Fraction Masters

A Chinese maths teacher’s approach towards cultivating metacognitive skills and mastery of fractions amongst 10- to 11-year olds. 

This semester I got the opportunity to observe a series of maths lessons given by a renowned maths teacher in Nanchang, China. With the detailed lesson data awaiting systematic analyses in a more academic fashion, I got the teacher’s permission to share with a wider audience some snapshots of the conclusive session on the unit entitled Definition and Properties of Fractions (Unit 4 in PEP, 2014a).

Mrs Q (pseudonym) often uses self-made animations with Flash and PowerPoint to represent mathematical concepts and interconnections between maths topics and between abstract knowledge and real world problems. Unfortunately, the projector in her class failed to work over the course of the last few lessons on the topic. Whilst the school was arranging someone to come over and fix the projector, she had to carry on teaching using the old-fashioned chalk and board without compromising the targeted lesson objectives. Since the total amount of time in a semester and the amount of maths content expected to be taught are both fixed, teachers in this country are very cautious about sticking to the timetable.

 

Prologue

This was the lesson before the last (a practicing session) on the unit. The purpose was apparent – reviewing the whole unit through thinking, discussing and making notes together. The teacher structured the lesson with an enormous amount of questions that were strongly mathematically interconnected.

Before the lesson started, Mrs Q tried to switch on the projector to see if there was any luck it would work, but it didn’t. Without a single hesitation, she announced that the class began. As in every classroom in China, the lesson started with the teacher and the class bowing to each other whilst exchanging greetings. The pupils bowed deeper to show their respect.

The first thing Mrs Q did was to ask the class, “What are the various sections of this unit we have just learnt?” The Class answered randomly, “Definition of Fractions (4.1). … Proper Fractions and Improper Fractions (4.2). … Basic Properties of Fractions (4.3). … Simplifying Fractions (4.4). … Common Denominators (4.5). … Mutual Conversions of Fractions and Decimals (4.6).”

The teacher then asked the class to talk about detailed aspects of knowledge that were covered in each section. She started with the first section and asked, “What have we learnt in the section, Definition of Fractions?” Pupils listed titles of each subsection in section 4.1, immediately after which the class became quiet. Mrs Q said, “Okay, what you have just summarised regarding the first section appeared to be fragmented, incomplete, and lacking details. This unit (Unit 4) contains the largest amount of knowledge amongst all units in this textbook (PEP 5B), so it is more difficult to grasp and therefore easier to make mistakes upon than other units. Thus, it is crucial for us to review it systematically. To give the knowledge a thorough summary, let’s make a revision outline for the unit.”

 

Whole-class Mind Maps Building

The remaining part of the lesson was structured by the teacher through whole-class Q and A interactions in which the entire class were brainstorming, discussing, drawing conclusions and writing notes together. Textbooks were referred to with specific page ranges, detailed content, typical examples discussed at the class level. Every time when the appropriate answer emerged from the class, the teacher would repeat (to approve the answer) what the pupils said whilst jotting down the corresponding notes on the board, knowing that everyone in the class would do the same thing (writing notes) simultaneously.

For some long statements, the teacher repeated (again, as an approval) what pupils had just said and then simply drew a long line to hold the space for the statements. It turned out to be a mutually understood convention that although the teacher omitted details of certain statements on the board, the pupils would never do so. They tried their best to write down every word in the notes. The duty of learning was apparently on the learners’ shoulder.

The teacher was there to guide/structure the learning. The guidance was systematic and well-prepared. Despite the guidance, children were the centre of the classroom throughout, since it was clearly their role to find all the answers to the questions arising in the class. This is what the Chinese educators termed the “teacher-guided and child-centred” approach (Miao and Reynolds, 2018, p.107). Everything seemed well anticipated beforehand, which might not be as easy as it appeared to be. Accurate anticipation of what can be achieved in the class requires a sound understanding of the maths content, the learners and optimal ways of maths teaching (Askew et al., 1997; Ball, 1988; Hill, Rowan & Ball, 2005; Shulman, 1986; Miao & Reynolds, 2018).

