using what they know

Students complain that Information Processes and Technology (IPT) course is too theoretical.  Ipso facto, boring, especially relative to the more hands-on and generally fun Information and Software Technology (IST) .  They’re right: there are a lot of concepts. BUT they’re wrong: it is not boring – though it can be.  That can be said of any subject/course, right?

The challenge (I love them, remember?) – as with any subject – is how to make it interesting and better yet, meaningful and relevant.

The irony is, in this day and age, we live and breathe the content of these courses. The problem is we don’t see the connection, much less tap into our experience and knowledge.

I’m not saying this is the best way to do it but this is how I’ve faced this challenge, for example, to teach the design tools in the Software Development Lifecycle. So, I started with this on the whiteboard which got their attention straight away:

Identifying the external entities was an interesting exercise, especially when we got to 3rd party apps and sites – so what are APIs? (I hope some of them try to find out). We had interesting discussions on what information flows in and out of the system (talking about digital citizenship stuff, it’s good).  We got talking about multimedia data types (revising, it’s good).  We got talking about apps they use as do I (building relationships, it’s good)…..and we’d only just started.

I got students to come up to the board and draw a Data Flow Diagram. My volunteer  called out for help and they came. Good, eh? They worked together and came up with something. It wasn’t right but it highlighted some misconceptions  (opportunities for learning, it’s good), questions about information processes (more revision, good stuff) and importance of assertiveness and social skills (all good stuff).

We also used facebook registration as examples for Decision trees and Decision tables – usually ‘boring’ stuff but a lot easier to understand using an example they’re all familiar with. They learned about the design tools and appreciate why we have them, in the first place (bonus).

I will also use facebook to look at the next core topic – Information Systems and Databases….they don’t know that yet. Dan Haesler will be happy – I’m driving down the social media way  😉

But this post really isn’t about facebook , social media or even IPT. This is about tapping into what students are familiar with or care about, to teach something new or to help them in the process of abstraction (recall I said that IPT content is mostly stuff we experience but not necessarily get into abstraction?).  It is also about taking a risk because it’s outside the syllabus and no curriculum support (Dr Sarah Howard will be so proud of me – haha) Truth is, the syllabus hasn’t kept up with technology (understandably so) in that social media itself is an abstraction – it does not strictly fall into the main info systems covered by the syllabus: transaction processing system, decision support system, multimedia system and automated manufacturing system. YET, it’s arguably the most prevalent info system there is now and a perfect example of how muddled or interlinked the different types of info systems really are – silos help us understand but we must remember to re-connect….I digress, erm, do I? The other risk is that as an info system, there’s a fair bit I don’t know about facebook – I’m saying “I don’t know a lot” in this class. The upside is that it helps teach how the tools learned in IPT help us learn a bit more about it….situated learning?

I’ve actually applied a fair bit of what I learned from PLANE’s Festival of Learning, within a few days of attending it. Good, eh?

I love a happy post full of learning. cheers all!

Revisiting the Vitruvian Man

As mentioned in my last post, Walking the walk, I recently completed a 2-week casual teaching block and agonised about walking MY walk. So, here’s another story…this time with Year 8s and on the topic of Rates and Ratios.

I had one lesson (yes ONE, the 3 with the year7s was a luxury in comparison) by the time I got my act together! Enter the Vitruvian Man again (yes, I’ve used him before, on Percentages and with more prep). Remixing my original idea was easier than starting from scratch! By the way, the previous time was with strugglers and this time with high achievers….note the differences in the activity, not just topic-wise but in also tasks…which also show how my teaching approach is evolving towards inquiry and reflective practice.

8mata1_vitruvian – have a squeeze (pdf) and remix if you like…the Vitruvian Man is a maths teacher’s friend!

