## Catering for student spread within mathematics classes

Three strategies and some misconceptions

Part 2 in a 3-part series

Part 2 in a 3-part series

Richard Andrew, 19th April 2017

Three strategies and some misconceptions

Part 2 in a 3-part series

Part 2 in a 3-part series

Richard Andrew, 19th April 2017

This, the second article in a 3-part series is again directed mainly at high school math/s teachers. The first article, Conceptual approaches to teaching and learning mathematics need to be student-centred, proposed that a conceptual approach to teaching and learning mathematics is most effective if it is also student-centred. It proposed that increased student spread, an inevitable by-product of a student-centred approach, is a major reason that many high school teachers experience difficulty when adopting a student-centred approach, or avoid the approach altogether.

In this article, I’ll outline three strategies which have proven to be effective when dealing with increased student spread.

In this article, I’ll outline three strategies which have proven to be effective when dealing with increased student spread.

In almost all cases, math/s units require a significant degree of progressive, sequential information. For example, students generally need to cope with two/three step equations before tackling four/five step equations. In right-angled trigonometry, students need to learn to find lengths of sides (given one acute angle and another side) before tackling problems involving double triangles. We can assume that most units will require at least 5-10 sequential teaching segments, each requiring some form of direct input from the teacher. The exceptions to this principle are units which are fully inquiry-based and/or open-ended. However, such units still require some degree of direct instruction from the teacher.

A likely misconception held by those unfamiliar with an effective student-centred approach, is that in a student-centred approach, the teacher does not utilise segments of direct instruction. The misconception is the view that students, via a student-centred approach are required to somehow ‘magically’ discover nearly ALL required information for a given unit by themselves. However, I don’t see how the teaching of typical math/s units without direct teacher instruction can succeed.

The type of student-centred approaches promoted by Learn Implement Share all contain a significant amount of direct instructional input from teachers. Unlike the traditional teacher-directed approach, however, direct instructional input via a student-centred approach occurs to an entire class intermittently, and mostly to small groups of students as the need arises.

A likely misconception held by those unfamiliar with an effective student-centred approach, is that in a student-centred approach, the teacher does not utilise segments of direct instruction. The misconception is the view that students, via a student-centred approach are required to somehow ‘magically’ discover nearly ALL required information for a given unit by themselves. However, I don’t see how the teaching of typical math/s units without direct teacher instruction can succeed.

The type of student-centred approaches promoted by Learn Implement Share all contain a significant amount of direct instructional input from teachers. Unlike the traditional teacher-directed approach, however, direct instructional input via a student-centred approach occurs to an entire class intermittently, and mostly to small groups of students as the need arises.

Within a teacher-directed approach, teaching segments are primarily delivered to the whole class at any one time. However, as soon as we encourage student spread – by adopting a student-centred approach – the opportunities for whole-class teaching segments all but disappear. Certainly, units will commence with a whole-class instruction, and there will be numerous whole-class segments where logistics are dealt with and summaries delivered. However, once the unit commences very few opportunities arise for whole-class, math/s instructional segments. The reason is obvious. For example, as 3 students reach ‘teaching segment 7’, several students are beyond this point, many are nearing ‘teaching segment 6’ and still others are progressing towards ‘teaching segment 5′. Therefore there is no opportunity for a whole-class instruction for ’teaching segment 7’.

How do we, therefore, effectively deliver instructional segments within a student-centred unit?

How do we, therefore, effectively deliver instructional segments within a student-centred unit?

To help solve this problem, peer teaching comes to the fore. Remember the adage ‘We learn 90% of what we teach and 10% of what we hear’. This statement describes the case for peer teaching. Enabling peer teaching requires some careful planning and management. However, if students are encouraged to naturally collaborate and help each other then peer teaching greatly reduces the pressure on the teacher when it comes to math/s instruction. I am not suggesting that peer teaching is a replacement for direct teacher instruction. I am proposing, however, that when peer teaching is functionally at work within a class of students then mathematical information is able to come from numerous sources rather than only from the teacher. The ‘3 before me’ mantra can powerfully operate here i.e. 'Students must seek information from three sources prior to asking the teacher for help'.

