At what age should we start teaching algebra?

5 May

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

2 Apr
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.

Growing Leaders – Developing Leaders (Session 6)

12 Mar
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.

    Why does character matter?

    15 Jan
    Bill Hybels defines character as: who you are when nobody else is looking.”

    I’ve been thinking about character a lot recently. It’s mainly because I’m about to deliver a reflection on ‘character’ at my church this morning for ‘Growing Leaders’ session 4. I’m doing the final exercise after the participants have learnt about character. The prezi I’m going to use is here: https://prezi.com/secure/fb63876cbf6a3511bdf57a95899a87ad879903b4/

     

    I was struck by a presentation at the UK Education leaders conference at BETT2011 yesterday (BETT is a technology for education conference that happens in the UK every year). Chris Gerry from the Future Schools Partnership talked about how they use IT smartly to get the best value for it. While a tiny part of his talk was about budget and systems, a large part was about emotions and self esteem – developing character in both pupils and teachers. Could developing character actually save money for the UK education system?
    The list that the participants came up with that are ‘essential’ for Christian leadership are:
    • love
    • holiness
    • servant-hearted
    • wisdom
    • encouraging
    • humility
    • perseverance
    • faithfulness
    • integrity
    • forgiveness
    • compassion
    • trustworthiness
    • integrity

    I wonder how much of that list applies to our secular leadership. To the likes of Gove (in Education), Cameron and Clegg. We’ve all heard of leaders who had a great gift, a great ministry, a great mission, whatever you want to call it, yet through some flaw in their character, their great task was cut short. Brought down by media revelations or a crisis in their private life. We all have flaws – the question is: do we brush them under the carpet and hope they go away or do we face them up and do something about them?

    As Christians we look at the character of Jesus as to be our guide for how we should shape our character. Find out more in Matthew, Mark, Luke and John in the Bible.
    Perhaps the significant characteristic to add the list above is that of authenticity. Jesus was all those things. He was servant-hearted, wise, encouraging, humble and all the rest. But he was authentic too. He lost his temper when he saw injustice. He challenged authority when he saw it was wrong. He became frustrated when his disciples were infeasibly dense. He was real with people.
    Later on this morning we will be looking at tools to help analyse character. One of the best I’ve found is the Johari window – a great tool for delving into who you really are, although you do need somebody to help you with it. To find out more, Google it.
    So, what shall I do with flaws when I find them?
    Augustine said: “Make sure your life sings the same song as your lips.” Authenticity – I think that’s the key for me.

    Life in the Red Zone

    27 Nov
    Rev_counter

    I was really interested in a discussion on #ukedchat last Thursday (#ukedchat is a Twitter group that meets every Thursday evening from 8-9pm to discuss issues in the UK education system). The issue in question was that of work-life balance. Some very interested points were raised, many with tongues firmly in cheeks (although it's often difficult to tell in Twitter). Some included "Question – do you think any less of your colleagues if they don't do the extra hours? <yes" and "Thank god I don't have kids I couldn't fit them and teaching in!"

    Now I happen to think that "Work-Life balance" is a really unhelpful term. It indicates that work is not a part of life. "Work-leisure balance" is a more helpful term because it indicates that work is part of life. And so is leisure.

    However I also think that the term 'balance' is a really unhelpful term. It indicates that things oppose each other, they are in tension with each other. I prefer the term 'blended life'. It indicates that work and leisure and other stuff can all blend together to make up your life. Sometime they overlap, sometimes they are distant from each other.

    The picture used in Growing Leaders is that of rev counters. When you're revving a car, you can under-rev it so the engine is not working as efficiently as it ought to; you can over-rev it so that the engine can get too hot and damages it; or you can rev it just right so that it is working as efficiently as possible. The over-revving is the point of most danger. It is life in the red zone.

    Growing Leaders identifies 5 ways in which this can happen:
    • Physically (red zone makes you depleted)
    • Emotionally (red zone makes you drained)
    • Relationally (red zone makes you distant)
    • Intellectually (red zone makes you stale)
    • Spiritually (red zone makes you disillusioned)
    Of course it is possible to cope with life in the red zone for a certain periods of time. But if you're in the red zone in all areas for a very long time, it is going to be damaging. So blend your life. See it as a blend of different things – sometimes one of those things will come to the forefront, sometimes it won't.

    I've found having a personal life statement a really helpful way to manage your red zones. Through it I've been able to define my life physically, emotionally, relationally, intellectually and spiritually. I've also been able to right down some key phrases that define how I am. One of them is 'Gives Hope to Children' – it reminds me that my calling as a teacher is about giving hope to those children in my school who have little hope. It helps me focus on what I should be doing and what I like to do. It is self-affirming and life-building. It enables me to have a bigger 'Yes' with which I can say 'No' to the worthy activities that otherwise might take over my life and take me away from my core purpose.

    Finally, as Christians, a blended life fits within the context of living a 'surrendered life'. Our first call is to growing closer to Jesus as his disciples. It is to surrender to Him. 'Surrender' can sound ominous and bleak, but Jesus is love – surrendering to Him makes blending life a real possibility. And it's good because He is good.

