An interesting question was posed to me today.
How do I subtract two number manually when the answer is negative??
For instance, 3896 - 4321 (to which the answer is -425).
I originally set up the problem in vertical columns
3896
4321
------
and tried to subtract..
3896
4321
------
?575
which obviously does not work.
So I thought about it.. the only obvious solution was to say, when subtracting always put the larger number on top.
4321
3896
------
*425
As this answer is positive, it is still incorrect. It requires an additional rule, that when the order is changed, the sign of the answer is negative. Thus the answer is -425.
I'm sure everyone knows this (and it's just one of those odd cases I haven't come across before), but it could be an interesting short investigation for upper primary or lower secondary doing directed number exercises.
.. and I have a stupid cold, my nose is dripping like a tap and I can't hug my daughter. It's made my day!
Thursday, February 25, 2010
Saturday, February 20, 2010
Evidence based education vs OBE
Educational trends tend to go in cycles. From ultra conservative, tried and true methods (such as direct instruction from defined syllabus) to ultra experimental (such as the whole of language approach).
Recovering from the ultra experimental 'OBE' we are now heading towards the ultra conservative 'evidence based' approach.
Although the evidence based approach has merits and is a very attractive alternative after OBE, I would suggest caution. The consequences of evidence based education is already starting to slow educational change through the inability of educational practices to change in time with social change (by the time evidence is gathered, social change has again occurred).
Current practice would be to identify an educational need, and then find a current practice (with evidence) to use to fulfil this need. The obvious issue with this is that where we have a new social situation, no evidence exists and with current research practices - no evidence will ever exist as typically research today does not seek to find a solution, only observe existing practice (existing practice which we know is flawed or wouldn't require research).
Has the pendulum swung too far, now stifling the innovative approaches that could be researched and widely implemented? To avoid this I think a middle ground needs to be found, where innovative practices are encouraged and then researched before extensive implementation. To have one without the other is to invite poor practices or stifling of positive change.
Recovering from the ultra experimental 'OBE' we are now heading towards the ultra conservative 'evidence based' approach.
Although the evidence based approach has merits and is a very attractive alternative after OBE, I would suggest caution. The consequences of evidence based education is already starting to slow educational change through the inability of educational practices to change in time with social change (by the time evidence is gathered, social change has again occurred).
Current practice would be to identify an educational need, and then find a current practice (with evidence) to use to fulfil this need. The obvious issue with this is that where we have a new social situation, no evidence exists and with current research practices - no evidence will ever exist as typically research today does not seek to find a solution, only observe existing practice (existing practice which we know is flawed or wouldn't require research).
Has the pendulum swung too far, now stifling the innovative approaches that could be researched and widely implemented? To avoid this I think a middle ground needs to be found, where innovative practices are encouraged and then researched before extensive implementation. To have one without the other is to invite poor practices or stifling of positive change.
Labels:
educational output,
evidence based education,
OBE
Friday, February 19, 2010
Time continued...
We were working on applying time calculations today, so I posed a question:
"If [student A] was given detention for 1.4 hours and [student B] was given detention 1 hour 25 minutes detention, who would be in detention the longest?"
Students had a guess and then they reviewed the caterpillar for converting between time units.
We then did a number of calculations with some templates to show how a calculation could be constructed.
Eg
3.4 hours = _______ x ________ mins
= ______________ mins
2 122 131 sec = ________ ÷ _________ ÷ _______ ÷ _______ days
= ____ days
1 hour 20 mins = ________ x _________ + ________ mins
= ______________ mins
After we did that, students were just given a range of questions to solve without the templates.
Eg.
2.8 hours = ___________ minutes
12 hrs 12 minutes = ______ hours
12 hrs 12 minutes = ______ days
Then we revisited our original detention problem and a range of similar problems.
Students then practiced with math-joke type connect-the-answer-with-the-question exercise (the old worksheet with a bad, bad mathematics joke at the bottom to solve). Students were able to solve the majority of problems.
yay!
There's nothing to say that with a stronger group I couldn't have taught the same topic by teaching basic time facts (such as 60sec = 1 minute) and then relied on their application of multiplication and division, but in this case I'm glad I didn't do that, the look on the faces of my students when they realised time calculations made sense (that they had found difficult over a long period) was priceless.