One of the more effective ways of dealing with increased student spread is to create comprehensive online units which use video to deliver the instructional segments. It needs to be said that this is very different to what is commonly coined a ‘Flipped Classroom’. The term Flipped Classroom refers to an online unit where students are required to watch the teaching segments – i.e. videos of the next lesson’s work – at home. This releases some of the (next day’s) lesson time for collaboration on what would have otherwise been homework exercises. Hence the teaching process is ‘flipped’ – teaching segments occur at home (the night before), and homework is done during lessons. In this Flipped Classroom model students are still being ‘herded’ in an attempt to keep their progress uniform. The approach is, therefore, still a teacher-directed approach, but certainly has potential advantages over the traditional, teacher-directed approach.

The Flipped Mastery Classroom, on the other hand, is student-centred . The Flipped Mastery Classroom is a system in which math/s units are prepared using an online format. These comprehensive online units of work allow students to progress at their own pace. Individual instruction is delivered by videos which can be watched and replayed anytime a student wants to watch them. The fact that students have the choice to watch a given video or not, and choice to replay a video or parts of a video is extremely powerful. And for these reasons, the instructional videos prove to be - in some ways - superior to face-to-face instruction!

Assuming the units are comprehensive and well designed, the teacher is freed up in class to spend extra time with those students who require the most assistance. This is because the majority of students are able to progress individually through the online unit without being reliant on the real-time teacher.

The Flipped Mastery Classroom, on the other hand, is student-centred . The Flipped Mastery Classroom is a system in which math/s units are prepared using an online format. These comprehensive online units of work allow students to progress at their own pace. Individual instruction is delivered by videos which can be watched and replayed anytime a student wants to watch them. The fact that students have the choice to watch a given video or not, and choice to replay a video or parts of a video is extremely powerful. And for these reasons, the instructional videos prove to be - in some ways - superior to face-to-face instruction!

Assuming the units are comprehensive and well designed, the teacher is freed up in class to spend extra time with those students who require the most assistance. This is because the majority of students are able to progress individually through the online unit without being reliant on the real-time teacher.

Creating comprehensive online units is time intensive and requires a wide skill set. The task also requires an appropriate level of technology for the teacher and for each student. However, there are some significant upshots. Firstly, once a unit is complete then that’s it – it is ready now, and with some minor tweaks, it is ready for each subsequent year. Secondly, the online approach can be extremely empowering for teachers and their students. Remember, this is not meant to replace classroom teaching. Rather, it can greatly improve direct math/s instruction through differentiation, enable student-centred learning and free the teacher up to give quality math/s instruction to small groups. It is, however, a big change from conventional teaching. Baby steps are recommended. And beware of expecting instant, overwhelming success. Great success can be expected, but only with time.

Obviously, using quality, comprehensive online units is an ideal way to both foster and deal with a wide spread of students. However, student spread can also be encouraged and handled without the use of technology and the 'mini lesson' proves to be instrumental here. It is not that the mini lesson is the strategy for handling student spread. Rather, it is the structure under which mini lessons are used. Allow me to explain.

If we accept that a student-centred approach is the way forward, and we accept that direct teacher instruction is also necessary, then the only option we have to deliver math instructional information – in the face-to-face-only classroom – is via mini lessons. A mini lesson is a math instructional segment, shorter than its teacher-directed, whole-class counterpart, but delivered to any small group of students requiring a given segment of math information at a particular time.

If we accept that a student-centred approach is the way forward, and we accept that direct teacher instruction is also necessary, then the only option we have to deliver math instructional information – in the face-to-face-only classroom – is via mini lessons. A mini lesson is a math instructional segment, shorter than its teacher-directed, whole-class counterpart, but delivered to any small group of students requiring a given segment of math information at a particular time.

“Everyone – stop work for a second. Paul, Jen and Mohammed have just reached (Section 6) so they require the next teaching segment. Is there anyone else who is ready for (Section 6)? Is there anyone who is nearly there? Is there anyone past this point who knows that they need to hear it again? OK, Grace and Jini. So you five watch closely. The rest of you … there’s no need to listen to this – just work ahead quietly for the next 5 minutes.”

The teacher then gives a short mini lesson to that group of students. The students do not need to be sitting together. They could be brought to the front if this helps. However, mini lessons often work best when students stay in their seats. After all, the teacher knows which students to direct the teaching segment to.

The real key in dealing with increased student spread, therefore, is not ‘The Mini Lesson’. The real key is using a strategy which incorporates the use of many mini lessons throughout a unit of work.

