“Hope is not a strategy”

But what is a strategy is doing the mathematics before the students.

In IT circles, it’s called ‘dog fooding’. And refers to developers and techies when, during the construction/development and planning phase, they work through their program through the eyes of the market consumers. As an average joe. As a high end firm. Through this process, developers see the product in terms of its usability, suitability, strengths and challenges. Michael DeFranco of Forbes wrote about dogfooding from an IT perspective, and Jennifer Gonzsales (from Cult of Pedagogy fame) wrote and spoke about it through an educators eyes.

(Side note: if you don’t follow Jennifer Gonzales/Cult of Pedagogy on all platforms and listen to her podcast, do yourself and your teaching a favour. Her approach and content is A+)

While I don’t wholly like the term ‘dogfooding’, in the teaching and learning world it might also be called anticipation.


I first heard about anticipation at a system learning day that was being facilitated by the Numeracy team here in the Parramatta Diocese. In amongst sharing awesome ideas about how to engage the wider community with the school’s Numeracy learning, the team introduced Five Practices for Orchestrating Productive Mathematical Discussions by Margaret S. Smith and Mary Kay Stein, published by NCTM and Corwin Mathematics. Boldly in 2018 I said there has been no book that has impacted my teaching and leadership practice than this one, and to this day I will stand firm on this call.

Smith and Stein define these practices as…

“The five practices were designed to help teachers to use student’s responses to advance the mathematical understanding of the class as a whole by providing teachers with some control over what is likely to happen…as well as more time to make instructional decisions by shifting some of the decision making to the planning phase of the lesson.” (p.9)

In addition to anticipation, monitoring, selecting, sequencing and connecting make up the five practices. Each is just as critical to the integrity as the next or the one prior.

Laura Wheeler created this amazing sketchnote summary of the article and graciously allowed me to use during our professional learning meeting.

5-practices-orchestrating-mathematical-discussions (1)

Laura’s thoughts and insights are very much worth following, and you can do so here.

At St Luke’s, anticipating can occur in the co-planning phase of task design and this was a concept introduced to teachers during the week. Using a simple, but high yield task called handfuls, teachers anticipated the possible student responses. In short, they did the maths through the eyes of a student in their class. Some prompting questions to help the teachers in this anticipation were:

  • What might you expect students to do with the equipment?
  • How would student x approach the task?
  • What about students who find mathematics challenging? What might they do?
  • What red herrings are present in the task that you would expect students to come across?
  • What misconceptions might a student bring to the activity?
  • What prompts (enabling or extending) will you be ready with to scaffold student’s understanding?
This group of Stage 1 teachers came up with 6 different types of responses. From left to right, they are: doubles, dice arrangements, a ‘blob’ , a row/line, pairs, and variations on the tens frame.

As a part of the co-planning and anticipation, staff collaborated on a Google Doc that represented teaching teams across Kindergarten to Year 4, representing the various abilities in these spaces. The result was a crowd sourced resource for use by all in Foundations to call upon when implementing handfuls as a warm up.

Why anticipate?

So as an instructional strategy, anticipating the mathematics in the co-planning stage is a key move. This is a much better strategy than hoping to be in the right place, at the right time and seeing the right thing to advance the learning. Anticipating tunes the teacher into the possibilities that may arise in the course of the activity, allowing for an awareness of what students might produce, where they might feel challenged, what instructional decisions to cue when the situation occurs and also to get a feel of if the task is suitable for the students. I acknowledge that by anticipating, you are likely not to imagine the entire range of responses to a particular task, it may mean you are less likely surprised by a response and more control in the moment to make choices that will positively effect the learning.

So for your next task, try anticipating in your planning. It certainly have the potential to value add to your task and ultimately student learning. What did you notice? Did you do the task through the eyes of a particular student or groups of students? Why did you choose that particular child? Was the task too easy or challenging? Let me know how you went by commenting below.


Early Years EMU Intervention-Day 6 Reflection

As a part of my EMU intervention reflection, I was asked to write about critical moment that has occurred in the past week or so related to the EMU program.

This week I heard my little EMUs self-correcting for the first time. I am sure they have done so before, but to hear them make an error, then correct themselves almost immediately leads me to believe that they have processes working inside their minds now that they may not have had occurring before.

The situation was skip counting along a bead string to prove a two-digit number. One less ten was said, and rather than continue with what obviously felt/looked/sounded incorrect for them, they stopped.


Then he said the correct amount of tens and continued on.


This doesn’t happen by mistake and is a result of honing the particular skip counting skills with experiences deeply rooted in strengthening the number triad and number sense itself. It also is a reflection of the external (often scaffolded by me) made internal. Many times towards the beginning of the program, when an obvious error (obvious to me) was made, I prompted the child with “Does that look right?”, “Can you prove that number to me?”, “You said x, show me x again”. Now that dialogue runs internally as a part of their self-regulation.

Self-correcting represents a shift in mindset of learning from mistakes and growth. The error does not define the student, and he/she can immediately bounce back with another effort. This is HUGE. At the beginning of the intervention program, a common response to an error was a passive appeal. Growth as a learner in this way is critical to learning across the board.