In a nutshell, the students were meant to investigate the “accuracy” of the Vitruvian Man ratios, extrapolating their own, apply some ratio skills AND reflect on their leaning. That’s not too much to ask in one lesson, was it? And by the way, they were going to use 2 technologies they’ve never used before: Wallwisher (which turned out to be blocked. gah!) and Dropittome.  They had to submit TO ME and they won’t see me again. CRAZY!

So, I completely forgot about this until a week later …. mostly, I didn’t think any of them would submit work to a casual teacher … I mean, really.  No grades attached.

I finally remembered to check my Dropbox and – surprise! I got submissions. I wasn’t even going to blog about this activity but, how could I not?  Here are some excerpts:

“New” ratios

  • A man equals 96 fingers – 1:96
  • A foot equals 16 fingers – 1:16
  • Palm to foot: 4:1
  • Length of outspread arms : 4 cubits


  • Is the theory 100% accurate?
  • Is the growth of your arms in proportion to the growth of your height?
  • Is the length of your foot, equivalent to the length of your forearm?
  • Do these ratios work for everyone?
  • What fraction of your body is one palm? (I love the use of the term fraction. yes!)
  • What is the ratio of the length of a hand to a full arm span?
  • When was the Vitruvian man first drawn/created?


  • Ratios are good because they make it possible to work things out rather than have to measure them every single time.
  • They may not always be 100% correct, as it was proven with my measurements.
  • Although this task was reasonably difficult in the beginning, I found that it was reasonably accurate and correct, as we further investigated Da Vinci’s theory. My results displayed similar findings to the theory; I further developed my rates and ratio skills. Prior to this maths topic, I didn’t completely understand it, however now I find that I can use ratios in many other places to simplify, numerals I am given or to find and approve theory’s such as the Vitruvian theory.
and the icing on the cake…

I really enjoyed this lesson. I liked how we got a chance to put our maths into practice.


So yeah, I’m happy.

That kid saying ‘reasonably difficult’ was being truthful! I nearly abandoned the activity! I’m glad I didn’t. Unlike my strugglers before who were keen to get straight into measuring, this high-achiever class had to be prodded away from their calculations and musings. I’m sure there’s something there but I’m not going into it for this blogpost.

Walking the walk

I just completed a 2-week block at another school, this time cover for mostly Maths classes. Lessons were already planned by the teachers so technically, all I had to do was deliver them (if keen) or leave the kids to do the assigned work and make myself available….this is what casual teachers do, right?

After a week though, I found myself literally sick to the stomach because I knew I could do more. Here I was harping on about to “Make room”,  “Student reflection”, “Inquiry Learning (1)” , IL(2)  and “Playing with different teaching styles and approaches” and then not doing them. Ha!

I could have hidden behind the curtain of ‘I am just a casual teacher, blink and I’m gone’ but deep down, I felt that I had to walk the walk….my walk. The question was whether I could pull this off in a Maths classroom, with students I didn’t even know the names of?


The year 7s were learning about Data and Graphs; identifying, reading, creating, critique-ing… that order! I’ve taught this unit before and that’s how we’ve always done it…just like most Maths teachers I know. I only had 6 lessons with year 7s and 3 of those where gone in the first week; so I had 3 lessons left for something different to indulge my ‘walk the walk’ thingy.

Here’s my mini-PBL; it’s not great but does tick the boxes of PBL essential elements (via, bar the public audience. There’s a “focus on inquiry, voice and choice and significant content”, as per starting with PBL article (via Edutopia)…relative to the constraints voiced previously.

[click to enlarge]

Students were allowed to work alone, in pairs or in groups of 3. Students had a choice of presentation mode/tools. Students had a choice of graphs.

Investigations were prompted by a couple of Olympic-themed infographics and a video on what is a typical person which aimed to raise questions on samples and populations, in particular.