The teacher then gives a short mini lesson to that group of students. The students do not need to be sitting together. They could be brought to the front if this helps. However, mini lessons often work best when students stay in their seats. After all, the teacher knows which students to direct the teaching segment to.

The real key in dealing with increased student spread, therefore, is not ‘The Mini Lesson’. The real key is using a strategy which incorporates the use of many mini lessons throughout a unit of work.

Correct, you will be repeating yourself! Each mini lesson will be repeated several times. And this flies in the face of everything most of us have learned as math/s teachers, namely that:

*‘The aim of the game is to give quality instructions ONCE – to the whole class – so that as many students as possible GET IT, thereby minimising the number of times we have to repeat the information.’*

I’m suggesting that the above mantra needs to change.

I’m suggesting that the above mantra needs to change.

The benefits of using mini lessons as a way of enabling student spread are significant yet subtle. In regard to mini lessons:

- Prior to as mini lesson one or more students have made a request to the teacher for the information, giving them ownership over their own learning. They have volunteered to be in the group to receive the teaching segment. i.e. there is an expressed ‘need to know’. This is significant, yet subtle, and rarely occurs in the teacher-directed scenario.
- Students receiving the mini teaching segment tend to be more engaged because they have requested to receive the information.
- The information delivered tends to be more targeted due to the small group size and also because the teacher has no doubt that the information IS exactly what these students require at this time.
- The set up to each mini lesson offers choice to students. The choice is subtle, however, any choice offers motivation.
- The set-up demands students to engage in some metacognition. Every time a teacher asks a question like, “Who else knows that they need to know this?” students are being asked to think about what they know, what they don’t know, what they need. They are being asked to think about their learning.
- Mini lessons give students who need the direct instruction another time, a second chance to hear it.

Naturally, a strategy such as this, which is a clear departure from the norm, will attract some scepticism. Upon reading the above many will adopt misconceptions about mini lessons. Below are several such misconceptions.

Mini lessons do sound inefficient, however the assumption that "In a teacher-directed model I only need to give the information once’ is false. The truth is, in a traditional teacher-directed approach, when delivering a whole-class teaching segment some students are not ready, at that time, for the information. Therefore the teaching segment can only create confusion and an overload of information for those students. Similarly, some students do not need the teaching segment because they already understand the information being delivered. There is no better way to turn off the more able students than to force them to listen to teaching segments they do not require. This I know to be true because I have done it too many times!

This, actually, is not a misconception, it is absolutely true! But I don’t see this as a negative. ‘Organised chaos’ is not a bad thing amongst an engaged group of students. Within a well-planned mathematics unit, many argue, ‘organised chaos’ at appropriate times, is educationally sound.

There’s no doubting the fact that, initially at least, managing student spread via mini lessons will be a challenge for those new to the strategy. It requires a different management style to the teacher-directed model. However, once embraced, this self-paced, independent, conceptually-based style of teaching (incorporating mini lessons) becomes second nature. Given the opportunity for self-reflection, sufficient time to implement and expert guidance, teachers find the transition to a student-centred approach with mini lessons to be relatively easy.

Tick. But shouldn’t this be the #1 aim of any teacher with any class – to have the students on-side? To win them to the teacher’s team? To create an engaged math/s classroom? It is true that if students are not (mostly) on-side then a student-centred approach will not be as effective as it could be. The same applies, however, to teacher-directed approaches. The difference is that it is easier to keep ‘the lid on the can’ when the approach is teacher-directed. But effective learning is not about students being inside a can with the teacher standing on the lid. Effective teaching is about students being engaged, wanting to learn, collaborating and effectively working with the teacher-as-facilitator. It's about students taking ownership over their learning.

If all this talk of student spread is causing you to wonder how you are going to cope, it might be time for you to spend some time observing some exemplary primary teachers who love teaching mathematics and who are comfortable with having different students working at different levels – and often on different tasks – simultaneously. Such teachers are incredibly skilful and well worth observing. We secondary teachers can take a lot from such exemplary examples of student-centred, primary pedagogy.

In the final article in this 3-part series, Let’s encourage student spread in math units, I’ll argue the case in favour of encouraging student spread, rather than seeing student spread as an inevitable, problematic by-product of a student-centred approach.

Comments anyone? We'd love your thoughts below! (Your email address will not be required)

Comments anyone? We'd love your thoughts below! (Your email address will not be required)

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