電子科技與 建構主義式數學解難

電子科技與

建構主義式數學解難

科技變革還是教學法變革？(3 of 3) May 12, 2021

Introductory Problems for Discussion

Problem 1: Is 643 a prime or composite number?

Problem 2: What is the 20

term of this sequence? 3, 9, 21, 45…

Problem 3: Draw the fifth level of the Sierpinski Triangle.

Recap: Internationalisation

“Internal reconstruction of an external operation”…

Construction of individual knowledge as generated by socially shared experiences

◦ As children discuss events/objects with a “more knowledgeable other”, they begin to incorporate this talk into their own thinking

◦ Internalization – process through which social activities evolve into internal mental activities (ex: self-talk to inner speech)

◦ Discussions, debates, arguments teach children that there are multiple ways to see same situation; process becomes

internalized

◦ Implications for teaching: facilitate tool-based math talk

3

Questioning with DGEs

• “what do you notice about…?”

• “what happens when (you drag)…?”

• “predict what will happen…”

• “explain why…”

電子科技與

建構主義式數學解難

Boundary crossing and

“learning as making”

STEM Integration?

Boundary crossing in STEM education Leung (2019, 2020)

◦ Instead of seeing a boundary as an obstacle, it

should be viewed as a potential for learning since a boundary contains common concerns on both sides.

◦ Boundary objects have different meanings in

different social worlds but their structure is common enough to more than one world to make them

recognizable, a means of translation.

◦ Leung (2020) proposes a boundary crossing

pedagogy in STEM education, featuring problem- solving, inquiry-based learning, and modelling among other boundary objects.

Boundary Object 1: Making

在「造」中學（Learning-by-Making）的過程中，亦包含了Dewey（1897）提出的「做」中學

（Learning-by-Doing）的元素。後者雖缼少了創造、構建和個人化的元素，但兩者也涵蓋了學生

動手、動腦等通過肢體協調表達和運用知識解決現實問題的過程，有助增進學習者的學習經驗 (Papert & Harel, 1991) 。而「造」的成果除了是其作品本身外，更是學生掌握和學習知識。

Learning-by-Making ++創造、構建和個人化

「造」

「做」

動手、動腦

Papert (1980):

Logo & Constructionism

“The essence of Piaget was how much learning occur without being planned or organized by teachers or schools. His whole point was that children develop intellectually

without being taught!”

Papert (1980):

Logo & Constructionism

“The essence of Piaget was how much learning occur without being planned or organized by teachers or schools. His whole point was that children develop intellectually

without being taught!”

Papert’s famous metaphors:

◦ Pencil lab vs computer lab

◦ Computer as mudpie or material

◦ Our schools: a modern surgeon visiting a hospital 100 years ago

From Logo to 3D CAD

Making with 3D CaD in Primary Schools

3D CAD in

Junior Secondary

A “3D Keychain” Project in junior secondary mathematics classroom:

• “Objective: Design a keychain with your name on it and have it printed in 3D; consider the time and cost of printing your project.”

• “Due to the time and cost for 3D printing, you must calculate the volume of your project

precisely and submit your calculation […]”

• “The volume of the entire 3D print must not exceed 2500mm³.”

• 3D CAD used: https://www.tinkercad.com/

“I felt like an architect designing a building”

Aspects of engineering design cycle and communication of design decisions

“After calculating the volume of my design, I realized that it was greatly exceeding the limit so I had to resize and recalculate the volume”

“I had to fix my design because […] it was exceeding the maximum volume so I had to make the whole keychain smaller and flatter.”

“I tried several designs […] then I came up with the idea of playing with depth”

“One problem was that I forgot about the 2500mm³so I had to lower my shapes height after I calculated the volume.”

“[…] to stay under the limit of 2500mm³. This taught me to remove items that were not needed and to resize objects.”

“I switched the sizing around 4 times just so I could make it the size I wanted. I kept on making each block smaller to achieve the size that I needed.”

“When you realise that it is equivalent to 2.5cm³ you realise that you have barely any room to work in.”

“Some challenges I had with this project have been to make it not too thin or small that it’ll break easily.”