Typical problems were given at points for the pupils to solve, but the teacher erased them afterwards just to leave on the board the key revision notes (the structure of knowledge). Near the end of the lesson, the teacher had completed her notes on the board; so did the pupils with a more thorough version in their notebooks.

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Mrs Q’s notes on the board

 

20180515_pupil notes 1
Pupil notes example #1 (partial)
20180515_pupil notes 3
Pupil notes example #2

 

An English version of the notes translated by me:

Mind Map translation_Definition and Properties of Fractions

 

Click on the following link to see the PDF version with better resolution:

Mind Map translation_Definition and Properties of Fractions

 

Hand-Drawn Mind Maps

One might imagine a lesson of maths and any other core subjects in a primary classroom without multimedia these days would mean that pupils are totally switched off, since children in this digital era are so used to all sorts of screens/interfaces. This was however not the case in Mrs Q’s class where pupils were making every effort to think and understand deeply the mathematics underneath the surface of numbers, representations and symbols.

Chalk and board, projector and interactive whiteboard are just superficial tools if they are not utilised effectively to mediate the teaching and learning of the subject. What really matters is how teachers teach to develop and deepen children’s understanding of fundamental mathematics.

When the lesson ended, everybody became an expert on the topic, which was evident in their structured notes and diverse ways of representing interconnected knowledge in this area through pictures that they drew over the weekend.

Looking at their pictures, we could easily imagine how everyone of them animated in mind their understanding of the knowledge, its structure and the connectedness between all elements of it, and then managed to map their imaginations onto the paper with coloured pens/pencils/crayons.

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Pupil hand-drawn mind map example #1
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Pupil hand-drawn mind map example #2
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Pupil hand-drawn mind map example #3
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Pupil hand-drawn mind map example #4
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Pupil hand-drawn mind map example #5
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Pupil hand-drawn mind map example #6
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Pupil hand-drawn mind map example #7
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Pupil hand-drawn mind map example #8
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Pupil hand-drawn mind map example #9
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Pupil hand-drawn mind map example #10
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Pupil hand-drawn mind map example #11
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Pupil hand-drawn mind map example #12

 

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Rote or Mastery?

James Stigler (March, 2002) once said in his speech on Lessons from Japanese and Chinese Education at Harvard:

Everybody told me before I ever studied teaching in Japan that teaching in Japan was rote. So, I went and studied elementary schools in Japan, and it doesn’t look rote. Whatever it is, it’s not rote.

Without looking into details of this lesson by Mrs Q, one might simply call it rote, since everyone was just talking about concepts and jotting down notes. However, whatever it was, it was not rote! Children were thinking and reasoning actively and collectively about mathematics throughout the lesson. In answering questions from the teacher, they had to explain mathematical concepts, solve problems posed by the teacher, talk about their problem solving strategies, and build connections between different parts of knowledge. Metacognitive skills in mathematics were constantly practised and enhanced. At each stage of whole-class Q & A, they wrote fine-grained ideas and concepts carefully down in their notebooks. With a mastery of Definition and Properties of Fractions (Unit 4 in PEP, 2014a), pupils were able to grasp the knowledge very quickly over the course of two lessons for the next unit on this topic – Addition and Substation of Fractions (Unit 6, ibid.). There is no reason to doubt that they will be proficient at Multiplication and Division of Fractions (PEP, 2014b) in the coming semester, because they have already mastered the essence of fractions.

 

References

Askew, M., Rhodes, V., Brown, M., Wiliam, D., & Johnson, D. (1997). Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency. London: King’s College, University of London.

Ball, D. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation, Michigan State University.   

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American educational research journal, 42(2), 371-406. 

Miao, Z., & Reynolds, D. (2018). The effectiveness of mathematics teaching in primary schools: Lessons from England and China. London and New York: Routledge.

People’s Education Press (PEP). (2014a). Mathematics (5B) [in Chinese 数学 (五下) ]. Beijing: PEP.

People’s Education Press (PEP). (2014b). Mathematics (6A) [in Chinese 数学 (六上) ]. Beijing: PEP.

Shulman, L. (1986). Those who understand: knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Stigler, J. W. (March, 2002). Lessons from Japanese and Chinese education.   Retrieved from http://forum-network.org/lectures/lessons-from-japanese-and-chinese-education/

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