## 1. views on math

### 1.1. Instrumental Understanding

### 1.2. Platonist (Relational understanding)

### 1.3. Inquiry - problem solving - social constructivist

## 2. Thinking caps/Labels/Pygmalion effects

## 3. Theories of math learning

### 3.1. associationism/behaviourism

3.1.1. Pavlov's dogs

3.1.2. times tables

3.1.2.1. rote learning

3.1.2.1.1. repetition and practice

3.1.3. learning outcomes in Aust. curriculum is very behaviourist (with an emphasis on observable outcomes).

### 3.2. gestaltism

### 3.3. constructivist

3.3.1. piaget

3.3.1.1. mental schemas

3.3.1.1.1. equilibrium

3.3.1.1.2. disequilibrium

3.3.2. e.g. conflict teaching (challenging students' pre-existing assumptions and misconceptions).

3.3.3. 1. identify where students are at

3.3.3.1. e.g. formative assessment

3.3.4. 2. scaffold the lesson for the student

### 3.4. social constructivist

3.4.1. vygotsky

3.4.1.1. Zone of Proximal Development (ZPD)

3.4.2. social interaction with people who know more than you

3.4.3. scaffolding

## 4. relational vs instrumental understanding (Skemp, 1976)

### 4.1. faux amis

### 4.2. instrumental

4.2.1. "rules without reason"

4.2.2. not transferable... leads to errors

### 4.3. Does it matter?

4.3.1. mismatch between the aims of teacher and students:- (1) teaching relationally to students who want instrumental knowledge, and (2) vice versa (e.g. 140 IQ kid struggling in school when taught instrumentally)

4.3.2. the word "mathematics" (Relational and instrumental, although covering the same content, can be considered so far apart that the word "mathematics" is misleading for many students. That failed distinction comes at a great cost to those who would otherwise find enjoyment from math).

4.3.2.1. Comparison

4.3.2.1.1. Instrumental

4.3.2.1.2. relational

4.3.2.2. Clearly relational understanding offers more benefits, so what barriers lead to instrumental teaching instead?

4.3.2.2.1. 1. relational understanding would take too long to achieve to succeed

4.3.2.2.2. 2. examination reasons

4.3.2.2.3. 3. relational understanding relies on expansion into other science disciplines

4.3.2.2.4. 4. peer pressure (e.g. junior teacher in a school where everyone teaches instrumentally)

4.3.2.2.5. **Many teachers do not even have relational understanding (and in fact the majority might lack this understanding)

4.3.2.2.6. 1. backwash effect of examinations

4.3.2.2.7. 2. overburdened syllabi

4.3.2.2.8. 3. difficulty assessing relational understanding. (It's very difficult to make valid inferences about the mental processes behind an answer).

4.3.2.2.9. 4. difficulty for teachers in restructuring their own schemas

4.3.2.3. practicum barriers

4.3.2.3.1. There exists no body of knowledge for teachers to draw upon. At the moment, teachers must learn from their own mistakes rather than from a shared fountain of experiences.

4.3.2.4. benefits of relational understanding

4.3.2.4.1. **analogy to a direction from point A to point B, versus a mental schema of a town

4.3.2.4.2. 1. means becomes independent of the end-goal

4.3.2.4.3. 2. building schemas in a subject area is itself very satisfying

4.3.2.4.4. 3. the complex a student's schemas, the more confident they are of expanding outside of the curriculum

4.3.2.4.5. 4. schemas are never complete, so the process of learning is always continuing

4.3.2.5. disclaimer: research gaps still exist on mathematical thinking in students

## 5. views on teaching math

### 5.1. transmissionist (aka. direct instruction)

5.1.1. gagne's teaching sequence

5.1.2. rule-example-practice

5.1.3. principle: reinforce correct associations

5.1.4. sequence: 1. careful sequencing 2. clearly defined learning outcomes 3. clear presentation by teacher 4. graded practice by students 5. feedback on students' answers

