What can mathematics education contribute to preparing students for our future society? Michiel Doorman The Global Challenges on The Development and The Education of Mathematics and Science

Michiel Doorman

Topics  IBL in science and mathematics education (the PRIMAS-project)  IBL and workplace contexts (the mascil-project)  The relation between IBL and RME  Discussion 3

Inquiry based learning (IBL)  Extending the repertoire of teachers towards IBL in daily classroom practice  PD modules containing: a session guide, handouts for teachers, sample classroom materials and suggested lesson plans,  And video sequences showing teachers trying these materials with their own classes 4

What is IBL? 5

Why IBL?  Students contribute actively to the learning process  Their role is related to phases of research (modelling) (questioning, planning, reflecting, …)  Students learn better, create ownership, learn research-related skills  Teachers have a better view on their capacities and can built on their ideas 6

PRIMAS: Promoting Michiel Doorman Fisme-IPN, 1/11/11

Aim of Primas (01/2010 – 12/2013)  A widespread uptake of inquiry-based learning in day-to-day mathematics and science lessons across Europe  Inquiry Based Learning approaches aim at fostering inquiring minds and attitudes in our students.

HOW DO WE TRY TO ACHIEVE THIS AIM?

Dissemination-plan 1. Classroom materials and their national adaptation 2. Professional development (long-term) 3. Information for teachers in one-day-events 4. Initial teacher training 5. Analyzing contexts and involving policy 6. Evaluation (to optimize strategies)

The focus of Primas  Extending the repertoire of teachers towards IBL in daily classroom practice  PD modules containing: a session guide, handouts for teachers, sample classroom materials and suggested lesson plans,  And video sequences showing teachers trying these materials with their own classes  PD Module 1: Student-led inquiry  PD Module 2: Tackling unstructured problems  PD Module 3: Learning concepts through IBL  PD Module 4: Asking questions that promote reasoning  PD Module 5: Students working collaboratively

One module in detail: 1. Student-led inquiry  Experience inquiry  Watch a lesson  Discuss teacher behavior and learning environment  What is new: from structured problems to structured lesson plans  Prepare a lesson

2. Tackling Unstructured Problems

4. Asking questions  Five principles for effective questioning: 1. Plan to use questions that encourage thinking and reasoning 2. Ask questions in ways that include everyone 3. Give students time to think 4. Avoid judging students' responses 5. Follow up students' responses in ways that encourage deeper thinking  Watch video of a lesson

5. Students Working Collaboratively

Experiences in NL

Teacher examples Add to get the nextMultiply to get the next “The students became owner of the mathematics”

Teacher examples “The new and open task triggered curiosity and inquiry. In one class it went much better than in the other. I don’t know why.”

Teacher examples “They didn’t find the formula for lenses themselves. But in the traditional setting that also doesn’t happen. They understood better why and how they operated the instrument. In my normal classes it is maybe too safe. In this way they are more challenged and have to think for themselves.”

Teacher examples Contrasting the science & math examples was helpfull for discussing: Student learning Ownership of main question Ownership of solution procedure Explicit attention for IBL processes Role of the teacher Importance of a lesson plan Role of classroom discussions

Review study [Kirschner e.a. (2006). Educational Psychologist]  Why inquiry-based teaching failed 1. Direct instruction and repeated practice effect long term memory 2. In “pure-discovery method’s” students get frustrated and mislead 3. IBL only works when students are educated and motivated enough to guide themselves  Respond: IBL works, when enough support is available 1. IBL ≠ ‘minimal guided instruction’ 2. Support and scaffolding is needed for:  Students’ development of discipline-related knowledge and strategies  Transfering expert knowledge  Structuring complex and open activities 3. Offer support with structured lesson plans, process support, worksheets,...

Island problem

The problem is the gap between the island of formal mathematics and the mainland of real human experience (Kaput, 1994) "Are all functions encountered in real life given by closed algebraic formulas? Are any?"

Mathematics education needs to contribute to preparing students for our future society Why? Become problem solvers in a quickly changing society Learn to deal with missing or superfluous data Learn to be mathematical creative Learn to (critically) reflect on the results Learn to communicate mathematical results … Giving students a sense of purpose and utility (<> mathematics as an isolated discipline)

Mathematics education needs to contribute to preparing students for our future society How? Include open problems and rich contexts that evoke inquiry Exploit students´ inquiry for learning mathematics

A traditional mathematics textbook task

 The task provides the main problem and exactly the information needed to solve it  The task asks for using a formula Neglect the context (e.g. dimensions) Try to find the numbers needed for an equation in x  The task does not support applying or learning to apply mathematics outside the mathematics classroom

Alternative tasks  The tasks provide a situation. Students need to identify the problem(s) that need to be solved and to collect the information needed  The tasks support applying and learning to apply mathematics outside the mathematics classroom

