Index Laws

I was at a presentation yesterday and was shown the following explanation for the "to the power of zero" law.  It used arrays and was a great visual for lower ability students that were unable to grasp the nature of the zero law through subtraction of powers.

It's a short video to help me remember the techniques involved, but thought it might be useful for others.  If I get motivated I will extend it to negative indices.

I can claim no credit for the idea, it was presented by the fantastic Pam Sherrard from Curriculum Support branch, who has collected some great ideas.  If you get a chance to go to her PD as a lower secondary or primary teacher I would highly recommend it - they're good fun.  I've noticed it's not working on my iPad.. I'll work on it.

Practicum teachers & Teacher instructors

Practicum is the gateway to the profession.  It ensures new teachers have the experience to have their own classroom and sets clear expectations of performance outside of an tertiary academic environment.

Practicum teachers are a resource that is relatively infinite.  Good practicum teachers are not.  Teaching is an attractive profession to students as it is a familiar environment and one where students see infinite room for improvement.  This gloss can come off to some degree during practicum.

Teachers instructors come in a range of colours and shapes.  Aggressive/passive, lazy/committed,  skilled/enthusiastic (grin),  assessment driven/programme driven and with a range of teaching styles.

Good/bad teacher instructors are relative judgements. They make and break practicum teachers.  Let's see some of the pitfalls for practicum instructors

• Providing insufficient opportunities to fail
• Giving insufficient guidance on expectations
• Providing inadequate encouragement
• Assuming complete content knowledge
• Providing inadequate written response
• Creating insufficient opportunities to show initiative
• Insufficient direction/focus
• Insufficient guidance on required scaffolding
• Overestimating diagnostic abilities
• Assumption of teaching skills (photocopying, marking, using equation editor, CAS calculator usage, fxdraw, whiteboard usage, interactive whiteboards)

When we look at an incomplete list such as this and then overlay behavior management (which is the usually the primary focus of practicum) we start to realise the hurdle that practicum teachers face.  When I hear about accelerated teaching programmes that reduce the time for undergraduate teaching degrees I fear that policy makers are underestimating the impact that good teacher instructors can make.  Furthermore they are underestimating the impost of the shifting of responsibility from the student during practicum, to the teacher during their first year of teaching.  This will be borne by experienced staff and administration dealing with issues typically dealt with on practicum.

Good teacher instructors are using their full range of abilities to ensure that practicum teachers are maintaining their confidence at appropriate levels and by ensuring student learning is continuing.  The ability of the system to identify good potential teachers is based around the practicum system and bypassing this system has potential for lowering teaching standards further.

Multiplying and dividing by powers of 10

I had my academic year nine class for the first time today and had a lot of fun. I had been warned about a few students, but they were arms deep in the trenches having a good go.

I took an experimental approach today with the class examining how to multiply and divide powers of 10. The idea was to create algorithms in student terms for solving simple equations.

We started with simple examples using whole numbers
5 x 10 = ...

"When multiplying by 10, 100, 1000... count the zeroes and put them on the end of the number you are multiplying."

25 ÷ 10 = ...

This time students considered the position of the decimal point:

"Count the zeroes in the number after the division symbol [divisor] and move the decimal point right of the other number [the dividend] that many digits"

We then looked at the case

2.5 x 10 =

and discovered our first algorithm didn't work as by our first algorithm 2.5 x 10 = 2.50

This lead us to a similar algorithm as for our division case:

"Count the zeroes in the multiplier and move the decimal point left of the other number [the factor] that many digits"

Using the whole number cases gave students an additional method of multiplying powers of ten than the messy loops moving decimal places method. The idea of this lesson was not to deny them mathematical language - but to give them an opportunity to explore a mathematical concept before formal language was introduced. It was a lot of fun for me and engaged them during the lesson.

We then looked at a few cases where the multiplier and divisor were not powers of 10. This exposed that students had difficulty with long division and long multiplication and were over dependent on calculators - which has the possibility of causing issues in non-calculator sections in upper school. We'll now go own to examining factor trees and ease into indices.

We also looked at 250 ÷ 10 where we converted the expression to a fraction and cancelled the zeroes - although we didn't consider why this works and will need to revisit it later.