26 August 2019 Computational Thinking Mathematical Thinking Two

26 August 2019 Computational Thinking & Mathematical Thinking: Two Sides of the Same Coin? Dr. Sylvia van Borkulo

Example CT & MT 231 + 492 (1) (2) Freudenthal Institute (3) Scientific and mathematical literacy for life (4) www. freudenthalinstituut. nl (5)

start Line up the numbers, draw a line below From right to left, is there a vertical row of numbers you haven’t added yet? no yes Add the numbers Put the outcome below the added numbers Subtract 10 from the outcome and put this number below the added numbers Freudenthal Institute no Outcome >= 10? yes Put 1 above the column to the left Scientific and mathematical literacy for life www. freudenthalinstituut. nl end

Content presentation 1. Introduction research project 2. Preliminary results Literature study Experts’ opinions: Delphi study Interviews teachers Pilot study algorithmic thinking 3. Hands-on: brainstorm on ideas Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Content presentation 1. Introduction research project 2. Preliminary results Literature study Experts’ opinions: Delphi study Interviews teachers Pilot study algorithmic thinking 3. Hands-on: brainstorm on ideas Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Consortium Utrecht University Paul Drijvers Sylvia van Borkulo Radboud University Erik Barendsen Maria Kallia SLO Jos Tolboom Teachers: Carina van Amerongen, Wim Caspers, Juan Dominguez, John Val, Fetske Zwaga Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Introduction Research Project • • • Computational thinking in mathematics education Consortium of two universities and five schools Three-year project Funded by NRO Context: 11 th grade (17 -year-old pre-university students) Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Computational Thinking … is “in the air”, both at a national and at an international level, … but also is a “container concept”. Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

http: //curriculumvandetoekomst. slo. nl/21 e-eeuwse-vaardigheden/digitale-geletterdheid/computational-thinking/

s c h e r m v a n h e t l e e r p l a n v International Society for Technology in Education, https: //www. iste. org/standards/for-students

Definition Wing (2006) Computational thinking involves solving problems, designing systems, and understanding human behaviour, by drawing on the concepts fundamental to computer science. […] Thinking like a computer scientist means more than being able to program a computer. It requires thinking at multiple levels of abstraction. Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

CSTA & ISTE (2011): Operational Definition CT a) formulating problems in a way that enables us to use a computer and other tools to help solve them, b) logically organizing and analysing data, c) representing data through abstractions such as models and simulations, d) automating solutions through algorithmic thinking (a series of ordered steps), e) identifying, analysing, and implementing possible solutions with the goal of achieving the most efficient and effective combination of steps and resources, and f) generalizing and transferring this problem-solving process to a wide variety of problems. Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

CSTA & ISTE (2011): CT Attitudes a) b) c) d) e) Confidence in dealing with complexity; Persistence in working with difficult problems; Tolerance for ambiguity; The ability to deal with open-ended problems; The ability to communicate and work with others to achieve a common goal or solution. Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Brennan & Resnick (2012) • computational concepts • practices • perspectives Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Mathematical Thinking Tall (1991, 1995): mathematical activity is perceiving objects, thinking about them, and performing actions upon them Schoenfeld (1992): develop a mathematical view Argyle (2012): four views of mathematical thinking: 1. thinking about mathematics 2. thinking when doing mathematics 3. habits of mind 4. thinking mathematically Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Dutch Curriculum Reform Committee (c. TWO, 2013) Central mathematical thinking activities are: • Modeling and algebraizing • Ordering and structuring • Analytical thinking and problem solving • Manipulating formulae • Abstracting • Logical reasoning and proofing Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Drijvers, P. , Kodde-Buitenhuis, H. , & Doorman, M. (2019). Assessing mathematical thinking as part of curriculum reform in the Netherlands. Educational Studies in Mathematics. https: //doi. org/10. 1007/s 10649 -019 -09905 -7 Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Mathematical Thinking Counting Arithmetic Algebra Geometry Calculus Computational Thinking Problem Solving Modeling Analyzing and Interpreting Data Statistics and Probability Simulation Data Mining Networking Automated Data Collection Gaming Set Theory Algorithmic Reasoning Topology Robotics Programming Sneider et al. (2014) Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Research Question How can a teaching-learning strategy, focusing on the use of digital tools, support 16 -17 years old pre-university students in developing computational thinking skills related to mathematical thinking in pure and applied mathematics courses? Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Research Sub-Questions 1. What are common aspects of computational thinking and mathematical thinking? 2. How can these common aspects be promoted in learning activities in which 16 -17 years old preuniversity education students use digital tools in pure and applied mathematics courses? 3. Do such learning activities lead to learning gains with respect to computational and mathematical thinking? Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Aimed Results • • • Theory-informed, practice-oriented list of key elements of computational thinking related to mathematical thinking; Empirically validated learning activities for upper secondary pre-university education students; Instruments to assess the related learning outcomes; Teacher guide on learning activities targeting computational thinking and mathematical thinking using digital tools; Policy document to inform upcoming curriculum reform. Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Content presentation 1. Introduction research project 2. Preliminary results Literature study Experts’ opinions: Delphi study Interviews teachers Pilot study algorithmic thinking 3. Hands-on: brainstorm on ideas Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Literature Study • 6 databases (ACM, IEEE, Web of Sciences, ERIC, Scopus, and Psyc. INFO) • query: "computational thinking" AND mathematic* • 479 papers, 66 included in review • both empirical and theoretical papers Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Delphi Study 9 teachers, 16 researchers Questions: 1. What characterizes computational thinking in mathematics education? 2. What aspects of CT can be addressed in mathematics instruction? 3. What aspects do computational thinking and mathematical thinking have in common? 4. Digital tools or unplugged? 5. Is programming essential? Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