    Valuing misconceptions on the way to explaining fractions

    26 Nov

    explaining fractions_0001.wmv
    Watch on Posterous

    I filmed this about 6 months ago, following an excellent session about fractions on the Mathematics Specialist Teacher Programme. The challenge that we were given, and then I in turn gave to the children, was given a 4-pint bottle of milk that gets 3/5 of a pint drunk each day, how many days does the milk bottle last for? Those of us with a formal background in maths would say:

    ÷ 3/5
    = 4 ÷ 3 x 5
    = 4 x 5 ÷ 3
    = 20 ÷ 3
    = 6 r 2.
    So the milk lasts for 6 and a bit days. If we wanted to be really fancy we would say the milk lasts for 6 and 2/3 days. And isn't it more practical to say the milk lasts for 6 days and there's 2/5 of a pint left over? Does our understanding of the algorithms let us say that?

    Also can children, who are without the drilled-in knowledge that when you divide a divisor you actually multiply, do this question?

    That's what the video explores – and there's some interesting misconceptions on the way.

    Learning Creativity in Maths at MaST HEI Day 5

    21 Nov
    MaST is the Masters level study programme I am on (standing for Mathematics Specialist Teacher). HEI merely stands for Higher Education Day.

    Creativity in Maths

    The Day begin with a lecture on creativity in maths. It's an interesting idea – creativity – because many teachers have the mental construct that creativity is all about thinking artisticly and creating things of aesthetic value. Derek Haylock went on to talk about about divergence and flexibility – a far different way concept of creativity in maths. One leads to trying to shoe-horn maths into a themed curriculum and doing lots of shape work that becomes artwork, the other leads to open-ended questions, good dialogue and child-centred learning. Here are my tweets:

    • About to hear Dr. Derek Haylock at #MaSThei5. http://derek-haylock.blogspot.com
    • #MaSThei5 creativity is not normally associated with mathematics (confusion between artistic and creativity)
    • #MaSThei5 find 2 numbers with a sum of 9 and a difference of 4? When we have the knowledge, what blocks us accessing it to solve a problem?
    • #masthei5 what are the processes that characterise creative thinking? How do we recognise creative product What kind of people are creative?
    • #masthei5 what conditions foster creative thinking? (all in maths context)
    • #masthei5 Derek Haylock demonstrate that we're all fixed, rigid thinkers by nature. We have to choose to think flexibly.
    • #masthei5. Equal pieces problem – will demonstrate on blog how we're all rigid by nature.
    • #masthei5 flexible thinking is the first step on a creative process in maths. Avoid rigidity an fixation.
    • #masthei5 2 kinds of fixation common in maths that limit creativity: algorithmic and content universe
    • #masthei5 ask children to draw a rectangle. What do most of them do?
    • #masthei5 creativity in maths includes thinking divergently: fluency (many), flexibility (kinds), originality, appropriateness.
    • #Masthei5 appropriateness is easy to define in maths (as opposed to art, writing, etc) so teachers fixate on this one part of divergence
    • #masthei5 how to develop divergent thinking in maths: problems with many solutions; problem-posing; redefinition.
    • #masthei5 redefinition – come up with lots of responses by redefining the elements, eg: what's the same as 16 and 36?
    • #masthei5 redefine by using lots of different ideas to create subsets of a given set of numbers
    • #masthei5 conflict between creativity an accuracy – do we value creativity as much as accuracy in maths?
    • #masthei5 graph of attainment vs. creativity (as Derek Haylock defines it) show 0 children in the high creativity, low attainment sector
    • #masthei5 factors associated with maths creativity include low anxiety, high self-concept, risk-taker, high attainer, being a boy. 
    • #masthei5 creative maths children are also 'broad categorisors'. They are good at identifying the same about numbers+ideas and make links.

    Writing Assignments

    Course Tutor, Mary McAteer gave us some top tips and hints to help us successfully write our first piece of level 7 writing.

    • #masthei5 Mary McAteer reminds us to demonstrate an understanding of ethical issues in essay and PLL
    • #masthei5 warns us against over use of Excel as a presentational tool for simple data

    Place Value

    Ian Sugarman definitely had the graveyard shift on the day. The last session after a big lunch on a 6 day week – on a Saturday when most would be out shopping, or slobbing in front of the TV – can't have been an easy lecture. And when the subject is the dry area of place value, it's always going to be a tricky one. The biggest thing I got out of this lecture is the warning against the indiscriminate use of number lines and the value of teacher column methods for securing place value when ordering decimals.

    • Context for place value #masthei5 getting things 10 times out can be at best expensive; at worst lethal…
    • #masthei5 misconceptions of place value after the decimal point are rife between ages of 7-11. Half-learned rules and over-generalisations
    • #masthei5 when pupils are given opportunities to explain their thinking, they often spot their own flaws.
    • #masthei5 to get place value it's helpful to sort and justify before ordering
    • #masthei5 talks about left-justifying decimals when I think it's helpful to justify by the decimal point
    • #masthei5 to get x10 relationship it's helpful to use pictures or Dienes apparatus to visualise place value
    • #masthei5 recommends http://nlvm.usu.edu – university of Utah website for good models and images.
    • Great activities advertised at #masthei5 at http://numbergym.co.uk (but not free)
    • At #masthei5 Ian Sugarman talks about standard algorithms can be a sledgehammer to crack a nut in questions like 81-78.
    • #masthei5 numberlines vs standard algorithms vs necessity of getting place value = conflicting interests
    • #masthei5 British children have been referred to as 'pathological splitters', as they partition numbers in both addition and subtraction.
    • #masthei5 Ian Sugarman advocates empty number lines, but not as another rote-learned method. Draw from 0 and emphasize progression.
    • #masthei5 maths in Holland always starts with a real setting, whereas in UK we start with pure maths.
    • #masthei5 can use 'same difference' method as alternative to empty number line for examples such as 83-37 (86-40 is much easier)