the process of computational problem solving

What is computational thinking?

Computational thinking (CT) is a problem-solving technique that imitates the process computer programmers go through when writing computer programmes and algorithms. This process requires programmers to break down complex problems and scenarios into bite size pieces that can be fully understood in order to then develop solutions that are clear to both computers and humans. So, like programmers, those who apply computational thinking techniques will break down problems into smaller, simpler fragments, and then outline solutions to address each problem in terms that any person can comprehend. 

Computational thinking requires:

• exploring and analysing problems thoroughly in order to fully understand them
• using precise and detailed language to outline both problems and solutions
• applying clear reasoning at every stage of the process

In short, computational thinking encourages people to approach any problem in a systematic manner, and to develop and articulate solutions in terms that are simple enough to be executed by a computer – or another person. 

What are the four parts of computational thinking?

Computational thinking has four foundational characteristics or techniques. These include:

Decomposition

Decomposition is the process of breaking down a problem or challenge – even a complex one – into small, manageable parts.

Abstraction

Also known as generalisation, abstraction requires computational thinkers to focus only on the most important information and elements of the problem, and to ignore anything else, particularly irrelevant details or unnecessary details.

Pattern recognition

Also known as data and information visualisation, pattern recognition involves sifting through information to find similar problems. Identifying patterns makes it easier to organise data, which in turn can help with problem solving.  

Algorithm design

Algorithm design is the culmination of all the previous stages. Like a computer programmer writing rules or a set of instructions for a computer algorithm, algorithmic thinking comes up with step-by-step solutions that can be followed in order to solve a problem.

Testing and debugging can also occur at this stage to ensure that solutions remain fit for purpose.

Why is computational thinking important?

For computer scientists, computational thinking is important because it enables them to better work with data, understand systems, and create workable algorithms and computation models.

In terms of real-world applications outside of computer science, computational thinking is an effective tool that can help students and learners develop problem-solving strategies they can apply to both their studies as well as everyday life. In an increasingly complicated, digital world, computational thinking concepts can help people tackle a diverse array of challenges in an effective, manageable way. Because of this, it is increasingly being taught outside of a computer science education, from the United Kingdom’s national curriculum to the United States’ K-12 education system.

How can computational thinking be used?

Computational thinking competencies are a requirement for any computer programmer working on algorithms, whether they’re for automation projects, designing virtual reality simulations, or developing robotics programmes.

But this thinking process can also be taught as a template for any kind of problem, and used by any person, particularly within high schools, colleges, and other education settings.

Dr Shuchi Grover , for example, is a computer scientist and educator who has argued that the so-called “four Cs” of 21st century learning – communication, critical thinking, collaboration, and creativity – should be joined by a fifth: computational thinking. According to Grover , it can be beneficial within STEM subjects (science, technology, engineering and mathematics), but is also applicable to the social sciences and language and linguistics.

What are some examples of computational thinking?

The most obvious examples of computational thinking are the algorithms that computer programmers write when developing a new piece of software or programme. Outside of computer programming, though, computational thinking can also be found in everything from instructional manuals for building furniture to recipes for baking a chocolate cake – solutions are broken down into simple steps and communicated clearly and precisely.  

What is the difference between computational thinking and computer science?

Computer science is a large area of study and practice, and includes an array of different computer-related disciplines, such as computing, automation, and information technology. 

Computational thinking, meanwhile, is a problem-solving method created and used by computer scientists – but it also has applications outside the field of computer science.

How can we teach computational thinking?

Teaching computational thinking was popularised following the publication of an essay on the topic in the Communications of the ACM journal. Written by Jeannette Wing , a computer science researcher, the essay suggested that computational thinking is a fundamental skill for everyone and should be integrated into other subjects and lesson plans within schools. 

