In the first post of this series exploring the IOI Process, I argued that pedagogical quality was a better measure than teacher quality. In this post I intend to explore how considering pedagogical effectiveness can enable us to understand, and measure the impact of our pedagogical approach.

While some researchers and teachers believe that certain activities have can be said to universally work or not work, for example Visible Learning suggests classroom behaviour matters while class size doesn’t. The IOI approach views understanding and measuring what works as being less universal and more contextual. While our unique definition of pedagogical quality does consider the unique needs of our students, having a more precise understanding of student development will enabled us to also measure the effectiveness of our pedagogical approaches in our unique context.

Following on from the previous blog post exploring learning mathematics, in this post we will explore how students develop as mathematicians.

So how do students develop as mathematicians?

Sure we could create a curriculum, identify progression points, number levels and use tick lists to try to understand how students development as mathematicians against these. Of course, we’d need to hope that we’ve covered them all of the important stuff, and hope that what is important doesn’t ever need to change. Sadly, that’s what we usually do. It’s beyond the scope of this post to debate how well our curriculum can assess student learning, what is for certain using a curriculum to measure pedagogical effectiveness is ~~almost~~ impossible.

A better way to understand the pedagogical effectiveness of our mathematical approach is to consider the developmental stages of learners, as they develop as mathematicians. Note this post is not intended to be an exhaustive exploration of the developmental stages of learning mathematics, rather it is intended to illustrate how educators can understand student learning developmentally.

Students arrive at school either as** non mathematicians** with little or no number sense or as **early mathematicians** who are able to identify patterns and use logic to make judgements and inform their questions. Over time **non mathematicians** develop as **early mathematicians **who then use concrete materials which enable them to use precision when sorting, selecting and classifying when completing puzzles and other maths based activities.

**Early mathematicians** further develop into **competent mathematicians** who use mathematical symbols, logic and reasoning to make decisions and solve increasingly complex problems and puzzles. The use of mathematical notation is often supported with concrete materials, as competent mathematicians often lack the confidence and the ability to deduce for themselves whether their thinking is correct.

Next **competent mathematicians** develop into **able mathematicians** who use mathematical symbols and language to represent and understand the logic of mathematical problems and puzzles. They are to recognise relationships between various mathematical concepts. They can work through the logic of their solutions and justify they the mathematical decisions that they make.

**Expert Mathematicians** develop a sense of mathematics that enables them to make predictions that are fairly accurate, as well as recognising the accuracy of these predictions. They use reasoning and mathematical knowledge to solve problems using new methods.

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By looking at the development of non mathematicians through to expert mathematicians, it is clear that the same pedagogical approach, for example, would be likely to be effective at helping students develop from non-mathematicians to early mathematicians as from competent mathematicians to able mathematicians. For example, stories, songs, play and other highly tactile experiences are typically used to help students develop from non mathematicians to early mathematicians. Obviously, using stories, songs, and play would not be the most effective pedagogical approach to help students develop from competent mathematicians to able mathematicians.

To understand pedagogical effectiveness we need to understand the development stages of our students. In the next blog post in this series I will demonstrate how a the IOI Learner Development Profile can be used to create development profiles for any learning and teaching approach and in turn how this understanding can help us measure pedagogical effectiveness.

**Teams at the IOI Weekend will use pedagogical effectiveness to understand, measure, discuss and design innovation learning and teaching practice.**

**A free fast paced three-hour taster on Friday night will provide you with a shorter experience of the IOI Weekend. This is a free event and will be held at:**

**May 15th 6PM – 9PM at New Era Melbourne**

**Level 2 141 Capel Street North Melbourne VIC 3051**

**Over three hours we will give you a taste the IOI Process highlighting:**

**IOI Pedagogical Quality Framework,****IOI Learner Development Profile,****the Modern Learning Canvas,****how pedagogical quality, effectiveness and capacity can be measured,****and get you on your way to develop an Innovation Thesis.**

**Please RSVP to if you intended to join us to help us with catering (light finger food and drinks.)**

Photo credit: Math Mania at East Chilliwack Elementary by University of the Fraser Valley CC By 2.0

Whether or not students arrive at school either as non mathematicians or as early mathematicians, they all arrive as problem solvers.I think that the pedagogical approach needs to engage the learners ability to problem solve first and foremost. This shifts the focus to more integrated, authentic curriculum that does develop the learners skills, knowledge and understandings of mathematics (along with other things).

I wonder whether it would be possible to understand the development stages of our students in relation to problem solving first and foremost? My gut feeling is that this could accelerate development of students from ‘novices’ to ‘experts’ across the curriculum. Any thoughts?

Hi Craig,

I totally agree. As these ideas, processes and tools have been developed we’ve made sure they are totally inclusive whether schools are using highly traditional approaches or not. What we’re attempting to do is provide educators with language and tools that enables them to more clearly understand and communicate what is powerful about they way they design learning and teaching. No matter their starting point.

What you describe is exactly how we hope schools and teachers will use these tools, in order to clearly describe, measure and justify learning design that impacts student learning beyond the metrics we currently have, namely test scores and rhetoric around teacher and instructional quality.

In the next post, that I’ve just published, I describe how these development profiles can be described, my hope is that educators will take these ideas to describe learning and development in terms of the things you mention… problem solving, creativity, curiosity, collaboration,… and y doing so make a much strong case for pedagogical approaches they use.