# When is a topic 'secure'?

Discussion in 'Mathematics' started by PaulDG, Dec 5, 2011.

1. ### PaulDGOccasional commenter

The problem here - and it's one I see a lot - is that isn't that teacher correct about the child "understanding collecting like terms"? - as they have correctly demonstrated it.

But what they've also demonstrated (assuming they got beyond stage 2) is that they don't understand either subtraction, directed number of both.

And that's something I see time and time again. Kids we are teaching "higher level skills" to who can and will be able to use those "skills" to hoover up some, perhaps most of the marks on a question of that type in GCSE, but who don't really have the "skill" because they still don't get directed number

I'm increasingly aware that those kids I see who "get algebra" are those who "got" directed number, order of operations and at the highest level, fractions.

Those who "don't get algebra" actually don't get directed number or order of operations.

2. ### kirbatron

I completely agree with this, students doing A level maths succeed or fail based on their algebra skill which is wholly dependent on their directed number ability and their order of operations ability.
I repeatedly correct students when they talk about 2x^3 and they say 2x (pause) cubed instead of 2 (pause) x cubed because they don't see the difference or understand the order of operations.

3. ### googolplexOccasional commenter

In the long term, no topic is completely secure unless students are in a position to revisit or extend their learning.
I would argue that topics are secure when it ceases to become a barrier to future learning at an equivalent or higher level. This doesn't necessarily mean that they don't need to be reminded of such concepts.

Secured or mastered?
In a spiral curriculum do we not keep returning to a topic?
so the child secures in some order
collecting alike terms
a + a + a = 3a
multiple instances of a single variable collect to form a simple term
then a mixture of single variables collect to form 2 or more simple terms
a + b + a + b
then combined alike terms that are already partially collected
2a + 3a
then a mix
then with
fractional coefficients
negative coefficients
mixed powers

I do not agree, it is not complicated enough, it is far too overcomplicated by the pointless idea of this question is level such and such and then this question is something else.
I have never seen any proper reference to a conceptual hierarchy tree in the NC. Until the time that I do I will continue to consider the NC a load of smeg developed by Civil Servants to pander to the far right ideologies of various governments.

You can teach the increasing order complexity in more than one way. You can have negatives or simply have subtractions and additions in. For me there is a trick to this.
For examples of one persons idea of a hierarchy
I refer you to
http://students.ed.uiuc.edu/mundell/eport/GeometryMathUnit.htm
as an example .
I have never imagined that the heirarchy progression is linear.
Building a brick wall is better than the ladder idea. Sometimes the "weight" of the 'top' is distributed far across the 'bottom'.
BTW sublevels are for annual assessment at their most frequent use.

5. ### TandyNew commenter

Arrggh! They don't exist! Never use them!
Smeg indeed. Though very little to do with Civil Servants and nothing to do with right or left... just basic misunderstanding of what learning actually is on both sides. The most damaging initiative in the 20 odd years of the NC has to be the National Strategies once they got over zealous and Capita driven - fiercely pushing all this sublevel tosh.
I like to think of mathematics as a giant version of Jenga, with new areas depending on the sound foundations below. Remove any of these and everything destabilises. This is why we get stupid systems that insist on teaching kids collecting like terms when they haven't yet mastered arithmetic.

6. ### emilyisobel

This quote should be on the wall of every maths classroom and office.
I just wish that I had the freedom to follow exactly this in my classroom, rather than having to follow a SoW that is the old NS.
And give a sublevel every term.

7. ### TandyNew commenter

Hello ei, long time, etc, etc...
Interesting that you feel you don't have the freedom to do this. Is that because of the department's decision or school or ...? Legally, you just need to provide access to the programmes of study - I think that leaves flexibility to do more or less anything you want

8. ### emilyisobel

Hi Tandy, yep long time
It's a departmental thing. I do my own thing to some extent but we test every half term or so and so I do need to follow. I tend to use algebra etc to reinforce arithmetic work, especially negative numbers and tend to treat solving equations as puzzles.
Personally I would like to see y7 spend at least a term on arithmetic reinforcement and understanding, with puzzles and problem solving thrown in to help in that. They also need to be able to use measuring instruments correctly. I would leave formal algebra and data handling work until they have the basics. The old NS tends to rush them through things too quickly and they don't have enough time to consolidate properly.

9. ### Subjectknowledge

It is possible that their problems are not simply with subtraction, or directed number, but also that they are reading the sign on the wrong side of the term. eg
4a - 2b + a + 5b
can be thought of as consisting of 4 terms (+ 4a) (- 2b) (+ a) (+ 5b) that will have the same total whatever the order. This means that any rearrangement is permissable as long as the + and - signs move with the terms as bracketed.
Thus we collect terms as (+ 4a) (+ a) (+ 2b) (- 5b) = 5a - 3b.
However, some pupils effectively bracket with the sign on the wrong side: i.e.
4a - 2b + a + 5b =
(4a -) (2b +) (a +) (5b) and rearrange as
(4a -) (a +) (2b +) (5b), getting 3a + 7b.
If all the terms are positive, this misconception remains hidden, so the cases with negative terms have to be done before you can be sure they are not using a incorrect method.

10. ### Subjectknowledge

This may be a realistic line to take, but I wonder if there is any point in doing 'collecting like terms' at all if you are going to stop there. After all, it arises as soon as you get an equation with an x on each side.

11. ### DMNew commenter

Edexcel 1380 Linear Foundation Paper 2 June 2011
11 a) Simplify 7x + 4x (1 mark)

12. ### Subjectknowledge

We do keep returning to a topic. However, sometimes they do not move forward enough on each visit to make progress. They keep being taught the same bits. (e.g. I see mean, mode and median of small data sets being taught in Years 4, 5, 6, 7,. 8, 9, 10). I would not want to collect terms of a single unknown on one visit, then positive cases with two unknowns on the next, then ones with a subtracted term the next etc. I'd rather wait until we can do several of these steps in the same week.

13. ### Subjectknowledge

I think Tandy's analogy is a very good one that can be taken further. In Jenga, some of the foundations can be taken out without the whole lot toppling, but the more holes there are, the more likely it is that that the structure collapses. I think this applies in learning mathematics. However, unlike Jenga, some bricks are more improtant than others. Taking out directed numbers or fractions makes algebra very unstable.

14. ### Subjectknowledge

DM, I hope you are not suggesting that Edexcel is any sort of authority on mathematics!
My point was not what is on the test, but what makes sense educationally.

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