This idea has been adopted in a number of different ways around the world, with a growing number of resources available to educators online. For example:

• the Computer Science Teaching Association (CSTA) partnered with the International Society for Technology in Education (ISTE) to share tools and resources to help teachers “prepare young learners to become computational thinkers who understand how today’s digital tools can help solve tomorrow’s problems”
• computational thinking pioneer Stephen Wolfram developed the Wolfram programming language with young learners in mind, making it easier to teach computational thinking skills to kids
• there are also resources available through websites such as CS Unplugged , which offers a collection of free materials to help teach computer science concepts to pupils

Become a computational thinker

Develop computational thinking skills with the online MSc Computer Science at the University of York. Through your taught modules, you will be able to apply computational thinking in multiple programming languages, such as Python and Java, and be equipped to engage in solution generation across a broad range of fields. Some of the modules you’ll study include algorithms and data structures, advanced programming, artificial intelligence and machine learning, cyber security threats, and computer architecture and operating systems.

This master’s degree has been designed for working professionals and graduates who may not have a computer science background, but who want to launch a career in the lucrative field. And because it’s studied 100% online, you can learn remotely – at different times and locations – part-time around your full-time work and personal commitments.

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• Applications
• Karel the Turtle
• Betty's Brain
• Game Creator by Cand.li
• Competences
• PILA for Research

Competency framework

Conceptual framework of the PILA Computational Problem Solving module

What is computational problem solving.

‘Computational problem solving’  is the iterative process of developing  computational solutions to problems. Computational solutions are expressed as logical sequences of steps (i.e. algorithms), where each step is precisely defined so that it can be expressed in a form that can be executed by a computer. Much of the process of computational problem solving is thus oriented towards finding ways to use the power of computers to design new solutions or execute existing solutions more efficiently.

Using computation to solve problems requires the ability to think in a certain way, which is often referred to as ‘computational thinking’. The term originally referred to the capacity to formulate problems as a defined set of inputs (or rules) producing a defined set of outputs. Today, computational thinking has been expanded to include thinking with many levels of abstractions (e.g. reducing complexity by removing unnecessary information), simplifying problems by decomposing them into parts and identifying repeated patterns, and examining how well a solution scales across problems.

Why is computational problem solving important and useful?

Computers and the technologies they enable play an increasingly central role in jobs and everyday life. Being able to use computers to solve problems is thus an important competence for students to develop in order to thrive in today’s digital world. Even people who do not plan a career in computing can benefit from developing computational problem solving skills because these skills enhance how people understand and solve a wide range of problems beyond computer science.

This skillset can be connected to multiple domains of education, and particularly to subjects like science, technology, engineering or mathematics (STEM) and the social sciences. Computing has revolutionised the practices of science, and the ability to use computational tools to carry out scientific inquiry is quickly becoming a required skillset in the modern scientific landscape. As a consequence, teachers who are tasked with preparing students for careers in these fields must understand how this competence develops and can be nurtured. At school, developing computational problem solving skills should be an interdisciplinary activity that involves creating media and other digital artefacts to design, execute, and communicate solutions, as well as to learn about the social and natural world through the exploration, development and use of computational models.

Is computational problem solving the same as knowing a programming language?

A programming language is an artificial language used to write instructions (i.e. code) that can be executed by a computer. However, writing computer code requires many skills beyond knowing the syntax of a specific programming language. Effective programmers must be able to apply the general practices and concepts involved in computational thinking and problem solving. For example, programmers have to understand the problem at hand, explore how it can be simplified, and identify how it relates to other problems they have already solved. Thus, computational problem solving is a skillset that can be employed in different human endeavours, including programming. When employed in the context of programming, computational problem solving ensures that programmers can use their knowledge of a programming language to solve problems effectively and efficiently. 

Students can develop computational problem solving skills without the use of a technical programming language (e.g. JavaScript, Python). In the PILA module, the focus is not on whether students can read or use a certain programming language, but rather on how well students can use computational problem solving skills and practices to solve problems (i.e. to “think” like a computer scientist).

How is computational problem solving assessed in PILA?

