di: Nice piece in ResearchEd on Connecting Maths Concepts

[Kerry Hempenstall]

https://researched.org.uk/cracking-the-learning-code/

[image: researchED] <https://researched.org.uk/>

(CMC) is a mathematics direct instruction (DI) programme. It is a packaged
resource which includes a teacher-scripted presentation book, additional
teacher guides for instructional strategies, pupil textbooks and workbooks
with an answer key, as well as additional placement tests to provide extra
worksheets for pupils who require more practice.

I used this textbook series as a remedial programme for intervention with
Year 7 and Year 8 while at Michaela Community School. The CMC textbooks
changed my understanding of mathematics and made me appreciate the
intricate and expertly designed structure of DI. More importantly, it
closed the most fundamental knowledge gaps the weakest pupils had and
accelerated their learning in their mainstream lessons.

CMC has been shaped through extensive field testing. It is different from
traditional study programmes because the field-test philosophy of CMC is
that ‘if teachers or students have trouble with material presented, the
*program* is at fault’.1 To ensure that there is no fault with the
programme, DI requires there to be a significant amount of attention to all
aspects of the teaching process.2 The programme strives to be faultless and
it is accepted that ‘if any one element of instruction is not done well,
high-quality instruction in other areas may not compensate for it’.3


CMC provides resources for the teacher and pupils which have been designed
so all aspects of the teaching process have been catered for.4 These
aspects consist of three main components of DI which allow all children to
learn effectively and efficiently:5

1.  Programme design

2.  Organisation of instruction

3.  Student-teacher interaction techniques

There are many great books and papers that eloquently discuss DI; this is a
brief summary of one of the three components of DI – programme design – and
its five elements.

*1) Analysing the content matter*

DI’s goal ‘is to teach generalised skills’.6 For this to be possible, the
concepts, rules and teaching strategies must be identified.

For example, a concept identified and taught in the Level D programme is
how to state a fraction from a diagram where one or more shapes are split
into an equal number of parts. This concept will provide a strategy to be
able to state a fraction from a number line. The concept has been taught in
both forms so pupils can gain a generalised strategy to apply to the widest
possible range of examples. This strategy will allow pupils to express a
mixed number on a number line or show that two fractions are equivalent
using a diagram, or be able to add fractions with common denominators which
sum to 1.

Identifying the content matter of a concept is the first step of programme
design.

*2) Clear communication*

Given that the content matter has been identified, the second aspect of
programme design is clear communication. This means creating an
instructional sequence that empowers pupils to apply a generalised strategy
in a wide range of examples.7 One part of this is called ‘general case
programming’, where instruction is designed to communicate one and only one
meaning, for all situations.8

For example, the Level D programme communicates how to state a fraction
from a diagram like this:

The top number is the total number of shaded pieces. The bottom number is
the total number of pieces in one unit.

This instruction didn’t change at any point throughout the textbook when
they were learning this skill, or a future skill which required pupils to
state a fraction from a diagram. More importantly, this strategy works for
all problem types: a proper fraction, an improper fraction, or a fraction
which simplifies to 1.

This is the same language which is used when stating a fraction from a
number line. The instruction deliberately uses the language ‘top’ and
‘bottom’ rather than ‘numerator’ and ‘denominator’ because it is
learner-friendly instruction. The same wording is used throughout. 9


*3) Instructional formats*

Next, instructional formats are created, based on the concepts, rules and
strategies to be taught, and clear communication used to teach pupils a
generalisable strategy. Format refers to the way a teacher presents each
question or explanation. The scripted teacher presentation book is very
helpful in providing each explanation for a concept which allows them to
use ‘effective, well-designed and precise language to communicate clearly
with all students’.10 In terms of the questions, the initial format of a
set of questions will be structured to support pupils but then the format
changes so pupils can apply their understanding independently.

For example, here is the transition between a sequence of exercises over
four lessons where pupils learn how to state a fraction from a diagram.

The format of the exercise has changed. The first set is focused on the use
of a shape. The second set includes questions where the fraction can
simplify to an integer. The third set is a mixture of number lines and
diagrams. The fourth set is a mixture of diagrams and number lines where
there is only one part between each integer.

The initial support is vitally important because it ensures a high level of
success and then with each exercise the process of ‘fading’ the format
comes into play: the format goes from ‘highly supportive to highly
independent’.12

*[image:
https://researched.org.uk/wp-content/uploads/2018/09/Practice-sets.png]*

*Practice exercises from Level D CMC textbook series (11)*

*4) Sequence of skills*

The sequence in which skills are taught can dictate how successful the
learning process is because skills are then practised continuously.13
Eventually,
the sequence also allows pupils to apply a generalisable strategy to deal
with exceptional situations too. For example, the skill of stating a
fraction from a diagram is covered in 40 consecutive lessons in one form or
another, ensuring a skill learnt in one lesson is used in subsequent
lessons. The continual review of one skill allows pupils to develop
automaticity, and so ‘re-teaching’ is unnecessary.14 The alternative is
teaching a skill which isn’t reviewed in the future, which means a pupil’s
understanding of that skill deteriorates and re-teaching is required.15

CMC provides resources for the teacher and pupils which have been designed
so all aspects of the teaching process have been catered for. These aspects
consist of three main components of DI which allow all children to learn
effectively and efficiently.

*5) Track organisation*

A track is an organisational framework where one skill is developed over
multiple lessons. For each skill practised there is a track, and this means
that in one lesson about 4–5 skills are included, instead of a narrow focus
on a single new learning objective occupying the entire lesson.16

This way DI ‘can extend the teaching and practice of a skill across many
lessons and weave prerequisite skill tracks into the tracks that integrate
these skills into more complex strategies’.17 Each skill is developed with
only one small change at a time to avoid pupils becoming overwhelmed with a
large quantity of new information.18 This allows pupils to learn new
concepts effectively and efficiently.

CMC is an extraordinary resource which has helped pupils learn more in less
time. CMC demonstrates that ‘higher-order thinking depends on the mastery
of more basic skills and involves the integration of concepts, rules and
strategies’.19 The beliefs that DI does not achieve this are most often due
to a misunderstanding of what DI is.
------------------------------

*References*

1.
Engelmann, S. (2003) *Connecting math concepts, teacher’s guide. *New York,
NY: McGraw-Hill.

2.
Watkins, C. L. and Slocum, T. A. (2003) ‘The components of direct
instruction’, *Journal of Direct Instruction* 3 (2) pp. 75–110.

3. Ibid.

4. Ibid.

5. Ibid.

6. Ibid.

7. Ibid.

8.
Engelmann, S. and Becker, W. C. (1978) ‘Systems for basic instruction:
theory and applications’ in Catania, A. C. and Brigham, T. A. (eds) *Handbook
of applied behavior analysis*. New York, NY: Irvington, pp. 325–377.

9. Ibid. 2.

10. Ibid. 2.

11. Ibid. 1.

12. Ibid. 2.

13. Ibid. 1.

14. Ibid. 1.

15. Ibid. 1.

16. Ibid. 1.

17. Ibid. 1.

18. Ibid. 1.

19. Ibid. 2.



-- 
Regards,

Kerry

Dr Kerry Hempenstall
Senior Industry Fellow,
School of Education,
RMIT University,
Melbourne Australia
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