TEACHING AND LEARNING INTEGERSTEACHING 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 ...

DIAGNOSTIESE SIKLUS KRITIESE UITKOMSTE

DATABASIS SPESIFIEKE UITKOMSTE

Wiskunde Gemeenskap Kinders

Ontleding Navorsing

Doelstellings Didaktiek

DIAGNOSE sterk intuïsies wankonsepte

HIPOTETISEERverklarings

ONTWERPaktiwiteite ONDERRIG

REFLEKTEER/ASSESSEER/EVALUEER

HEELGETALLEWatter DRIE van die volgende dink jy souleerlinge VOOR onderrig in bewerkings(maar NA ’n bekendstelling aan die konsepen die notasie) die MAKLIKSTE die MOEILIKSTE vind?

1. ¯7 + ¯52. 10 + ¯33. ¯4 + 74. ¯8 + 35. ¯12 – ¯36. ¯5 – ¯127. 3 – 88. ¯7 – 49. 8 – ¯310. 6 ¯411. ¯7 512. ¯3 ¯4

100°C: water kook

¯5°C: minus 5 grade(5 grade ONDER vriespunt)

% of learners correct

GRADE 8

before teaching GRADE 9

after teaching GRADE 10

after experience

¯7 + ¯5 61 74 94 ¯12 – ¯3 57 63 76 6 ¯4 54 84 97 ¯8 + 3 52 78 93 10 + ¯3 51 75 94 3 – 8 49 69 86 ¯4 + 7 48 77 92 ¯7 5 45 74 94 ¯5 – ¯12 34 55 58 ¯3 ¯4 33 85 96 ¯7 – 4 21 50 55 8 – ¯3 17 46 48

RESEARCH RESULTS:RESEARCH RESULTS:

1 . V u s i h e t 4 3 b e e s te . H y v e rk o o p 1 8 b e e s te .H o e v e e l b e e s te is o o r?

2 . V u s i h e t 4 3 b e e s te . J o h n h e t 1 8 b e e s te .H o e v e e l m e e r b e e s te h e t V u s i a s J o h n ?

3 . V u s i h e t 4 3 b e e s te . J o h n h e t 1 8 b e e s te .H o e v e e l b e e s te m o e t J o h n k ry o m n e ts o v e e l a s V u s i te h ê ?

x = 43 – 18

DIFFERENT MEANINGS OF SUBTRACTIONDIFFERENT MEANINGS OF SUBTRACTION

Neem weg??

18 + x = 43

Onderrigteorie en -praktyk in Wiskunde: 'n KortbegripOnderrigteorie behels pogings tot

identifisering en beskrywing van die verskillende opsies (alternatiewe) wat daar ten opsigte van Wiskunde-onderrig en -leer bestaanidentifisering 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, enverklaring 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?So, wat is die alternatiewe?

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, endie 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.

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

Grayson Wheatley (1991)Grayson Wheatley (1991)

ENGINEERING METAPHORE

SENDER

RECEIVER

MESSAGE

MEDIUM

Leerkrag

LeerderLeerstof

INKLEDING WISKUNDE: Intuïtief tot formeelKONKRETEMODELLE

TemperatuurSkuldFilmAtoomkern

SEMI-KONKRETEMODELLE

GetallelynGrafieke

PATRONEDISKRETEOBJEKTE

INVERSES(OORLOG)

WISKUNDEAKSIOMAS REËLS

¯7 + ¯510 + ¯3¯4 + 7¯8 + 3¯12 – ¯3¯5 – ¯123 – 8¯7 – 48 – ¯36 ¯4¯7 5¯3 ¯4

KONTEKS VIR MOTIVERING HULPMIDDEL OM ANTWOORD TE KRY (VOORLOPIGE ALGORITME) REÊL AANVAARBAAR TE MAAK (VERKLAAR/BEWYS) HULPMIDDEL OM TE ONTHOU

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

6 4387

421 8

6 43870

42018

BRING DOWN! SUBTRACT!