Musings – positives and negatives

Inquiry was a challenge. The students struggled with the list of questions, i.e. what were good questions to ask. My biggest concern here was that most of them aimed to please me, the teacher, to pick the right questions and answer them. What I really wanted them to do was list the first questions that came to mind, e.g. “What is this graph about?”, “Why did they use this type of graph?”, “Is this a population or representative sample?”, “Is the scale correct?”, etc. It took several instances to assure them that there were no right or wrong questions, as such, and that some questions were impossible to answer, e.g. such as if the sample size and profile were not given.

Timing was tight. Ideally, the students could have been given more guidance to look for more complex and diverse examples. I’m inclined to think that this would have been better at the start of the unit, i.e. I “wasted” the first 3 lessons doing direct instruction and socratic questioning with the whole class.

Presentations helped uncover misconceptions. This was gold! Students were also learning graphs in Science and were talking about the line of best fit and scatter plots. No wonder they were confused when I was teaching them line graphs the week before; some of them thought line graphs were connected scatter plot dots. This misconception came out when a couple of groups presented about line graphs as being bad because joining the dots was “a bad idea”. This misconception would have been practically impossible to uncover using the traditional method of teaching mentioned above because they would have had little opportunity to “make that mistake“.

Big picture approach helped put things in perspective. I’m a fan of big-picture teaching, (re: post on algebraic equations). This mini-PBL got students looking at different graphs all at once, not one-at-a-time as you would in traditional teaching. So when a student asked a fantastic question of how to represent 0.7 in a picture graph using a scale of 10, it allowed opportunities to discuss options such as using a different scale and (the most obvious they didn’t see it) – use a different type of graph….because they can.

This sacrificed Knowledge and Skills for Working Mathematically. This bugs me because I feel I have let the students down because we didn’t tick all the boxes in the syllabus document for Knowledge and Skills such as “using line graphs for continuous data only”. I am biased towards focusing on working mathematically, e.g. “generate questions from information displayed on graphs”, something I believe transcends usability beyond standardised tests and high school years.  Balancing these is a challenge most Maths teachers face. I made my choice and certainly hope that I didn’t do students a disservice.

So, in the end, I did walk the walk….and gained from it.

Multiplication, Multiples and Factors

My daughter is in Year 5 and yes it is NAPLAN year; even if I’ve forgotten this fact, her homework would have reminded me.  She has just finished worksheets on multiples and factors – including concepts of highest common factor, lowest common multiple.  She told me they haven’t been taught in class – which, if it boils down to it, is going to be her word against the teacher’s.  And that’s not really what I’m posting about.  Nor is this post about worksheets and homework.

This post is about multiplication and using the times table.

As a high school teacher, I know that some kids struggle multiples and factors conceptually.  Once applied in fractions and Algebra, the confusion mounts.  A good foundation built up through primary/elementary school years cannot be under-rated.

One of the problems I think is that kids really don’t know, i.e. memorised, their times tables.  Though they conceptually know of multiplication, or even have several models of it, many still have to calculate often by repeated addition through to college/university as @angrymath tells us (and I can believe it).  If you think Cognitive Load Theory makes sense, it follows that knowing one’s times tables reduces the cognitive load when doing maths.  There are also strategies – even ‘tricks’ – to help.  One of the strategies I teach my kids (my daughters and students alike) is what I call the “goalpost multiplication” approach.  Briefly, estimate to the nearest goalpost (x2, x5, x10) and work up or down.  For example, 8×7 = 8×5 + 7 + 7.  Note that even in this strategy, there is a need for conceptual understanding (and review) of certain concepts such as associative property and number sense via estimation.

That was a dense paragraph and there are things I’d like to qualify:

  1. I don’t advocate rote learning at the expense of conceptual understanding
  2. I think Cognitive Load Theory makes sense and I have noted that students who expend much effort (use of working memory) calculating relative to effort on mathematical thinking (e.g., application of calculations) tend to have lower self-concept with regards to their numeracy
  3. Memorising the times table is not necessary but useful, if just to reduce cognitive load and corollary, improve self-concept with regards to maths
  4. If you really cannot bear to memorise the times table, at least have some strategies to get you by.  As a teacher, help your students learn these strategies

Ok. so here’s what I did with my daughter.

