I gathered up some practice for students and was thinking about how I could get them to do some revision. I hated revision as a student as I had a quick memory and remembered things fairly easily.
It's not true of all students though. So I found 300 questions on the topic (simplifying and balancing equations) and made up an A5 booklet of 12 pages. Then I made up some little reward packs and said that the first three students that completed page 1 with 100% accuracy would get a pack. Whatever revision work was left at the end of the period would be done for homework over the next week (to give encouragement for those that for a second considered loafing).
In the past marking of each page has been an issue. To get over this I combined two of students favourite things - writing on the white board and finding errors in each others work. Students wrote their name on the board and had to mark the work of the previous name on the board. Five students (randomly chosen from the rest) that had completed a page of work and had marked another students work would also get a reward.
We all had a laugh when the last and hardest question was repeatedly incorrect so that the 3rd place prize was ultimately won at student 15. The random draw was good incentive to keep going.
All in all students completed about 75 questions each in an hour (writing the question and answer for each sum). At the end of the lesson we talked about how it was important to develop concentration for the full 60 minutes in preparation for 2 hour exams later in the term and the need to strengthen muscles in the hand to withstand the onslaught of essay writing.
I tried it with both 9's and 10's and had success in both classes with 90% of students engaged and only a couple of students needing to "have words" with at the end of the lesson. Many students asked if they could complete the remaining questions over the weekend and I supplied answers for them to check as they progressed.
Friday, April 8, 2011
Monday, April 4, 2011
Teaching Linear Equations and Functions
Linear "anything" can send chills down the spines of many adults. For many students it is an exit point from mathematics. The inability to grasp the connection between an equation and its graph can mean a student languishes in any but "maths for living" type classes.
Yet there seems to be different reasons why students don't like linear algebra and linear functions. My top ten suspicions why students don't understand linear topics is listed below.
Mum says its hard
We should not estimate the impact we have as parents. By placing the kernel that we found it hard, our students will have to face the likelihood that they have the potential to know more than the most respected person in their lives. It's ok for it to conquer them because it conquered you. As an adult it really is rather easy to learn! Before passing on our prejudices, we need to find time to grab a text and figure it out from a worked example. It will make you feel good and your student will benefit from someone that can help too. Excel books can be found at booksellers for around $15 and can be a good starting point.
Girls can't do Maths, Boys can't be neat.
BS. I don't accept this from students and nor should you. Girls have outperformed boys for many years in mathematics, (esp. up to year 10). We have to be careful to walk softly when girls start noticing boys and don't want the nerd slur. Similarly, boys seem to think that sloppy work is acceptable - it's not and they can do better when monitored and prompted. It also improves their accuracy and notation.
Lack of primary algebra & directed number knowledge
This is not a dig at primary teachers, but it is a dig at the Curriculum Council. The lack of a syllabus has harmed education in WA and the implementation of OBE failed our students. In saying that, the CC is trying to make amends with the new courses in senior school and if the do-gooders don't get started again, we may have some reasonable curriculum reform. The trick will now be to get year 7 out of primary and get students into the hands of specialists in mathematics, whilst upskilling secondary teachers in ways to deal with younger students.
Lack of sufficient practice and connections to context
Many students grasp the major concepts quickly (like finding an equation for two points) but lack scaffolding in their understanding to establish lasting recall. Those eloquent in eduspeak will know the edubabble for this concept but the idea is sound. The motivation for this blog entry was a group of year tens currently struggling with remembering how to create a linear equation. In after school classes we have worked to connect the idea to shooting aliens (with an equation driven gun), distance time graphs, ice cream sales (using tables and difference patterns), intersection points, changing slope, y intercepts and x intercepts over a three week period. With a solid understanding of linear, extending concepts into quadratics and other functions is considerably simpler. These simple (but growing in numbers - we're now over 30 students) after school classes are leaving students enthused and ready to work once classes start.
Limited value seen in abstract knowledge
Sadly, many students are unable to see value in abstract algebra in year 10 and this limits their development. Without rudimentary skills in linear algebra much of the senior courses in mathematics are inaccessible by our students. A lack of rote learning and a focus on problem solving has reduced the ability of students to value skills based work.
Lack of connection between reward and effort
This is a huge concern not limited to linear algebra. The year 9 C grade standard lists linear algebra requiring fluency by year 9. If students don't meet this standard - their grade in year 10 will be a D or worse, even if developmentally they are finally able and work hard to understand abstract algebra. This lack of reward for effort will start to be seen throughout the mathematics course if we (and our regulators) are not careful.