3D CAD in

Junior Secondary

Balloon-Powered Cars with 3D CAD

Recall: Embodied making/learning with “3D Pens”

Artefact construction Inquiry-

based learning

tools for New thinking

Constructionist learning with “3D Pens”

Artefact construction Inquiry-

based learning

tools for New thinking

◦ Artefact construction

◦ Active learning through hands-on production

◦ Personal, purposeful, and flexible

◦ Low entry, high ceiling tasks

◦ Inquiry-based learning

◦ Experimentation and modelling

◦ Inductive reasoning and generalization

◦ Engineering design cycle

◦ New tools for thinking

◦ Embodied and material interactions with concepts

◦ Artefacts as tools

電子科技與

建構主義式數學解難

Problem-based Digital Making

“What mathematics do we teach when computers do all mathematics?”

◦ Wolfram (2010): When “machines… do all calculations”, the

mathematics that we need to be doing in the real world involves…

1. Recognizing where mathematics is applicable

2. Translating practical problems into mathematical problems 3. Solving the mathematical problems

4. Interpreting and evaluating the outcomes

Modelling Cycle (Blum & Leiss)

Modelling cycle: Example

Modelling cycle: Example

y=1.18x+5.3 100=1.18x+5.3

x≈80

9:02+0:80=10:22PM

What does 1.18 and 5.3 mean in the context of the phone-charging scenario?

Boundary Object 2:

Computational Thinking (CT)

◦ CT is the thought process involved in formulating a problem and expressing its solutions in such a way that a computer can effectively carry out (Wing, 2006);

◦ Computational tools offer students a context in which students can reify abstract constructs and explore and model with mathematics concepts in a dynamic way (Wing, 2008).

◦ As Papert (1980) argued, “computer presence could contribute to mental processes not only instrumentally but in more essential, conceptual ways, influencing how people think even when they are far removed from physical contact with a computer” (p. 4).

◦ Strong connection with STEM, especially in mathematics.

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Computational (mathematical) concepts (Kong, 2019; Brennan & Resnick, 2012)

Computational (mathematical) concepts Sequences and iterations:

identify and repeat a series of steps for a task Boolean logic:

support for mathematical and logical expressions Conditionals:

make decisions based on conditions Events/Functions:

one thing causing another to happen Variables:

name variables descriptively to make them distinguishable from each other

https://scratch.mit.edu/projects/528053195/

1. Move the cat 2. Use variables

Example in Scratch

Boundary Object 3:

Problem Solving

◦ Transdisciplinary Problem Solving in STEM

◦ Engaging K–12 students with real-life phenomena and problems.

◦ Understanding that real-life problems are open-ended, ill-structured and lack simple solutions reflects the

global, 21st-century competence of a problem solver (OECD, 2018).

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Form of

integration Features

1. Disciplinary Concepts and skills are learned separately in each discipline.

2. Multidisciplinary Concepts and skills are learned

separately in each discipline but within a common theme.

3. Interdisciplinary Closely linked concepts and skills are learned from 2+ disciplines with the aim of deepening knowledge and skills.

4. Transdisciplinary Knowledge and skills learned from 2+

disciplines are applied to real-world problems and projects, thus helping to shape the learning experience.

Computational Problem Solving

◦ CT is a powerful cognitive tool for problem-solving across all spectra of human inquiry.

◦ Defining CT for Maths and Science Classrooms (Weintrop, et al., 2015)

◦ Ten core computational thinking skills (see also, Polya, 1945):

Computational Problem Solving Practices (Kong, 2019; Brennan & Resnick, 2012)

Computational problem-solving practices Tinkering and modelling:

use tools and representations to explore concepts creatively Algorithmic thinking:

articulate a problem’s solution in well-defined rules and steps Abstracting and generalizing:

see a problem at different levels of detail Testing and debugging:

ensure that things work; find and solve problems when they arise Remixing:

make something by building on existing projects or ideas

Revisting Problems for Discussion

Problem 1: Is 643 a prime or composite number?

