Juggling As A Way of Building Deep Understanding

“The true mathematician is not a juggler of numbers, but a juggler of 𝘤𝘰𝘯𝘤𝘦𝘱𝘵𝘴.” – Ian Stewart

More & more, I'm appreciating this idea- 

That to truly understand maths, you need to see the connections between concepts  (not just know a definition, or a series of steps).

When you put mathematical connections in the centre, the purpose of what we do changes:

• From looking for answers, to searching for meaning
• From finding LOTS of practice questions, to looking for examples that will uncover new insights
• From aiming for ticks and avoiding crosses, to aiming for explanations
• From climbing a ladder of skills, to looking back and forth – and juggling concepts.

But what does this look like in practice?

Here are three examples…

Problems that put connections in the centre

Example 1. Cross Sums

If we're focused on answers, we might ask students to do this:

Calculate the following:

1. 1+3+5
2. 4+8+3
3. 7+2+3+6
4. etc.

In contrast, Cross Sums is connections-focused:

By being connections-focused, students necessarily use more skills and concepts, including:

• Arithmetic laws
• Number properties
• Odds and evens
• Managing data.

Example 2. Target Number

If we focus on answers, we might ask students to do this:

Calculate the following:

1. 2 × (2 + 8) + 75 – 50 + 3
2. (2 + 8) ÷ 2 + 75 ÷ 3 + 50
3. (75 – 50) ÷ (2 + 3) + 8 ÷ 2
4. etc.

In contrast, Target Number is connections-focused:

By being connections-focused, students will naturally focus on 

• The relationship between what they calculate and the number it produces
• Using estimation, and benchmarking calculations
• Which solutions are possible and which aren’t
• Managing data.

Example 3. Six Shapes

If we focus on answers, we might ask students to do this:

Calculate the perimeter of this shape:

In contrast, Six Shapes is connections-focused:

By being connections-focused, students will think about:

• What counts as a side and what doesn’t
• What counts as a corner and what doesn’t
• Using symmetry to simplify the possibilities
• Managing data.

What’s the big deal about data?

In each example above, ‘managing data’ is an important feature for making connections.

Data refers to:

• Sums that make 23 and sums that don’t
• Calculations that make 407 and calculations that don’t
• Polygon combinations and the total number of sides

In these instances, students aren’t just consuming data – they’re analysing it. 

And that analysis of data allows students to step back and look at the bigger picture of the problem, making it possible for them to:

• observe,
• question,
• conjecture,
• conclude,
• reason and
• connect.

All of which are crucial actions for deeper, long-lasting conceptual understanding.