I whipped out the Times Table (here’s one you can grab, too)


I used this multiplication grid as a visual aid to explain/highlight the following points:

  1. Any number multiplied by 1 is itself;  one day, she’ll learn that this is called multiplicative identity property
  2. The green line is the square of numbers showing, for example that 3 x 3 = 9 “boxes”
  3. The green line is like a mirror, a line of symmetry, so one side is the reflection of the other showing 3 x 4 = 4 x 3; one day, she’ll learn that this is called the commutative property of multiplication
  4. Rows or columns are multiples of that number (in orange)
  5. The factors of a number are the numbers you multiply to get that number, i.e. the headings in orange
  6. A number that only shows up in row/column 1 (numbers in blue) means it is a prime number, i.e. its factors are only 1 and itself
  7. A number repeating across the table left-to-right, say, means it is a common multiple – a multiple shared by the numbers with it as a multiple  (e.g. 12 in purple is a common multiple of 1, 2, 3, 4, 6, 12)<most kids this age understand the concept of common – use this to connect to maths; this highlights literacy as well as help build confidence, i.e. common multiple is not such an alien concept after all>
  8. A number repeating across the table means it has several factors (e.g. 12 in purple shows up in 1, 2, 3, 4, 6, and 12 and these are its list of factors)
  9. A common factor is a factor that is shared by 2 or more numbers. <this is harder to show using the times table but build on concepts of common and factor list>

One could reasonably argue that teaching all points above is risking cognitive overload.  Teaching one-on-one, I could tell that she was keeping pace especially using the times table as visual aid.  My first attempt failed just using the definitions on the worksheets; the times table definitely helped.

I also found out that the tables they have at school where the other kind designed for rote learning: 2 x 2 = 4, 2 x 3 = 6, 2 x 4 = 8, etc.  She was quite pleased with this multiplication grid and using it for finding patterns (pre-Algebra skills) and not just as direct reference.

To be honest, I don’t mind if she forgets all of the “definitions” above if she can work to find them again.  Given the iterative nature of the Maths syllabus, she undoubtedly will have lots of opportunities to re-visit this.

As an aside,  creating a multiplication grid using a spreadsheet with formulas and auto-fill would make a good little tech-integration task and exercise of Algebraic skills; there’s more than 1 algorithm.


I just read this story-telling post by @billgx which inspired me to write this post; it’s’ been a while, I know!

Bill’s post has just highlighted the power of stories – something I’ve been tossing around in the old brain for a while.  I thought it best to capture just a few of these thoughts for future use in teaching and learning.

We are always telling stories. Stories don’t have to be oral or written narratives. Just yesterday in my digital photography workshop, I alluded to photos as telling (or capturing) stories. It is sometimes difficult to glean the story depending on the medium used but the story is definitely there.

We enjoy listening to stories. Stories can be affirming with a by-product of connecting the audience to the story-teller – or the content itself.  I still remember when I told a class about Descartes before launching into the Cartesian plane. As a teacher, listening to our students’ stories afford us a glimpse to who they are and knowing students is important for good teaching and learning to happen.  Ditto for students.Here is a story-writing exercise I once did with a class just to get the creative juices flowing:

  1. Give each student a piece of paper.
  2. Everyone starts off with the same line, e.g. “Today as I got to the school gates….
  3. Then, everyone passes the paper to the left (or right) and then write a sentence to continue on.
  4. Continue the round robin, with prompts for introducing the ‘problem’ or ‘resolution’ or ending. 

Make sure you make time to read a few out or perhaps publish the stories. Depending on the age group, it might be necessary to establish some ground rules as you would for anything ‘anonymous’.

This might even be useful in a maths class, e.g. get students to do round-robin solution – each one just does one step. This may drive home the point of the importance of reasoning and ability to follow someone else’s line of thought.