## What is a ‘Key Concept’?

In the Maths Pathway Learning Model, students spend some time working individually on ‘modules, with the support of their teacher and peers. These modules are constructed using an atomised approach, honing in on a single, specific part of the curriculum at a time.

A Key Concept is a mathematical idea which underpins numerous modules, tying them together. To fully grasp a Key Concept, students should go beyond the atomised approach of modules; they should think critically and deeply, reason with their ideas, and discuss the concept with other students. This is the purpose of the ‘mini-lesson’ in the Maths Pathway Learning Model.

## Don’t students need explicit teaching module-by-module?

Targeted explicit instruction is a very important part of the Maths Pathway Learning Model, as is independent learning where students attain new knowledge by forming connections with existing knowledge. The learning model balances these modes of learning (among others) in a way which is both pedagogically appropriate, and practical in a real-world classroom.

When learning autonomously, students are working on modules which are targeted very carefully to their point of need. For a student to access a module, they must first have shown they are ready to learn it – by either completing or mastering a set of prerequisite modules. Each module is scaffolded very carefully, such that students build their new knowledge by forming connections and extensions from existing knowledge. Each part of a module is a natural extension of the part before, and can be successfully attempted by a student without the need for explicit instruction.

This is quite different to a “chalk-and-talk” approach, which provides explicit teaching first and autonomous practice afterwards. Instead, the exposure to new concepts is interwoven with the practice, such that students are always learning as an active participant, and reasoning and forming connections as they go. In addition to the pedagogical drivers behind this approach, there are also practical considerations here: in reality, the spread of student entry-points is so great within a single class that it would be impractical to rely on the teacher to provide explicit instruction for students on the module level. Attempting this approach would leave the teacher with very little time to spend on the more highly impactful proactive explicit teaching modes (see below).

The autonomous learning approach comes with its own challenges, and students require safety nets along the way. Worked solutions within a module are written with the explicit intention of helping a confused student clarify and understand each question. Videos with each module are designed to work as a safety-net as well; it is not intended that students watch these before starting the module, but access the videos as a resource in case they are stuck. The final level of safety net is explicit instruction; either from a peer or from the teacher. One of the most rewarding parts of teaching in the Maths Pathway Learning Model is in developing students’ capacity to make intelligent use of those safety nets. By the time a well-prepared student puts up their hand for help, they are primed and ready for an ‘ah-ha’ moment with the teacher.

So day-to-day module completion requires only *reactive* explicit teaching when students require extra help. The learning model also includes three *proactive* explicit teaching modes, without which the students’ overall learning experience can become unbalanced. The first of these is data-informed targeted intervention, where a set of individuals are identified as having a particular need – usually a learning skill, a mindset or a habit – which the teacher can actively address. The second proactive teaching mode is leading rich lessons (including project work), where the teacher guides a whole class through a meaningful mathematical problem with multiple entrance and exit points. The third type of proactive targeted teaching is leading mini-lessons, where Key Concepts in mathematics are explored and developed.

## How are students grouped for Key Concept Mini-Lessons?

A student is included in a mini-lesson group report when they are ready to learn at least one module within a Key Concept, or have recently completed such a module.

Some students may be standing on the shore – ready to wade out into the concept but with feet still dry. Others may have waded in already, and be familiar with the shoreline. Still others may be swimming further out. The point of the mini-lesson is to take all these students and fly together above the surface to get a sense of the shape and feel of the big idea (i.e the Key Concept) as a whole.

## All the students in my Mini-Lesson group are at really different points. What should I do?

Try zooming out a level.

A trap some teachers fall into is trying to use mini-lessons to perform the function which modules already perform; that is, understanding a new piece of mathematics, or becoming proficient with a particular process on its own. At that level of granularity, the learning intention for the mini-lesson is likely to be really specific and require a particular entry point for students to benefit. The good news is that such mini-lessons generally aren’t necessary; this is already covered by students’ autonomous learning, supported by just-in-time explicit teaching instigated by the student.

What students *do* need is a chance to explore, discuss and get the shape of a mathematical concept (or a set of closely related concepts). Such discussions aren’t focussed on one definition, process, representation or method. Instead they’re looking at how these are connected together into a broader mathematical idea (i.e. a Key Concept), and how that idea connects to other parts of mathematics.

Whether a student has seen only 1 module in this key concept, or has seen 8 modules, they can still get a feel for the shape and flavour of this mathematical concept. The first student may be seeing much of this for the first time, and may not fully understand everything which is discussed – but will be well primed for things to click into place when they complete the modules later. The second student may already be familiar with the pieces, but may not have considered their connections in this way before, or stopped to think about where this idea fits in the broader schema of mathematical knowledge.

Leading mini-lessons of this type requires skill and practice, but is hugely rewarding – and is key for students developing a well-rounded understanding of mathematics as a whole.

## That all sounds wonderful in theory, but can you give me some examples?

If you are new to running Key Concept mini-lessons in this way, it can be quite daunting to start. Happily, many of the teachers in the Maths Pathway Community have shared their experiences with us, which has allowed us to produce a set of full lesson plans for particular Key Concepts. These provide excellent examples of the pedagogical approaches which can be taken for mini lessons.

**Physical model to connect with the abstract.** This lesson plan about Understanding Whole Number Place Value exposes students to some physical, tactile models of tens and units, to form connections with the visual and numerical models within modules. This serves both to prime a novice student with an experiential anchor prior to encountering such modules; and to reinforce a more expert student’s understanding by looking at a familiar idea in an unfamiliar way.

**Physical model for patterns and conjectures.** This lesson plan about Single-Digit Multiplication also uses a physical, hands-on embodiment of a mathematical concept – with students working together actively to seek for patterns and form conjectures. The presence of that model gives enough scaffolding to allow all students to be involved in that pattern-seeking. In lessons of this type, students can uncover some mathematical facts, but the primary aim and focus is to form mindsets and broad ideas via the discussion itself. Similar examples are this lesson plan about Operating with Even and Odd Numbers, and this lesson plan about Rounding and Estimating Multiplication.

**Visual model for reasoning and justification.** This lesson plan about Multiplying Fractions draws on a visual rather than a physical model. The mathematical idea which this visualisation exposes may be more familiar to some students than to others, but the lesson focusses on discussion about the ‘how’ and the ‘why’ of that idea. This gives a hook for all students to reason collectively, with careful guidance from the teacher in the small group setting.

**Connecting multiple representations.** This lesson plan about Linear Inequalities gives students a task which novice and expert students can work on together, to stimulate the sorts of discussions which make connections across the key idea. This is achieved by having students translate between multiple representations / embodiments of the same mathematical idea. A similar example is this lesson plan about Linear Graphs.

**Examples and counterexamples.** This lesson plan about Number Lines also has students compare and contrast different items, but with a different emphasis. They explore examples and counterexamples of a mathematical visualisation – including real-world examples – and collectively discuss the reasoning behind their categorisation. This acts to repair and head off misconceptions, and build an appreciation for the way this embodiment fits with real world applications.

**Comparison and relationships.** This lesson plan about Quadrilaterals has students collect and organise information about mathematical objects - on the basis of their mathematical properties - and form comparisons and contrasts on this basis. This highlights the connections and also the differences which can exist between different mathematical objects, which feeds into the way in which students approach learning about these objects later, or using information about such objects in later problem-solving. A similar example is this lesson plan about Area and Perimeter of Rectangles.

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