Computational problem solving is assessed in PILA by asking students to work through dynamic problems in open-ended digital environments where they have to interpret, design, or debug computer programs (i.e. sequences of code in a visual format). PILA provides ‘learning assessments’, which are assessment experiences that include resources and structured support (i.e. scaffolds) for learning. During these experiences, students iteratively develop programs using various forms of support, such as tutorials, automated feedback, hints and worked examples. The assessments are cumulative, asking students to use what they practiced in earlier tasks when completing successive, more complex tasks.

To ensure that the PILA module focuses on foundational computational problem solving skills and that the material is accessible to all secondary school students no matter their knowledge of programming languages, the module includes an assessment application, ‘Karel World’, that employs an accessible block-based visual programming language. Block-based environments prevent syntax errors while still retaining the concepts and practices that are foundational to programming. These environments work well to introduce novices to programming and help develop their computational problem solving skills, and can be used to generate a wide spectrum of problems from very easy to very hard.

What is assessed in the PILA module on computational problem solving?

Computational problem solving skills.

The module assesses the following set of complementary problem solving skills, which are distinct yet are often used together in order to create effective and efficient solutions to complex problems:

• Decompose problems

Decomposition is the act of breaking down a problem goal into a set of smaller, more manageable sub-goals that can be addressed individually. The sub-goals can be further broken down into more fine-grained sub-goals to reach the granularity necessary for solving the entire problem.

• Recognise and address patterns

Pattern recognition refers to the ability to identify elements that repeat within a problem and can thus be solved through the same operations. Adressing repeating patterns means instructing a computer to iterate given operations until the desired result is achieved. This requires identifying the repeating instructions and defining the conditions governing the duration of the repetition.

• Generalise solutions

Generalisation is the thinking process that results in identifying similarities or common differences across problems to define problem categories. Generalising solution results in producing programs that work across similar problems through the use of ‘abstractions’, such as blocks of organised, reusable sequence(s) of instructions.

• Systematically test and debug

Solving a complex computational problem is an adaptive process that follows iterative cycles of ideation, testing, debugging, and further development. Computational problem solving involves systematically evaluating the state of one’s own work, identifying when and how a given operation requires fixing, and implementing the needed corrections.

Programming concepts

In order to apply these skills to the programming tasks presented in the module, students have to master the below set of programming concepts. These concepts can be isolated but are more often used in concert to solve computational problems:

• Sequences

Sequences are lists of step-by-step instructions that are carried out consecutively and specify the behavior or action that should be produced. In Karel World, for example, students learn to build a sequence of block commands to instruct a turtle to move around the world, avoiding barriers (e.g. walls) and performing certain actions (e.g. pick up or place stones).

• Conditionals

Conditional statements allow a specific set of commands to be carried out only if certain criteria are met. For example, in Karel World, the turtle can be instructed to pick up stones ‘if stones are present’.

To create more concise and efficient instructions, loops can communicate an action or set of actions that are repeated under a certain condition. The repeat command indicates that a given action (i.e. place stone) should be repeated through a real value (i.e. 9 times). A loop could also include a set of commands that repeat as long as a Boolean condition is true, such as ‘while stones are present’.

• Functions

Creating a function helps organise a program by abstracting longer, more complex pieces of code into one single step. By removing repetitive areas of code and assigning higher-level steps, functions make it easier to understand and reason about the various steps of the program, as well as facilitate its use by others. A simple example in Karel World is the function that instructs the turtle to ‘turn around’, which consists of turning left twice.

How is student performance evaluated in the PILA module?

Student performance in the module is evaluated through rubrics. The rubrics are structured in levels, that succinctly describe how students progress in their mastery of the computational problem solving skills and associated concepts. The levels in the rubric (see Table 1) are defined by the complexity of the problems that are presented to the students (simple, relatively complex or complex) and by the behaviours students are expected to exhibit while solving the problem (e.g., using functions, conducting tests). Each problem in the module is mapped to one or more skills (the rows in the rubric) and classified according to its complexity (the columns in the rubric). Solving a problem in the module and performing a set of expected programming operations thus provide evidence that supports the claims about the student presented in the rubric. The more problems at a given cell of the rubric the student solves, the more conclusive is the evidence that the student has reached the level corresponding to that cell. 