Poor environment to complete assignment work
Many students in low socioeconomic schools do not have home environments conducive to homework. This is especially prevalent in at risk students. Schools need to encourage usage of safe areas to complete such work either under punitive (which can be more socially acceptable) or extra curricular environments.
Lack of study
An average student will not gain a lasting understanding linear algebra if they do ten questions and then move to the next topic. Given that the key concepts need some level of memorisation (how to collect like terms, establishing the equation of a line, the connection between an equation and a plane, creating ordered pairs, plotting them, difference tables etc), students needs to spend some time considering what they know and what they would like to recall freely.
Lack of in class revision
It is a topic that must be revisited over and over again throughout the year until it is as fluent as order of operations or times tables. It is the next key plank after basic numeracy is established.
A reluctance to start early
We need to ensure that linear algebra is introduced as soon as directed number, fractions and place value beyond thousands is understood. Those capable of dealing with abstract knowledge need it and we should not delay because heterogeneous classes typically teach to the middle. We need to challenge ourselves and seek to find when students are capable of starting algebra and find ways to provide opportunities to these students to advance.
There we go.. It's everyone's fault - students, parents, teachers, administration, regulators. Now let's get out there and fix it!
Yet there seems to be different reasons why students don't like linear algebra and linear functions. My top ten suspicions why students don't understand linear topics is listed below.
Mum says its hard
We should not estimate the impact we have as parents. By placing the kernel that we found it hard, our students will have to face the likelihood that they have the potential to know more than the most respected person in their lives. It's ok for it to conquer them because it conquered you. As an adult it really is rather easy to learn! Before passing on our prejudices, we need to find time to grab a text and figure it out from a worked example. It will make you feel good and your student will benefit from someone that can help too. Excel books can be found at booksellers for around $15 and can be a good starting point.
Girls can't do Maths, Boys can't be neat.
BS. I don't accept this from students and nor should you. Girls have outperformed boys for many years in mathematics, (esp. up to year 10). We have to be careful to walk softly when girls start noticing boys and don't want the nerd slur. Similarly, boys seem to think that sloppy work is acceptable - it's not and they can do better when monitored and prompted. It also improves their accuracy and notation.
Lack of primary algebra & directed number knowledge
This is not a dig at primary teachers, but it is a dig at the Curriculum Council. The lack of a syllabus has harmed education in WA and the implementation of OBE failed our students. In saying that, the CC is trying to make amends with the new courses in senior school and if the do-gooders don't get started again, we may have some reasonable curriculum reform. The trick will now be to get year 7 out of primary and get students into the hands of specialists in mathematics, whilst upskilling secondary teachers in ways to deal with younger students.
Lack of sufficient practice and connections to context
Many students grasp the major concepts quickly (like finding an equation for two points) but lack scaffolding in their understanding to establish lasting recall. Those eloquent in eduspeak will know the edubabble for this concept but the idea is sound. The motivation for this blog entry was a group of year tens currently struggling with remembering how to create a linear equation. In after school classes we have worked to connect the idea to shooting aliens (with an equation driven gun), distance time graphs, ice cream sales (using tables and difference patterns), intersection points, changing slope, y intercepts and x intercepts over a three week period. With a solid understanding of linear, extending concepts into quadratics and other functions is considerably simpler. These simple (but growing in numbers - we're now over 30 students) after school classes are leaving students enthused and ready to work once classes start.
Limited value seen in abstract knowledge
Sadly, many students are unable to see value in abstract algebra in year 10 and this limits their development. Without rudimentary skills in linear algebra much of the senior courses in mathematics are inaccessible by our students. A lack of rote learning and a focus on problem solving has reduced the ability of students to value skills based work.
Lack of connection between reward and effort
This is a huge concern not limited to linear algebra. The year 9 C grade standard lists linear algebra requiring fluency by year 9. If students don't meet this standard - their grade in year 10 will be a D or worse, even if developmentally they are finally able and work hard to understand abstract algebra. This lack of reward for effort will start to be seen throughout the mathematics course if we (and our regulators) are not careful.
Poor environment to complete assignment work
Many students in low socioeconomic schools do not have home environments conducive to homework. This is especially prevalent in at risk students. Schools need to encourage usage of safe areas to complete such work either under punitive (which can be more socially acceptable) or extra curricular environments.
Lack of study
An average student will not gain a lasting understanding linear algebra if they do ten questions and then move to the next topic. Given that the key concepts need some level of memorisation (how to collect like terms, establishing the equation of a line, the connection between an equation and a plane, creating ordered pairs, plotting them, difference tables etc), students needs to spend some time considering what they know and what they would like to recall freely.
