Big Ideas
Big Ideas
Decomposition helps us solve difficult problems by managing complexity.
- dividing complex problems into parts that are easier to conceive, understand, and program
- Sample questions to support inquiry with students:
- How do we break down a problem into several smaller, simpler pieces?
- How do we know if a problem should be decomposed further?
- Is there a better way to break a problem into smaller pieces and reuse code?
Algorithms are essential in solving problems computationally.
- sets of rules or instructions that precisely define a sequence of operations
- Sample questions to support inquiry with students:
- How does acting out a solution help us to develop an algorithm?
- How is an algorithm formulated?
- What makes one algorithm better than another algorithm?
- How do we know that our algorithm is correct?
- Can all problems be solved by a series of predefined steps?
Programming is a tool that allows us to implement computational thinking .
- a thought process that uses pattern recognition and decomposition to describe an algorithm in a way that a computer can execute
- Sample questions to support inquiry with students:
- How do we decide which programming language to use in solving a specific problem?
- Why is code readability important?
- What factors affect code readability?
- How much source code documentation is enough?
- Are there patterns in the problem that can be generalized?
- How do we recognize patterns that can be translated into rules?
Solving problems is a creative process.
- Sample questions to support inquiry with students:
- How many different ways can this problem be solved?
- How do we approach solving a problem in different ways?
- Without knowing a solution, how do we start to solve a problem?
Content
Learning Standards
Content
ways to represent basic data types
- number systems (e.g., binary, hexadecimal)
- strings, integers, characters, floating point
basic programming concepts
- variables, constants, mathematical operations, input/output, generating random numbers
variable scope
- local versus global
ways to construct and evaluate logical statements
- logical operators (AND, OR, NOT)
- relational operators (<, >, <=, >=, ==, !=, or <>)
- logical equivalences (e.g., De Morgan’s laws), simplification of logical statements, truth tables
use of control flow to manipulate program execution
- decision structures (e.g., if-then-else)
- loops (e.g., for, while, nested loops)
development of algorithms to solve problems in multiple ways
- step-wise refinement, pseudocode or flowcharts, translating between pseudocode and code and vice versa
techniques for operations on and searching of arrays and lists
- append, remove, insert, delete
- searching algorithms (e.g., linear and binary searches)
problem decomposition through modularity
- use of methods/functions to reduce complexity, reuse code, and use function parameters
- return values
uses of computing for financial analysis
- time value of money, appreciation/depreciation, mortgage amortization
- modify the variables of a financial scenario to run a “what-if” analysis on them (e.g., compare different monthly payments, term lengths, interest rates)
ways to model mathematical problems
- estimate theoretical probability through simulation
- represent finite sequences and series
- solve a system of linear equations, exponential growth/decay
- solve a polynomial equation
- calculate statistical values such as frequency, central tendencies, standard deviation of large data set
- compute greatest common factor/least common multiples
Curricular Competency
Learning Standards
Curricular Competency
Reasoning and modelling
Develop flexible thinking to analyze and create algorithms
- understanding that different algorithms can be used to solve the same problem
Explore, analyze, and apply mathematical ideas and computer science concepts using reason, technology, and other tools
- examine the structure of and connections between mathematical and computer science ideas (e.g., demonstrating the connection between theoretical and experimental probability through simulation)
- inductive and deductive reasoning
- predictions, generalizations, conclusions drawn from experiences (e.g., with coding)
- graphing technology, dynamic geometry, calculators, virtual manipulatives, concept-based apps
- can be used for a wide variety of purposes, including:
- exploring and demonstrating mathematical relationships
- organizing and displaying data
- generating and testing inductive conjectures
- mathematical modelling
- integrated development environments (IDE)
- third-party libraries
- visual code comparison tools to view code differences (e.g., Meld)
Model with mathematics in situational contexts
- use mathematical concepts and tools to solve problems and make decisions (e.g., in real-life and/or abstract scenarios)
- take a complex, essentially non-mathematical scenario and figure out what mathematical concepts and tools are needed to make sense of it
- including real-life scenarios and open-ended challenges that connect mathematics with everyday life
