1. SNE4210 - Arithmetic acquisition and barriers: Teaching and learning (10.04.07) Guri A. Nortvedt

2. Contact information g.a.nortvedt@isp.uio.no 22 85 48 77

3. 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!”

4. What is the role of learning in teaching? Teaching = learning Teaching learning

5. “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

6. 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

7. 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”

8. 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)

9. 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?

10. 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)http://pub.uvm.dk/2002/kom/

11. Components of mathematical competency: To be able to pose and answer questions in, with and about mathematics To be able to use the mathematical language and tools Mathematical thinkingRepresentations Problem solvingSymbols and formalism ModellingCommunication ReasoningTools like calculators, computers, rulers …

13. We are not really addressing mathematics – only arithmetic Think about it – in this model – what does arithmetic consist of?

14. An example: 15 + 3 Representations: symbol, concrete, tallies/ drawings/ fingers Problems solving or fact retrieval or standard algorithm or…. Language - sum – addend – add…

15. 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?

16. 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

17. Important message from Askew et al What teachers say and what they do! Found in other research projects –

18. 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)

19. What then should teachers do?

20. 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

22. 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

23. Hundred boards (hundred board on over head projector to illustrate patterns and regularities, different strategy use among participants…)

24. 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?