Sick Day

So today was my first sick day - after lying on the floor after my last lesson yesterday I thought perhaps a day off might be required! Never one for complete rest, here are a collection of the notes from the PGCE scrapbook:

Useful Apps

No Hands

http://www.ehyde.com/No%20Hands/

This allows you to type in the names of all your students in each class you teach, add what time you teach them, then can be used as an overlay in the corner of a PPT to randomly select names for questions - great when you're not sure of everyone's names!

If you have any trouble setting it up, Matt Kennedy is also a whizz :-)

Bomb Countdown

http://www.online-stopwatch.com/bomb-countdown/full-screen/

Scares the students to death but gets their attention back! It won't stop bombing of it's own accord so make sure you can easily get to the computer to turn it off!

Starting Off on the Right Foot

An extract from a newspaper article in PDP:

Don't start with the "rules lesson." Instead of lectureing them, do something that makes them go "Whoa!" Take a risk. Surprise them. Delight them.

This (sort of) fitted in with watching Sister Act 2 today (cringe!) She does this 3 lesson in, the one after they glue her bum to her chair :-S

Red, Amber, Green Trays

Following a lesson or assessment - ask students to place their books / papers into one of three trays depending on how well they think they understood the lesson.

Dish Up the DIRT

D = Dedictaed

 I  = Improvement (&)

R = Reflection

T = Time

Find time for students to work on their own or together to improve their work (provide a checklist for this to happen!)

Learning Objectives

Separate learning objective from context and include "success criteria" so that students know how they can achieve those objectives!

Just purchased labels so they can stick them into their books - one attempt at getting them to write it down turned out to be terrible!

Also, after a colourful display of language from a girl I had otherwise considered to be very well spoken, I'm going to try to give them cards if they say a swear word and get them to write down 5 alternative words they could have used - may go horrendously wrong but I thought it was worth a try!

Click View

http://online.clickview.co.uk/MyLibrary/Play?Id=9dee390d-f1e1-a7ae-b5be-bf8b365f0cc5&m=False

Inset Day

A whole plethora of advice was offered to us throughout workshops on Inset Day. As a pupil I always wondered what these days were about and now I finally know!

Feedback
I chose this as the first workshop to attend, as I have found myself spending far too long marking books!
There was a significant emphasis placed on self-assessment and peer assessment  and I felt one of the most effective ways to do this was by use of the rubistar matrix website that we were introduced to:

Plenaries

Another area in which I feel my teaching is weak - I often run out of time or can't think of an appropriate plenary. However, the idea of mini plenaries suited my teaching style very well, so I will use them in the future.

My Name! and The Role of Assessment

This morning's curriculum session saw one of the lead developers of GeoGebra come in and show us just a smidgen of the things it can do - amazing! One of the things we did was create functions to make our names, woohoo! Here is my attempt: http://www.geogebratube.org/student/m23067. Many more examples can be found on GeoGebra Tube.

This afternoon's PDP was on The Role of Assessment; both summative and formative assessment were discussed.

• secondary schools are being accused of "teaching to the test"
• "how do you spell catalyst" - "doesn't matter, you'll get the marks anyway"
• techniques for making the student responsible for their own development
• verbal feedback stamps: "Sir said..."

WWW & EBI from Pupils

I asked my Class Teacher to run a brief activity with my class when I wasn't there to ask for WWW and EBIs about my teaching. Some were really uplifting and some were things I hadn't realised I was / wasn't doing so definitely some useful points to work on. Full notes on this are here, but below I have written some of my most pressing thoughts.

So it seems that in general they like the variety of activities, particularly when they are doing group work (including when they are teaching their peers) and the silent lesson went down a treat.

Questioning

A big concern is choosing which students to ask questions to, and something I remember feeling frustrated about at school.

I want to ask the students who have their hands up, as they are so excited that they know the answer, but obviously then you’ll probably only hear from ¼ of the class throughout the entire lesson. Instead, I was advised to use the ‘hands down’ policy, which has served to get many of the pupils interested in the task, but I then received this feedback from a pupil “I don’t like it when you pick on people when they don’t know but other people are desperate to answer,” which goes back to the original problem of frustration, so I am again questioning the technique.

A possible solution to this lies in the use of whiteboards, getting students who think they know the answer to write things down on whiteboards and hold them up so that I am aware that they know it, but in the meantime use the random name generator to select a student to answer the question initially. I think this would work best if I first asked the students why they think it is important that I find out that some people don’t know the answer.