1. What characterizes computational thinking in mathematics education? Agreement on: • abstraction • decomposition • pattern recognition • algorithmic thinking • modelling • logical thinking Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

2. What aspects of CT can be addressed in mathematics instruction? Agreement on: • abstraction • decomposition • pattern recognition • algorithmic thinking • modelling • data analysis • data representation • automation • generalization • evaluation Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

3. What aspects do computational thinking and mathematical thinking have in common? Agreement on: • abstraction • decomposition • pattern recognition • algorithmic thinking • modelling • logical thinking • generalization • evaluation • problem analysis • analytical thinking Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

4. Digital tools or unplugged? A mix of both digital tools and unplugged activities is ideal in mathematics education: • Digital tools and unplugged activities are useful for developing computational thinking skills in mathematics education • Unplugged activities are useful for conceptual understanding Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

5. Is programming essential? Programming and the way of thinking it involves are useful in mathematics education (e. g. , as a skill for today’s society, debugging, visualization). Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Interviews teachers 7 teachers Semi-structured with questions about: - their ideas on CT aspects - relation CT and MT - practicing CT in their class - digital tools/programming Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Themes • • • Aspects from Delphi study are relevant and important. CT and MT are closely related. Examples are needed. Time is needed. Programming should be learned outside of math class. The exam standards should include CT. Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Some quotes 1. 2. 3. 4. "Everything is math" "CT requires more insight of students" "Both math and CT are formal thinking" "Programming is an enrichment of mathematics class, but should be a separate subject. " 5. "Not everyone is willing to change. If CT is a real part of the curriculum, everyone has an interest in doing CT. Then it will be more than your own hobby. " Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Pilot Study Algorithmic Thinking Master thesis project by Ivana den Hartogh Four lessons to develop algorithmic thinking - Unplugged - Flow charts - Excel, Java. Script, or graphing calculator Piloted in two school, three classes (15 -17 yr). Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Example task Check and possibly improve the following steps: Step 1: Take strawberry from box Step 2: Wash strawberry Step 3: Remove top strawberry Step 4: Put pieces back in box Step 5: Check for untopped strawberries in box Step 6: If yes, repeat step 2. Step 7: If no, ready Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

start Take strawberry from box Wash Remove top yes Put back in box Untopped strawberry in box? no end Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Example Excel Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Example Graphing Calculator Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Results Pilot Study Algorithmic Thinking Students showed: • • Ability to read and execute an algorithm Ability to specify a problem precisely Ability to construct an algorithm Ability to understand improve an algorithm Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Content presentation 1. Introduction research project 2. Preliminary results Literature study Experts’ opinions: Delphi study Interviews teachers Pilot study algorithmic thinking 3. Hands-on: brainstorm on ideas Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Hands-on Brainstorm in small groups • • How can CT be practised in math? What are possible topics? Can you think of a concrete example? What are requirements/restrictions? What are difficulties or opportunities? Are digital tools necessary? Is programming necessary? Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl

Conclusion • Two sides of the same coin? … Maybe! • Further research is needed • Hopefully our research will clarify relation CT & MT Freudenthal Institute Scientific and mathematical literacy for life www. freudenthalinstituut. nl