Why We Need To Use Less Block Practice And More Interleaving When Teaching Mathematics
In this article, I explain why the over-use of ‘Block Practice’ in mathematics teaching is problematic, and why ‘Interleaving’ is advantageous.
What is Block Practice?
Block Practice, when applied to the mathematics classroom, is what most of us experienced at school - practice the one type of question multiple times before moving onto the next. Each block of questions requires students to use only one procedure.
Block Practice in Action - Example #1
1. Find the mean of the following sets of scores:
a) 2, 5, 3, 7, 4
b) 4, 7, 9, 2, 2, 4
c) 2, 5, 3
d) 8, 4, 6, 2, 0, 3, 0
a) 2, 5, 3, 7, 4
b) 4, 7, 9, 2, 2, 4
c) 2, 5, 3
d) 8, 4, 6, 2, 0, 3, 0
Block Practice in Action - Example #2
In a typical, right-angle trigonometry unit, we teach, at some point during the unit, the procedures for calculating unknown side lengths and angles of right-angled triangles. Typically, we set our students to work consolidating the procedures via Block Practice. This means that when students are practising the tangent ratio for calculating the length of an unknown side, THAT is all they are doing. And when practising the sine ratio for calculating the length of an unknown side, THAT is all they are doing. Block Practice is also applied to the practice of the cosine ratio for calculating the length of an unknown side, as well as for calculating unknown angles.
The problem With (Too Much) Block Practice
When we compartmentalise instruction into Block Practice and have students drill each part, we force students into rote learning - replicating a procedure we have demonstrated on the board. This requires very little cognitive effort and sits at the lowest rung on Bloom’s Taxonomy.
As a result, this tends to not only cause students to disengage but also ‘does not effectively enable students to develop a deep understanding of the necessary mathematical ideas involved’ (Silver et al., 2009, p. 503) At best, we see many students compliantly going through the motions. At worst, we see students disengage, throw in the towel, and adopt as their main aim to walk out that door!
‘Blocked practice provides students with a crutch. If students don't learn to solve problems without it, they will struggle during a test when their crutch is snatched away’ (Rohrer, Dedrick, Agarwal, 2017, p.6)
As a result, this tends to not only cause students to disengage but also ‘does not effectively enable students to develop a deep understanding of the necessary mathematical ideas involved’ (Silver et al., 2009, p. 503) At best, we see many students compliantly going through the motions. At worst, we see students disengage, throw in the towel, and adopt as their main aim to walk out that door!
‘Blocked practice provides students with a crutch. If students don't learn to solve problems without it, they will struggle during a test when their crutch is snatched away’ (Rohrer, Dedrick, Agarwal, 2017, p.6)
Is Block Practice Always Detrimental?
Again, from Rohrer, Dedrick, Agarwal © 2017, p11, 'Some blocked practice is useful, especially when students encounter a new concept or skill.'
If Block Practice, when overused, is an ineffective pedagogy, then what is the alternative?
The alternative is Interleaving.
If Block Practice, when overused, is an ineffective pedagogy, then what is the alternative?
The alternative is Interleaving.
What is Interleaving?
Interleaving, when applied to the mathematics classroom, prevents students from being able to draw upon a single procedure to answer a block of questions. This is because Interleaved questions are mixed questions - usually, questions that are mixed just enough so that the same concepts are being explored, but different procedures are required for successive questions.
Interleaving - An Example
The screenshot below was taken from the guide referenced above.
Interleaving Explained
The aim of Interleaving, in simple terms, is to intentionally have students working on several related ideas simultaneously so that the opportunity to blindly follow a procedure is removed. Setting tasks that require students to engage with several connected concepts results in students reasoning, thinking cognitively and choosing appropriate procedures. Interleaving is inherently more engaging because it creates in students an innate sense that they are tangibly involved in the process, thinking things through. It pushes the practice higher up Bloom’s Taxonomy triangle.
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How much interleaved practice is enough?
'The ideal amount depends on the student and the material, but studies suggest that at least a third of the practice problems should be interleaved.’ Rohrer, Dedrick, Agarwal © 2017, p11.
However, we need to be very careful about the ⅓ Interleaved recommendation because this means we will use a whopping ⅔ of Block Practice! The way I see it, to advocate such a high portion of Block Practice implies that a Procedural-first, Understanding-second mindset is being used, and as is explained here, such an approach can be problematic.
