Temporary Blog

This is my temporary space to put posts while I import from Posterous.

The King of shapes: the stellated icosahedron

There's nothing quite as good as mathematical toys for Christmas. After I had wrested this 'geomac' off the children, I made my very favourite shape – the stellated icosahedron.

I just love adding points to a platonic solid.

With 60 faces, 90 edges and 32 vertices, Euler's formula still holds true: 32+60-90=2 (vertices+faces-edges=2 for all solids without holes in them).

The question for young mathematicians is "do all the 3D shapes you know follow this rule?" and following on from this "can you make a 3D shape that doesn't follow this rule?" [clue: try making a donut out of geomac].

Good teaching decreases mathematics anxiety

This weekend, I found myself doing something I've not done before – disagreeing with Professer Derek Haylock. Giving his second lecture to Edge Hill MaST cohort 1, the man who's seminal work "Mathematics Explained for Primary Teachers" has pride of place on my shelf, said some things that didn't quite hang together for me.

His lecture was on the subject of mathematics anxiety – something that most adults have either experienced or can empathise with. His main point was this: if you teach mathematics well, you don't get students who are anxious about maths. As someone tweeted on the day "My God I never thought of that. I hope the person giving this advice is paid a fortune." Given that the audience was a room full of primary maths specialists, or 'maths champions', the advice is more purposeful if given a more negative slant: don't allow bad maths teaching in primary – you'll just get adults who are anxious about maths.

Briefly I will sum up what I thought were his main points and then I'll say where and how I disagreed with him.

• Many adults experience anxiety in maths when they are afraid to make mistakes in public, or given a mathematical challenge they cannot think clearly to carry it out.
• These adults can trace their feelings of anxiety back to a single experience usually between the ages of 9-11 at primary school.
• This experience is always a negative interaction with a teacher – Prof. Haylock quoted adults saying that their teacher had shouted things like "why can't you just get it right?" There was a real emphasis on the negative experience being when maths is thought of as either right or wrong.
• Many of these adults reported they could only learn maths by learning a rule by rote and couldn't master any conceptual learning.
• Some of these adults become primary teachers.
• Teaching styles are to blame for mathematical anxiety – 'traditional methods' create more anxiety; a 'problem-solving / relational approach' creates less anxiety. Quoting from Newsted, he described a traditional approach as one of direct instruction, followed by practice and application, whereas in the 'problem-solving approach' the teacher acted as a facilitator, with the children suggesting their own methods and strategies for solving problems.

Aside from the dangers of telling rooms full of teachers that 'rote learning is always bad' and 'this is the only way to do it', my main disagreement was the way he linked the single negative experience with a given teacher to the traditional teaching method. It doesn't take the room being in rows or table groups for you to have a bad experience with a teacher. Neither does it mean that you if are using a 'problem-solving approach' then teachers can't lose their tempers and make everyone frightened of maths.

In my own experience I've tried both traditional and 'problem solving approaches'.

I would call them using a rigid scaffold and using a negotiated scaffold. In the former, the teacher plots the course through the learning (the scaffold) and takes the students through that course through direct instruction, practice and intervention; in the latter the student and teacher negotiate the path through the learning.

Both approaches work.

In fact this time last year I did an experiment where I did 6 weeks of negotiated scaffolding in maths, then 6 weeks of rigid scaffolding in maths. The children made progress in both periods.

Delving a bit deeper into the Newstead report I see that the traditional approach includes: "The teacher decides what is right or wrong and intervenes in the case of mistakes. Later word 

sums may be used as application of methods. Social norms are more static and involve more discipline, rewards and teacher authority." Now to me that's not traditional teaching. Traditional teaching is where direct instruction is followed by practice, yes, but then appropriate intervention from the teacher. And so now it leaves me thinking that Haylock, quoting Newstead isn't comparing 'Problem Solving' with 'Traditional', but is comparing 'Problem Solving' with 'Bad Teaching'.

I'll go on to say that Haylock is right by saying that for a student to have one-to-one negative interactions with an authority figure such as a teacher will cause anxiety, in any subject. The teacher that chooses 'traditional teaching methods' but can avoid the negative interactions can still teach a class without causing anxiety amongst the students. And a teacher that attempts to be a 'facilitator' but then loses their temper when the students don't choose a method they were anticipating will also cause anxiety. It's not about the style, or dare I even say it the teaching, it's about the teacher themselves.