Please note: the rubric is updated as feedback is received from teachers on the clarity and usefulness of the descriptions.

Table 1 . Rubric for computational problem solving skills

Learning management skills

The performance of students on the PILA module depends not just on their mastery of computational problem solving skills and concepts, but also on their capacity to effectively manage their work in the digital learning environment. The complex tasks included in the module invite students to monitor, adapt and reflect on their understanding and progress. The assessment will capture data on students’ ability to regulate these aspects of their own work and will communicate to teachers the extent to which their students can:

• Use resources

PILA tasks provide resources such as worked examples that students can refer to as they build their own solution. Students use resources effectively when they recognise that they have a knowledge gap or need help after repeated failures and proceed to accessing a learning resource.

• Adapt to feedback

As students work through a PILA assessment, they receive different types of automated feedback (e.g.: ‘not there yet’, ‘error: front is blocked’, ‘try using fewer blocks’). Students who can successfully adapt are able to perform actions that are consistent with the feedback, for example inserting a repetition block in their program after the feedback ‘try using fewer blocks’.

• Evaluate own performance

In the assessment experiences designed by experts in PILA, the final task is a complex, open challenge. Upon completion of this task, students are asked to evaluate their own performance and this self-assessment is compared with their actual performance on the task.

• Stay engaged

The assessment will also collect information on the extent to which students are engaged throughout the assessment experience. Evidence on engagement is collected through questions that are included in a survey at the end of the assessment, and through information on students’ use of time and number of attempts.  

Learn about computational problem solving-related learning trajectories:

• Rich, K. M., Strickland, C., Binkowski, T. A., Moran, C., & Franklin, D. (2017). K-8 Learning Trajectories Derived from Research Literature: Sequence, Repetition, Conditionals. Proceedings of the 2017 ACM Conference on International Computing Education Research, 182–190.
• Rich, K. M., Strickland, C., Binkowski, T. A., & Franklin, D. (2019). A K-8 Debugging Learning Trajectory Derived from Research Literature. Proceedings of the 50th ACM Technical Symposium on Computer Science Education, 745–751. https://doi.org/10.1145/3287324.3287396
• Rich, K. M., Binkowski, T. A., Strickland, C., & Franklin, D. (2018). Decomposition: A K-8 Computational Thinking Learning Trajectory. Proceedings of the 2018 ACM Conference on International Computing Education Research  - ICER ’18, 124–132. https://doi.org/10.1145/3230977.3230979

Learn about the connection between computational thinking and STEM education:

• Weintrop, D., Beheshti, E., Horn, M., Orton, K., Jona, K., Trouille, L., & Wilensky, U. (2015). Defining Computational Thinking for Mathematics and Science Classrooms. Journal of Science Education and Technology, 25(1), 127–147. doi:10.1007/s10956-015-9581-5

Learn how students apply computational problem solving to Scratch:

• Brennan, K., & Resnick, M. (2012). Using artifact-based interviews to study the development of computational thinking in interactive media design. Paper presented at annual American Educational Research Association meeting, Vancouver, BC, Canada.

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• Introduction to Computation and Programming Using Python

Introduction to Computation and Programming Using Python , third edition

With application to computational modeling and understanding data.

by John V. Guttag

ISBN: 9780262542364

Pub date: January 5, 2021

• Publisher: The MIT Press

664 pp. , 7 x 9 in , 140

ISBN: 9780262363433

Pub date: March 2, 2021

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• Description

The new edition of an introduction to the art of computational problem solving using Python.

This book introduces students with little or no prior programming experience to the art of computational problem solving using Python and various Python libraries, including numpy, matplotlib, random, pandas, and sklearn. It provides students with skills that will enable them to make productive use of computational techniques, including some of the tools and techniques of data science for using computation to model and interpret data as well as substantial material on machine learning.