Lack of in class revision
It is a topic that must be revisited over and over again throughout the year until it is as fluent as order of operations or times tables. It is the next key plank after basic numeracy is established.
A reluctance to start early
We need to ensure that linear algebra is introduced as soon as directed number, fractions and place value beyond thousands is understood. Those capable of dealing with abstract knowledge need it and we should not delay because heterogeneous classes typically teach to the middle. We need to challenge ourselves and seek to find when students are capable of starting algebra and find ways to provide opportunities to these students to advance.
There we go.. It's everyone's fault - students, parents, teachers, administration, regulators. Now let's get out there and fix it!
Labels:
daily reflection
Wednesday, March 30, 2011
Developing deeper understanding
Progress maps and outcomes have damaged mathematics in WA. By making distinct learning points without a web of links to outcomes, mathematics in WA has become disjointed and subsequently students lack fluidity between topics.
I doubt this is a new complaint and has been a fault of many attempted curriculum reforms, but it has been exacerbated by a renewed focus on assessment and the lack of credible assessment performed in early years. In many cases a year 10 student can perform a percentage calculation if (and only if) it is preceded by 10 examples of exactly the same type. A student can get 80% in their test by teaching study skills for a percentages test and by creating decent notes... but do they have an understanding of proportion and how it applies to percentages? In many cases they do not.
As a teaching group we have been talking about percentages (as OBE pushed many decimal concepts into high school and they are now being pushed back by national curriculum). It is important to learn how to teach it more proficiently in lower school and to our lower ability upper school students. One of the more successful ways we have encountered is to use relationships with ratios.
Problem: Find 50% of 50.
Using a ratios approach
100% of an object is 50
50% of an object is x
To get from 100% to 50% we have to divide by two (100% ÷ 50% = 2)
100% ÷ 2 = 50%
thus to stay in proportion
50 ÷ 2 = 25
Using a paper strip it is easy for students to see the proportions in action.
They can readily see that 50% is between 0 and 50. It's easy to experiment with a wide variety to proportions and it readily extends to percentages greater than 100%, percentage increase, percentage decrease, finding percentages given two amounts and negative percentages.
Using a formulaic approach
Take the percentage, divide by 100 and multiply by the amount.
or
Take the amount, divide by 100 and multiply by the percentage.
I know which of the two approaches is quicker and easier to teach.. but to extend the formulaic approach to other types of problems requires new sets of rules to remember and apply. Without a basis of understanding it becomes difficult to know which formula to apply and when to apply it (unless it was proceeded by a worked example - which leads us back to the original concern).
Using ratios and an algebraic approach
x ÷ 50 = 50 ÷ 100 (rewrite ratios as an equation)
x = 50 x 0.5 (multiply both sides by 50)
= 25
Once students understand some basic algebra and proportion, the solution becomes trivial (as it is for many of us). Sadly many students today do not reach this level of proficiency. I'm sure there are other more effective and efficient ways to teach proportion and percentages (and even some that don't use pizzas) but I think my point is fairly obvious.
I think sometimes we can get carried away by the need to meet an outcome and teach the how (as is driven by a packed curriculum) rather than using an exploratory approach that provides students with understanding which can have lasting consequences (often unseen by those that don't teach senior school topics). I originally saw the paper strip approach (or something similar) done by Keith McNaught at Notre Dame university. It has stuck with me throughout my teaching. When I am tempted to get curriculum dot points completed and tested (disregarding deeper understanding), it is always a good reminder of what should be done.
As a final note.. I do believe that nothing replaces practice and students need skills based work that requires rote learning (such as what is done with the formulaic approach). Which means as teachers we have to get better at providing pathways through the why (such as via the ratio method and with formal proofs) into the how (such as with formulaic approaches) and then making connections to other techniques (as seen with the algebraic approach) - always remembering that students shouldn't have to re-invent wheels which in many cases took millenia to form.
I doubt this is a new complaint and has been a fault of many attempted curriculum reforms, but it has been exacerbated by a renewed focus on assessment and the lack of credible assessment performed in early years. In many cases a year 10 student can perform a percentage calculation if (and only if) it is preceded by 10 examples of exactly the same type. A student can get 80% in their test by teaching study skills for a percentages test and by creating decent notes... but do they have an understanding of proportion and how it applies to percentages? In many cases they do not.