Problem 2: What is the 20

term of this sequence? 3, 9, 2, 45…

Problem 3: Draw the fifth level of the Sierpinski triangle.

4. Detect composite or prime; display all the factors.

3. Detect composite or prime; # of factors if composite 1. Detect if composite

2. Detect composite or prime

https://scratch.mit.edu/projects/528068550/

https://scratch.mit.edu/projects/528072708/

https://scratch.mit.edu/projects/528079252/

https://scratch.mit.edu/projects/528082348/

Problem 1: Is 643 a prime or composite?

Problem 1 (Related):

The prime or composite game

See also:

Kong, S.-C. (2019). Learning Composite and Prime

Numbers Through Developing an App: An Example of

Computational Thinking

Development Through Primary Mathematics Learning. doi:

10.1007/978-981-13-6528-7_9

A202_Carrie and Ken's balance A203_Ken and Carrie problem (II)

Problem 2:

Number sequence of 3, 9, 21, 45, …

Problem 2:

Two approaches to modelling

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Model the balance (B) after n days in the form of a geometric series: B = d¹ + d² +d³ + …, where dⁿ=3(2^n-1), i.e. 3, 6, 12, …

Problem 2 (Related):

Fibonacci Machine

Problem 3: Sierpenski Triangle

Problem 3 (Related): Fractal tree

https://scratch.mit.edu/projects/528110669 https://scratch.mit.edu/projects/493292640/

Others: Base 2 and 8 Number Generator

Discussion

◦ From CT to algebraic thinking

◦ Students dealt with arithmetic sequences, and geometric sequences and series, without using any algebra, instead using block-based programming codes of as little as one line:

‘[set deposit to][deposit + 222]’ (Ng & Cui, 2020).

◦ Interesting interplay between CT and mathematical thinking (Cui & Ng, 2021)

◦ CT can serve as a bridge for children to advance their arithmetic thinking toward building coherent and meaningful learning of more advanced algebra in later years.

◦ Action – Process – Object – Scheme (Dubinsky, 1991)

◦ Evidence for integrating and leveraging computer science to solve problems in other disciplines (e.g., mathematics).

◦ mathematics content and problem-solving practices can be purposefully synergized in a programming context.

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Problem-based digital making:

computationally-enhanced mathematical problem solving

Recap

◦ Boundary crossing in STEM education:

1. Making

2. Computational thinking

3. Problem Solving

電子科技與

建構主義式數學解難

Problem-based Digital Making:

Evidence-based Practice

數學解難中的「造中學」：

預視香港中小學中融合計算思維的數學課程

Funded by the General Research Fund, Research Grants Council (Hong Kong) Ref No. 14603720

Principal Investigator: Dr. Oi-Lam Ng, Department of Curriculum and Instruction

The Chinese University of Hong Kong

Mathematical Problem Solving through Digital Making:

Envisioning a Computationally Enhanced Mathematics

Curriculum in Hong Kong's Primary and Secondary Schools

INNOVATIONS

1. Desmos to engage students and collect data

◦ Facilitate reflection and self-paced learning

INNOVATIONS

2. Use of videos to present mathematical problems

Move away from paper-and-pencil problem solving

3. Low-floor, high-ceiling tasks

Use of extension problems to differentiate learning

Lesson 1:

Algebra and Iteration

Lesson 2:

Counting and Probability

Lesson 3:

Fractal Geometry Maths Arithmetic sequence;

geometric sequence and series

Counting and List,

Experimental & theoretical probability

Fractal Geometry Programming Variables and Iteration Nested loops Recursion

Integration of

PROGRAMMING and

MATHS

Research objectives

To study students’ development of computer science concepts upon engaging in a series of problem-based dM activities.

To examine students’ computational problem-solving practices, mathematical thinking, and any CT- and mathematics-related challenges that emerge during problem-based dM activities.

To observe the impact of CT on students’ perspectives about themselves as computational thinkers and problem solvers, and about the role of programming in various facets of life.

To develop evidence-based accounts of implementing a computationally enhanced mathematics curriculum in formal education settings and a possible learning trajectory.

We are recruiting participating schools!

THANK YOU!