The book is based on an MIT course and was developed for use not only in a conventional classroom but in a massive open online course (MOOC). It contains material suitable for a two-semester introductory computer science sequence.

This third edition has expanded the initial explanatory material, making it a gentler introduction to programming for the beginner, with more programming examples and many more “finger exercises.” A new chapter shows how to use the Pandas package for analyzing time series data. All the code has been rewritten to make it stylistically consistent with the PEP 8 standards. Although it covers such traditional topics as computational complexity and simple algorithms, the book focuses on a wide range of topics not found in most introductory texts, including information visualization, simulations to model randomness, computational techniques to understand data, and statistical techniques that inform (and misinform) as well as two related but relatively advanced topics: optimization problems and dynamic programming. The book also includes a Python 3 quick reference guide.

All of the code in the book and an errata sheet are available on the book's web page on the MIT Press website.

John V. Guttag is the Dugald C. Jackson Professor of Computer Science and Electrical Engineering at MIT.

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Introduction to Computer Science and Programming edX Course

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Programming for Problem Solving

This book provides an introduction to computer programming using Python as a way to solve problems. It focuses on programming concepts and fundamentals within the context of solving real world problems.

Table of Contents

• Attributions
• Acknowledgments
• Learning Objectives
• Introduction
• Computational Thinking
• An Example Algorithm
• Verifying your Algorithm
• The Process of Computational Problem Solving
• Values and Variables
• What is a Program?
• Computational Problem Design Using the Basic Programming Constructs
• The Role of Programming in the Field of Informatics
• Unit Summary
• Practice Problems
• Computer Hardware Architecture
• Digital Computing: It’s All about 0’s and 1’s
• Operating Systems—Bridging Software and Hardware
• Software Development Tools
• Learning Programming with Python
• Writing a Python Program
• The Python Interactive Shell
• The Basics of Python Programming
• Example: Using Variables and Literal Constants
• Operators and Expressions
• Practice with Operators & Expressions
• Evaluation Order
• Input/Process/Output Pattern
• Type Converter Functions
• Python’s Standard Library
• More on Strings
• Object Oriented Programming
• Simple Graphics Programming
• Graphics Windows: Coordinate Systems
• GraphWin Objects
• Text Methods
• Entry Objects
• Displaying Images
• Generating Colors
• Interactive Graphics
• Boolean Expressions
• Logical Operators
• Conditional Execution
• Exception Handling
• Practice with Handling Exceptions in our Programs
• The for…in Statement
• Nested Loops
• Basic File Processing
• Installing Python 3 and IDLE
• Using Python and its IDE 
• Appendix B: Python Cheat Sheet

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Computational Thinking: Its Purpose & Importance

by Lcom Team | Nov 14, 2023 | Blogs

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Computational thinking is more related to math and algorithms than it is to digital technology. It refers to “computing” a solution by breaking down a problem into its separate parts and discovering the effective steps that reliably and effectively resolve the problem. In math, this might look like the “ order of operations ,” which defines how to decompose and solve a linear math problem.

What is the Purpose of Computational Thinking?

The purpose of computational thinking is to be able to solve complex problems in a structured, effective and repeatable way. Computational thinking, while drawing on principles from computer science and mathematics, can be applied not only to mathematical or technology-related problems, but to real-world problems as well. Therefore, computational thinking provides an effective and repeatable process for solving complex issues regardless of whether they are technologically dependent.

How Does Computational Thinking Work?  

In a previous article defining computational thinking , we discuss how computational thinking identifies a clear, defined step-by-step solution to a complex problem. But how, exactly, does one utilize computational thinking to define this solution?