5.1.5. assumption: teaching relies on transmission of knowledge from an expert to a learner

### 5.2. constructivist

5.2.1. Wright’s guiding principles for classroom teaching

5.2.2. e.g. tutes, problem based learning, conflict teaching (teachable moments)

5.2.3. principle: challenge students to construct their own ideas

5.2.4. components: 1. carefully selected challenges 2. students expects to find their own solutions 3. students communicate with each other 4. emphasis on relating and explaining ideas

5.2.5. assumptions: consists of helping learners make sense of their experience

## 6. Boaler (1998)

## 7. mathematical development

### 7.1. - children learn at different rates and in different ways - development moves from informal notions to concrete/symbolic, abstract and generalised ideas - math concepts can develop simultaneously - children who do not develop abstract concepts do not recognised underlying patterns and structures - children learn when they are challenged beyond existing concepts (vygotsky)

### 7.2. views of math development

7.2.1. social-constructivist (group theory) (collective understanding) (negotiation and consensus on meaning) (socio-cultural theory)

7.2.1.1. predominant theory in Australian teaching

7.2.1.1.1. small groups, problem solving

### 7.3. process

### 7.4. views of math learning

7.4.1. embodiment

7.4.1.1. perceptuo-motor activity and imagination (Nemirovsky and Borba, 2003)

7.4.2. piaget (constructivist)

7.4.3. vygotsky (scaffolded)

7.4.4. children's math development can be abstract and generalised earlier than previously believed (fMRI/neuropsychology)

7.4.5. social constructivist

7.4.6. concept development through children's imagery

7.4.6.1. drawings classified as static or dynamic

7.4.6.1.1. dynamic imagery was associated with higher achievement

## 8. technological tools: impact on math learning and teaching

## 9. Promoting meaningful early literacy

### 9.1. .

9.1.1. Piaget's conservation task

9.1.2. subitising

9.1.3. CountMeIn

9.1.4. recognising and creating patterns

9.1.5. counting processes

9.1.5.1. some year 6 students go through grades without even understanding conservation (VERY IMPORTANT TO CHECK AND ASK FOR EXPLANATIONS FROM KIDS)

9.1.5.2. Some Piagetian tests

9.1.5.3. one to one counting

9.1.5.4. benchmarking?

9.1.5.5. seriation (order forwards and backwards)

9.1.5.6. structuring (cubes)

9.1.5.7. "benchmarking"?? tasks

9.1.5.8. .

9.1.5.9. equivalence

9.1.6. Relational understanding

### 9.2. (Research still in progress, especially in neuroscience)

### 9.3. Research Findings

9.3.1. In light of this research, educational institutions all over the world are now moving away from "Numeracy" and towards "Big Math for Little Kids"

### 9.4. Current situation in Australia

## 10. The Elephant in the Classroom

### 10.1. Emily Moskam

### 10.2. We all need math because of the increase in technology E.g. Engineers, etcs

10.2.1. 20 million more jobs for mathematical problem solvers in the future

### 10.3. two types of maths. The boring dull math in most schools, and the interesting engaging real life math of work, play and life.

10.3.1. aversion to math is reflected in pop culture

### 10.4. Problem solving instead of rote learning

10.4.1. employers require problem solvers, continuous learners and team players... not knowledge bases.

10.4.1.1. communication

10.4.1.2. sheer persistence

10.4.1.3. flexibility

10.4.1.4. team working

### 10.5. .

10.5.1. traditional vs problem solving approach to math teaching

10.5.1.1. traditional math is often disconnected from real life applications of math

## 11. The power of a single game to address a range of important concepts in fraction learning

### 11.1. concepts

11.1.1. equivalence

11.1.2. fractional language

11.1.3. improper fractions

11.1.4. addition of fractions

11.1.5. problem solving

11.1.6. visualisation

11.1.7. probability