## Three Ways to Authentically Engage Your Students

Steps To Winning Over Your Mathematics Class

A disengaged student cannot learn

An engaged student always learns!

I am yet to find a teacher who disagrees with the notion that student engagement is a fundamental prerequisite for the learning of mathematics. However, when it comes to implementing approaches to teaching, student engagement is rarely given the attention it deserves. Possibly, this is because student engagement is not only difficult to measure but is sometimes confused with the teacher ‘entertaining students via games and jovial dispositions’.

To overcome these limitations and to help bring student engagement to the forefront of any teaching approach, I prefer to use the term ‘Authentic Student Engagement’ – where students want to be in the room, understand what it is they are working on, and take ownership over their learning.

Below are three keys that can help you to authentically engage your students, i.e. help you to ‘win over your Mathematics class’.

To overcome these limitations and to help bring student engagement to the forefront of any teaching approach, I prefer to use the term ‘Authentic Student Engagement’ – where students want to be in the room, understand what it is they are working on, and take ownership over their learning.

Below are three keys that can help you to authentically engage your students, i.e. help you to ‘win over your Mathematics class’.

The first essential prerequisite for a student to become authentically engaged in Mathematics is ‘to understand the task they’re working on’, an idea that has been supported by scholarly research spanning over 60 years. (1) Although this might seem obvious, school-based mathematics has traditionally been oriented towards a ‘procedures first, understanding follows with practice’ approach.

If we want to ‘win over our maths class’, then we need students to be engaged and enjoying mathematics - we want to use practices that help students enjoy maths rather than despise it! And one guaranteed way to have students dislike maths is to force them to work on ‘maths’ that they don’t understand for extended periods. (Try working for 30-minutes at a time on anything that you don’t understand and see how you fare!)

In other words, the first step in winning over your maths class is to structure activities in such a way that students understand what it is they are doing; i.e. the work has students using their thinking rather than trying to recall the way the teacher showed them to gain the answer.

The problem with using a ‘procedures-first, understanding-second’ approach is that, by default, it requires students to work through activities they don’t understand, at least for a while, until they gain sufficient practice and hopefully understand it.

Rather than using a ‘procedures-first, understanding-second’ approach, wouldn’t it make more sense to use an approach that promotes understanding first and procedures second? After all, when done well, an ‘understanding first, procedures second’ approach is much more likely to:

When a classroom is comprised of students who use their own thinking and take ownership over their learning, then we can claim to have ‘won the class over’!

If we want to ‘win over our maths class’, then we need students to be engaged and enjoying mathematics - we want to use practices that help students enjoy maths rather than despise it! And one guaranteed way to have students dislike maths is to force them to work on ‘maths’ that they don’t understand for extended periods. (Try working for 30-minutes at a time on anything that you don’t understand and see how you fare!)

In other words, the first step in winning over your maths class is to structure activities in such a way that students understand what it is they are doing; i.e. the work has students using their thinking rather than trying to recall the way the teacher showed them to gain the answer.

The problem with using a ‘procedures-first, understanding-second’ approach is that, by default, it requires students to work through activities they don’t understand, at least for a while, until they gain sufficient practice and hopefully understand it.

Rather than using a ‘procedures-first, understanding-second’ approach, wouldn’t it make more sense to use an approach that promotes understanding first and procedures second? After all, when done well, an ‘understanding first, procedures second’ approach is much more likely to:

- Have students use their thinking rather than struggle to remember ‘what it is the teacher told me to do’.
- Give students a feeling of empowerment - (”Hey, I can do this!”)
- Enable students to develop a sense of ownership over their learning.

When a classroom is comprised of students who use their own thinking and take ownership over their learning, then we can claim to have ‘won the class over’!

**Disclaimer:**Some educators disagree with this ‘concepts-first, procedures-second’ idea, arguing that the approach should be one of ‘concepts and procedures together’. I agree that some situations allow for the two to work together. However, other situations lend themselves more so to allowing students to work with the concepts prior to seeing the procedures. Either way, we are suggesting a departure from the ‘procedures-first, understanding will (hopefully) follow with practice’ paradigm. This is related to Key #1. ‘Maths Tricks’ can be defined as ‘mathematical shortcuts that do not make sense’. Most of us use ‘tricks’ in our teaching, possibly because we think they save time, possibly because we think they help students or perhaps because we’ve never thought deeply about whether or not there is a need for an alternative.

For example, when ‘Cross Multiplication’ is taught (as most of us were taught it) it cannot be understood mathematically as a standalone procedure because it makes no mathematical sense. Therefore it can be classed as a trick. However, this is not to say Cross Multiplication shouldn’t be taught. A suggestion would be to instead start by teaching students to multiply algebraic fractions the long way, i.e. ‘multiply both sides by one denominator, then by the other denominator’. Then prompt students to recognise a shortcut. Cross Multiplication can, therefore, be introduced as a ‘shortcut’ - as opposed to a ‘trick - a shortcut that makes mathematical sense.