Task Explanation

A further concern is in how I am explaining my tasks – one student commented that I am “good at explaining what we need to do” yet another said “make the task more clear so people understand better.” Unfortunately I think part of the confusion may lie in the fact that they are not always concentrating when the task is being explained, but I am also aware that many of my explanations have been rushed, especially when I am excited for them to get onto the task!

Perhaps a bit more time spent on explaining with bullet points on the board, and leaving them up, might help. Alternatively, more detailed task explanations could be given to some students if they are continually struggling, but this would involve publicly identifying them, which I would like to avoid.

Relationships

My final, and most upsetting, concern, was a student who said “you talk to other people on my table more than me.” Due to a poor scribbling out on the paper, I have been able to identify the table on which this student sits and will make a concerted effort to speak to all of them equally until the relationship has been mended. However, the student who they felt I spoke more to, was the student on the table who needed more attention in general as he is one of the weaker students in the class. Arrrgh!

Misconceptions

By listening for what you want you don't hear what they say.

Read What is a Fraction. Ask what the denominator is telling us.

Collecting my First Homework

Today I collected in the homework from the lesson I had taught on Interior and Exterior Angles just before half term.
Their task was to write a letter to the cousin telling them about the lesson - this one was my fave!

Children cling onto length x width like it's a lifebelt

Anne began by addressing a point that was brought up last week - that some people felt they couldn't take down everything that was said in sessions. She reiterated that she didn't expect us to do so, but to take note of the points we thought were most relevant to us. We were directed to her texts for further information:

Raising Achievement is Secondary Mathematics - Watson
Mathematics as a Constructive Activity - Watson and Mason
Watson, Pratt, Jones [to be release Jan 13]

Marking Issues

• time constraints
• didn't know the student and know whether the effort / attainment / understanding was 'usual'
• sometimes didn't have classroom context with which to compare
• looking for a specific thing in marking
• giving useful comments and suggestions to students
• thinking about what is actually shown, not assuming understanding that isn't demonstrated

Assessment Methods

A variety of assessment methods were examined during the session, with the following being of particular interest:

• APP: assessing pupil progress
• know it or don't questions are not suitable for hands down questions
• group discussion followed by questioning means all students can be asked as they have had a chance to 'air ideas' with peers
• each student will have their own definition of red, yellow and green in the traffic light system

Gabriel's focus in the afternoon session was on mathematical language, considering where and when it was necessary and / or appropriate to use technical mathematical language.

It was suggested by one of the interns that getting the class to say a new word together may prevent issues with mispronunciation. Additionally a glossary at the back of pupils' exercise books combined with a reward system for using technical terms seemed to work well in a school where there were a large number of EAL students.

We finished on definitions of odd and even numbers, after which I wonder whether we were more confused than to start! It was concluded that the word divisible meant that an integer could be divided by the number in question to give a whole number result, whereas can be divided by did not necessarily imply this.

Rich Tea or a Hobnob?

Our first useful PDP session - a truly fantastic lecture.

I think we were all relieved to hear that the guy who was lecturing us was "crap" when he first started out, but has clearly improved an awful lot since - he had our undivided attention for the entire session!

It was suggested that we had "just enough control for you to be able to teach," rather than aiming for a palpable air of fear when we walk into the room.

We were directed to read Charlie Taylor's Behaviour Checklist (notes here), use gold stars with all age groups and learn names as quickly as possible.

And finally, it was time for Peter Kay...

I've got a £60,000 car out there and I can't read

So I desperately tried to make the most of this PDP lecture on Meeting Individual Learning Needs - Reading and Writing; after I'd got over the fact that a headteacher couldn't open a powerpoint presentation, or in fact any document at all, there were a lot of things that provoked some thought.

Displays

• avoid putting up long pieces of students' writing as other pupils are unlikely to read it
• replace with brightly coloured posters so when students' attention drifts it is likely to fall on something useful

Mazes

We were given a set of 'maze clues' with which to navigate which focussed on the story of Romeo and Juliet. I've tried to find a way to incorporate this into maths; a draft attempt can be found on this website, focussing on a cake problem to be solved using simultaneous equations. I liked the way in which it was suggested that tasks could be approached from multiple angles and so have tried to use this idea in solving the cake problem.

