Week+3+-+How+children+learn+mathematics

//**What are things I can do as a teacher to help children learn mathematics?**//
 * __Learning Activity 3.1:__**
 * Help my students to construct their own meaning, instead of teaching them to 'parrot'
 * Focus less on 'typical' maths ideas such as numbers, geometry, multiplication. Instead, emphasise the versatility of maths skills, and how they relate to 'everyday' activities. For instance, basic angles can assist in driving skills (parallel parking, reversing etc.)
 * Give adequate support to struggling students. This may take the form of additional resources, group work (to allow students to teach one another) or other effective scaffolding techniques
 * Choose varied activities and a wide range of resources. Utilize the outdoors, libraries, local museums, or any other resource that may enrich the learning process.
 * Continually update my own knowledge. This may be through 'curriculum days,' discussion with colleagues, or other resources readily available to teachers.

**__KEY IDEAS FROM THE READING (Reys, p. 14-33)__**
 * __Create a positive learning environment__: rewards or good behaviour (intrinsic/extrinsic), seating arrangement, positive atmosphere
 * __Avoid negative, anxiety-ridden experiences__: focus on meaning instead of memorization, emphasise problem-solving strategies rather than aiming towards finding a solution, model a positive attitude, create fun challenges for students, encourage self-reflection (metacognition)
 * __Establish clear expectations__: clearly outline values and unanimous class rules regarding respect, 'do only what makes sense to you,' encourage metacognition
 * __All students are as likely as one another to have an aptitude for maths:__ regardless of race, culture, gender, socio-economic status etc. Expect high achievement from everyone, use a wide variety of role models from different backgrounds, re-iterate equality in ability to parents
 * __Retention:__ meaningful learning, active participation (measuring a circle with a ruler, rather than being told), establish connection to well-known ideas, period re-iteration of key ideas
 * Behaviourism** - the idea that learning means producing a particular response to a particular stimulus (a teacher instructs class on find average, when presented with a group of number, class finds average). Excludes cognition (thought processes). Provides instructional guideline, short-term progress, emphasises accountability (clear goal at start of lesson, i.e. multiplying fractions). Behaviourism teaches that behaviours can be shaped by re-enforcement.


 * Quotes:**
 * 'Students with conceptual knowledge can link ideas in networks of connected meaning, incorporate new information into those networks, and see relationships among different prices of information.' (Hiebert and Grouws, 2007)
 * 'Learning is active and internally monitored; it is a process of acquiring, discovering and constructing meaning from experience.' (Reys et al, 2009)
 * 'A student who learns a procedure without meaning will have dificulty knowing when to use it, remembering how to do it, applying it in new situations, and judging if the results are reasonable' (Reys et al, 2009)
 * 'Students learn mathematics with understanding by actively building new knowledge from their personal experiences and prior knowledge.' (Reys et al, 2009)
 * 'Teaching occurs only to the extent that learning occurs' (Reys, 2009)


 * //__3 BASIC TENETS OF CONSTRUCTIVISM:__//**
 * 1) ** Knowledge is not passively received; rather, knowledge is actively created or invented (constructed) by students (Piaget, 1972) **
 * 2) ** Students construct new mathematical knowledge by reflecting on their physical and mental actions. THey observe relationshipsm recognize patters, and make generalisations and abstractions as they integrate new knowledge nto their existing mental structures (Dienes, 1960) **
 * 3) Learning reflects a social process in which children engage in dialogue and discussion with themselves as well as with others (including teachers) as they develop intellectually (Bruner, 1986)

__**How children learn (Reys, 2009):**__
 * Several characteristic and identifiable stages of thinking exist, and children progress through these stages as they grow and mature
 * Learners are actively involved in the learning process
 * Learning proceeds from the concrete to the abstract
 * Learners need opportunities for talking about or otherwise communicating their ideas with other.

__Specific points for students of low socio-economic status__ (Lubienski, 2007):
 * __Recommendations (taken from NCTM's Learning Principle:__**
 * Teach to the developmental characteristics of students
 * Actively involve students (exploration, sense-making, reflection, metacognition)
 * Move learning from concrete to abstract (manipulatives, models, multiembodiment, formal representation follows conceptualization)
 * Use communication to encourage understanding (oral, written, use of models to provoke discuss)
 * 1) Emphasise meaningful learning instead of rote memorization
 * 2) Ensure that student learning is achieving the intended curriculum goals of the teaching (that the intended learning is occurring)
 * 3) Reflect on achievement results, identify problem topics, and thus, they can be remedied
 * 4) Teacher should promote the highest quality of teaching and curriculum for all students, regardless of socio-economic status (or anything else, for that matter! DW)

Procedural: follows a sequence eg. a²=b. No meaning attributed to steps. Conceptual knowledge: Meaning and understanding of mathematical ideas i.e. when you multiply a number by any other number than 1, you always get a bigger number than you started with.  ** ** Steve uses both conceptual and procedural knowledge to divide the sweets amongst his friends. He understand the concept of dividing equally, so everyone ends up with the same amount. He comprehends the procedure involved in dividing the sweets (23/4 can be worked out by knowing that 4x5=20, with a remainder of 3). Steve also knows
 * Learning Activity 3.2:
 * __ __Slide 1 – Steve__

__Slide 2 – Jill__** __**

** __Slide 3 – Nancy__ **