Good teachers reduce anxiety.

Good for the fractions learning; bad for the coffee mug

Sometimes children hear the word 'fractions' and they turn off.

I saw it on Wednesday when I started my lesson on comparing and ordering fractions. I had barely uttered the words when I saw a few heads drop. A few children joined in when I asked them what they knew about fractions – one knew the word 'third'; someone else knew 'part'; yet another one knew they have something to do with division. But quite a few heads with dropped.

So while the keen had their hands up, and others were looking to avoid eye contact, I slid an empty coffee mug into an empty plastic bag. Then, for security, whilst the conversation continued, I placed the first plastic bag into a second one.

Then I smacked it against the wall. Really hard.

All the children looked – some jumped.

I proceeded to pull pieces out of the bag and estimate how much of the mug each piece had been, from the large chunks (1/3 or 1/5) to the tiny chips that were only 1/1000 or maybe even smaller.

The children were engaged and by the end of the lesson all of them had made some progress about ordering and comparing fractions. Even the special needs group children who, according to their data, struggle to order numbers 1-100.

As a bonus, we even specified that the bottom of the fraction was called the denominator and the top number the numerator – I love it when children learn proper maths words, although it was amusing to hear one child call the top number the nominator and the bottom number the dominator.

So, if you're stuck with teaching fractions – break something. At least you'll stop the heads from dropping…

Growing Leaders Session 9 – Staying Fresh

Today the participants have been reflecting on staying fresh in leadership.

What can you put in place to maintain your leadership role without becoming weary and disillusioned?

We discussed 5 stages of training:

1. Personalise your training
2. Stretch yourself
3. Work out a rhythm
4. Keep close relationships
5. Complete the course

Each participant came up with suggestions on how to improve in these areas, and hear are some of their suggestion:

Personalise your training

• Continue to meet with mentor after the course (if they are willing) – to discuss ongoing issues.
• Is there anything you need to stop doing to give the time to get where you are going?
• Get more biblical knowledge
• Focus on God
• Practice a spiritual discipline
• Plan rather than simply just letting things happen
• Regular fellowship meeting
• Pray and push at doors
• Listen to Him for my plan
• Words of encouragement and motivation
• Be open to new ideas including ongoing discussions with mentor
• Be prepared to share ideas and thoughts.
• Write the plan down.
• Share with people who can support you.
• Review personal life statement.
• Discuss issues with mentor.
• Prayer
• Use personal life statement to develop priorities.
• Take stock of what you have.

Stretch yourself

• Find out from God which areas He wants to grow me in and be prepared to delay others for the 'Greater Yes'
• Don't stretch myself unnecessarily but trust God to give me the strength to grow.
• Be open to new opportunities – take risks.
• Seek God to help recognise areas for growth
• Seek God to smooth the 'rough' edges.
• Struggle to remain open to further change.
• Be willing to step out / try new things.
• Make time to reflect on day every evening
• Be open to requests for help from the church.
• Use journalling to remember the positive experiences and lessons learned from not-so-positive ones.
• Learn not to be afraid of vision.
• Be brave enough to practice gifts God has given you in opportunities you have.
• Share thoughts about gifts / direction your going in with others
• Pray / ask God for guidance / opportunities.
• Change and development are part of growing.

Work out a rhythm

• Put things around you (at home, work, in the car, etc.) that will help you direct thoughts towards God.
• Keep a rhythm of spiritual discipline.
• If you don't manage time, then it will manage you.
• Daily Bible study
• Prayer at same time / place each day.
• Have time for myself at work – be willing to delay dealing with non urgent matters, until I have time to reflect.
• Keep a rhythm of spiritual discipline.
• Never say "I'm too busy for…" or "I don't have time for…" you always do have time but need to make decisions about how to spend it. (Editor: Gandalf himself said "All you have to decide is what to do with the time that is given to you.")
• Pray
• Have a plan to follow but be flexible to change if need to.
• Complete day off – totally away from work – be accountable to this.
• Meet deadlines rather than seeing them as flexible – don't expect too much – give yourself more time than you think you need.
• Plan in rewards.