By the way, for a Dan Meyer-recommended, ‘must-have’ resource containing many common tricks and their shortcut alternatives, try googling ’Nix the Trix’. You’ll find a downloadable version as well as a short book.

For example, when ‘Cross Multiplication’ is taught (as most of us were taught it) it cannot be understood mathematically as a standalone procedure because it makes no mathematical sense. Therefore it can be classed as a trick. However, this is not to say Cross Multiplication shouldn’t be taught. A suggestion would be to instead start by teaching students to multiply algebraic fractions the long way, i.e. ‘multiply both sides by one denominator, then by the other denominator’. Then prompt students to recognise a shortcut. Cross Multiplication can, therefore, be introduced as a ‘shortcut’ - as opposed to a ‘trick - a shortcut that makes mathematical sense.

By the way, for a Dan Meyer-recommended, ‘must-have’ resource containing many common tricks and their shortcut alternatives, try googling ’Nix the Trix’. You’ll find a downloadable version as well as a short book.

As has been mentioned, one of the indicators of an authentically engaged student is ownership over learning. Traditionally, maths teaching has been teacher-centric. However, the limitation of a teacher-centric approach is that it is not conducive to students owning their learning. For students to own their learning, the learning ideally needs to be (at least somewhat) student-centred. This is a tricky thing to explain on a page - it’s a bit like trying to explain to parents with ‘toddler issues’ that reading to their toddler will greatly help them to bond and that the improved bond will have numerous flow-on effects re their relationship with and the behaviour of their toddler. Such things need to be experienced in order to be understood. The point is, anyone experienced with student-centredness will know, without doubt, that one of the default outcomes is greater student ownership over learning.

Disclaimer: By ‘student-centredness’ I’m not suggesting we ‘throw a bunch of equipment at our students and ask them to see what they can discover about ‘trigonometry’!’ What we are referring to are highly-structured activities that place the students at the centre through guided inquiry, exploration and that require them to use their thinking more than their memory of ‘what it was the teacher said’.

Disclaimer: By ‘student-centredness’ I’m not suggesting we ‘throw a bunch of equipment at our students and ask them to see what they can discover about ‘trigonometry’!’ What we are referring to are highly-structured activities that place the students at the centre through guided inquiry, exploration and that require them to use their thinking more than their memory of ‘what it was the teacher said’.

## Uh-Oh … what about the increased spread of students?

One aspect that must be addressed by any teacher adopting a student-centred approach is the increased degree to which students spread out within an activity or unit of work. Effectively dealing with a wide student spread does not come naturally to many teachers. Some strategies are required:

A wider student spread results in less opportunity for whole-class lectures. However, utilising mini-lessons to impart necessary explicit information to small groups of students as they require it can be highly effective. Yes, this means you teach each ‘bit’ several times, but the significant advantage with mini-lessons is that you are imparting information to students in response to them asking for the information rather than, as is the case in whole-class lectures, imparting information to students whether or not they are listening and/or ready for it.

This involves delivering comprehensive online units to students, enabling them to work in a student-centred manner and hence with increased engagement. Quality online units, by default, can deal very effectively with the increased student spread. However, Flipped-Mastery Learning is not appropriate for some classrooms/schools due to technology limitations.

Peer teaching is valuable. In line with the adage, ‘We learn 10% of what we hear and 90% of what we teach’, not only does it help the receiver, it arguably helps the ‘peer teacher’ more. Peer teaching can be structured, where the teacher dictates who teaches what, when and to whom. However, it can also be ‘organic’. Organic peer teaching is the result of a classroom culture where the teacher allows peer teaching to occur as a natural part of most lessons with clear guidelines and checks in place. Organic peer teaching, in particular, helps to promote authentic engagement as it draws on students’ desire for social learning; a desire to collaborate and help each other.

We all want to win over our maths class, which means we want our students to be authentically engaged – i.e. wanting to be in the room, understanding what it is they’re working on and taking ownership over their learning. This article attempts to demonstrate that by prioritising the understanding of mathematical concepts, refraining from using ‘Maths Tricks’, and transitioning to a more student-centred approach, we can move towards the authentic engagement of our students.

We all want to win over our maths class, which means we want our students to be authentically engaged – i.e. wanting to be in the room, understanding what it is they’re working on and taking ownership over their learning. This article attempts to demonstrate that by prioritising the understanding of mathematical concepts, refraining from using ‘Maths Tricks’, and transitioning to a more student-centred approach, we can move towards the authentic engagement of our students.

^{1}Brownell & Sims, 1946; Brownell & Moser, 1949; Carpenter, Fennema, Peterson, Chiang, & Leof, 1989; Fuson & Briars, 1990; Cohen, McLaughlin & Talbert, 1993; Hiebert & Wearne, 1993; Carpenter, Fennema, & Franke, 1996; Hiebert et al., 1996; Newmann & Associates, 1996.

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