It also reflects an internalisation of general number sense for that student. Using the analogy from explicit reading instruction, what he did to produce the error must have felt incorrect, looked incorrect and mustn’t have made sense. Therefore I can fix it. The students are constantly honing their number sense through the tuning in task or rich task/investigation. All of these experiences are teased out, stretched, proven, conjectured, disproven and finally become the narrative to play over in their imagination at the next challenge.

Needless to say that I am very proud during moments like this because ultimately it’s less of me and more of them. And that’s the goal. I am there to provide the specific instructional intervention, but the longer they are with me, the more they have become agents of their own learning with less and less prompting from me.


Ladder of Abstraction

Today in PL, K-2 had an awesome talk about Bruner’s CPA theory and the impact on our Mathematics teaching. While we were unpacking the Singapore Bar Model method as a way for our K-2 students to visualise number problems, we stopped to acknowledge the sequence of learning experiences that scaffold the documents we program with.

We noticed that the documents encourage tangible and concrete experiences, but the more familiar we become with the content we begin to remove the concrete and rush children to the abstract (such as symbolic representations).

Here lies the stumbling block. In our eagerness, we hustle the learning to quickly and don’t allow for enough time where students are sitting in a representational or pictorial zone. Students require opportunities to record their thinking in many ways, including in picture format. Strong connections between concrete and pictorial encourage smoother transition between pictorial and abstract representations of the same concept.

The ladder of abstraction analogy was a way that we could meet our children’s needs by moving flexibly up/down the ladder. We don’t want to rush our children too fast up the ladder until they have been challenged enough at the concrete and pictorial level and been able to play with, test, conjecture, prove and disprove their understanding of concepts.

It was a real flow moment for myself and the K-2 team.

Times as Many

As mentioned, I am a early year’s numeracy intervention trainee. I have been working in 2018 with three ‘at risk’ students in Year 1. The EMU (Extending Mathematical Understanding) Intervention Program designed by Ann Gervasoni “is an approach for assisting students who are vulnerable in aspects of whole number learning and at risk of not learning school mathematics successfully” (Gervasoni, p.15, Extending Mathematical Understanding). The intervention program sits within a whole school, and diocesan, approach for ensuring that all children build their understanding through engaging experiences, purposeful reflection and explicit teaching.

My current group of EMUs have had significant experiences building their understanding of the array as a window into the multiplicative structure. They have developed some significant skills in counting larger collections and have worked very hard at bolstering their place value knowledge to inform further learning.

Their biggest growth, and one that makes me proud as both their Year 1 teacher and intervention teacher, is that the students now see themselves as Mathematicians. They approach challenging tasks with a tool kit they didn’t dip into before. They demonstrate grit in the face of difficulty. They test their ideas and re-shape them accordingly. They collaborate and share their thoughts with others. As I wrote, this I believe is their greatest growth. It may not show in the cold-cut data, but I can see it in their development as learners.

However, we are stuck on ‘times as many’

Essentially, I am asking these children to use the multiplicative structure of the situation to find an answer when objects are partially modelled or perceived. What I have noticed is because these students have found it difficult to generate mental images of number quantity in their head (hence one of the reasons they may require intervention in the first place), tasks that test their ability to create the mental images will challenge them incredibly.

My goal in the coming weeks is to scaffold ways to use physical items just enough for these kids to develop series of constructed generalisations that they can call upon when required. Then once these generalisations are established, they’ll be able to recall them like “a movie playing in their head” to use what they can perceive to find an answer.

The more I speak to other intervention teachers, both literacy and numeracy, I hear them say “What didn’t you do yesterday to set them up for success today”. Perhaps in my rush from the array situations I didn’t allow the children to see enough partially covered arrays to bridge the gap between modelled and abstract.

Either way, this is one of the biggest sticking points we’ve had in this journey so far. I look forward speaking to fellow Numeracy mentors in the coming weeks to further unpack the understanding behind the concept.


G’day, my name is Mark and I am a father of two young girls (3 y.o toddler and 3.5 month old baby) and husband to an amazing and supportive wife. Our little family lives a faith filled life, in pursuit of a loving and wondrous God.

I have been teaching ten years now across Catholic schools in Western Sydney, NSW Australia. Over the years I have been teaching, I have been given many opportunities to lead my colleagues and currently hold a leadership team position.

I presently teach Year 1 and also am training to be an early year’s numeracy intervention specialist. I am half way through the first of two years into the training and am finding it a challenge that has developed my PCK in Mathematics.

So why blog? Well it’s not my first rodeo. I’ve experimented with blogging in a few different formats. I participated in some Web 2.0 courses where blogging with students was in vogue. I tried to blog as a teacher about a class and our daily experiences. And then I tried to blog about teaching. Each of these experiences fell flat on the floor as I felt a brief flurry of excitement then it eventually stopped.

This time though l, I think I’m on the cusp of finding my ‘why’ teaching, learning and leadership, and I want to share it with colleagues, peers and a greater community.