Just follow these steps and you’ll be OK at Enlargements

Here’s a statement which I don’t think will be too controversial – I would have thought maths teachers the world over would agree with this:

I want my students to gain a deep understanding of the mathematics, not just follow a procedure to get the right answer.

This is our aim. We don’t always get there. We have different ways of getting there. I have recently re-read this seminal paper by Skemp from 1976.

Screen Shot 2017-06-01 at 13.18.19

In it he talks about Relational Understanding, which I have generally thought of as understanding of concepts, and Instrumental Understanding, which I think of as understanding of procedures.  In my teaching I have been inclined to build conceptual understanding first, and then see what methods make sense from there.  However, I’m starting to think that it’s not that simple.  There are some situations where some instrumental understanding might come first and act as a foundation on which to build relational understanding.  Ultimately we want both, but the order in which we achieve this is not always the same.  We should not dismiss a didactic approach that provides a clear sequence of steps and worked examples as a part of the journey to a deeper mathematical understanding.

I recently observed a colleague teaching the topic of Enlargements to a Year 11 revision class, who are entered for the Foundation paper.  He used a method which I don’t think I had thought of before.  It was heavy on the Instrumental Understanding, but it worked and the students were doing some more tricky fractional and negative enlargements with centres of enlargement not at the origin.  About as hard as it gets for these types of exam questions (thanks for Maths Genie for these examples)

Screen Shot 2017-06-01 at 13.36.31.png

So, what was this wonderful method?  Well, it might be nothing new to you dear reader, but it used vectors.  It relied on students being secure with describing the translation between two points using vector notation.  Given that this is how Translations are described and that we typically teach Translations (along with Reflections and Rotations) before Enlargements in a topic called Transformations, this should be a build on / consolidating of what was learned a few lessons ago.

The steps go something like this.

  1. Label the vertices of the shape you’ve been given (say A,B,C,D, something like that)
  2. Circle the centre of enlargement, CoL (helpful to distinguish it from a vertex later)

Screen Shot 2017-06-02 at 14.57.56.png

3. Find the vector that moves you from the CoL to each vertex.
4. Multiply each vector by the scale factor

Screen Shot 2017-06-02 at 14.26.07

5. Now use those vectors to plot the new points, starting from the CoL again. Connect the vertices to form the shape.

6. Finally draw in some ray lines to convince yourself that you have not made any mistakes.

Screen Shot 2017-06-02 at 15.14.58.png

I think that’s quite a neat method that enables us to go straight to what might be seen as the most difficult example. The only real difficulty here, however is multiplying a (pretty simple) negative fraction by an integer.  Something which should be secure by the time this topic is being taught. And if it’s not secure, well here is an opportunity to practise it without detracting too much from the main objective.

Another benefit, is that it might be easier for students to plot points by counting squares rather than draw accurate, extended ray lines, as pointed out by Mr Blachford on Twitter.

Once students have done a few examples, we can draw attention to some things, for example:

  • If the scale factor is >1, it gets bigger, if it’s <1 it gets smaller
  • The scale factor applies to each side length of the shape (but not the area…)
  • Negative scale factors always place the image the “other side of the CoL”
  • All ray lines must go through the CoL. This is how it is constructed.

I would use Geogebra (as in fact I did to create these images) to examine what happens when we move the CoL. (Note: Geogebra uses the US terminology Dilation rather than Enlargement – which actually is more descriptive of what we are doing, isn’t it?!)

Jun-02-2017 15-24-44.gif

And from there we can go onto questions that show the enlargement and ask for a description.  I would use Geogebra for that bit as well as it is very easy to create images that can then be used to ask learners to describe what they see.

Just for fun, I had a go at a 3D version in Geogebra.  I’l leave you to decide whether or not it adds anything. You should be able to access it here. This is what it looks like:

3D dilation.gif

This is an example.  It’s one way of doing it. For other topics, for example fractions, I would prefer to spend a decent chunk of time on building conceptual understanding before focussing on algorithms to get right answers. But that’s for another post…

 

 

 

 

Interleaving Algebra and Perimeter

Over the last couple of years we have organised our Year 7 curriculum so we do some introductory algebra early on.  Forming expressions from words and collecting like terms, would be topics that I would put in the introductory bucket. The benefit of this is that it can be interleaved into various other topics to extend thinking and promote generalising.  Perimeter is an example of this, when we can give side lengths letters instead of numerical values.

An alternative to presenting a bunch of text-book type questions is to investigate a simple 4 piece Tangram, as described in this task from Mike Ollerton.

Screen Shot 2017-05-31 at 12.26.06

(Click on the image to access the full Word document)

The task as presented is primarily an exercise in shape, but I might use a slightly modified version of the Tangram to focus purely on perimeter.

