To proportion and beyond!

Today's session with Anne was all about linking shape, algebra and number.

We began by first considering Task 6, where we are to mark a set of books and were offered the following suggestions:

1. Use post-its so you don't "commit" yourself to the feedback

2. Strategies to ensure you have a life:

• glance at all books and cover key mistakes next lesson
• get students to peer mark in class for quick questions
• have a key question to look at for every student during the lesson, which will show their understanding well

We then looked at various samples of poor marking and decided on what comments / feedback would be more appropriate. The comments can be found here.

Next we moved onto looking at the number of minimal paths that could be taken between a point and the origin on a graph...

and found that Pascal's Triangle popped out!

By the final task, I think we were all feeling very Friday afternoon-like and therefore struggled to understand what was going on with our formula for a T...

... when one intern remembered the session was meant to finish at 4pm!

Baffled by Addition

Gabriel's session this morning began by looking at the Sushi Problem, a self-differentiating problem which could be solved by a variety of methods. The mathematical ideas in this problem covered:

• fractions
• problem solving
• addition
• trial and error
• manipulating fractions
• writing equations
• solving equations
• visualising
• changing whole
• reasoning and proving

where the pink writing represents non-content related mathematical skills.

In order for this to be an effective task, Gabriel circulated around the room, looking at different solutions, to allow him to lead the lesson by choosing certain students to give their methods of solution. Slides from this session are here.

Dealing with Misconceptions

There are two ways to do this:

• choose questions to avoid these misconceptions coming into play
• choose situations to bring these to the surface and then quash them forever more!

Shortly after lunch came the baffling multiplication and addition...

Anne gave us a bottle of lemonade to drink between two of us?

Have they shared the drink?

Have they divided the drink equally? 

Following an activity involving equally segregating various items between our tables, we came up with the following list of division models:

• simultaneous counting
• angle division
• division including congruency
• measurement
• division by folding and cutting

It was then noted that there are two main types of division, division resulting in stuff and division resulting in pieces. Additionally, division can often be made easier by looking for common factors, but this is rarely taught until Year 7!

Next came a lot of playing with blocks - bedlam.

"Don't slap them about for playing"

 a + b = c           c = a + b           b + a = c           c = b + a

These relationships showed that the equals sign doesn't mean work something out, but rather that equality works both ways. In particular, they show 8 ways to represent the same relation, without any reference to numbers. This could also be used an introduction to algebra.

           

Multiplication is not repeated addition!

This is the way that almost all students are taught at primary school, but is stretching a rubber band not also multiplication?

Remember: learning your times tables and long division algorithm by heart is not the answer!

And finally, we briefly discussed exponentials (which is most definitely not repeated multiplication!) but we struggled to find a definition other than scaling by a changing scalar.

An Inspirational Day

Today was an incredibly inspirational day, led by Mike Ollerton, http://mikeollerton.com/index.html

My main conclusion is that I am going to have the saddest wish list for Christmas ever seen, but all of very valuable items for the classroom!

Novel Mathematics

The session began with the use of playing cards, totalling them, arranging them into sums, then moving onto magic squares (though this was led by first getting rows, then columns, adding up to the same number). 

Other ideas for playing cards were also given (see ATM). 

Next was a fantastic way to visual fractions. By folding a piece of paper one way to make a wardrobe, then the other way to make thirds, we formed twelfths of the shape (lines were drawn on with pen). The task evolved as follows:

Dividing a single sheet up into fractions to use in
equivalent fractions and fraction manipulation

• hold up 1/3
• hold up 1/4
• what is 1/3 + 1/4 ? (can see immediately from counting without being taught method)
• continue with subtraction
• what is 1/3 of 3/4? - take 3/4 size and fold it into 3
• division: using a 'division arm', count how many twelfths on top over how many underneath - interesting to show that division can still be done by first going to equivalent fractions
• repeat with 3 x 5 grid
• finally lead on to finding a method
• generalise algebraically

What does an art student take with them?

      -  a portfolio

What does a maths student take with them?   

