes at the start of year 11. It will be interesting to see what they will do. I know though that I can't in all honesty tell them that they will pass if they think the grass is greener elsewhere. I'd much rather have those students that are enjoying themselves in the new courses.
The pall cast by students wishing to leave really dampens my enjoyment of classes as I was really looking forward to working positively with them. I suppose it's just the ups and downs of working with adolescents.
The main course affected seems to be the year 11 3ab MAS course. The introduction of new content seems to have spooked a few students. I am concerned that the MAT only students this year will struggle in 3C MAT next year without the additional practice provided by 3AB MAS. It is only guesswork at this stage.
Subscribe to:
Posts (Atom)