Whether the problem to be solved is in a technological environment or is “offline,” (that is, not related to technology), computational thinking helps to approach, understand, analyze and resolve the problem in an effective and efficient manner. The process is as follows:

• Decomposition. First, the problem is decomposed into smaller, more manageable parts. This helps the problem-solver more effectively understand the problem while being able to eliminate those parts that are irrelevant.
• Pattern Recognition. In the next step, pattern recognition , the problem solver identifies patterns or connections between the different parts identified during decomposition—or even to other previously-solved problems. The purpose of this step in computational thinking is to further simplify the problem as well as to begin identifying areas of the problem that may be solved similarly.
• Abstraction. Decomposition and pattern recognition empower the problem solver to use abstraction to identify the most relevant information within the problem while eliminating that which is either repeated elsewhere or irrelevant. This simplifies an otherwise complex problem and creates a more efficient environment for the individual to identify how the different parts of the problem may be solved.
• Algorithmic Thinking. Algorithmic thinking is the process of defining a step-by-step solution to the problem. The key to an algorithmic solution is that it should be able to be replicated for a predictable and reliable outcome (in other words, for those familiar with billiards, “ slop shots ” don’t count). The benefit of having a replicable solution is that it is more certainly a reliable outcome if the result can be repeated. In addition, a well-defined replicable solution may be more effectively used in part or in whole to resolve other issues.

Why is Computational Thinking Important?

Computational thinking is an important future-ready skill for students and adults alike. This sophisticated process for problem-solving empowers the learner with more effective tools to solve complex problems as well as to produce more effective processes in the future.

• Problem solving. The most well-known benefit of computational thinking is the increased ability to solve complex problems. Just like how computational thinking provides effective steps to solve a complex problem, the process of computational thinking, itself, is a computational solution for solving complex problems.
• Automation and efficiency. Computational thinking is essential in the automation of tasks and processes, which means it’s critical for such applications as coding and automation. The applications of these are far-reaching, from science and engineering to marketing, sales, social sciences, big data and more.
• Data Analysis. In the age of big data , computational thinking is essential for processing and interpreting vast amounts of information. It helps in extracting meaningful insights and making data-driven decisions.
• Innovation. Computational thinking is a driver of innovation. At its core, computational thinking helps to solve complex problems, which is the same basis that inspires innovative solutions to these problems. Without the ability to problem-solve using computational thinking, it would be difficult to define and replicate innovative solutions to modern problems.
• Career opportunities. The ability for an individual to use computational thinking to problem-solve empowers the individual with a “soft skill” that is highly valued in most industries and leadership positions. From manufacturing to finance, technology to healthcare and beyond, these industries actively seek individuals who can solve complex problems and drive innovation.

Final Thoughts

Learning.com Team

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Founded in 1999, Learning.com provides educators with solutions to prepare their students with critical digital skills. Our web-based curriculum for grades K-12 engages students as they learn keyboarding, online safety, applied productivity tools, computational thinking, coding and more.

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Four computational thinking strategies for building problem-solving skills across the curriculum

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Two decades into the 21st century, educators are still tackling the question of how to help young people prepare for a rapidly evolving work landscape . Industry leaders have long called for more emphasis on skills such as critical thinking , communication and problem-solving , though the definitions and methods for teaching all of these can vary widely. At the International Society for Technology in Education conference in July, a number of education leaders and teachers discussed a framework that can help build students’ problem-solving skills in any subject: computational thinking.

Much of the research and discussion on computational thinking in the last twenty years has focused on computer science contexts . Harvard’s Karen Brennan , for example, has led studies and developed resources on computational thinking with Scratch . But several advocates argued that these skills are not just applicable to coding and should be integrated across the curriculum. They outlined four strategies that make up the computational thinking process:

Decomposition - breaking a complex problem into smaller parts or questions

Pattern recognition - identifying trends, differences or similarities in data

Abstraction - removing unnecessary elements or data to focus on what’s useful in solving a problem

Algorithmic design - making steps and rules to solve problems

Most problems will require students to employ multiple strategies. Julie Evans , CEO of the education nonprofit Project Tomorrow, illustrated that point by asking attendees at one session to draw a cat in less than 30 seconds. No drawing looked exactly the same, but the participating educators had to quickly break their mental image of a cat into important parts, such as a tail and whiskers (decomposition). They discarded unnecessary data; for instance, a cat can be conveyed by drawing its head and body or just its face (abstraction). And they envisioned and executed steps to get from a blank page to a completed drawing (algorithmic design).