As a teaching group we have been talking about percentages (as OBE pushed many decimal concepts into high school and they are now being pushed back by national curriculum). It is important to learn how to teach it more proficiently in lower school and to our lower ability upper school students. One of the more successful ways we have encountered is to use relationships with ratios.
Problem: Find 50% of 50.
Using a ratios approach
100% of an object is 50
50% of an object is x
To get from 100% to 50% we have to divide by two (100% ÷ 50% = 2)
100% ÷ 2 = 50%
thus to stay in proportion
50 ÷ 2 = 25
Using a paper strip it is easy for students to see the proportions in action.
They can readily see that 50% is between 0 and 50. It's easy to experiment with a wide variety to proportions and it readily extends to percentages greater than 100%, percentage increase, percentage decrease, finding percentages given two amounts and negative percentages.
Using a formulaic approach
Take the percentage, divide by 100 and multiply by the amount.
or
Take the amount, divide by 100 and multiply by the percentage.
I know which of the two approaches is quicker and easier to teach.. but to extend the formulaic approach to other types of problems requires new sets of rules to remember and apply. Without a basis of understanding it becomes difficult to know which formula to apply and when to apply it (unless it was proceeded by a worked example - which leads us back to the original concern).
Using ratios and an algebraic approach
x ÷ 50 = 50 ÷ 100 (rewrite ratios as an equation)
x = 50 x 0.5 (multiply both sides by 50)
= 25
Once students understand some basic algebra and proportion, the solution becomes trivial (as it is for many of us). Sadly many students today do not reach this level of proficiency. I'm sure there are other more effective and efficient ways to teach proportion and percentages (and even some that don't use pizzas) but I think my point is fairly obvious.
I think sometimes we can get carried away by the need to meet an outcome and teach the how (as is driven by a packed curriculum) rather than using an exploratory approach that provides students with understanding which can have lasting consequences (often unseen by those that don't teach senior school topics). I originally saw the paper strip approach (or something similar) done by Keith McNaught at Notre Dame university. It has stuck with me throughout my teaching. When I am tempted to get curriculum dot points completed and tested (disregarding deeper understanding), it is always a good reminder of what should be done.
As a final note.. I do believe that nothing replaces practice and students need skills based work that requires rote learning (such as what is done with the formulaic approach). Which means as teachers we have to get better at providing pathways through the why (such as via the ratio method and with formal proofs) into the how (such as with formulaic approaches) and then making connections to other techniques (as seen with the algebraic approach) - always remembering that students shouldn't have to re-invent wheels which in many cases took millenia to form.
Labels:
daily reflection
Monday, March 21, 2011
Review of material written
Well, one thing was obvious.. the 3A MAS kids aren't quite at the level expected yet. We barely reached unit vectors which meant that we didn't get to the meat of the topic. This was a shame as the helicopter example is a great example of how vector topics fit together. It has indicated next time I need to go a bit further backward and put a few more examples in for unit vectors. We also need to look at the difference between adding and finding the difference between two vectors. Possibly also looking at examples of each in action. Easy fixed. The year 9's and 10's were comfortable with Linear functions and could use difference tables capably according to the tutor, if anything the work was a bit easy! This is good news and unexpected!
Unfortunately the 2C finance EPW was as expected and underlines that the group is a bit weak.. the students stopped after they thought they had learned something, which meant that they didn't get to the meat of the assignment (rookie mistake!). I think in more than a few cases social life and sporting interests come first. One student had done the work.. the rest were a bit of a shambles. My feeling is that the EPW is right, we should be able to make an assumption that year 10's have done compound and reducible interest and (with a bit of revision on their own) should be able to answer reducible interest problems with a calculator. One in the 80's, a couple of high forties and that's about it. Very disappointing result but hardly surprising given the incomplete take home sections. Hopefully what they have done will help them understand it properly when the topic arrives. These are students entering 3A and they can't be spoonfed and expect to do well.
Unfortunately the 2C finance EPW was as expected and underlines that the group is a bit weak.. the students stopped after they thought they had learned something, which meant that they didn't get to the meat of the assignment (rookie mistake!). I think in more than a few cases social life and sporting interests come first. One student had done the work.. the rest were a bit of a shambles. My feeling is that the EPW is right, we should be able to make an assumption that year 10's have done compound and reducible interest and (with a bit of revision on their own) should be able to answer reducible interest problems with a calculator. One in the 80's, a couple of high forties and that's about it. Very disappointing result but hardly surprising given the incomplete take home sections. Hopefully what they have done will help them understand it properly when the topic arrives. These are students entering 3A and they can't be spoonfed and expect to do well.
Labels:
daily reflection
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