UTRECHT SUMMER SCHOOL- August 2008

The Utrecht Summer school is a program organized by Freudenthal Institute for Science and Mathematics Education (FIsme) in the Netherlands. The aim of this program includes bringing participants up-to-date in curriculum development and research in the field of science and mathematics education. It also includes refreshing and deepening the knowledge in core subjects of various fields of Science and mathematics. (For more information of the Utrecht summer school, see http://www.utrechtsummerschool.nl/ ) .
In the 2008 Summer School it was really a pleasure for me and colleagues from different part of the world (mainly Asia, Africa and Europe) to discuss the principles of Realistic mathematics Education (RME) to think about ways in which students can learn mathematics meaningfully. For pictures of the Summer School 2008, see: http://www.fi.uu.nl/en/summerschool/images/index.html. In this page i post summaries of activities of various workshops, presentations and interesting concepts in mathematics teaching particularly for the primary and secondary schools.

REALISTIC MATHEMATICS EDUCATION (RME): What is it?

Lesson on limits

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KNOWLEDGE CAN NOT BE TRANSMITTED, KNOWLEDGE CAN ONLY BE OBTAINED IN AN ACTIVE WAY…

RME: An approach towards the learning and teaching of mathematics developed by the Freudenthal Institute, Utrecht University, the Netherlands(www.fi.uu.nl).

Started around 1970
* Not affected by “New Math”-movement
* Freudenthal said: “New math” uses anti-didactical inversion: the endpoint of the work of mathematicians (e.g. set theory as organizing tool) is used as a starting point for instruction.
* Alternative: mathematics as a human activity:
- organizing subject matter from reality
- ‘guided’ opportunity to ‘re-invent’ by doing
* Focus: on math as an activity, on the process of mathematization, not on math as a closed system . You can find examples in the clip following.

WHAT RME IS?

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TEACHING STRATEGIES

Teaching-strategies should involve:

- COMMON SENSE

- CONNECTIONS
Do not introduce new formula’s were they are not needed! Relate to the students’ common sense and to what they know already; use what they know rather than adding a new formula to their knowledge…..

- CONTEXTS/APPLICATIONS/EXPLORATIONS
Start a new topic with preferably a context in which the mathematical topics get a meaning. Give time for exploration!

- E-C-R: Estimation – Calculation – Reflection
A very useful teaching-technique, that can be used for a lot of topics. By using it, a ‘feeling’ for numbers and measurement will be developed.
If this technique is used all the time, a student will know that something went wrong if he wrote:
47.01 - 0.65 = 22.1, or 6.25 – 4 = 6.21 . Some examples in "Distance between two points" and "Angles" can be found here and here.

QUESTIONING-TECHNIQUES TO ENCOURAGE REFLECTING ON AN ACTIVITY:
To encourage reflection on activities done by students, there are three ways you can use being the teacher:

1) by asking different kinds of questions:
(i) reflective questions: “what happened here?”
(ii) predicitve questions: “what happens next?”
(iii) open questions: “can you do this in more ways, you think?”
(iv) mirroring questions: “So if I understand you correctly, I hear you say: ….. Is that what you mean?”

2) by having the students report orally: the teacher requires the students to report back from what they have done. The students then think about their activity for the best way to talk about it. (And you will still use your questions during their reports!)

3) by having the students report in writing: the teachers directs the students to record things that come up from questions. The students draw and write (using their own words) in their notebook what they think has been discovered .

You use these types of questions to help the students structurize their way of working in a way that is understandable for more people. So an actual situation would be that students are working on assignments in class, you are walking around and you want to know what the students are doing, or how they are thinking; you want them to reflect on their thinking.

VERY GOOD TO REALIZE:
If you are really interested in what your students are doing, you will ask ‘the right’ questions without realizing what kind of questions they are!

QUESTIONING-TECHNIQUES TO ENCOURAGE REFLECTION ON AN ANSWER:
WHY do you think that is the answer? HOW did you come to that answer?
Actual situations in which these types of questions accour is when you are interactive with your students, discussing exercises, or questions about a (new) topic, etc.

RECOGNIZING DIFFERENT TYPES OF QUESTIONS
(f.i. in textbooks)

Low-level questions:
one-level questions, all information needed is in the question, there is only one way to solve it.

Example:
Problem 1: “We went for a ride in the car. We drove 231 miles and had to fill up at the petrol station. The tank took 14.3 gallons. How many miles per gallon did we get?”

High-level questions:
More steps have to be taken before you can find the answer, you can probably derive all information from the given but it is possible that you have to do some reasoning before you have that information, there are different strategies to get to the answer.

Example:
Problem 2: “We have driven 2/3 of the journey and the tank is still ¼ full. Do we have a problem if we don’t fill up?”

There is also the difference in WHAT-HOW-WHY-questions.
WHAT (WHEN, WHERE) – questions are called “fact-questions”
HOW-questions are called “partly fact, partly thinking-questions”
WHY (WHAT IF) - questions are called “thinking-questions”

In mathemaitcs-textbooks however, these words are not used all the time, so you should really take a look at what kind of activity is asked from the student in answering the question.

MIS CONCEPTIONS

In the post below i present some common examples teachers experience in mathematics lesson as were discussed in this summer school.

Some misconceptins in maths teaching

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For more information on how you can participate in this school in the coming years you can contact:
Jaap den Hertog

Coordinator Summer School

Utrecht University
Freudenthal Institute
for Science and Mathematics Education
Email: jaapdh@fi.uu.nl