1. Divide2. Multiply3. Subtract4. Birdie falls out of nest

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

A helpful mnemonic – a memory aid - for remembering the definitions of the trigonometric functions is given by "OH, AH, OA," or "SOH CAH TOA", i.e., Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, Tangent equals Opposite over Adjacent

Mnemonics for remembering SOH CAH TOA include:

Sex On Holiday Can Add Highlights To Our Adventures

Sex On Holidays Can Always Have The Odd Advantage

THE AFFECT OF RULES

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

A c t i v i t y 1 : B u i l d i n g b r i d g e sW h e n b u i l d i n g a b r i d g e , e n g i n e e r s h a v et o l e a v e s m a l l g a p s i n t h e r o a d b e t w e e nb r i d g e s e c t i o n s t o a l l o w f o r h e a te x p a n s i o n .F o r a c e r t a i n b r i d g e t h e s i z e o f t h e g a p i s2 ,3 c m a t a t e m p e r a t u r e o f 0 C . F o r e a c h 1 C t h a t t h e t e m p e r a t u r er i s e s , t h e g a p b e c o m e s s m a l l e r b y 0 , 0 5 c m .

1 . C o m p l e t e t h e f o l l o w i n g t a b l e s h o w i n g t h e s i z e o f t h e g a p a td i f f e r e n t t e m p e r a t u r e s :

T e m p e r a t u r e ( t C ) 0 1 2 3 4 2 0 3 0 ¯ 5 ¯ 1 0

G a p s i z e ( d c m ) 0

d

d = 2,3 – 0,05t d = 2,3 – 0,05(¯1)

REAL WORLD MATHEMATICS

DEFINITION The curriculum for Mathematics is based on the following view of the nature of the discipline. Mathematics enables creative and logical reasoning about problems in the physical and social world and in the context of Mathematics itself. It is a distinctly human activity practised by all cultures. Knowledge in the mathematical sciences is constructed through the establishment of descriptive, numerical and symbolic relationships. Mathematics is based on observing patterns; with rigorous logical thinking, this leads to theories of abstract relations. Mathematical problem solving enables us to understand the world and make use of that understanding in our daily lives. Mathematics is developed and contested over time through both language and symbols by social interaction and is thus open to change.

The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach. . . . a teacher wishing to impart the right attitude of mind toward problems to his students should have acquired that attitude himself. -George Polya, How to Solve it, 1945

WANTED: A SWIMMING-TEACHER WHO CAN SWIM HIMSELF

Freudenthal was fiercely opposed to what he called a didactical inversion, where the end results of the work of mathematicians were taken as starting points for mathematics education.

He said this was anti-didactical!

As an alternative, Freudenthal advocated that mathematics education should take its starting point in mathematics as an activity, and not in the teaching of mathematics as a ready-made-system.

CONCEPTSFIRST PRINCIPLES

SEMANTIC MEANINGPRELIMENARY ALGORITHM

SYMBOLSRULES

SYNTACTIC MEANINGFINAL ALGORITHM

GRADUAL SOPHISTICATION

0,2 0,03 = ?

1000)32()10010()32()1003()102(

)10010003,0()10102,0(03,02,0

FRONT

FINISHED MATHEMATICS

006,01000

6100

3102

03,02,0

BACKMAKING MATHEMATICS

Number of decimal places …

212

332

3)3(2

362

xx

xx

xx

23

)3(2362

xx

xx

FRONT

FINISHED MATHEMATICS

BACKMAKING MATHEMATICS

TRANSPOSE!

2x + 3 – 3 = 5 – 3

2x + 0 = 5 – 3

2x = 5 – 3

2x + 3 = 5

2x = 5 – 3

Solve for x: 2x + 3 = 5

Learners can themselves gradually shorten the real thing from back to front!

Why the surface, face-value interpretation of “taking over”??

