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.
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.

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.

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.

Thursday, March 17, 2011

CAS calculators

The importance of using calculators appropriately cannot be underestimated.  Percentages and compound interest are two of the most misunderstood topics in year 9 and 10 and many student errors could have been prevented with effective use of calculators.  This year my year 9, 10 and 2C classes all did compound interest at about the same time.  All three classes were able to use the CAS calculator to construct the equations required for reducible interest. 

Teaching calculator usage in year 9 should prevent some of the errors in year 10 and 2C because:
a) they will not be struggling with "how to use the calculator" next year (modes, cell referencing & formulas)
b) they will be able to calculate percentages of amounts with or without a calculator
c) they will be able to work with the idea of a period of time and know that this needs to be consistent across an equation
d) they will be able to work with interest periods other than annually
e) they will be able to identify simple and compound interest problems

There are many times calculators are inappropriate but in this context it is an engaging tool and the novelty helps focus students on a fairly dry topic.  It is unfortunate that the 2C class did not have this benefit as they are struggling with remembering what compound interest is and how reducible interest relates to it.  Finding time in the curriculum to promote appropriate usage is well worth the effort as this is one of the occasions where a calculator/spreadsheet is used in a real life context over pen/paper.  A good series of worksheets can be found at under the intermediate tab.  It does take some patience but students will quickly learn how to create spreadsheets well.  I would also show students how to use the fill range tool (under the edit menu) to make the process a little quicker.  It may be worthwhile to use MSExcel first in a computing lab.

It is obvious that many students have not seen how spreadsheets can be used in computing classes or are not making the cross curricular connections of how that knowledge could apply in mathematics.

A byproduct of the classes is that it was a good assessment checkpoint to see if they understood how to apply percentages of amounts and whether students could see how it fits within a multistage question.  The tens did very well making the transition from spreadsheets to the compound interest formula and I now anticipate that it will be an easy transition to finance mode for more complex worded questions.

Monday, March 14, 2011

Calm before the storm

Crossed the half way point of the term and things finally calmed down for a few hours.  Most of my nine's have now completed their NAPLAN revision and we have a few lessons up our sleeve for the end of term.  They're settling down now that they are starting to realise

a) don't come to lessons unprepared or you will have to sit down the back doing lines and redo the lesson at lunch time.
b) don't do homework or you will stay in until it's done
c) refuse punishment and the number of lunchtimes double - the first with me and subsequent ones with the team leader.

It's old fashioned but the results speak for themselves.  Students that do their work feel good about themselves and students that would otherwise have fallen through the cracks are slowly coming online.  The next lesson is using the CAS calculators - so at least it will be a break from NAPLAN preparation and book work.

Our academic extension class started this week and the first five year 9 and 10 students experienced linear algebra ala aliens. We created bullets using linear equations and shot aliens with them.  Using CAS calculators made this quite fun experimenting with different spots on the hill (the hill was the y axis and we modified c for different points on the hill) and changing the angle of the laser (modifying m). Next lesson we'll use a series of linear equations to reflect bullets off mirrors.  I hope to extend this to matrices later as it is an obvious fit (even if it is only linear equations).  We'll do four lessons of this and then do some isometric and oblique drawing outside to help them visualise objects in 3D before starting some ballistics using quadratics and calculus before revisiting linear equations (with the ice cream example) and optimising some finance solutions.  At the end they were asking whether we could go for two hours instead of one (groan!).

A number of EPW's went out for my 11 and 12's including the Finance EPW I wrote over the last four weeks.  It seems common that 2C students don't know how to use their calculators and teachers are not confident an investigative approach is the best way to learn them.  Three teachers in my small group have all raised concerns about the EPW (seemingly without reading it) but we shall see how it goes.  Given that the answers are provided, online links to assistance has been given and they have a week to investigate, I lack understanding why this is so hard.  We shall see.

My tens are confidently using spreadsheet and finance mode on the CAS to solve a variety of compound interest and repayment problems.  I hope they don't face the same issues as the current year 11's when doing 2C and 3A with regards to using the calculator.

