Hands, Head and Heart (3H) framework for curriculum review: emergence and nesting phenomena

Abstract

In this work, we report the emergence of the Hands, Head and Heart framework that arose within the curriculum review for subject knowledge courses for primary school pre-service teachers in the National Institute of Education, Singapore. Through an initial grounded analysis of a survey of pre-service teachers and faculty focus group meeting data, the responses were broadly categorised into hands, head and heart domains and these formed an initial framework for discussions in the review committee meetings. By revisiting the data from the survey, an analysis through a complexity lens revealed the emergence of a characteristic nested self-similarity of the framework. Over the course of several committee meetings, further self-similarity was discovered. We conjecture that the Hands, Head and Heart framework and its self-similarity property provide a potential basis for a holistic approach to curriculum review. We used this framework to revise the learning objectives of the subject knowledge curriculum by resolving perspectives which previously seemed contradictory.

Appendices

Appendix 1: Survey questions for the PTs

This section provides the survey questions for the Further Mathematics course that were administered to the PTs (questions for other courses were identical except for the course name). The survey was on a voluntary basis and had full anonymity in their online submission. They were also allowed to choose not to answer any question if they did not wish to, although in practice, all the 69 respondents at least answered all the Likert scale questions, but were more selective in answering the open-ended questions.

Statement/Question	1 – strongly disagree, 5 – strongly agree
The Subject Knowledge course, ASM40A: Further Mathematics Topics, builds up my confidence in teaching Primary Mathematics.	1	2	3	4	5
I think it is necessary to have a deeper understanding of the mathematical concepts (beyond the level covered within Curriculum Studies courses) in Primary Mathematics.	1	2	3	4	5
Why do you think that it is necessary/not necessary?
Further Mathematics Topics help me to understand the mathematics taught in Primary Mathematics at a deeper level.	1	2	3	4	5
I find that the contents in Further Mathematics Topics are relevant to my teaching in Primary Mathematics.	1	2	3	4	5
Which topics in Further Mathematics Topics do you find relevant to your teaching in Primary Mathematics? Why do you think they are relevant? (N.A. if not applicable) Which topics in Further Mathematics Topics do you find irrelevant to your teaching in Primary Mathematics? Why do you think they are irrelevant? (N.A. if not applicable)
There is too much content to cover in the Further Mathematics Topics course for one semester.	1	2	3	4	5
The course notes for Further Mathematics Topics:
• are enjoyable to read;	1	2	3	4	5
• enhance my understanding of concepts taught in class.	1	2	3	4	5
The exercises for Further Mathematics Topics:
• are manageable;	1	2	3	4	5
• enhance my understanding of the topics.	1	2	3	4	5
The further exercises for the Further Mathematics Topics are:
• interesting to work with;	1	2	3	4	5
• challenging.	1	2	3	4	5
The tests conducted in ASM40A: Further Mathematics Topics:
• are fair ways to assess my performance in the course;	1	2	3	4	5
• determine how I study for the course;	1	2	3	4	5
• require a lot of memory work.	1	2	3	4	5
If you are given a choice for alternative forms of assessment, what would you suggest? Please give a reason for your suggestion(s).
I find that the Subject Knowledge courses (ASM Series) have helped me to better understand my Curriculum Studies (ACM Series: Teaching and Learning of Primary Mathematics) courses.
Please elaborate, perhaps with an example, your answer above.
In your own words, recall the stated objectives of Subject Knowledge (ASM40A) courses. (What did the tutors say the course objectives were?)
What would be your desired objectives of Subject Knowledge (ASM40A) courses? (You may write down the stated objectives above if you agree with them.)
What aspect(s) of the courses do you like and why?
What aspect(s) of the courses do you think may be further improved and how may they be improved?
Any additional feedback and/or suggestions on the courses?
If you have taken courses from Subject Knowledge English and Subject Knowledge Science, are there any good practices from the SK English and SK Science that may be applied to SK Math courses?

Appendix 2: PT responses on perceived learning objectives

The following were the responses of the PTs to the question of why it was ‘necessary to have a deeper understanding of the mathematical concepts (beyond the level covered within the curriculum studies courses) in primary mathematics.’ In total, there were 48 responses to this question. Twenty-nine responses were considered to be in direct response to the question and they are listed below.

• 1. Viewing the course as a step towards being a better teacher and/or improve their pedagogy

As the PTs had a distinct career path as a teacher in their immediate future, it was obvious that their identity as a teacher-to-be would largely influence their consideration as to whether a particular activity would be relevant to their professional development. As a result, there was a total of 17 responses of this kind. The paragraphs in italics were the actual responses by the PTs. For easy reading, we categorised the results based on our initial coding scheme, in italics and bold. We have further categorised the responses through the 3H lens.

1.1 Hands

Delivery of explanations. Four of the responses indicated that it would be necessary for them to have deeper knowledge so that they would be able to explain to their future students better.

‘Deeper understanding allows us to deliver lessons with more clarity and it reaches the pupils better too.’

‘One has to have a deeper and broader understanding of concepts one intends to teach, in order to teach it well, such as being able to explain and provide alternative solution methods.’