Keep close relationships

• Increase our openness to being forgiven – it will help us in the area of forgiving others.
• Keep meeting with mentor
• Pray for and look for opportunities to serve others.
• Be yourself with people and don't be judgemental.
• Value relationships
• Show others that you care.
• Learn to make time for others.
• Seek and take opportunities to invest in others.
• Be more disciplined in prayer life
• Prepare more for mentoring sessions.
• Be intentional in all that I do.
• Plan time to spend with others regularly – put it in diary so other things don't crowd.

Complete the course.

• Be honest with God about our good bits and bad bits.
• Remember how God has blessed me in the past and expect that in the future.
• Invite the Holy Spirit into my day EVERY day for guidance and strength.
• One day, we will all be part of the most beautiful worship service ever! For all eternity!
• Time to reflect / listen to God;
• to soak in God's presence and hear what he may be saying for the longer term.
• Be faithful in the things (little or Big) that God gives me to do as I run the race.
• We are like children going to work with our Dad for the day. He doesn't need us – chooses to involve us because he loves spending time with us. Enjoy it.

The session ended with prayer.

The book all primary / elementary teachers should read. #mathchat

I was just engaged in a conversation on #mathchat about skill levels in primary teachers, when I realised that the book all teachers of young children should read was sitting right next to me: ‘Mathematics Explained for Primary Teachers’ by Derek Haylock.

Do’s and Don’ts of Primary (Elementary) level Algebra

In my last post I argued that we should be teaching the thinking that becomes algebra from as early an age as possible. But what are those skills? What are the Dos and Don'ts? Many of the don'ts stem from the place of arithmetic thinking in our curriculum. Thinking arithmetically is all about getting a right answer, it's not always about being able to use that right answer to get more right answers in the future, and I think this is at the heart of what follows:

To develop algebraic thinking:

Don't:

1. Don't use the equals sign as an operator. Many children see the equals sign and think Do something; Work that out; Add those. The equals sign represents balance, equivalence. Children need to learn that in arithmetic to support their algebraic thinking.
2. Don't represent things with the same initial letter as the problem, like 'a' for apples and 'b' for bananas. All it does is reinforce the misconception that the letter stands for an object or a specific number, rather than a variable.
3. Don't get tied up in knots about BODMAS (the order that operations are carried out). The context of the given problem will sort that out. It needs to be made explicit when algebraic notation is introduced – you can explain how different calculators work those our sequentially or using an algebraic precedence of operators.
4. Don't limit thinking about sequence to the next number. See if the children can see the rule or the pattern.

Do:

1. Teach patterns from an early an age as possible. Here's Marylin Burns fantastic lesson.
2. Do give children plain paper for them to represent their maths graphically.
3. Tabulate patterns and sequence so children can move from seeing the 'up-and-down rule' (the sequential generalisation) to the left-to-right rule (the global generalisation).
4. Follow the previous step by asking 'what's my rule?'
5. Use empty box problems (e.g. 4+□=11)
6. Do encourage children to represent the problem, not just solve them. Then the numbers can be changed and children can use the same representation to solve harder problems (perhaps by using a calculator and a spreadsheet).
7. Do use a trial and improvement approach. This is especially powerful when it can be done using a spreadsheet.
8. Do use the fantastic free materials that exist free all over the internet. Here's some that help children to find rules and describe patterns that the UK government produced a few years back, stored on the website of Dudley LA.

If there are anymore do's and don'ts, or any that you disagree with, please leave a comment.

At what age should we start teaching algebra?

Like many people, algebra is a slightly painful word. Rows and rows, indeed columns of columns of x's and y's attacked me at secondary school. I didn't really get what they meant, even though I was actually quite good at solving equations.

Now as a primary school teacher I still have a blind spot when it comes to algebra, there's something about it that I don't quite get.

But I've had a revelation today. I think I know what I've not been quite getting all this time.

I've just read a chapter in a wonderful book by Derek Haylock: "Mathematics Explained for Primary Teachers" (4th Edition). I've been able to access the book through the MaST programme I'm on at Edge Hill University – but it was so good that I bought the whole book from Amazon. It starts with a question that illustrates why I don't get question. I don't want to steal Haylock's thunder, so here's a different version of the same concept:

On a school visit, 6 students are can go for every 1 teacher. There are t teachers, s students can make the visit. Describe the relationship between s and t.