As always 1B's seem to underestimate the difficulty of the course and seemingly need to fail a test before they realise that they need to study.  I'm pretty sure my bunch are not going to top the three groups this time - but I have hope yet that some in class revision will turn them around.

Saturday, March 12, 2011

A profession that consumes the individual

One of the things to consider as a teacher is how isolating the career can be. As someone responsible for 100 students and their individual well being, it can be easy to fall into the trap of allowing the job to consume all of your available time to effectively respond to their needs.

The better a teacher you become, the more you realise you can do. The more pressure there is to perform.

Focusing on one class leads to deficits in other classes. These deficits are then questioned and you start to doubt your ability and there starts a downward spiral difficult to arrest on your own.

Then there are personal considerations when faced with students that relate directly to your life story. The child that is facing issues that you faced as a child and believe you can make a difference to their lives. A laptop computer given on loan, buying a student text, giving a few minutes extra tuition, making sure they have enough money for an excursion, advocating for a student - I know teachers regularly do these things. Knowing that it would be difficult to enjoy your weekend and satisfy your conscience if you didn't act when you had the opportunity.

Another trap is allowing a deficit of time to let you lose your support network. Being consumed by teaching can lead to a one dimensional person, having only one interest and thus having limited interest to others. This can make it a lonely profession especially when the majority of conversation you have is with minors.

It doesn't just affect you, it affects those around you. Supporting a teacher is a full time occupation. You come home tired and spent. Events of the day can overwhelm you. It can be a real pressure cooker at times, especially around TEE and reports or when the playground is on fire.

Somebody told me about the monkey analogy and how if someone passed you the monkey - it was important to pass the monkey to another (yes it was an admin person). As a metaphor for problems I think as a teacher, the tribe of monkeys needs a support network capable of dealing with them. Admin sometimes needs to remember this.

Maybe I'm a bit old fashioned. Maybe I have to look at it a bit more like a job and less like an opportunity to make a difference. I wonder if I would be able to do it anymore if I thought about it that way.

It's no wonder many teachers are a little bit more than strange.

A bigger worry is that you fail to notice it after a while :-)

Tuesday, March 8, 2011

Good Day

After the issues with the 2C test it was nice to have a good day. My 10's were responsive and worked well whilst our Principal was in the room for a whole hour doing his impromptu visits. It's good that he does them, but it can be a bit harrowing. We investigated how to use our CAS calculators to build spreadsheets and will now start looking at the results to investigate compound interest further.

There were lots of things I would do differently with the lesson itself but I can't fault the kids in that they followed instruction, were able to use formulas and solve a compound interest problem using technology by the end of the lesson. After replacing most of the batteries in the morning, only two failed during the lesson which was ok.

I checked my 9's homework and that was a different story. I used some old fashioned "I will do my homework when my teacher asks me to otherwise I will have to write this." x 100 to ensure that students had some encouragement to do their homework in future. Those that did their homework enjoyed it if nothing else.

My 1B's are going ok, they finished the exercise but are not fully understanding cumulative frequency, so we will need to redo that lesson. I must remember tomorrow morning to hunt out a worksheet that will reinforce the connection between cf and median (and xf and mean).

A nice change from Friday.


Friday, March 4, 2011

Making mistakes

You know.. it would be nice to not make mistakes. It's even better when your mistakes aren't distributed to multiple schools for scrutiny. I had the wonderful opportunity of writing three assessments for moderation groups all at the same time, two tests (one for 3A MAT and one for 2C MAT) and an EPW (for 2C). Tests did not exist that could be pulled off the shelf and I didn't want to use a Curriculum Council EPW as they have been widely leaked (yes I'm looking at you Curtin University!).

Anyhow, the 2C paper had an error (three circle Venn diagrams aren't part of the curriculum) and it was one of my complex questions along with another question that I changed at a teacher request to set notation. Unfortunately by doing so it also reduced them to non complex questions. The test (although broadly covering key concepts) did not have the required complexity.

Once marked the curve for my class was badly skewed. It's a bit embarrassing as it's the first time I've taught 2C and really wanted to do the right thing by my moderation group. The test had an error in it and I had to re-issue the marking key as well as the original one had mistakes in it too.