      -  a result 

This lead us onto thinking about using written work - getting students to write about what they have learnt and the process that they went through is a far clearer way of gauging their mathematical understanding than a simple test. It was also suggested that children, occasionally, need to stand up at the front and give a very short presentation on something about mathematics. And idea for this presentation could be: "where is the mathematics in..."

Criteria for Assessing Written Work

• communication - steps taken
• extension of the task - and doing that extension
• mathematical understanding
• mathematical vocabulary
• working systematically

A lot of the emphasis of this session was on the culture of the classroom.

A key thing to come out of this was "being able to deal with stuckness." Teaching the students how to approach a problem was key to life and exams. When Mike was told by a student, "I'm stuck," he replied:

"Hi Stuck, I'm Mr. Ollerton, nice to meet you."

Other activities using algebra involved looking at Fibonacci's sequence, choosing any two numbers as starting points. We then gave our partner the starting and the ending digit of a 5-box Fibonacci sequence and were challenged to find the missing numbers - a self-differentiating task as it can be tackled either by trial and error or by algebra. This can then be further extended to 7-box, 9-box etc and finding the middle value.

Geoboards.
Made from 12mm x 160mm x 160mm
MDF and escutcheon pins.

Geoboards are without doubt the most versatile resource I've seen so far, here are just some of the ideas of topics we had for their use: 

• vectors
• gradients and equations
• co-ordinates
• symmetry
• reflection and rotation
• shapes and their properties
• area and perimeter
• trigonometry
• Pythagoras

As a way to establish prior knowledge, paired work was suggested, e.g. write down everything you know about the number 6, then go round class writing all ideas up on the board.

With all of these "rich tasks" it seems the most important thing is:

"everyone can participate if the starting point is simple enough"

Another activity was using a "decimal grid" and the variety of tasks that could lead on from there, seen around the outside in blue. It was particularly noted that joining up 'isovalues' gave the gradient of the line.

The final activity was to look at angles and algebra - essential making two crease lines on a piece of paper, drawing over them, then following the tasks on the sheet below.

Handouts on Ofsted criteria for Outstanding Teachers can be found here.

First Day in School

Today we had our first day in a "real average school." (Quote from Professional Tutor!) A wealth of advice was offered during the day, with the following key points:

Contract Activity

• do an icebreaker with your class - get to know them
  (possible collage for homework)
• contract activity       ===============>
• seating plans: look at class IEPs
• raid the shared drive for all it's worth!

With everything I saw as well, I have started to make a Wish List of items I would really like to have as teaching aids!

A few other things to think about were:

Giving Useful Feedback

Feedback

Making sure it's actually relevant, and subscribe to the WWW EBI theory, seen right.

Teachers TV

This has now been closed by the government (grr...)  but the clips can be found on a mirror site at:

http://www.schoolsworld.tv/taxonomy/term/64,87,98

We then spent an hour after school using the whiteboards and commenting on the effectiveness of each others' use (Task 1).

Organisation Central

With so many things to think about, most of this evening was spent organising my tasks, room, food etc!

Here is a summary of the key points of today:

Preparing to Teach a Topic
Nick gave us a framework to use when planning to teach a topic, to help us think about all the different aspects which will guide us to plan tasks and lesson structure. Currently working on a database for this: Lesson Planning and Evaluation.

The essential questions are:

Motivation
1. Why is this topic worthy of study?
2. What is the relevance of this topic to life / maths?

Thinking
3. What images of experiences come to mind?
4. Have you ever thought about it differently?

Doing
5. What do you 'do' when you 'do' this topic?
6. What do you 'say' when you 'do' this topic?

We answered these questions for the topic of Negative Numbers. An important thing to consider with this topic is terminology, i.e. subtract, negative, minus etc.

Novel Mathematics
Nick gave out two sets of numbers and asked us to organise ourselves into groups of 3 or 4 where by adding or subtracting our numbers gave zero. 

TLC Meeting
We briefly got together as a TLC to decide on our observations for this week, the framework of which is in the Question Observation Template.


The afternoon was a PDP session on how race, gender, ethnicity, disability etc. affect your view of children as learners.