Aktiwiteit 4: Vermenigvuldig met negatiewe getalle

1. Wat dink jy is die antwoord van ¯5 ¯7?Hoekom dink jy so? Bespreek!

2. Voltooi hierdie patrone en bespreek:

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 =

. . . 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

THE CASE OF DECIMALSTHE CASE OF DECIMALS

Grade 6: three decimal places:Grade 6: three decimal places:Arrange from the smallest to largest:0.234 0.725 0.483

Grade 5: two decimal places:Grade 5: two decimal places:Arrange from the smallest to largest:0.23 0.72 0.48

Grade 4: one decimal place: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

2 12 12 + 12

10 + 10 =202 + 2 = 420 + 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:A MULTIPLICATION EXAMPLE:

’’N AANBIEDINGSTRATEGIEN AANBIEDINGSTRATEGIE

1. DIAGNOSE VAN INTUÏSIES/WANKONSEPTE1. DIAGNOSE VAN INTUÏSIES/WANKONSEPTEDiagnostiese toets, klasbespreking

2. KONSEPONDERSTEUNING

• VERGELYK 4 vs 2, ENS

• VERGELYKINGS: 4 + x = 3

•TEMPERATUUR

3. DISKRETE OBJEKTE /ANALOGIE MET POS GETALLE7 + 5 ...

6 4 ...7 – 5 ...

Laat kinders hul intuïsies gebruik en formaliseer!

Activity 1: Working with negative numbers1. You may have heard on the weather forecast that

they sometimes say the temperature is "3 degreesbelow freezing point". This is often indicated as"negative 3 degrees" and written as ¯3.If the temperature is 5°C and it becomes 10 degreescolder, what will the temperature be?

2. In a quiz, contestants can choose questionscounting 5, 10 or 20 points. If the contestantanswers correctly, the points are added to his score,but if the answer is wrong, the points are subtractedfrom his score.John has 10 points and answers a 15 point questionwrong. What is his score now?

3. Do the following calculations as you think theyshould be done.(a) ¯8 + ¯6 (b) ¯12 – ¯5(c) ¯4 + ¯4 + ¯4 + ¯4 + ¯4 (d) 4 ¯6

4 4 = 16 4 4 = 16

4 3 = 12 3 4 = 12

4 2 = 8 2 4 = 8

4 1 = 4 1 4 = 4

4 0 = 0 0 4 = 0

4 ¯1 = ¯1 4 =

4 ¯2 = ¯2 4 =

4 ¯3 = ¯3 4 =

WAT VAN ¯3 4 ? PATRONE

Formuleer eie reëlsFormuleer eie reëls

(hulpmiddel om te (hulpmiddel om te onthou; nodig vir onthou; nodig vir spoed …)spoed …)

4. OORLOG-OORLOG4. 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 INDUKSIEEie reëls via INDUKSIE

Verdere oefening waar Verdere oefening waar leerlinge hul REËLS gebruikleerlinge hul REËLS gebruik

5. ATOOM?5. ATOOM?

7 – 5 =

9 – 4 =

1 – 6 =

...

Voorlopige algoritme:

7 – 5 = 12 + 5 –5

= 12 + 0

= 12

4 – 4 = 04 – 3 = 14 – 2 = 24 – 1 = 34 – 0 = 44 – ¯1 = 4 – ¯2 = 4 – ¯3 =4 – ¯4 =4 – ¯5 =

Eie reëls via Eie reëls via INDUKSIEINDUKSIE

Refleksie: Aftrek maak nie kleiner nie!

6. PATRONE/AKSIOMAS?6. PATRONE/AKSIOMAS?¯3 ¯4 = ?

4 ¯4 = ¯16 ¯4 4 = ¯163 ¯4 = ¯12 ¯4 3 = ¯122 ¯4 = ¯8 ¯4 2 = ¯81 ¯4 = ¯4 ¯4 1 = ¯40 ¯4 = 0 ¯4 0 = 0¯1 ¯ 4 = ¯4 ¯1 = ¯2 ¯4 = ¯4 ¯2 = ¯3 ¯4 = ¯4 ¯3 =

Voorlopige algoritme:

Eie reëls via induksieEie reëls via induksie

Kliek vir aktiwiteit:

Deduktiewe oortuiging?¯3 0 = ¯3 (4 + ¯4) = 0

¯3 4 + ¯3 ¯4 = 0

¯12 + ?? = 0