Numeracy: General Mathematics Content and Processes

Purpose: To draw on findings from college & adult mathematics education research in relation to mathematical and pedagogical content.

References: 17, 23, 25

1. Does the activity develop an appreciation of mathematics by developing:
1. a qualitative understanding of some of the big ideas of mathematics such as infinity, symmetry, structure, recursion, proof, chaos, randomness?
2. an understanding that there are multiple views of the nature of mathematics and that there is controversy over its philosophical foundations?
3. an awareness of how and the extent to which mathematical thinking permeates everyday and shopfloor life and current affairs, even if it is not called mathematics?
4. a critical understanding the uses of mathematics in society: to identify, interpret, evaluate and critique the mathematics embedded in social and political systems and claims, from advertisements to government and interest-group pronouncements?
5. an awareness of the historical development of mathematics, the social contexts of the origins of mathematical concepts, symbolism, theories and problems?
6. a sense of mathematics as a central element of culture, art and life, present and past, which permeates and underpins science, technology and all aspects of human culture?

2. Pedagogical activities :
1. Does the activity model the use of appropriate technology in the teaching of mathematics?
2. Does the activity foster interactive learning through writing, reading, speaking, and collaborative activities?
3. Does the activity actively involve learners in meaningful mathematics problems that build upon their experiences, focus on broad mathematical themes, and build connections within branches of mathematics and between mathematics and other disciplines? [Will learners then be able to view mathematics as a connected whole relevant to their lives?]
4. Does the activity model the use of multiple approaches — numerical, graphical, symbolic, and verbal — to help learners to learn a variety of techniques for solving problems?
5. Does the activity provide learning activities, including projects and apprenticeships, that promote independent thinking and require sustained effort and time? [Will learners have the confidence to access and use needed mathematics and other technical information independently, be able to form conjectures from an array of specific examples, and draw conclusions from general principles?]

3. Learners doing mathematics:
1. Are learners engaging in substantial mathematical problem solving?
2. Are learners learning mathematics through modelling real-world situations?
3. Are learners expanding their mathematical reasoning skills as they develop convincing mathematical arguments?
4. Are learners developing the view that mathematics is a growing discipline, interrelated with human culture and connected to other disciplines?
5. Are learners acquiring the ability to read, write, listen to, and speak mathematics?
6. Are learners using appropriate technology to enhance their mathematical thinking and understanding and to solve mathematical problems and judge the reasonableness of their results?
7. Are learners engaging in rich experiences that encourage independent, nontrivial exploration in mathematics, develop and reinforce tenacity and confidence in their abilities to use mathematics in related studies and elsewhere as workers/citizens?

4. With respect to transfer, does the activity:
1. show the learners how to perform a detailed analysis of the shared or similar components — and the specifically different aspects — of the initial and target tasks?
2. include the ability to transfer as a specific and explicit goal, by establishing inter-relations between the two situations and the related discourses and practices, by translating between the terms/languages used, and by generalising the methods used across contexts?
3. in teaching the initial task, seek to incorporate a balance of generality and situational features — i.e., to anticipate what will come to be seen as similarities across situations, and differences, respectively?
4. teach the initial task in more than one context?
5. allow practice in recognising the cues that signal the relevance or ‘applicability’ of an available skill? [In mathematics these cues are recurrent features of pattern, structure, or relationship.]
6. allow repetition or practice on the target task? [This is to help the learner to appreciate the possible range of generalisation, and the constraints resulting from crucial differences in discourse/areas of application.]

5. Does the assessment address particular mathematical ideas & techniques, namely:
1. reasoning processes (e.g., deduction & induction, making inferences, proving) and mathematical problem solving & modelling?
2. conceptual knowledge and computation?
3. the ability to interpret and critically react to quantitative and statistical information embedded in print or media messages?
4. the transfer of mathematical problem solving across life and work contexts?
5. simulations of problem-solving situations that are typical of those in which mathematics is useful outside of school?
6. a small number of big ideas   (powerful constructs or conceptual tools — e.g., number, pattern & order, chance & data, space & shape, change & approximation) that are accessible to virtually all students?
7. deeper and higher-order understandings of these big ideas?