Back to Exams!

One of our tasks for the week was to complete a self assessment on the topics we felt we were less confident in, so to do this I decided to do some A Level exam papers, which highlighted the following topics:

• C1: factor theorem
• C2: trapezium rule, binomial expansion, revise arith and geom series esp summing, LOGARITHMS! check transformations, tricky trig identities
• C3: iterations, logs, complex trig graphs, inverse functions, composite functions

Time to get studying!

Very Confusing Decimals

Our prior reading for this lesson gave us an insight on what this lesson would be about - a variety of epic misconceptions about decimals and place value amongst students.

Childrens Understanding of Mathematics 11-16 [B2]

 We were encouraged to think about the importance of carefully selected questions to test understanding. This lead onto Task 3: Decimal Interviews. Within this task we are to conduct short interviews with Year 7 and Year 8 students, focussing on specific misconceptions with this topic and trying to find out why these misconceptions may occur.
Our TLC group have selected "Instrumental and Relational Understanding of Place Value" as a topic to examine. Our planning for this interview can be found in the Task 3 folder.

At the end of the lesson, Gabriel asked us for anonymous feedback about the lesson - might be useful to do to find what the students enjoy / appreciate / find annoying etc.

For the final part of today - UNION TIME! Having little idea about what separated the unions, I asked a few questions and after finding out that one of the unions had many male members that didn't think that women deserved equal pay, I was fairly sure that wasn't the union for me(!) so went for National Union of Teachers - we'll see what they're like!

Multiplying Multi-Digit Numbers by 7

Title: Multiplying Multi-Digit Numbers by 7
Learning Objective: Multiplying Multi-Digit Numbers by 7
So what do you think the lesson is about today? Multiplying Multi-Digit Numbers by 7

So today began with the lesson from hell, Anne maxed us all out by putting us in an actual lesson situation, which we were then to tear apart after she'd finished! We came up with a whole load of ways in which it was "wrong" - how long until I accidentally make one of those mistakes :-S The main issues were:

• insufficient number of examples
• no examples written down or left on board
• blamed a mistake she made on us
• didn't qualify what "show working" meant
• didn't allow us to use the method we were used to
• no space on worksheet
• completely unenjoyable!

I also found out something very interesting about dyslexia, that students struggle to read and write at the same time - something to consider. Another intern was particularly helpful in highlighting some other points to think about to make it easier.

We were then introduced to our first task:

Task 1: Writing on a Whiteboard

More notes on this task can be found on the Maths Tasks document.

A random idea that came to me was for a possible plenary activity: "This maths will be useful to me in the future because..." perhaps to encourage students that the topic will actually be useful!

We were then looking at ways to be articulate in mathematics; our words are below.

Sticker Match up Lesson with Nick

The class began with each student being given a sticker with a shape on it and sent outside - we were to get into groups of similar shapes, then split into two according to a criteria we decided upon.

Back in the classroom we were given a sheet of all the shapes in our group and asked to think of different ways to group them.

N.B. These were on stickers which we stuck onto post its so they were "easier to move around."

A possible extension to this, would have been to have each shape labelled with a prime number, then multiply those in each group together, give it to the other "half" of our group and get them to do prime factor decomposition to work out the groups. They would then need to look at the pattern to find out which group one belonged to.

Two types of knowledge:

Subject knowledge

Pedagogical subject knowledge  (articulating the ideas to a class)

Novel Mathematics
Kaprika's Problem 

Any three digit number, providing it had at least two different digits that were more than one apart would follow this pattern.

a) Arrange digits into largest possible number
b) Arrange digits into smallest possible number
c) Subtract from each other
d) Take answer and repeat
It would always converge to 9. 

Extension: Think about convergence in different bases. Doing this in base 8 made us consider what it might be like for struggling student to do it in decimal!

Another thing to consider is what do you do when some of the class get it and some don't?

• extend for those that understand, repeat to improve others
• bring whole class together

The day finished with our Friday Reflections Assignment, a retrospective look back at this week for Monday and deciding on our topic for next week. This is within our TLC, topic discussion can be found in the Friday Reflections W/B 24/09/2012 document.

