Module 1 Thinking Mathematically Flashcards
Continuum of Learning
overview of the progression of learning for all the strands and includes the substrands and outcomes from Kindergarten to Year 10.
Content
is presented in stages and represents the knowledge, skills and understanding that are acquired by a typical student by the end of that stage.
Constructivist theory
‘constructivist’ approach to teaching – ie we approach teaching mathematics in the same way children learn mathematics – by actively constructing their own knowledge via both social mechanisms and active engagement.
Piaget
emphasised the active role of the learners in the construction of their own knowledge… the major tenet of constructivism”.
first major tenet of constructivism
Knowledge is not passively received, but actively construed by the learner.
A second major tenet of constructivism:
Students can construct new knowledge through reflection upon their physical and mental actions.
A third major tenet of constructivism
derives its origins from the work of socio-cultural theorists such as Vygotsky (1978): Learning is a social process.” (Bobis et al., 2009, p. 8)
Bruner
Talks about ways of ‘Knowing’:
- Enactive (using concrete manipulatives)
- Iconic (using pictorial images)
- Symbolic (either mathematical words or symbols).
We move children through 3 stages: Concrete → pictorial →symbolic.
3 Models for teaching
ELPSARA, the Language model and Polya’s four step method for problem solving.
ELPSARA
Experience, Language(list of appropriate words that students should be able to use to communicate their mathematical knowledge and understanding. Any words that appear in bold are new terms that will need to be introduced during that unit of work.), Pictorial representation (concrete/hands), Symbolic representation (usually numbers and written words), Application (either background or new knowledge about a concept within a meaningful context can occur during any of the phases in this sequence.), Reflection, Assessment- improvement-how you could improve the lesson for future students.
Language model (4 stages of language)
Children’s language (3 lollies, but I took away two) Material language (take 1 from 3) Math language (subtract 3 from 1) Symbolic language (3-1)
Polya’s Problem Solving Steps.
Read and understand the problem: Who can explain to the rest of us what this problem is about? Who thinks they can work this problem out? Which words are needed to answer the problem? Which ones are not? What are you being asked to do? Can you underline the key words?
- Devise a strategy/action plan: What strategy are you going to use? How do you think we can work this out? Is there a better or different strategy that you can use if you get stuck? What are you going to do first? 3. Carry out the strategy/action plan. What patterns do you see? Are there any more solutions? Is there another way to find the solution?
- Check your answers Do you agree with David? Why not? What do you think the answer is? Does the answer actually answer the problem?
What are you doing when Working Mathematically
Communicating: describe, represent and explain mathematical situation, concepts, methods and solutions to problems, using appropriate language, terminology, tables, diagrams, graphs, symbols, notation and conventions.
Problem solving: students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively.
Reasoning: students are reasoning mathematically when they explain their thinking, deduce and justify strategies used and conclusions reached, adapt the known to the unknown, transfer learning from one context to another, prove that something is true or false, and compare and contrast related ideas and explain their choices.
Understanding: Students build understanding when they connect related ideas, represent concepts in different ways, identify commonalities and differences between aspects of content, describe their thinking mathematically, and interpret mathematical information
Fluency: Students develop fluency when they calculate answers efficiently, recognise robust ways of answering questions, choose appropriate methods and approximations, recall definitions and regularly use facts, and manipulate expressions and equations to find solutions.
Problem Solving Strategies
Trial and error Working backwards Making a diagram Using a table or chart Looking for a pattern Making a model Making a list Thinking logically Systematically trying all possibilities
Assessing a good inquiry lesson
Align with intended aim, syll outcomes, stage, assumed prior knowledge, learning experience and assessment
Quality of learning experiences and teaching strategies
Quality of investigation used in lesson
Ability to communicate ideas clearly
Creativity and originality of lesson
differentiated instruction
different learning experience or different learning performance
4 inputs teachers struggle with
-student aptitude
Difficulty of reading materials
Time devoted to instruction
Coverage of curriculum
Language developed during Stage 1
should include number line, number chart, odd, even, missing number and number sentence.
Language developed during Stage 2
odd and even numbers,
Additive number patterns.
Informal language such as “goes up by” and/or “goes down by”.
specific “term” in a pattern, for example “The first term is 2.”
Language-rows, digits, multiplication facts, is the same as, and equals.
Language developed during Stage 3
children should begin to use the language: increase and decrease. They should also be exposed to number sentences involving more than one operation. Students should be encouraged to find their answers by working backwards and using inverse operations, rather than trial and error. Providing them with number sentences that do not have whole numbers as answers will assist this
Review Questions:
Question 1
How would you support children as they work through a problem-solving lesson? What key questions might you ask them?
Enabling Prompts
Questions asked such as What have you tried?, What do you know? What are you trying to find out?
The Australian Curriculum Mathematics identifies ‘Proficiency Strands’ to describe mathematical activities required of students. The strands are actions. What do they include?
Understanding:
building robust knowledge of adapatable and transferable mathematical concepts.
the making of connections between related concepts,.
the confidence to use the familiar to develop new ideas
The why as well s the how of mathematics.
Fluency
Includes skill in choosing appropriate procedures
Carrying out procedures flexibly accurately, efficiently and appropriately
Recalling factual knowledge and concepts readily
Problem Solving
Includes the ability to make choices, interpret, formulate, model and investigate problem situations and communicate solutions effectively.
Reasoning
Includes the capacity for logical thought and actions eg analysing proving, evaluating, explaining, inferring, justifying and generalising.
Relational Understanding:
the ‘why’ of mathematics
Not only know how to follow a procedure but also understand why the procedure worked.
Instrumental Understanding:
the ‘how’ of mathematics
Follow a procedure to complete an exercise
Conjecture
A place in mathematical thinking which happens when when we are generalising. Conjectures are mathematical ideas that seem reasonable but have not been proven.
Enabling prompts
Tasks to make the task accessible so that all students may make a start.
Exercise
A question posed to a person who has already solved a similar problem or been taught a solution to a method. Known as knowledge or procedures and can be thought of as learning for problem solving.
Extending Prompts
For students who may finish task early. (Extension Activities).
Generalising
Step after spotting patterns where you are thinking about the mathematical process. Enables us to notice patterns in the data and to spot aspect of the problem or investigation that stay the same when other things change.
Investigation
A type of mathematical problem. A task is given to a person for exploration in any direction they choose.
Justification
An important part of mathematics which ensures you are convinced of the answer to your problem and can prove it.
Problem
Is a question posed of a person who initially does not know what direction to take to solve the problem. There may be possible paths to the solution. Once the problem has been solved the solution path becomes known. If the same person is given the problem again, it will now be an exercise for them.
What are Heuristics?
Problem Solving Strategies
Specialising
an important part of the problem solving process which allows us to gather data about the problem. It enables specific examples to by trialled to gain a sense of what we are trying to do to solve the problem.
