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! :-)