Finding out our Schools!

Presentation: Learning as a PGCE Student
The day began by highlighting our tutors in different positions:

• Curriculum Tutor: Gabriel
• General Tutor: Roger Firth
• School Mentor: ?
• Professional Tutor: Chris Deakin
• College Tutor: ?

We were advised to "get reading" so after setting up an account and password to log on to the computer, I was finally able to search for

Getting the buggers to behave [B1]

find the coding and meander around the library until I found it. As I'm expecting a lot of reading this year, all of the references will be put on the document below.
Course References

Trevor made a very interesting comment about feedback on pupils' work:

Pupils never intentionally make a mistake; most children are trying to get it right

So when looking at their mistakes, try and think about why they've got that misconception and think about how you can correct it next lesson. Additionally, when helping a particular student, position yourself so you can still see the majority of the class.

Things to think about in observations:

• classroom organisation
• spend some time in reception
• opening and ending routines
• transitions between phases
• behaviour management
• see the same class / student in different subjects
• targetted observation is more useful than general
• continue observations throughout the programme

When teaching, think about what you are aiming to achieve in that lesson.

The presentation culminated with a rather depressing quote:

Just when you think you see a light at the end of the tunnel, it's just some bugger with a torch bringing you more work.

Fabulous. 



Meeting with General Tutor
Yesterday we found out which school we are in:

Today we met with our general tutor in school groups, where I have found 9 other students placed at my school including one in mathematics. Note - this tutor assess our PDP Assignment (Personal Development Programme).

We'll be given a travel subsidy of £37 for the journey, a nice bonus! Contributions to petrol from other interns is found here.

Presentation: Schooling and Public Policy

Although rather difficult to hear the lecturer over the air conditioning, some key points were made:

• 1944: Secondary School education became available for all
• Governments have a big impact on the education system
• Teacher standards have now been reduced and made more concise, so there is now more room for interpretation
• Introduction of academies means they are now reportable to the Minister of Education rather than their Local Authority

This final note was interesting:

You can have equality or equality of opportunity but not both. Equality means holding back brighter children.

NCETM Tests

Somehow managed to get my brain to function this morning so am doing the NCETM tests to gauge general subject knowledge. Definitely some key points to work on!
https://www.ncetm.org.uk/self-evaluation/summary/101/121?all#topOfNextSteps

My results are in the document: 20120919 NCETM Curriculum Knowledge Notes

First Day of Maths

So after all the overwhelming amount of information we've been given and the number of things I need to keep track of, I can't think of any other way to do this except keep a blog!
So here it is, my diary of this apparently hellish year, all compacted into one place. Horrah!

The day begin with an effective icebreaker, which had some other underlying aims, a theme we were to see repeated throughout the day. The first idea I particularly liked, so will code this under my "novel mathematics" colouring:

Novel Mathematics
Each student was given a post-it to put on their head. We were then to ask questions to other students to allow us to guess our post-it, e.g. am I a prime number, am I a function etc.

A phrase to remember was:

Work out what delayed you today, so you can ensure you arrive on time tomorrow.

When students arrived late, the teacher asked others on their table to bring them up to speed. We are advised to consider management of resources to ensure they were used most effectively.
When asking questions, it became important to ask students who would have something to say, e.g. after group work. Avoid shaming pupils who aren't paying attention but instead use a phrase such as:

Matthew you've gone off somewhere, can you come back to us please.

Shown that movement of pupils is not necessarily a bad thing but can be used to carry ideas around the room; avoid it seeming like a punishment.

The session culminated in discussing our observations from the previous week in our TLC group, notes of which can be found here:
20120918 Contrasting Techniques for Behaviour Management between Primary and Secondary Schools

The plenary was an exercise in class counting down in fractional increments - harder than it sounded!

Friday Reflections Post Observation Week

After three days spent in a primary school and two in a seconday school, our first task was to write up a Friday Reflections exercise, summarising our experiences of the week, which can be found below:
20120917 Friday Reflections on Week 1 Observations.docx