I propose that if we are presenting mathematics conceptually via an Understanding-first, Procedures-second approach where students work through activities that have them exploring and thinking mathematically BEFORE they are taught procedures then very little Block Practice will be required. However, if our major focus in on the teaching of procedures then perhaps we will require ⅔ Block Practice; however, I would have thought 50 percent Block Practice would be the maximum that could reasonably be advocated.
However, we need to be very careful about the ⅓ Interleaved recommendation because this means we will use a whopping ⅔ of Block Practice! The way I see it, to advocate such a high portion of Block Practice implies that a Procedural-first, Understanding-second mindset is being used, and as is explained here, such an approach can be problematic.
I propose that if we are presenting mathematics conceptually via an Understanding-first, Procedures-second approach where students work through activities that have them exploring and thinking mathematically BEFORE they are taught procedures then very little Block Practice will be required. However, if our major focus in on the teaching of procedures then perhaps we will require ⅔ Block Practice; however, I would have thought 50 percent Block Practice would be the maximum that could reasonably be advocated.
The Power of Interleaving
‘The Benefits of Interleaving to the Learning of Maths’ (Belham, 2018). helps to further explain Interleaving and Block Practice,
Interleaving improves learning ... it helps students to differentiate between two concepts. It is easier to understand the difference between an elk and a moose if you see them side by side than one after the other. The same happens with other concepts … it (also) helps students to figure out the right strategy or formula on the basis of the problem itself. For example, imagine that students are learning how to calculate the volume of different shapes. Usually, they would learn the formula for a cylinder and apply it to several problems involving cylinders. Then, they would lean the formula for a sphere and apply it several times to problems involving spheres ... The problem with this routine is that, even before they read the question, students already know which formula to use! That’s simply because they know that block of problems will be of the same kind and will require the same strategy (Block Practice). If, however, you present them with a set of questions that can be about spheres or about cylinders in a random order (Interleaving) ... students will need to understand how to figure out the right formula based on the problem and nothing else. Doing this will massively help them on cumulative examinations, as questions can be about any topic. That way, interleaved practice not only boosts learning but also prepares students for future examinations.
Other research supports the use of strategies that require students to operate at higher cognitive levels:
‘The regular use of cognitively demanding tasks in ways that maintain high levels of cognitive
demand can lead to increased student understanding and the development of problem-solving and reasoning (Stein & Lane, 1996) and greater overall student achievement (Hiebert et al., 2005).’ (Silver et al., 2009, p. 504).
Interleaving improves learning ... it helps students to differentiate between two concepts. It is easier to understand the difference between an elk and a moose if you see them side by side than one after the other. The same happens with other concepts … it (also) helps students to figure out the right strategy or formula on the basis of the problem itself. For example, imagine that students are learning how to calculate the volume of different shapes. Usually, they would learn the formula for a cylinder and apply it to several problems involving cylinders. Then, they would lean the formula for a sphere and apply it several times to problems involving spheres ... The problem with this routine is that, even before they read the question, students already know which formula to use! That’s simply because they know that block of problems will be of the same kind and will require the same strategy (Block Practice). If, however, you present them with a set of questions that can be about spheres or about cylinders in a random order (Interleaving) ... students will need to understand how to figure out the right formula based on the problem and nothing else. Doing this will massively help them on cumulative examinations, as questions can be about any topic. That way, interleaved practice not only boosts learning but also prepares students for future examinations.
Other research supports the use of strategies that require students to operate at higher cognitive levels:
‘The regular use of cognitively demanding tasks in ways that maintain high levels of cognitive
demand can lead to increased student understanding and the development of problem-solving and reasoning (Stein & Lane, 1996) and greater overall student achievement (Hiebert et al., 2005).’ (Silver et al., 2009, p. 504).
Interleaving infused through a unit of work
If you would like to see a unit of work that strongly features Interleaving as part of a conceptually-based approach, then check out the free tutorial A Conceptual Approach To Teaching Right-Angled Trigonometry.
If you came to this article from that tutorial then, upon your return you will see Interleaving cleverly infused into a trig unit. (If you lose the link, just sign up for it again!)
If you came to this article from that tutorial then, upon your return you will see Interleaving cleverly infused into a trig unit. (If you lose the link, just sign up for it again!)
Call to Action
Read about a 2019 randomized controlled trial by Rohher et al that Daniel Willingham that showed the significant positive effect of Interleaving compared to Block Practice.
A much shorter commentary on that randomised trial can be accessed here.
We would love to hear from you about your own experience with Block Practice and Interleaving - feel free to leave a comment below.
A much shorter commentary on that randomised trial can be accessed here.
We would love to hear from you about your own experience with Block Practice and Interleaving - feel free to leave a comment below.