Screen Shot 2017-05-31 at 12.41.59

Before asking students to cut the triangles out of the shape, we might agree on some labels for the side lengths.  If we focus just the shorter sides, we could call Triangle A’s short lengths a, Triangle B’s short lengths b, and Triangle C’s short lengths c, so we end up with something like this.

Screen Shot 2017-05-31 at 12.41.39

Again, before getting to work with the scissors we could discuss how we might describe the hypotenuse lengths in terms of a, b and c.  And in fact, if we really need c at all…

Using this notation for the side lengths, we can then cut up the 4 shapes and generate many other shapes and find their perimeters.  Using 1 shape, 2 shapes, 3 shapes, all 4 shapes: what is the shape with the longest perimeter? The shortest perimeter? What is the difference between longest and shortest?  What shapes are different but have the same perimeter – can we prove this using algebra?

It hopefully presents a need for “collecting like terms” as well as some introductory practice in using the technique.

9-pin Geoboard

A second task that links algebra to perimeter uses a 9-pin geoboard.  This sounds fancier than it is. You don’t need the actual boards, students can create their own in their books or you can give them some dotty paper. First, we tell students that we are going to make triangles using only these lines a, b, c. This is a key image that we will need to refer to either on the board or on a handout.

Screen Shot 2017-06-01 at 07.38.07

The next task is to construct triangles using various combinations of these lengths. Each triangle must fit within the 3×3 array.  Depending on the class and your objectives for the lesson, at some point you show them that there are only 8 “different” triangles. An opportunity here for a discussion on what we mean by different and what congruency is.

Screen Shot 2017-06-01 at 07.43.57

Once we have these, we can go through the process of calculating the perimeter for each one using the a, b, c notation we introduced earlier.

There are some options from here.  Mike’s suggestion is that we order these from smallest to largest.  We could do that by just looking at the shapes and having a guess. It’s pretty obvious that C is smallest although some of the in-between ones are harder to see. We could establish that a<b<c in the first picture (again by looking).  Then we would also need to decide, for example which is bigger: 2a or c.  With an older group, you might even use Pythagoras and express c and b in terms of a using surd form.

By looking at the differences between each shape’s perimeter, we start dealing with negative quantities of a, b, and c.  If we then sum up all those differences, we should end up with an expression the same as the difference between the smallest and largest, with the c’s cancelling out. Which is quite satisfying and obvious if you think about the expressions lined up on a number line.

Screen Shot 2017-06-01 at 08.25.32

There is a fair degree of flexibility within tasks like these and I believe that as teachers we need to select carefully what routes we expect to go down in a lesson. There is a danger that we try to encompass too many different topics in one go and if all of these topics are new to a class then they (and you!) are likely to lose track of what they are actually meant to be learning in this lesson. However, if you are confident that the learners in your class are secure with certain concepts (in this case collecting like terms with negative coefficients) then it is a good way to consolidate and practice this knowledge whilst pushing into new territory.

 

 

 

Birth Date Values

One of the great things about my role(s) this year is that I have had the opportunity to meet some fantastic Maths teachers and educationalists and last week I hosted Mike Ollerton for two separate events.  Mike has made many significant contributions to Mathematics education over the years and he has kindly permitted me to write about the ideas he shared with us last week. This is the first of a series of posts.

This is a simple activity that feels quite fun and personal but could lead to some rich discussions. Mike’s description of it is here:

Screen Shot 2017-05-31 at 08.44.08.png

After going round the class, asking several children for their BDVs, there are many questions which might present themselves. Can you ask children to work out someone else’s birthday given their BDV?  Mike suggests lots more questions:

  • Which BDVs only have one birth date?
  • What are the minimum and the maximum BDVs in a class?
  • Which BDVs have the most dates?
  • What is the smallest BDV which cannot be made?
  • What is the largest unique BDV?
  • Which dates are square BDVs?
  • Which dates are triangular BDVs?
  • In a group of people who has the average BDV?

What other problems can you devise based upon BDVs?

Simultaneous Equations, refining the procedure

Going over Simultaneous Equations today with Year 11, we all agreed that the thing that is most confusing about solving equations like these are the negative numbers:

4 - 2x = 9 + 6y
6 - 2x = 7 - 2y

We also agreed that we much prefer these types of questions when we heave a sigh of relief realizing that we can add instead of subtract.

4 + 2x = 9 + 6y
6 - 2x = 7 - 2y

Adding is easier. We have one less choice to make and we don’t need to keep track of which is the minuend and which is the subtrahend.

This is subtle.  I would always start teaching simultaneous equations using some real-life examples (the ideas in this post).

screen-shot-2017-02-15-at-18-04-42

I think real-life examples work in this case, because it is apparent what “extra” you are getting for the extra money.  These problems create the need for the algebra – it becomes a way to represent what is going on by writing less.  With these problems, everything is positive so it makes sense to think of the difference between the two situations or the two equations.

