Dobbs Ferry School District

The Dobbs Ferry School District consists of three schools: Dobbs Ferry High School, Dobbs Ferry Middle School and Springhurst Elementary School. It is located in the quiet hamlet of Dobbs Ferry, NY, just 20 miles north of New York City. I visited Springhurst for a day and taught three classes, made a presentation on mathematics teaching to the teachers, and talked about the math program in the school to the parents.

Grade 5 Lesson

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Scarsdale Teachers Institute

STI Course A Closer Look at Singapore Math by Yeap Ban Har

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This set of slides document the content of the weekend course on the use of anchor problems in enhancing teaching and learning of mathematics.

SingaporeMath in Action

Based on teacher education program in Singapore, about 90 Filipino educators made up the first class of Singapore Math in Action. Slides from four sessions are available here. The next Singapore Math in Action is scheduled for 24-25 September 2010.

For participants of the first class, these are the readings that you should complete to deepen your learning:

Read http://www.simplypsychology.pwp.blueyonder.co.uk/bruner.html for Bruner`s ideas
Read http://en.wikipedia.org/wiki/Jerome_Bruner also for Bruner. There is a link to　Spiral Curriculum
Read http://www.skemp.org.uk/ for Skemp`s idea on understanding. This is his website and the paper on relational and instrumental understanding is found under the section titled Papers.
Read http://www.zoltandienes.com/ for Dienes` six-stage theory. And try to locate some reading, books or on-line, about perceptual and mathematical variability.

For participants who want to have a documentation of their learning beyond the certificate each has received, you are invited to submit something that you have tried out in your own classrooms based on Singapore Math in Action that you have attended. Details are in the e-mail sent to all participants of the course.

SingaporeMath in Action 1

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Questions posted at the seminar are found at www.askbanhar.blogspot.com

SingaporeMath in Action 2

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This session is on mathematical problem solving.

SingaporeMath in Action 3

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This session is on all things related to fractions.

SingaporeMath in Action 4

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Ateneo de Manila Summer Institute

This is the third year I have been here to teach the summer institute. This year I am teaching the middle school program where the teachers are teaching Grades 4 to 7. We began after a brief opening ceremony.

In the opening session, I tried to discuss Bruner's enactive, iconic and symbolic representations using division of whole numbers by fractions and long division algorithms. The teachers asked questions related to teaching of division of fractions by fractions, multiplication of decimals and operations involving negative integers. We ended the day with a game of 'either one or three'- in a game where two players take turn to remove exactly one or three paper clips from apile of clips and the winner is the person who remove the last clip(s) - what is the winning strategy?

What if the rule is changes to wither 'one or two'?

We begin Day Two with Dienes' Theory of Variation ... but not before I pose the class the Fido Problem. Tell you more later.

Problem-Based Approach

BBS April 2010 Singapore Math in Indonesia by BBS Maths Consultant Dr Yeap Ban Har

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The open-ended approach common in the Japanese mathematics classroom uses a single problem in a mathematics lesson. In training teachers at Bina Bangsa School - a family of schools in Indonesia that uses the Singapore curriculum - I introduced the idea of teaching new concepts, doing drill-and-practice as well as getting students to apply what they learn using a well-selected or well-crafted problem. Other than workshops and lectures, teachers also get to see such problem-based lessons - there was one on learning area (primary three) and one applying Pythagoras theorem to find distance bewteen points (secondary two). Teachers saw examples, they worked through afew such problems, I modelled a few of such lessons with them and one with students,they worked with other colleagues to design one such lesson which hopefully they will try out and see how studenys respond to the lesson).

In designing such lesson, teachers select a problem and solve the problem themselves to understand the processes and challenges involved, as well as to see the mathematics in the problem. They decide if the problem can be used to introduce a topic or to provide drills-and-practice or to provide opportunities for students to apply what they have learnt.

They also considered how students will solve the problem and how to use the solutions to help students construct knowledge.

Fraction Addition: Dienes' Theory of Mathematical Variability

In the final lesson with my first year diploma students (they will go on to read at least two more mathematics education courses as part of their pre-service training), we had a chance to review the test they took and key learning theories.

I used examples from fraction additions to discuss theory of variability (Zoltan Dienes). How is 2/5 + 2/5 different from 2/3 + 2/3? Both are symbolic representation of fraction additions. But mathematically they are not the same. Dienes refer to this as mathematical variability. In 2/3 + 2/3, students need to have the knowledge of mixed numbers and improper fractions. Such knowledge is not needed in doing 2/5 + 2/5. These two tasks possess mathematical variability.

In Singapore, these are taught in different grade levels. 2/5 + 2/5 is taught in Primary 2 while 2/3 + 2/3 are taught in Primary 4.

How about 1/8 + 1/8? This is taught in Primary 3. Why is addition of like fraction (2/5 + 2/5, 1/8 + 1/8 and 2/3 + 2/3) are considered not the same and are taught in different grade levels? According to Dienes, there is mathematical variability among the tasks. The Singapore approach seems to suggest the need for students to be systematically introduced to each variation.

In 1/8 + 1/8, students need to know equivalent fractions. Mathematically the three tasks are not the same and this is mathematical variability that Dienes wrote about.

Is 5/8 + 5/8 a variation of the three tasks given above?

Fractions: Equal Parts

The mathematics education course focuses on curriculum structure, teaching strategies including ICT and students' responses (errors). For example in the Diploma programme, the first course (of three or four) includes Teaching of Whole Numbers, Teaching of Fractions, Decimals and Percent, Teaching of Ratio and Teaching of Rate and Speed.

Student teachers also learn the Singapore curriculum framework, learning theories, scheme of work and lesson planning, Ministry of Education initiativces including the role of calculators and ICT.

In a lesson on Teaching of Fractions, teachers were presented with this problem - is a rectangle cut into four equal parts by its two diagonals?

There were student teachers who said that the parts are not all equal because the triangles are not 'the same' meaning not congruent. Subsequent whole-group discussion with paper rectangles and a pair of scissors led to four responses that are shown at the top of this page.

There was a response that suggested cutting each the the four parts into two triangles which are equal parts as they are congruent triangles. Each of the four parts is then clearly 2 eighths. See Photograph 1.

A second response is based on using the formula to calculate the area of triangle. On Photograph 2, it can be seen that B = 2h and b = 2H. The area of the two non-conguent triangles can be shown to be the same.

Another response includes cutting each of the four parts into two congruent triangles and rearranging the pieces to form congruent triangles. Thus, the triangle shaded black and the one shaded red in Photograph 3 can be rearranged to form congruent triangles.

Finally there was another response that is based on using two congruent triangles to form a rhombus. This resulted in two congruent rhombii. If the two rhombii are equal, it follows that half of one rhombus (one of the triangles) is equal to half of the other (the other triangle).

The student teachers were told that in mathematics classrooms, teachers use different models to show fractions. In the example, we use the area model - where we use area to represent fractions. There is also the length model (e.g. the bar model or line model) where length is used to represent fractions. Other models include the volume model (e.g. using a cylinder of water to show fractions).

In terms of curriculum structure, we learn that equal parts is an important concept in understanding of the fraction notation. In terms of common misconceptions, we see that some students may have the wrong idea that equal parts refer to congruent parts rather than parts with equal area.

In Singapore curriculum, fraction is introduced formally in Primary 2.