…for a learning outcome.

Hello and welcome to the 46th edition of our fortnightly newsletter, Things in Education.

The choice of teaching strategy plays a crucial role in shaping students' learning experiences. It's important to recognise that different teaching strategies cater to different learning outcomes. Or in other words, the choice of a teaching strategy depends on the learning outcome to be achieved. Earlier this week, one of our team members came across this tweet (or is it called a post, now?) on lattice multiplication. And we thought this was a great example to make the point about how there are different strategies for different types of learning outcomes.

In this article, we will explore how lattice multiplication can be a powerful tool for deep exploration of what multiplication is, how it works and what the role of place values is while multiplying. We will also try to make the argument that lattice multiplication may not be the most effective choice for practice.

What is lattice multiplication?

Let’s first understand what lattice multiplication is.

Say you want to multiply 314 by 81. This means you want to multiply a three-digit number and a two-digit number. So the first step is to create a 3x2 grid of squares, as shown below.

Then draw diagonal lines across the entire grid, as shown below.

Then put in the digits so that they align to each square, as shown below:

Now go on and multiply each column with each row and record the result in the corresponding diagonal sections. For example, 3 x 8 = 24.

Fill in the rest of the grid.

Now, add up the numbers in each of the diagonals. For example, the dark grey diagonal will be 4, the light brown will be 3, the green will be 14 (so you write the four and carry the 1 into the yellow diagonal, which will be 5 and the red one will be 2.

And voila! 314 x 81 = 25,434.

Teaching Strategies: Practice vs. Deep Exploration

Teaching strategies can be broadly classified into two categories: practice strategies and exploration strategies. Practice strategies focus on skill-building and procedural expertise, while exploration strategies encourage students to delve deeply into a topic for a comprehensive understanding. So, that brings us to the question – would lattice multiplication work better as a practice strategy or an exploration strategy?

In our opinion, using the lattice multiplication tool will work great as part of an exploration strategy. Using this tool:

Teachers can lead students through the lattice multiplication process, helping them grasp the procedure just like we did earlier in this piece.

Teachers can then ask students to explore the lattice multiplication method with different combinations of numbers. They can ask students to understand what the diagonals mean. If there is one three-digit number and another two-digit number, how many diagonals appear? What happens when you need to multiply two two-digit numbers?

Teachers can help students see if they can spot patterns with the number of digits and the number of diagonals formed.

And finally, teachers can encourage students to find any exceptions where the above patterns fall apart, or where this method cannot be used at all.

We encourage you to pause reading here and do some of the things that we could ask of the students.

The tool of the multiplication lattice along with the guided questions from the teacher becomes a great strategy to get students to deepen their understanding of multiplication. They will explore ideas of multiplication being repeated addition; ideas of what a place value is, ideas regarding how when multiplying, place value is very important; and why the effect of place value becomes greater as we move from ones to tens to hundreds to thousands. Now, we understand that which of these ideas will be explored deeply will depend on the grade of the students, and that is where teachers will have to take a call on what aspects of the lattice multiplication smorgasbord to focus the students’ attention on. The possibilities are probably endless.

So, why do we not recommend using this method for practice? Well, one may, but it is not the optimal choice. Just as we wrote earlier, when lattice multiplication offers such a multitude of concepts and avenues to explore for a student, why only use it to practice something? Practice can be done with the lattice multiplication method, but it is like using the cricket bat signed by a very famous cricketer to beat your clothes with. Also, this tool will become cumbersome and inefficient for larger numbers. Think 12,344 x 34,806. Too much space needed, and too much preparation needed to do one multiplication.

Finally, we firmly believe that the choice of teaching strategy should align with specific learning outcomes for that class. Lattice multiplication is a fun and exciting way to understand multiplication. It does excite students, and motivates them to try out different combinations to multiply. As good as it may be, we would not use it if the learning outcome for the given day is to establish mastery over the procedure of multiplication. It is most effective when the student is exploring what multiplication means in a deep manner.

For a learning outcome which involves getting students to master the procedure of multiplication, it would be nice to have some practice strategies, like straight vertical multiplication problems, real-world problems and pictorial problems which need multiplication. Such practice strategies for procedural mastery are very important in skill building and should be used for the relevant learning outcomes.

If we have to leave you with a final thought it would be this – that teaching methods, tools and resources should be selected with a clear understanding of the desired learning goals, enabling students to build procedural expertise and/or a deep comprehension of mathematical concepts.

We are working on a fantastic new teaching tool called TEPS, which will provide you with research-backed teaching strategies. Watch this space! If you are interested in getting early access to TEPS, please sign up here.

If you found this newsletter useful, please share it.

If you received this newsletter from someone and you would like to subscribe to us, please click here.

Edition: 2.20