‘Necessary because having deeper content knowledge will help in my explanation beyond instructional understanding which I think would help students develop deeper conceptual understanding of the subject. If we only know the steps, and if students were to ask how certain formulae or steps are derived, we will not be able to answer them.’

‘It is necessary because this is a degree course. Furthermore, only with knowledge beyond the curr[iculum] studies in primary math[ematics] would one be able to explain mathematical concepts.’

Simplifying explanations. While the previous category was only concerned about the ability to present mathematical ideas and concepts, this category involves the decomposition of the mathematical idea, identification of the correct representation for the students and/or to develop useable definitions for the students.

‘I believe that before teachers teach about the basic concepts of mathematics, they should understand what is beyond it. That way, they could help their students understand better when they break it down to chunks for them.’

‘You need a much deeper understanding to explain a certain concept as simple as possible to students and relate it to something they can connect to.’

‘We need to know more in depth in order to find the right method to simplify explanations to our students.’

Longitudinal coherence. ‘It allows us to understand the pre-requisites better and what is the next higher learning for my students which will facilitate the prior knowledge I can tap on and what level of scaffolding I need to be mindful of.’

Working with high ability students. ‘For our own knowledge and also improves our ability to help/challenge the higher ability students.’

Effective teachers. Two of the responses used the word ‘effective,’ but they were not further elaborated.

‘In order to be effective in teaching Primary Mathematics, I feel that it is important to have a firm foundation in content knowledge’

‘So that we can be more effective in teaching Maths’

1.2 Head

Making connections. Here, the responses illustrate how PTs associate deeper knowledge with connections.

‘You need a much deeper understanding to explain a certain concept as simple as possible to students and relate it to something they can connect to.’

‘A deeper understanding provides us the basis to help students to make connections and real world applications.’

There were also diverse reasons PTs have in viewing why a deeper understanding was necessary.

Understanding heuristics and problem solving. ‘Since primary school students learn heuristics which require deeper thinking, it may be useful to have a deeper understanding of mathematical concepts which may be useful in helping students to understand heuristics and problem solving.’

Understanding difficulties of the topic. ‘In order to teach, the teacher will need to know the topic and understand the possible difficulties of the topic to better help the students.’

1.3 Heart

Emotional preparedness. ‘The courses help me to understand and upgrade my content knowledge on Mathematics which made me feel prepared for practicum.’

General appreciation. ‘Helping students think like a mathematician, i.e. use abstract thinking, helps them appreciate better why they are learning it (not for exams).’

• 2. Gaining affective sensitivity to their perspective students

Another set of interesting responses showed that a few of the PTs viewed having deeper understanding as a means to understand their future students’ viewpoint. While the responses above have already hinted at some level of nesting by considering their future students in their responses, the below responses were explicit about it:

‘Similar to how we deliver according to the ZPD of our children, it should be of interest to us educators to be a +1 to be able to sustain interest.’

‘Great in helping me understand why students would struggle in understanding certain concepts.’

‘Helping students think like a mathematician, i.e. use abstract thinking, helps them appreciate better why they are learning it (not for exams).’

• 3. To understand formula or method

A few responses took an instrumental view of having a deeper mathematical knowledge, whereby they see it as a means to understand the formula or method. One would have expected that such responses to be overwhelming, but only three responded in this manner.

‘If we only know the steps, and if students were to ask how certain formulae or steps are derived, we will not be able to answer them.’

‘We learn better when we understand the concept behind certain formulas we have to use.’

‘So that I can understand how this method came about instead of just memorizing and doing the questions’

1.4 Other positive responses

There were a number of other responses that, while they could not be classified directly, gave us a hint on what the PTs thought about why a deeper understanding would be necessary.

‘Especially important for non-math majors like myself... one module right before final posting did not feel like it was sufficient at all.’

‘To be more equipped with subject content’

‘So that you would be equipped with more maths knowledge’

The above responses highlighted a possibility of how the PTs perceived themselves as requiring more subject knowledge to fill the inadequacy in their knowledge.

1.5 Negative responses

On the other hand, of the responses that explained why they do not find having a deeper understanding a necessary endeavour, they all pointed to the fact that the content was not taught in primary mathematics:

‘We will not be teaching it to our students in the future.’

‘Would rather have a course which focuses on primary level pedagogies’

‘Don’t see the relevance between how to teach well and knowing such knowledge’

1.6 Responses that are considered not meaningful

Here we will list a sample of the responses that are not considered to be directly relevant for completeness. While they may be useful to understand their sentiment (usually frustration), such responses do not directly respond to the question that was asked.

‘I can’t even do those questions on my own. I don’t even know how to solve them without help. How am I supposed to teach upper primary mathematics?’

‘It’s not the content, it’s the way the tests and exams are done. We are doing PGDE.’

‘I would rather have a course which focuses on primary level pedagogies; I found it difficult to catch up with the material on the course and the content seemed to be skewed towards people with a mathematics background.’

‘Define deeper. Sometimes, at different levels, there is no need to go into such a deep level that it confuses the trainee teacher who have no 'DEEP' mathematical background. Think engineering for example, some things can just be 'taught' without going through the grind... what are these things?’