The temptation is to say 6s=t. That is exactly what I did in the equivalent problem that Haylock set me. But then, say 30 students make the trip, then according to the equation I just wrote, I need 6*30 teachers. 180 teachers for 30 students? Slightly over-powering! The answer is s=6t

Haylock makes the point that I'm getting confused between 'things' or 'objects' and variables.

In arithmetic, which dominates primary teaching, I use letters as abbreviations – hence 't' for teachers. There's also m for metres, kg, mm, l, and many more. In algebra, letters never represent abbreviations for measurements, they represent variables – they stand for whatever the number you've chosen. An amount that can be changed. It is precisely for this reason that it is unhelpful to use 't' for teachers and 's' for students, because it provides the illusion that you are representing the actual teachers as a tangible thing., rather than the number of them.

I think many of us in teaching younger children think of algebra as a nice extension to do when the children have really got their arithmetic sorted. But I'm seeing now that if we only ever train children to think arithmetically, than we are doing them a disservice. Algebra is a branch off the same mathematical tree that Arithmetic grows on, it is not a branch that nicely extends from Arithmetic. Algebra develops from recognising and playing with patterns, investigating sequences and seeing how things can be represented as bigger or smaller. Many of us teachers, especially in schools were standards are low, look at these lessons and wonder 'how will this help the children's maths?' And by maths we are thinking of arithmetic and doing well in tests (which for 11 year olds are about 50% arithmetic). We are not thinking of developing the children's brains so they can generalise patterns and represent problems.

I can hear the question being posed. So what? Why should children have to generalise patterns and represent problems?

Well the answer comes down to being able to solve problems with much bigger numbers and larger degree of complexity. I might be able to solve a problem with my arithmetic skills, but if I can represent it I can use a spreadsheet or a scientific calculator to solve it for any number. Likewise I might be able to work out the 15th term of the triangular number sequence, but working out the 77th is a rather harder challenge – I can save loads of time by generalising the pattern, representing it with algebra and calculating from there.

I wonder how many software developers, games designers, app creators and the like can get away with only thinking arithmetically? I don't know anything about how those kinds of jobs work, but I'm sure that some level of algebraic thinking is required for those jobs.

So. An answer to my question: as young as possible. In my next post I'll start to explain how…

Growing Leaders session 7

Growing Leaders Session 7 began with an activity where participants thought about the groups they are part of and what metaphor best suits their group. Car, Animal, Chocolate Bar were the categories chosen, and participants thought about what type of car, animal and chocolate bar would be most appropriate for their particular group.

Some of the metaphors were:

Car

• Range Rover sport
• Old Bus
• Battered Lada
• Something that's difficult to get started.

Animal

• Camel
• Cow
• Tazmanian Devil
• catterpillar
• Octopus

Chocolate Bar

• Revels
• Galaxy Caramel
• Quality Street
• With Fruit and Nuts

Alison followed on to talk about how teams work together to achieve three purposes – to meet the needs of the task, the needs of the individual and the needs of the group. She used illustrations from the Growing Leaders leadership team (Rob, Alison, Emma and Steve) to demonstrate some of these points and how the different strenghts of these individuals work together to make a team that kind of works.

The participants then split into groups of their own ministry areas to talk about how their teams are working to achieve these goals. The ministry areas represented included:

• Youth
• Whole Church
• International
• Worship
• Evangelism / Alpha
• Synergy (20s and 30s)

Some reflections from this group included:

• It's easy to focus so much on one of the areas (like the needs of the group) to the detriment of the other two areas.
• We need to learn the skills of being able to challenge individuals.
• How do you cope with different individuals having different visions within the group?
• Often we think it's all about us as individuals, when actually it's all about God – the team can help lift the burden.
• Using people's individual strengths – play to those that get them done.

We continued to look at the leader's role, with people looking at how leader's develop the individual; ensure the group achieves the task; build and maintain the unity of the group. We proceeded to study John 17:20-26 where Jesus teaches about his heart for community and people work well together.

Question: If only have a few hours left, what would you say to them about their working together.