From there we can introduce the need to “multiply up” to get the same co-efficient for one of the terms.   In my experience, most students don’t struggle too much with this concept.

What gets tricky is subtracting one equation from the other when negative signs are involved.  We might identify the problem as being one of a lack of mastery of a fundamental concept – in this case negative numbers.  So before teaching these types of simultaneous equations we could do lots of practice and drills on negative numbers.  But I’m not sure that is always a helpful approach.  Some things are just more confusing (present a higher degree of cognitive load if you like).   There is a lot going on, and lots of it needs to be done mentally. So, maybe there is a case here for explicitly teaching an easier technique which is less prone to error.

A simple way to avoid subtraction is by always ending up with equations with opposite coefficients of one of the variables.

So, for example when solving the following:

4x - 3y + 1 = 0  (A)
3x - 7y + 15 = 0  (B)

multiply (A) by 3 and (B) by -4. A valid shortcut / rule to remember here is that multiplying by a negative simply “flips the sign” of any term it is being applied to. So we end up with:

12x - 9y + 1 = 0
-12x + 28y - 60 = 0

It’s easy to check that the sign of each term has been flipped before then proceeding to add the equations.

As any Maths teacher would, I believe strongly in teaching for deep understanding of concepts, not blind memorisation of rules (Nix the Tricks is a frequent reference point). But there is also a place for remembering certain procedures to reduce cognitive load (times tables being the most obvious example). Dani Quinn has written a great post on this in relation to “moving the decimal point” here.   Manipulating Simultaneous Equations like these is another example of when there is a case for a bit of explicit teaching of a method.

 

 

Praise where it’s due

For regular readers of this blog, I’m afraid this post has nothing to do with maths. Writing helps clarify thoughts and that is my main reason for doing it. If others read it and find these clarified thoughts have some resonance, that makes me happy. If you feel moved to comment on these thoughts and add your own experiences, then that is really powerful as it hopefully moves us all forward.

This post is about school-wide systems for praise and sanctions. I am not SLT, and this year I don’t have a Pastoral leadership role. The reason I am involved in this is because I am a member of a Teaching and Learning Focus group.  For the last couple of years, my school has run these groups in the directed time after school that traditionally might have been used for one-size-fits-all twilight INSETs.  At the beginning of the year, teachers self-select which group to join, each group being led by a member of SLT. There are about 6 meetings throughout the year with the idea being that a particular issue is discussed, researched and some form of action taken.  It’s a form of Action Research. From my perspective this has worked well this year.  I see this as the leadership of the school saying, “OK, we do lots of good things here, but it’s not perfect. What can we improve? And how? Work as a team, do some research, consult with other staff and come back with a proposal.”

As a team, we are now at the proposal stage.  I think it still needs some work and more consultation with staff.  We haven’t done the “hearts and minds” bit yet.  If we don’t do a good job of convincing the entire staff body that this is a good idea, the whole thing will have been a waste of time.  My school is not the sort of organisation that issues diktats from senior management, I’m glad to say.  Organisations that do rarely achieve the change they require. People may pay lip service the new initiative for a while, but if you cannot convince professional adults (i.e. teachers!) that this is something worth doing, it simply won’t happen.

So first, let’s look at our current “Praise and Concern” system. I’m going to start with “Concern” because that is the bit we are not proposing to change, but it’s important to understand the context.  As all schools do, we have a behaviour policy.  I won’t go through the detail of it here, but it basically involves warnings, detentions and removal from the classroom to a “referral room”.  Each of these “concerns” is to be recorded on SIMS by the teacher who issues the sanction.  There is a workload implication here, but it is generally viewed as something worth doing.  Recording the data centrally is important because it enables other staff (i.e. form tutor, head of year, etc.) to get an overview of issues pertaining to an individual student occurring across different subjects.  It’s particularly useful for highlighting frequent low-level transgressions which might not result in detentions but which could be impacting learning.

At the moment, the same system is used for Praise.  These are also recorded on SIMS. There are a bunch of categories and, as with Concerns, the teacher is expected to write a sentence or two of comments.  Every few weeks, a Praise and Concern report is sent to tutors. This report lists, by student, each Praise and each Concern.  How form tutors use this information varies widely, with some displaying it all in front of the form, some displaying just the praise bit, some having individual conversations and some not sharing it directly with students but just taking note of it for themselves. The raw numbers are also shared in end of term reports that go home to parents.