Think creatively and with curiosity and wonder when exploring problems
- by being open to trying different strategies
- refers to creative and innovative mathematical thinking rather than to representing math in a creative way, such as through art or music
- asking questions to further understanding or to open other avenues of investigation
Understanding and solving
Develop, demonstrate, and apply conceptual understanding through experimentation, inquiry, and problem solving
- includes structured, guided, and open inquiry
- noticing and wondering
- determining what is needed to make sense of and solve problems
Visualize to explore and illustrate computer science concepts and relationships
- visualize data structures pictorially
- use flow charts
- use code visualization tools or websites (e.g., http://pythontutor.com/)
Apply flexible and strategic approaches to solve problems
- using different algorithms to solve the same problem
- designing algorithms that solve a class of problems rather than a single problem
- deciding which programming patterns to use to solve a problem
- choosing an effective strategy to solve a problem (e.g., guess and check, model, solve a simpler problem, use a chart, use diagrams, role-play)
- interpret a situation to identify a problem
- apply mathematics to solve the problem
- analyze and evaluate the solution in terms of the initial context
- repeat this cycle until a solution makes sense
Solve problems with persistence and a positive disposition
- not giving up when facing a challenge
- problem solving with vigour and determination
Engage in problem-solving experiences connected with place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures
- through daily activities, local and traditional practices, popular media and news events, cross-curricular integration
- by posing and solving problems or asking questions about place, stories, and cultural practices
- through cryptography (e.g., Navajo Code Talkers from WWII)
Communicating and representing
Explain and justify mathematical ideas and decisions in many ways
- use mathematical arguments to convince
- includes anticipating consequences
- Have students explore which of two scenarios they would choose and then defend their choice.
- including oral, written, pictures, use of technology
- communicating effectively according to what is being communicated and to whom
Represent computer science ideas in concrete, pictorial, symbolic, and pseudocode forms
- using models, tables, flow charts, words, numbers, symbols
- connecting meanings among various representations
- using concrete materials and dynamic interactive technology
Use computer science and mathematical vocabulary and language to contribute to discussions in the classroom
- partner talks, small-group discussions, teacher-student conferences
Take risks when offering ideas in classroom discourse
- is valuable for deepening understanding of concepts
- can help clarify students’ thinking, even if they are not sure about an idea or have misconceptions
Connecting and reflecting
Reflect on mathematical and computational thinking
- share the mathematical and computational thinking of self and others, including evaluating strategies and solutions, extending, posing new problems and questions
Connect mathematical and computer science concepts with each other, other areas, and personal interests
- to develop a sense of how computer science helps us understand the world around us (e.g., daily activities, local and traditional practices, popular media and news events, social justice, cross-curricular integration)
Use mistakes as opportunities to advance learning
- include syntax, semantic, run-time, and logic errors
- by:
- analyzing errors to discover misunderstandings
- making adjustments in further attempts (e.g., debugging)
- identifying not only mistakes but also parts of a solution that are correct
Incorporate First Peoples worldviews, perspectives, knowledge, and practices to make connections with computer science concepts
- by:
- collaborating with Elders and knowledge keepers among local First Peoples
- exploring the First Peoples Principles of Learning (http://www.fnesc.ca/wp/wp-content/uploads/2015/09/PUB-LFP-POSTER-Princi…; e.g., Learning is holistic, reflexive, reflective, experiential, and relational [focused on connectedness, on reciprocal relationships, and a sense of place]; Learning involves patience and time)
- making explicit connections with learning mathematics
- exploring cultural practices and knowledge of local First Peoples and identifying mathematical connections
- local knowledge and cultural practices that are appropriate to share and that are non-appropriated
- Bishop’s cultural practices: counting, measuring, locating, designing, playing, explaining (http://www.csus.edu/indiv/o/oreyd/ACP.htm_files/abishop.htm)
- Aboriginal Education Resources (www.aboriginaleducation.ca)
- Teaching Mathematics in a First Nations Context, FNESC (http://www.fnesc.ca/resources/math-first-peoples/)