1. TEACHING AND LEARNING INTEGERS First, we agree, that learning/understanding means connecting it to previous knowledge. SO, THE FUNDAMENTAL, MOST IMPORTANT PRINCIPLE IS THAT PLANNING TEACHING SHOULD BE BASED ON WHAT WE KNOW ABOUT CHILDREN. YES, CHILDREN IN GENERAL, BUT ESPECIALLY THESE CHILDREN IN FRONT OF ME IN CLASS! SO, WE MUST KNOW WHAT CHILDREN KNOW!!!!! Use a diagnostic approach...

4. RESEARCH RESULTS:

6. x = 43 – 18 DIFFERENT MEANINGS OF SUBTRACTION Neem weg?? 18 + x = 43

7. Onderrigteorie en -praktyk in Wiskunde: 'n Kortbegrip Onderrigteorie behels pogings tot  identifisering en beskrywing van die verskillende opsies (alternatiewe) wat daar ten opsigte van Wiskunde-onderrig en - leer bestaan  identifisering en ontleding van die implikasies van die uitoefening van verskillende opsies ten opsigte van die aard en gehalte van leeruitkomste sowel as van die produktiwiteit (spesifiek tydseffektiwiteit) van Wiskunde-onderrig, en  verklaring van die verskille tussen die implikasies van verskillende opsies. Onderrigpraktyk behels die rasionele keuse tussen opsies vir spesifieke inhoude en spesifieke leerlinge. So, wat is die alternatiewe?

8. Om Wiskunde te leer behels die konstruksie van wiskundige "begrippe" (in die mees algemene sin van die woord) deur leerders. Leer is 'n individuele konstruktiewe sowel as 'n sosiale interaktiewe proses. Wiskunde-onderrig behels  die inisiëring van leergeleenthede, d.w.s. geleenthede waarbinne leerders wiskundige begrippe kan konstrueer, sowel as  die bestuur van hierdie geleenthede, en  die monitering van die leeruitkomste. 'n Basiese opsie wat telkens in Wiskunde-onderrig uitgeoefen moet word, is of leerders geleentheid gegee word om hul kennis na aanleiding van die uitvoering van take/die oplos van probleme te konstrueer, of by wyse van vertolking van beskrywings (uiteensettings, verduidelikings) wat aan hulle verskaf word. Indien dit d.m.v. take/probleme is, is daar die opsie om die probleme individueel of in kleingroepe op te los.

9. Ideas and thoughts cannot be communicated in the sense that meaning is packaged into words and "sent" to another who unpacks the meaning from the sentences. That is, as much as we would like to, we cannot put ideas in students' heads, they will and must construct their own meanings. Our attempts at communication do not result in conveying meaning but rather our expression evoke meaning in another, different meanings for each person. Grayson Wheatley (1991)

10. ENGINEERING METAPHORE SENDER RECEIVER MESSAGE MEDIUM Leerkrag LeerderLeerstof

11. Kan jy ’n verduideliking gee vir elke bewerkingsgeval vir elke konteks? Motiveer as dit onmoontlik is!

12. 6 438 7 42 1 8 6 438 70 420 18 BRING DOWN! SUBTRACT! 1. Divide 2. Multiply 3. Subtract 4. Birdie falls out of nest

13. DO NOT CONFUSE A MNEMONIC – A MEMORY AID WITH UNDERSTANDING! The steps for long division are Divide, Multiply, Subtract, Bring Down: Dad Mom Sister Brother Dead Monkies Smell Bad Dracula Must Suck Blood

14. Sex On Holiday Can Add Highlights To Our Adventures Sex On Holidays Can Always Have The Odd Advantage

15. THE AFFECT OF RULES

16. THOU SHALT NOT DIVIDE BY ZERO! THOU SHALT NOT ADD UNLIKE TERMS! FIRST MULTIPLY, THEN ADD

17. d = 2,3 – 0,05 t d = 2,3 – 0,05( ¯ 1) REAL WORLD MATHEMATICS

19. WANTED: A SWIMMING-TEACHER WHO CAN SWIM HIMSELF

21. CONCEPTS FIRST PRINCIPLES SEMANTIC MEANING PRELIMENARY ALGORITHM SYMBOLS RULES SYNTACTIC MEANING FINAL ALGORITHM GRADUAL SOPHISTICATION

22. 0,2  0,03 = ? FRONT FINISHED MATHEMATICS BACK MAKING MATHEMATICS Number of decimal places …

23. FRONT FINISHED MATHEMATICS BACK MAKING MATHEMATICS

24. TRANSPOSE! 2x 2x + 3 – 3 = 5 – 3 2x 2x + 0 = 5 – 3 2 x = 5 – 3 2x 2x + 3 = 5 2x 2x = 5 – 3 Solve for x : 2 x + 3 = 5 Learners can themselves gradually shorten the real thing from back to front! Why the surface, face-value interpretation of “taking over”??