We feel that the main problem with this system is the idea that Praise and Concern are seen as two sides of the same coin. They are not.  Often students look at their net score (i.e. Praises minus Concerns) and I know of students who have asked for Praises to offset a certain number of Concerns. An evident issue is that it is often the students with the poorest behaviour who end up with the most praises “Well done, you’ve got your pen out, have a praise”. That sort of thing.  And the quiet ones who do what is expected of them day in, day out get ignored.  That is borne out in the data and from focus groups with students – they know what’s going on.

It is also apparent that there is a lack of consistency about how often and how many teachers issue Praise.  Some ignore it completely and use their own systems. Others use it frequently including issuing whole-class praise, which is nearly as bad as whole-class detentions in my view.

In parallel with Praise and Concern, we also have a system of housepoints.  This culminates at the end of the year with the inter-house Sports Day (still my favourite day of the year!).  Lots of points are awarded on Sports Day but these add to points collected throughout the year by individual students.  Housepoints are more personal to students.  They write them in their planners when they are awarded and each term the form tutor “collects” them and enters them into a central system.  Our hypothesis is that because students write the reason for the Housepoint in their planner and then keep a record of them, these are more meaningful and motivating.  From focus groups with students we believe that this is true for lower years (Year 7-8) but house points are less valued by older students (Year 10 up).

In looking for something better, we have reviewed educational research on the role of praise in teaching.  We were looking at the role of extrinsic rewards impacting on intrinsic motivation (here and here), and on how to make praise “purposeful” (here).  We defined purposeful praise as praise which would motivate a love of learning and challenge our students.

This is closely linked to ideas of growth mindset.  How, when and what praise is given for can impact on a students’ mindset but it’s a highly complex picture and difficult to draw general conclusions to apply in the classroom.

However, among our group there was a fairly close consensus on the following:

  • We should praise the process and the effort observed in the moment, not the individual.
  • Extrinsic rewards (e.g. stickers, housepoints, postcards home, etc.) have a place but need to be valued by students and need to be issued for something specific, for going “above and beyond”
  • Narrative, personal feedback given to students is more likely to motivate and challenge them than extrinsic reward
  • We need to be more fair and notice and acknowledge those who are quietly engaged in the struggle of learning.
  • The current housepoint system should be relaunched but there should be no attempt to track students’ individual scores.  You are collecting them for your house!
  • Centrally collecting data on praise issued is not valuable (although collecting it on Concerns is).

I would be really interested to hear from any others on their perspectives.  How do the praise and sanctions systems sit together in your school? What do you do that works particularly effectively?  Either comment below, or get in touch by e-mail (mark.horley@gmail.com)

Mathematical debates, and Bounds

It’s possible that what I am about to explain is already completely obvious to many a maths teacher.  But after 5 years of teaching, it’s only just dawned on me (a recurrent theme of this blog by the way!) so I thought I’d share.

I like teaching the concept of Bounds.  It highlights a fundamental difference between concepts that exist in Maths and the real world.  We measure things using numbers as continuous variables.  When we measure the length of a table, say, as being 120cm, we are rounding to the nearest centimetre.  We could be more accurate and say it is 120.4cm or even 120.4187234cm but this is still an approximation.   It reminds me of my engineering days, looking at milling machines that were capable of cutting to the nearest 0.1mm. I used to wonder about how those milling machines were made and the tolerances required in the dimensions of their components.  What machines were used to make the components? And what about the machines that were used to make those machines? Where does it stop?!

But one of the stumbling blocks with bounds is always with the Upper bound.

Screen Shot 2017-05-06 at 11.04.20

“But you told us that 5 rounds up!” is the usual complaint.  And then we get into a heated arguement about whether:

Screen Shot 2017-05-06 at 11.30.16

Which of course it is, for the same reason:

Screen Shot 2017-05-06 at 11.30.09

(these images from Don Steward can be useful for this one).
Sometimes that argument can be fun, but I think that as teachers we need to beware of going down rabbit holes. The more inquisitive students might find this philosophical debate stimulating but it can be a turn-off for others. And even the ones that do actively engage in that discussion may not be convinced at the end of it.  It’s just another one of those decisions that we make in the moment as we read the mood in the class.  In making that decision we should ensure that we are taking into account the feelings and needs of all students, not just the more vociferous ones.
So, to the point of this post, how to move this debate on.

Earlier this week, I was marking this GCSE question, which most of my students, including those sitting Foundation were answering no problem.

Doc - 3 May 2017 - 14-10 - p1

I’m pretty sure that I have previously suggested that students state the “Lower Bound” and the “Upper Bound” almost as though they are two separate answers.  But expressing the range of possible values in this way makes more sense. It is also entirely consistent with the well-understood rule that “5 rounds up”. In the question above, if the actual length was 13.5, it would be rounded up to 14, but a value below 13.5 would not.  Whereas a measured value of 12.5 would be recorded as 13.

All makes perfect sense now!

 

 

Ideas for better maths teaching