SNE4210 - Arithmetic acquisition and barriers: Teaching and learning (10.04.07) : SNE4210 - Arithmetic acquisition and barriers: Teaching and learning (10.04.07) Guri A. Nortvedt
Contact information : Contact information g.a.nortvedt@isp.uio.no
22 85 48 77
Today’s and next week’s lectures : Today’s and next week’s lectures Outside and in: - raise some questions to think about: about mathematics teaching and learning - something about the present state - good general principles - some examples from special education “Everybody knows what it means to KNOW mathematics, but there is no consensus on how to TEACH mathematics!”
What is the role of learning in teaching? : What is the role of learning in teaching? Teaching = learning
Teaching learning
Teaching learning
“Traditional teaching” : “Traditional teaching” Believing knowledge can be transferred from one person to another
Students seen as empty boxes – to be filled
Telling or explaining
Black board
Text books
Independent work
Silent work
The case of Norway …(Solli, 2004) Traditional special needs teaching in mathematics: : The case of Norway …(Solli, 2004) Traditional special needs teaching in mathematics: Inclusive schools
Norwegian students really like their schools
0,4 % in special needs schools
5,5 % of the students
Gender difference: 70 % boys – 30 % girls
Late
Not as successful as wanted
Traditionally – one teacher – one student
Doing over and over again
The same procedure as last year
Fragmenting and isolating knowledge
The case of Norway – teacher’s and teacher student’s reluctance to teaching mathematics : The case of Norway – teacher’s and teacher student’s reluctance to teaching mathematics I can do it if you tell me the formulae, but I do not understand why it works….
Mathematics viewed as instrumental
I prefer not to….
Many teachers lack formal training
An official view? “If you teach young children you need less formal training that if you teach secondary school children”
Young children : Young children Comes to school eager to learn to read and do mathematics
Already have much knowledge and competencies: - can count - simple sums and subtractions - can compare - can describe - can sort - have an understanding for time and space (- have a language)
The squirrel task – a division problem: (1 squirrel mom, 3 squirrel children and 13 nuts) (Alseth, 2003)
Core questions : Core questions 1. What is mathematical competency?
Mathematics for all
Skills for life long learning
Back to basics movements
2. How can mathematics be thought?
3. What obstacles in learning mathematics should teaching be sensitive towards?
Mathematical competency – one model – Mogens Niss (2002) : Mathematical competency – one model – Mogens Niss (2002) … to be mathematically competent is to be prepared to act with consciousness and insight in situations containing a specific kind of (mathematical) challenges
* To be able to pose and answer questions in, with and about mathematics
To be able to use the mathematical language and tools
(http://pub.uvm.dk/2002/kom/) (full report – but in Danish, so not for all of you)
Components of mathematical competency: : Components of mathematical competency:
Slide12 :
We are not really addressing mathematics – only arithmetic : We are not really addressing mathematics – only arithmetic Think about it – in this model – what does arithmetic consist of?
An example: 15 + 3 : An example: 15 + 3 Representations: symbol, concrete, tallies/ drawings/ fingers
Problems solving or fact retrieval or standard algorithm or….
Language - sum – addend – add…
Number sense – example 1 (Griffin, 2003) : Number sense – example 1 (Griffin, 2003) Can you see 3/5 of something?
Can you see 5/3 of something?
Can you see 3/5 of 5/3?
Can you see 2/3 of 3/5?
Can you see 1 + 3/5?
Three teacher orientations towards teaching numeracy : Three teacher orientations towards teaching numeracy Askew et al (King’s College) (1997, 2000, 2001): - What characterizes effective teaching? - Who is at the centre? Responsibilities? Roles?
Transmission
Discovery
Connectionist
Important message from Askew et al : Important message from Askew et al What teachers say and what they do!
Found in other research projects –
The NCTM vision (Standards, 2000) (The National Council for Teachers of Mathematics) : The NCTM vision (Standards, 2000) (The National Council for Teachers of Mathematics) Constructivism is a theory about knowledge and learning; it describes both what “knowing” is and how one “comes to know”. Based on work in psychology, philosophy, and anthropology, the theory describes knowledge as temporary, developmental, monoobjective, internally constructed, and socially and culturally mediated. Learning from this perspective is viewed as a self-regulatory process of struggling with the conflict between existing personal models of the world and discrepant new insights, constructing new representations and models of reality as a human meaning-making venture with culturally developed tools and symbols, and further negotiating such meaning through cooperative social activity, discource, and debate. (Fosnot, 1996)
What then should teachers do? : What then should teachers do?
A counting example (Fuson, 1991) : A counting example (Fuson, 1991) String level
Unbreakable list level
Breakable chain level
Numerable chain level
Bidirectional chain level
Donlan and Hutt (1991)
Susie – age 8
Wants to learn to count to 100
Slide21 :
54 : 2 = (National tests project – Norway – Nortvedt, unpublished) : 54 : 2 = (National tests project – Norway – Nortvedt, unpublished) Grade 4 student – girl
54 : 2 =
Divides worksheet in two – makes a tally on left hand side – right hand side – left hand side – a total of 54 times
Counts each side – gets totals of 27 on both sides
Concludes – 54 : 2 = 27
Hundred boards : Hundred boards (hundred board on over head projector to illustrate patterns and regularities, different strategy use among participants…)
Counting : Counting For next weeks lecture, I would like you to think about how teachers in your country views counting in the mathematics classroom. What would a teacher do if a student use counting to solve a mathematics problem?