• Pray for them
• Stay focused
• Keep encouraging / Well done
• Talk and listen to each other
• Remember the impact you have on each other's lives.

Jesus, with 24 hours to live talked (in John 17:20-26) about his passion for unity – holding shared belief; passing on the glory of the Father, to Jesus, to his disciples and so on; ministry of Jesus; presence of Jesus through the Spirit.  The purpose of this unity is mission.

Steve followed by talking about the 5 dysfunctions of a team and the participants reflected on these negatives and how they might affect each group.

Effective teams:

• Trust on another;
• Engage in healthy conflict;
• Commit to decisions, plans and vision;
• Are accountable to each other;
• Achieve collective results.

The session finished with prayer and reflection.

See the full gallery on Posterous

Growing Leaders – Developing Leaders (Session 6)

Previous Session

In session 6 we spent the first few minutes reflecting on what had been helpful or puzzling from the previous session.

Helpful things included:

• Diagrams,
• Lifeboat story,
• C.S Lewis,
• Acts 2 comparison.

Puzzling things included:

• The Quiz,
• Transition Curve
• and Planting the seed.

Developing Leaders

Introduction

Looking at ingredients needed in a leader who can develop others, the group came up with these ideas:

• God's vision for them;
• willingness to take risks;
• commitment;
• ability to listen;
• setting a good example;
• encouragement;
• prayer.

After a brief reflection on how Samuel debated over Jesse's sons in 1 Samuel 16 v6-13, we looked at the passages in the Gospels that indicated how Jesus chose his disciples. Each group looked at a different passage and then fed back to the main group:

Identify

Matthew 4 v18-22: the context was Jesus just returning from the wilderness: He chose ordinary people in mundane jobs; he was inspired to pick them.

Mark 3 v 13-19: Jesus was definite and deliberate, choosing people with different skills and from different backgrounds.

Luke 5 v 1-11: Jesus just did it, choosing his disciples with authority and without arrogance.

Luke 5 v 27-31: At Levi's banquet he gave an open invitation, getting involved with people who were classed as sinners.

John 1 v 35-42: John the Baptist declared: "Behold the Lamb of God" – got some interested and then one gets another to listen. Jesus demonstrated a charismatic presence.

John 1 v 43-51: Jesus says "Follow me" with charismatic confidence. He then used Philip to reach Nathaniel removing skepticism and using discernment.

Invest

• takes intention
• takes time
• takes its toll

Entrust

Why develop leaders?

• if we didn't then there wouldn't be any more leaders.
• it encourages church growth
• because Jesus told us to (Great Commission)

How did Jesus develop his leaders? – Looked at Mark 3-6

Chapter 3 –

Chapter 4 – Exposes them to new experiences + v33 "as much as they could understand" Jesus gave them as much as they cope with. Jesus stretched them, without straining them – he didn't push them too hard. He taught them – taking them aside and giving them ongoing input.

Chapter 5 – Modelled how to behave. Disciples followed Jesus around learning how to deal with different situations. He demonstrated no fixed patterns but modelled dealing with individuals.

Chapter 6 – Jesus gave a lot of authority – casting out demons. He didn't let them be isolated. He gave them the responsibility to fix the problems like in the feeding of the 5000.

What are the blockages to developing leaders?

• middle-class models

• basing decision on leadership based on middle-class credentials
• logic before creativity.
• not what we know but who we know (PLUs – People Like Us)

using only those who've already led. It's self-limitating if we only use the people we always use.

Our own insecurity

• they might not do it like we do
• they might be better than us
• they might make a mistake
• time – it's quicker to do jobs ourselves – but then others won't develop their independence.

Harvey shared an experience of finding it difficult to lead others without having a relationship with people. Matt J and Dave both demonstrated good coaching by asking open-ended questions like

• "What would you do differently now?"
• "Why do you think that that happened?"

We commented on the quality of this questioning and compared it with how Emma, Steve and Alison had all tried a mentoring approach – giving advice to Harvey. Richard pointed out that just becomes something feels awkward, it's not necessarily wrong.

Other common blockages include:

• a critical atmosphere – "failure not success takes you to the core of Christian ministry."
• limited view of leadership – e.g. David, Gideon, Saul / Paul
• No sense of call
• Heirarchy

We finished the session with time to reflect.