Temporary Blog

11 Apr

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

The King of shapes: the stellated icosahedron

1 Jan
2012-01-01_14

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

21 Nov

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

6 Oct
Fraction_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

11 Jun
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

30 May

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.

Photo

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

5 May

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.

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