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!
Monday, April 4, 2011
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.
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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
Sunday, March 20, 2011
Writing lasting material
It makes me laugh that we invest time in our teachers, but rarely invest time in the resource bank of a school. This causes a massive information loss each time a staff member leaves the school and requires significant effort to regain capacity back to the previous level.
We are at present putting material together for our after school classes and the lack of extension resources is amazing. The most common response is that extension classes after school are usually just repackaged classroom material at a higher level.
This can't be right. If a student seeks extension it's because they want material not found in the classroom - this is one aspect of summer school success we have. We don't just teach year 11 material to year 10's, we repackage it such that it is context specific, timely and interesting. One of the joys of an after school class is that you are not confined by syllabus and delivery points and you can delve into topics in a little more detail if students are interested. Hopefully students that didn't quite get it can now see where the majority of students are. Students that have a solid understanding can draw connections to other areas of mathematics and other learning areas.
I believe the resources I seek have been written and are sitting in drawers around WA. I understand why teachers are proprietary about their resources. Little time is given to developing resources and they have to be done in your own time. DOTT is taken up with marking, meetings, behavioural resolutions, recouping sanity time and parental contacts. It leaves little time for planning and developing of resources. If schools were better able to value what after school programmes could achieve, monitored what they did achieve, set goals to maximise future achievement and provided time to prepare resources to meet these goals then just maybe a few more students in the middle would find success and a few more high achieving students may be able to seek the stars.
Given the changes in curriculum, I'm not writing material to fit state or national curriculum, IB or NCOS. I'm sticking to topics that can be used across year groups and ability levels. The first two topics students have asked for are Linear functions (lower school) and Vectors (upper school). I've designed a written format and a method of delivery and I have some material on Finance that I can bend into this format. We'll see how it goes tomorrow and Tuesday.
There are opportunities "beyond the classroom" where schools can and do make real differences. It's a shame that all too often it is because of individuals rather than by initiatives by the school itself.
We are at present putting material together for our after school classes and the lack of extension resources is amazing. The most common response is that extension classes after school are usually just repackaged classroom material at a higher level.
This can't be right. If a student seeks extension it's because they want material not found in the classroom - this is one aspect of summer school success we have. We don't just teach year 11 material to year 10's, we repackage it such that it is context specific, timely and interesting. One of the joys of an after school class is that you are not confined by syllabus and delivery points and you can delve into topics in a little more detail if students are interested. Hopefully students that didn't quite get it can now see where the majority of students are. Students that have a solid understanding can draw connections to other areas of mathematics and other learning areas.
I believe the resources I seek have been written and are sitting in drawers around WA. I understand why teachers are proprietary about their resources. Little time is given to developing resources and they have to be done in your own time. DOTT is taken up with marking, meetings, behavioural resolutions, recouping sanity time and parental contacts. It leaves little time for planning and developing of resources. If schools were better able to value what after school programmes could achieve, monitored what they did achieve, set goals to maximise future achievement and provided time to prepare resources to meet these goals then just maybe a few more students in the middle would find success and a few more high achieving students may be able to seek the stars.
Given the changes in curriculum, I'm not writing material to fit state or national curriculum, IB or NCOS. I'm sticking to topics that can be used across year groups and ability levels. The first two topics students have asked for are Linear functions (lower school) and Vectors (upper school). I've designed a written format and a method of delivery and I have some material on Finance that I can bend into this format. We'll see how it goes tomorrow and Tuesday.
There are opportunities "beyond the classroom" where schools can and do make real differences. It's a shame that all too often it is because of individuals rather than by initiatives by the school itself.
Labels:
daily reflection
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