Labels:
number
Wednesday, February 17, 2010
Teaching Elapsed Time
Teaching time is always a little problematic with a class, as some students will have this well and truly conquered by year 10 and others will struggle.
Elapsed time is a difficult topic for many as it drags in a lot of sub topics. With each step it is important to draw student's attention to possible mistakes and also to any parallels with an analogue clock.
A common method is to find the number of hours elapsed and then add the remaining minutes on either side (eg. for 2.14 to 4.15: 2.14 -> 3.00 -> 4.00 -> 4.15 would be 46min + 1 hour + 15 mins = 2 hours 1 minute)
The usual approach is to
a) draw a number line
Issues: Students don't relate a number line with time, and commonly place decimal marks (eg. 10 between each hour) rather than 12 (for 5 minute intervals).
b) place the start and finish time on the number line.
Issues: Students don't realise that the start time and end time have to be placed in that order. Eg. if the start time is 8am and the end time is 7am they want to put 7am first on the number line.
c) mark on the hour after the start time and the hour before the start time
Issues: Students have difficulty adding the two times inside the interval. If 7.30am is the start time, they might add 7.00am instead of 8am or for a 4.30 finish time they might add 5.00pm or 3pm.
d) mark on midday and midnight if they lie between the start and finish time
Issues: This is problematic especially with times over 12 hours where both midday and midnight are involved. Students are often not sure whether 12pm or 12am is midday or midnight. They also get confused moving from 12am to 1 am (counter-intuitive).
e) calculate the time between each number on the timeline
Issues: This is the bugbear of the exercise. Students are not sure of the answer counting up to the nearest hour and counting back to the previous hour. Eg Finding the time between 1.17am and 2am or 4.00pm and 4.55pm. Many issues here are related to issues in part c)
f) add the elapsed times
Issues: Students write times such as 7hrs 85 minutes not realising 85mins is greater than an hour.
An alternate approach is to go up in hours and add the remainder (eg. 2.14-> 3.14 -> 4.14 -> 4.15 = 2 hours 1 minute). This may help struggling students and reduce the amount of calculation required.
Elapsed time is a difficult topic for many as it drags in a lot of sub topics. With each step it is important to draw student's attention to possible mistakes and also to any parallels with an analogue clock.
A common method is to find the number of hours elapsed and then add the remaining minutes on either side (eg. for 2.14 to 4.15: 2.14 -> 3.00 -> 4.00 -> 4.15 would be 46min + 1 hour + 15 mins = 2 hours 1 minute)
The usual approach is to
a) draw a number line
Issues: Students don't relate a number line with time, and commonly place decimal marks (eg. 10 between each hour) rather than 12 (for 5 minute intervals).
b) place the start and finish time on the number line.
Issues: Students don't realise that the start time and end time have to be placed in that order. Eg. if the start time is 8am and the end time is 7am they want to put 7am first on the number line.
c) mark on the hour after the start time and the hour before the start time
Issues: Students have difficulty adding the two times inside the interval. If 7.30am is the start time, they might add 7.00am instead of 8am or for a 4.30 finish time they might add 5.00pm or 3pm.
d) mark on midday and midnight if they lie between the start and finish time
Issues: This is problematic especially with times over 12 hours where both midday and midnight are involved. Students are often not sure whether 12pm or 12am is midday or midnight. They also get confused moving from 12am to 1 am (counter-intuitive).
e) calculate the time between each number on the timeline
Issues: This is the bugbear of the exercise. Students are not sure of the answer counting up to the nearest hour and counting back to the previous hour. Eg Finding the time between 1.17am and 2am or 4.00pm and 4.55pm. Many issues here are related to issues in part c)
f) add the elapsed times
Issues: Students write times such as 7hrs 85 minutes not realising 85mins is greater than an hour.
An alternate approach is to go up in hours and add the remainder (eg. 2.14-> 3.14 -> 4.14 -> 4.15 = 2 hours 1 minute). This may help struggling students and reduce the amount of calculation required.
Labels:
Elapsed Time
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