Speed Writing

Apparently one of the hardest things to do is to generate text; once we have something on paper it is easy to go back and improve it. Hence another task was to start with the sentence "what makes writing difficult to me..." after which we had 2 minutes to write. If anyone stopped writing, we had to start the timer again (though this didn't happen). The idea of this is that pupils don't get hung up on whether the sentence is the best they could possibly write, or whether things are spelt perfectly, but that they follow Nike's slogan and

"Just Do It"

I'm not sure yet how this might be useful in maths, but perhaps as a mini-plenary, starting with "Today in maths I ..." and seeing what they write about in 1 minute. To make them more confident it might be best to do it anonymously. It could even be done as a snowball activity...

Japanese Writing

One of the activities we did was to copy down some japanese writing. Here I found myself to be copying it down bit by bit, rather similarly to the way that a dyslexic student whom I had a shadowed was doing during a science lesson - copying words down letter by letter rather than as chunked words. The text can be found here.

And finally, it was noted that parents can have a big impact. The speaker reflected on a time when she had called in a parent to discuss their child's attendance at school. The parent asked the speaker what type of car she had, to which she proudly replied "Audi A3." The parent retorted

"well I've got a £60,000 car out there and I can't read"

So not much motivation for the child to go to school coming from there then!


Slides from this lecture are split between OUTSTANDING READING AND WRITING and READING AND WRITING

Speed-Dating with Year 9s

In our PDP session today, our professional tutor organised us to questions Year 9 students in a 'speed-dating' style. My notes from this exercise are here.

Who hasn't yet aired their armpits ...

Duval states that in order to understand underlying concepts, a change of representation is required. His full paper discussing this is: Duval, R., A COGNITIVE ANALYSIS OF PROBLEMS
OF COMPREHENSION IN A LEARNING OF MATHEMATICS

Anne selected some activities focussing on these different representations, that are shown below.

The following key points were raised:

• use of colour as a checking device
• arrows imply from and to
• careful with 33.3% = 1/3, but  44.4% =/ 1/4
• pictorial representations of money on pink cards
• spatial representation is determined by the students
• may have been better to have physical piles of money
• money could have been replaced with weights

We then looked at a crossing the river problem.

Eight adults and two children need to cross a river. A small boat is available that can hold one adult, or one or two children. 

The task was very structured, encouraging us to act out the scenario, then predict, plot graphs, explain, and generalise. The representations we used were:
ENACTIVE: moving people physically across the river, moving objects around
ICONIC: drawing arrows to represented movement
SYMBOLIC: numbers (data), variables (expression), graph

It was noted that in this exercise using tracking arithmetic, i.e. writing 4 + 1 instead of 5 was useful as it could lead to the equation.

In terms of this detailed structure, Anne suggested that it was not required and more to the point unadvisable. We were encouraged to think about taking children to a high ropes course:

You want to help them feel safe, but they only need ad-hoc hints after that.

 The next activity was another matching activity, this time with graphs.

We were encouraged to find multiple representation software.

The final task was to consider the multiple uses of a number line.

To the Pub!

With CA1 happily handed in this morning, it was time for another session on geometry.

We began by considering a "real-life" problem, which was presented to us by handout:

We discussed the problems in groups and thought about the different ways in which it could be solved, what you would do as the starter activity in this lesson and how you would extend the task.

Possible extensions discussed involved finding the minimum length of fence needed to separate the land; turning a quadrilateral into a triangle of the same area and then proving by induction that any shape could be turned into a triangle of equal area by using the same technique.

This problem was posed in a Japanese classroom; the full lesson can be seen on the website. The key points to take from this lesson were:

• ensuring personal reading and thinking time before group discussion
• motivation by selecting 2 students for Bando and Chiba
• students drew solutions on board before group discussion
• hint cards for differentiation
• verbal explanation of problems avoids problems with different reading times
• second problem was still a high level problem
• hint cards are not read aloud

Later in the pub we discussed the idea of reading each others' Curriculum Assignments anonymously, so I have created a folder here:

https://drive.google.com/?tab=mo&authuser=0#folders/0BxnnS5919b_XNThHc2R4MDNKNlE

Each intern will need to request permission to access this folder.

Ratio and Proportion are Everywhere!

Monday morning again began with Task 2: Teaching Task, this time on shifts of attention. It involved writing algebraic expressions using clouds instead of x, which I personally found rather difficult and made me start to appreciate how difficult it must be for students learning about algebra for the first time. It also made me consider that I should try not to use x wherever possible and get students used to using a variety of letters in their expressions.
A detailed description of the task can be found here.