An example The parking problem

The try-out of the task by an experienced teacher  The task was used by an experienced teacher  We discussed in advance the structure and aim of the task  She adapted the lesson plan for her context

The lesson plan a 60 minutes lesson 5 minutesCreate groups of 4 and introduce the problem 10 minutes Students work in groups and identify what further information is needed 5 minutesWhole class discussion to collect questions, share ideas, distribute responsibilities and give feedback on final product 25 minutes Students continue and finalize designs and underpinning 10 minutes Presentations of (some of the) groups 5 minutesReflection

Notes during whole class discussion

Final products

Teacher’s noticing  Students worked on the task seriously and with enthusiasm  You need to be able to switch between group work and whole class discussions  Difficult to emphasize the mathematics (working with scale, being precise, …)  Difficult to evaluate the mathematics in the final product (process, creativity, underpinning were easier to judge)

Teacher’s noticing: students need scaffolds  Emphasize the importance of adopting a structured lesson plan structure out of the task -> structure into the lesson plan  Use meta-cognitive prompts ( Wijaya, 2015 ) Rephrase problems, known and unknown, … underline information reflect

Characteristics of open problems and rich contexts that evoke inquiry  Evoking inquiry: Students need to plan inquiry  The task does not structure the solution procedure  Scaffolds are needed (e.g. lesson plan, prompts, …) Students need to collaborate and communicate  Creating sense of purpose & utility: Contexts relate to a (workplace) practice Tasks set students in a professional role Tasks ask for a product

Open problems and rich contexts that evoke inquiry  Using open problems in rich contexts…  Is that RME?

Realistic Mathematics Education  Freudenthal: anti-didactical inversion = endpoint of the work of mathematicians (e.g. set theory as organizing tool) is used as a starting point for instruction  Alternative: mathematics as an activity - organizing subject matter from reality - organizing mathematical subject matter

Realistic Mathematics Education constructivist approach to learning mechanistic approach to learning target applications (rich contextual problems) sourcetarget applications

“Rather than beginning with abstractions or definitions to be applied later, one must start with rich contexts that ask for mathematical organization; or, in other words, one must start with contexts that can be mathematized” Hans Freudenthal (1905 – 1990)

non-routine problems and rich contexts support the learning of mathematics Driving activity: mathematizing

Students reinvent mathematics by mathematizing Organizing subject matter from reality (horizontal mathematizing) Organizing their own mathematical activity (vertical mathematizing) Freudenthal on learning mathematics

Mathematizing ‘Realistic’ context Mathematical model Mathematical objects, structures, methods Horizontal mathematizing Translate Vertical mathematizing Abstract

Contexts that evoke the need to grasp change and that support reasoning with intervals

Modelling motion

Exponential growth Growth factor 2

 Growth(2) is growth after 2 days: 4 times as much  Growth(4) is growth after 4 days: … times as much  Days(8) is number of days to get 8 times as much: 3 days Growth(4) = Days(64) = Days(10) =

Explain:  D(50) + 1 = D(100) “the number of days to get 50 times as much plus one day in which the amount doubles equals the number of days to get 100 times as much”  D(32) = D(16) + D(2)  D(24) – 3 = D(24/8) = D(3)  Growth(2) is growth after 2 days: 4 times as much  Growth(4) is growth after 4 days: … times as much  Days(8) is number of days to get 8 times as much: 3 days

Voronoi diagrams  Five wells in a desert. Imagine you and your herd of sheep are standing at J. You are very thirsty and you have this map. To which well would you go for water?  Color the region of positions that all have well 2 as the closest place to find water.

Voronoi diagrams

Questions  Which possibilities with 4 centers?  What are the diagrams with the following centers?  Explore: ?

Mathematizing ‘Realistic’ context Mathematical model Mathematical objects, structures, methods Horizontal mathematizing Translate Vertical mathematizing Abstract

Mathematizing ‘Realistic’ context Mathematical model Mathematical objects, structures, methods Horizontal mathematizing Translate Vertical mathematizing Abstract Log(ab) = Log a + Log b D(50) + 1 = D(100)

Mathematizing ‘Realistic’ context Mathematical model Mathematical objects, structures, methods Horizontal mathematizing Translate Vertical mathematizing Abstract Theorems on perpendicular bisectors, triangles, circles, …

Characteristics of open problems and rich contexts that evoke inquiry  Evoking inquiry: Students need to plan inquiry  The task does not structure the solution procedure  Scaffolds are needed (e.g. lesson plan, prompts, …) Students need to collaborate and communicate  Creating sense of purpose & utility: Contexts relate to a (workplace) practice Tasks set students in a professional role Tasks ask for a product

Characteristics of open problems and rich contexts for learning  Non-routine problems in rich contexts evoke informal representations Horizontal mathematizing  Tasks and teacher change focus from informal context-related strategies to formal mathematics Vertical mathematizing  Introduce models to support level raising

Thank you Michiel Doorman Freudenthal Institute