Hopefully the 3A paper is ok (it's harder than the 2C paper and I think my students are going to get a little wake up call) and I must say - the amount of work required to write a 2C EPW should not be underestimated. If you're interested in an original 2C Finance EPW based on spreadsheets leave a comment with your DET email address and I'll forward it to you (Your email address is safe, - I moderate all comments before release and I'll delete the comment before it goes online so that the email address is not made public).

I've been flat out trying to get it all done (and interim reports) and bed down my classes. Hopefully now it will settle as all of my NCOS assessments for term one have been done and I can start enjoying myself again working on the lower school courses. Ten year 9/10 students approached me today to run an afterschool extension class again. They're fun but a lot of work when you and the kids are hot and tired.

We'll see how it goes. Bring on the long weekend!

Wednesday, March 2, 2011

Solving Venn diagrams where the intersection is unknown


Today in 2C MAT we came across that old chestnut, the Venn diagram with the missing value in the intersection with a number in A, B and the outside region.

In many cases the easiest way is to use a guess and check approach and a lot of the time the answer will fall out by substituting into the intersection and revising your result based on the values
A union B + the outside region = n.


Another approach is to name the segments and solve a series of equations:

a = 20-b
c = 30-b
a + b + c + 5 = 40

By substitution (20-b) + b + (30 - b) + 5 = 40
Therefore b=15

Once the intersection(b) is known, finding "A only"(a) and "B only"(b) is trivial.

I was asked the question "why teach this technique?" and my response was that it was not formally taught, it was a logical answer for a question given. We have some unknowns, we have some equations, why not solve for them? This sort of problem solving "setting up of equations" technique is common in optimisation and linear programming - why not use it in a probability setting?

I remember a particular student that was renowned for having solutions of this nature where his answers always deviated from the answer key and he had the right answer (or was on the right track) more often than not. We still call intuitive answers like this after "that" student as they forced the marker to find the underlying logic rather than application of a given method (if that student is reading this - get offline and study for your uni courses, scallywag!)

Anyhow, a third and more common approach is to rearrange the property:
A U B = A + B - A intersection B

By rearranging the equation
A intersection B = A + B - AUB

Since we know that:
AUB = U - (the outside region)

to find AUB is fairly simple:
AUB = 40-5
= 35

A intersection B = 20 + 30 - 35
= 15 (as before)

This approach does have the advantage that you can talk about the intersection being counted twice when the union is calculated by adding A + B where A and B aren't mutually exclusive.

I can't really see how this problem could be classed complex given the second method exists. Perhaps, if combined with a wordy explanation, a question of this sort could be made complex but to my mind that would defeat the purpose of the syllabus points in defining complexity. After all, why should something be classed a "complex question" if the only reason was that the question was worded to be understood by students with strong English comprehension?

Further exploring the properties of one

To find an equivalent fraction of a decimals, one way to explain it is to take the decimal part of the original number and place it over the lowest place value. Leave any whole numbers in front. (This only works for non-recurring decimals)

eg 0.123

The lowest place value is thousandths, the decimal part is 123.


0.123 = 123/1000

An alternative way to explain it is using properties of one. The idea is that
a) numerators of fractions should be whole numbers and;
b) the fraction should be equivalent to the decimal.

We can ensure the fraction is equivalent if we only multiply or divide by 1 or more importantly a fraction that is equivalent to 1.

To satisfy part a)
To make 0.123 a whole number we have to multiply it by a power of 10 - 1000 (10^3). This was a concept we had investigated earlier.

..but if we multiply by 1000 we will change the original number from 0.123 to 123 - it will no longer be equivalent.

So to satisfy part b)
We multiply by 1000/1000 (or 1!)

.0123 = .123/1 x 1000/1000
= 123 / 1000

I like this because it continues to explore how fractions are constructed, the connection between decimals and fractions and why decimal conversion works. I wouldn't try it in classes with low ability due to the possibility for high levels of confusion if understandings of multiplication and commutative properties are not properly understood.

An earlier article exploring one and fractions can be found here.