We began the session by discussing Task 3: Decimal Interviews which we had carried out in pairs at school in previous weeks. I think a lot of us had been astounded by the number of misconceptions that students had; some of the key points are below:

• doing investigations on a calculator to try to find the rule
• sometimes needed to use a closed question to elicit a response
• in multiple choice questions, pupils tried to find the 'teacher pattern' rather than the 'mathematical pattern'
• something about a zero made pupils believe a number was much smaller
• introducing a conflict can elicit response and thinking
• correct answer may not necessarily mean correct thinking
• anticipate certain misconceptions and plan probing questions
• wait time: need to give students the time to think before asking
• some students feel the need to answer every question, even when the question is not directed at them
• choosing example is important: it appeared a lot of the students had been exposed predominantly to decimals that had a value of less than 1

We then had a long discussion about a number of activities that may relate to the concepts of ratio and proportion. One particular question stirred a lot of responses:

Student A got 9 out of 10 for this test. Student D sat the same test but his teacher marked it on a different scale. Student D got 93 out of 100 for his test. Did someone do better? If so, who?

It was also discussed that introducing conflict may serve to increase motivation of the students and forcing them to address their misconception, e.g. are 2/3 and 3/4 the same?

After this session we were encouraged to read further literature on the topic:
Resnick R. B. et al. (1989), Conceptual Bases of Arithmetic Errors: The Case of Decimal Fractions, Journal for Research in Mathematics Education, 20 (1) 8-27

A Session on Task Design - Thank You Gabriel!

Today's session with Gabriel was incredibly helpful as it focussed on Task Design; the subject of our CA1 Assignment. Slides can be viewed here.

We began by considering interior and exterior angles in a polygon, then following the principles of needing to convince:
a) ourselves
b) a friend
c) a sceptic

We then looked at the different ways in which the task could have been designed:

It was noted that no formulation is necessarily "better" than another, but they each can be considered in terms of the outcomes desired and the students themselves.

The following ideas were discussed:

Option 1

• question suggests it should be easy to find out

Option 2

• invites investigation
• like proving your answer to a friend
• students might responds with 'no'! - but then ask why not

Option 3

• what does the word investigate mean
• difficult to lock program to ensure focus on task
• self-differentiating

Option 4

• may not get this exact formula, therefore might assume they are wrong
• when give the formula, students may not feel the need to prove it themselves
• takes away the opportunity for discovery

At the end of this session, Gabriel asked us to fill in a questionnaire to help for a section that he is writing for a book on teacher education. This prompted me to discuss the following ideas:

• considering my own proofs: think about how I am going to introduce a concept to the class; do I want to prove it beyond all doubt or do I want to leave that as an exercise for the students to do?
• possible lesson idea: introduce an idea and then have 5 minutes silent thinking about the concept - find out who is convinced and who is not then get the convinced people to prove to the 'sceptics'
• task design: think carefully about the outcomes desired from the task and how to structure it to achieve these outcomes

A whole lesson of printing...

The last two days at school have been manic!

On Tuesday I taught my first small group lesson, which went well but there were lots of things that could be improved! My template for lesson plans and their reflections can be found here. As I only taught 6 pupils the topic, the teacher took the opportunity to use this the following day and had each of those students teaching a table each - it worked really well!

The majority of observations this week have been for CA1, so looking at task design. I have also managed to get hold of photos of the students I am teaching, so have been able to add them to my database and learn names as quickly as possible! 

Helping out with netball club after school has also helped me to get to know some of the students, but I'll definitely bring a whistle for umpiring next week!

Teaching with Bayley

Our School PDP session this morning focussed on Behaviour Management where we looked at a lot of the clips from the Teaching with Bayley series.

Do you want a room full of smelly armpits?

The day began rather nervously as we were the first TLC to present Task 2: Teaching Task. After initially feeling completely bewildered by the tasks we were to present, I felt it went relatively well today - our full plans can be found in the Task 2: Teaching Task folder.

Anne then began an initially bizarre session with number grids but then got us to consider the thought process she had gone through to decide on how to present each part of the lesson to us. Here are some of my thoughts:

• began by determining the terms involved: domino, cover etc.
• allowed us to choose our own shape: gives students a sense of personal engagement as well as meaning each student may be working on slightly different questions; deliberately disallowed the simple bar to avoid laziness!
• initially gave us the 100 multiplication grid then made it easier
• chose 7 addition grid next as it avoided any erroneous multiplication patterns that may have been found from even number grids
• the progression to a 8 addition grid provided an incentive to generalise, which was then forced when we were presented with a problem of an n addition grid
• we were tasked with writing our own questions similar to the ones that Anne had posed to us: this could be used to write a homework, using the pupils own ideas, or potentially a match up of shapes and co-ordinate sets

Van Hiele Levels of Geometrical Thought
Starting from the very basic thought process, Van Hiele define the levels as:

• Visualising: seeing whole things
• Analysing: describing, noticing same / different
• Abstraction: distinctinons, relationships between tasks
• Informal deduction: generalising, identufying properties
• Rigour: formal deduction, properties as new objects

Full definitions of the Van Hiele levels.