Bryan Cox, who works in the Georgia Department of Education to broaden computer science education, offered practical and pedagogical reasons for integration. Not all schools offer computer science and even at schools that do, not all students take those classes . For elementary school teachers, stand-alone computer science lessons can feel like one more thing to add to an already packed curriculum. “Integration is less disruptive,” Cox said. He also said integration mirrors how computational thinking occurs in the real world in fields like medicine, automotives, law and sports.

Over the past two years, Project Tomorrow trained 120 teachers in New York City elementary schools to integrate computational thinking into their classrooms. In one example from a second and third grade writing unit, students wrote a realistic fiction story and created a movie to bring the story to life. That may sound like a pretty typical language arts project, but the difference was in the approach, according to Project Tomorrow instructional coach David Gomez. Rather than being told how to write a realistic fiction story, students developed an algorithm for the process, with steps such as making up a pretend character, giving the character a name, imagining the setting and so on. In this example and others, Gomez said that algorithms help students acknowledge the steps they are following during a task and increase their awareness of their work processes.

Gomez works with teachers to help students recognize when they’re using other computational thinking strategies, too. One second grade teacher, for example, used a poster with sticky notes for students to reflect on which strategies they’d used in different subjects throughout the day.

Evans said she loves hearing kids identify the strategies in discussions with each other. She’s heard questions like “Did you try abstraction?” and “Why didn’t you do pattern recognition ?” from students chatting with classmates. “Those little tykes in second grade are already developing their problem-solving muscles, and they’ve got the vocabulary to have that be a sustainable skill for the future,” she said.

Crafting computational problems

Not every question or problem is a computational one. Carolyn Sykora, senior director of the ISTE Standards programs, shared three characteristics that teachers can use to identify a computational problem:

• It’s open-ended with multiple potential solutions. “How can we design a car to get from point A to point B?” is an example that meets this criteria, whereas “How does a self-driving car work?” is a knowledge-based question.
• It requires using or collecting data. Data doesn’t just mean numbers. It could, for example, be the lines in a poem or the notes in a musical composition.
• It includes an opportunity to create a procedure or algorithm. In some cases, such as an engineering challenge, it’s easy to identify where this opportunity will arise. But often that’s not so clear. “Sometimes you don’t understand where the algorithm design comes into play until you do your problem decomposition,” Sykora said.

Using these characteristics can help teachers rethink curriculum, rather than trying to add something new. “We have our tried and true lessons and the things that we want our kids to learn,” Sykora said. The next step is to look at those lessons and ask, “How can we take something that’s knowledge-based and turn it into a computational problem?”

A computational approach based on the Legendre-Galerkin method for solving a distributed optimal control problem constrained by the biharmonic equation

• Original Paper
• Published: 24 April 2024

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• Manoochehr Khasi 1  

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This paper presents a Legendre-Galerkin spectral method to compute the solution of a distributed optimal control problem (OCP) constrained by the biharmonic equation on regular and irregular domains. First, the optimality system is obtained by the Karush–Kuhn–Tucker optimality conditions. Next, it is discretized by the proposed method, and a coupled matrix equation is obtained. Finally, the arising system is solved by using the Kronecker product. Using the appropriate base functions causes a system with sparse matrices. Also, one of the advantages of this approach is its high accuracy for solving fourth-order problems. The error estimate is also investigated theoretically. Many numerical examples are included to demonstrate the accuracy and efficiency of the proposed approach.

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Manoochehr Khasi

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Khasi, M. A computational approach based on the Legendre-Galerkin method for solving a distributed optimal control problem constrained by the biharmonic equation. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01832-w

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Received : 02 July 2023

Accepted : 27 March 2024

Published : 24 April 2024

DOI : https://doi.org/10.1007/s11075-024-01832-w