26. ... the research brings Good News and Bad News. The Good News is that, basically, students are acting like creative young scientists, interpreting their lessons through their own generalizations. The Bad News is that their methods of generalizing are often faulty. Steve Maurer, 1987 The symbolism of algebra is its glory. But it is also its curse. William Betz, 1930

27. THE CASE OF DECIMALS Grade 6: three decimal places: Arrange from the smallest to largest: 0.234 0.725 0.483 Grade 5: two decimal places: Arrange from the smallest to largest: 0.23 0.72 0.48 Grade 4: one decimal place: Arrange from the smallest to largest: 0.2 0.7 0.4 Arrange from the smallest to largest: 0.23 0.7 0.483

28. 2  12 12 + 10 + =20 2 + 2 = 4 20 + 4 = 24 = 24 3  12 = 36 4  12 = 48 5  12 = 510 Should develop a mathematical culture! Check answers. Does it make sense? Is it always true? A MULTIPLICATION EXAMPLE:

29. ’N AANBIEDINGSTRATEGIE 1. DIAGNOSE VAN INTUÏSIES/WANKONSEPTE 1. DIAGNOSE VAN INTUÏSIES/WANKONSEPTE Diagnostiese toets, klasbespreking 2. KONSEPONDERSTEUNING VERGELYK  4 vs  2, ENS VERGELYKINGS: 4 + x = 3 TEMPERATUUR 3. DISKRETE OBJEKTE /ANALOGIE MET POS GETALLE  7 +  5... 6   4...  7 –  5... Laat kinders hul intuïsies gebruik en formaliseer!

31. 4  4 = 16 4  3 = 123  4 = 12 4  2 = 82  4 = 8 4  1 = 41  4 = 4 4  0 = 00  4 = 0 4  ¯1 =¯1  4 = 4  ¯2 =¯2  4 = 4  ¯3 =¯3  4 = WAT VAN ¯3  4 ? PATRONE Formuleer eie reëls (hulpmiddel om te onthou; nodig vir spoed …)

32. 4. OORLOG-OORLOG 10 +  3 = 8 +  5 = 12 +  2 = Voorlopige algoritme om antwoorde te ontwikkel as data vir induksie, bv. 10 +  3 = 7 + 3 +  3 = 7 + 0 = 7 1 +  3 = 4 +  5 = 2 +  8 =  3 + 7 =  8 + 5 =  6 + 9 = Eie reëls via INDUKSIE Verdere oefening waar leerlinge hul REËLS gebruik

33. 5. ATOOM? 7 –  5 = 9 –  4 = 1 –  6 =... Voorlopige algoritme: 7 –  5 = 12 +  5 –  5 = 12 + 0 = 12 4 – 4 = 0 4 – 3 = 1 4 – 2 = 2 4 – 1 = 3 4 – 0 = 4 4 – ¯1 = 4 – ¯2 = 4 – ¯3 = 4 – ¯4 = 4 – ¯5 = Eie reëls via INDUKSIE Refleksie: Aftrek maak nie kleiner nie!

34. 6. PATRONE/AKSIOMAS? ¯3  ¯4 = ? 4  ¯4 = ¯16 ¯4  4 = ¯16 3  ¯4 = ¯12 ¯4  3 = ¯12 2  ¯4 = ¯8 ¯4  2 = ¯8 1  ¯4 = ¯4 ¯4  1 = ¯4 0  ¯4 = 0 ¯4  0 = 0 ¯1  ¯ 4 = ¯4  ¯1 = ¯2  ¯4 = ¯4  ¯2 = ¯3  ¯4 = ¯4  ¯3 = Voorlopige algoritme: Eie reëls via induksie Kliek vir aktiwiteit: Deduktiewe oortuiging? ¯3  0 = ¯3  (4 + ¯4) = 0  ¯3  4 + ¯3  ¯4 = 0 ?  ¯12 + ? = 0