This way of thinking is not restricted only to top set with other groups just using arithmetic!

Adolescents are very concerned with identity (writing name on objects), belonging, being heard (but don't always know how to get your attention), being in charge (even of giving advice to the person next to them), being supported, feeling powerful (feeling that their idea was really important), negotiating authority (maths is the authority in the classroom), arguing in ways that makes adults listen.

"There are very few things in maths that can't be checked if the right tools are given."

Finding tasks can have either one or multiple answers, and maybe even define a class of these answers. A closed question can still open up your thinking!

Aside on powerpoints ... might be useful to have dots on a page where things will turn up.

Controlling parameters in examples helps students to see patterns and generalise, ie. introduce sequences as:
2, 4, 6
2, 5, 8
2, 6, 10
Rather than just choosing random examples, like the mish-mash that seems to be found in textbooks.

Gradient exercise: might have to know p-q-r-s question to do earlier questions even if it hasn't been explicitly written down, kind of like catching a ball and parabolas.


Lesson Planning
There is no research that shows that starter, main and plenary is the best way to structure a lesson. A plenary doesn't need to be at the end, as it is coordinating the whole class discussion and summarising ideas.
Lots of stuff about Japanese styles of teaching was discussed, the notes on it can be found at https://weblearn.ox.ac.uk/portal/hierarchy/socsci/education/pgce/maths/page/resources
at some point in the very near future!


The day concluded with a PDP lecture on Adolescents which was infinitely more engaging than the last!

Multiplication Madness

Today Nick threw a number of addition, subtraction and multiplication sums at us, to be done both mentally and written down. The variety of methods we used to do this was quite large!

Without doubt though, the new psycho method of multiplication was the most interesting even if it would take some getting used to ... but in involved no carrying! (similar to the Chinese method in that regard)

A Day of Nothing

Just in case I read this back and wonder what I did ... I did in fact, do nothing. Apart from a PDP lecture.

Much Decimal Confusion

Today was another day in school, this morning seeing a Year 13 lesson in mechanics. They literally worked like angels and the teacher could have just done boring textbook stuff and they would have got on with it, but she still created a thoughtful lesson. In particular I noted the task for the starter, which is detailed in Task Design Observations - Amy.

Much of the rest of the day was spent interviewing Year 8 pupils about their understanding of decimals, where a variety of mahoosive misconceptions were seen:

• tenths column is in fact the unitth column
• 1 - 8 = 7 and 0 - 4 = 4
• 4087 is 4 million and 87

Challenge accepted.

Why don't you get Miss to teach so you can rest your voice?

Our PDP session within school today involved a minibus tour around the local area, where we saw a real variety of housing standards and from this could gain certain insights into the range of background from which the students may originate.

Within our TLC, after reading sections and discussing our finding in meeting at G & D's yesterday, we have decided to focus on Task Design. My observations on this theme from today can be found at Amy - 20121002 - Task Design Observation.

The following period was with my mentor who gave us lots of interesting information and showed us where to find education needs information for the classes we will be teaching.

Fourth period was spent with a Year 11 class, looking at vectors. They began by doing an exercise in the worksheet (sadly the teacher had almost no voice so was unable to do much interactive teaching.) When the teacher went up to explain an exam question, one of the pupils shouted out

"Sir, why don't you let Miss teach so you can rest your voice?"

So rather than pansying out, I decided to accept the challenge and after a kind student showed me how to use the interactive whiteboard we were away. Not my finest performance as it was completely unplanned (note to self: don't ever intentionally try and "wing-it") but we got through it and overhearing a few

"Oh, I get it now!"

was rather rewarding!

We conducted some of the Decimal Interviews with a Year 8 class this afternoon with some higher ability pupils, which still threw up some interesting misconceptions! Our preliminary notes are here.

A brief chat with the PE department at the end of the day enabled us to plan for tomorrow - netball club after school! :-)

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