teaching and learning integers first, we agree, that learning/understanding means connecting it to...
TRANSCRIPT
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
HEELGETALLE
Watter 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 + ¯5
2. 10 + ¯3
3. ¯4 + 7
4. ¯8 + 3
5. ¯12 – ¯3
6. ¯5 – ¯12
7. 3 – 8
8. ¯7 – 4
9. 8 – ¯3
10. 6 ¯4
11. ¯7 5
12. ¯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 Kortbegrip
Onderrigteorie behels pogings totidentifisering 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 e
r 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
3
10
2
03,02,0
BACK
MAKING MATHEMATICS
Number of decimal places …
2
123
32
3
)3(23
62
x
xx
xx
x
23
)3(23
62
x
xx
x
FRONT
FINISHED MATHEMATICS
BACK
MAKING 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 GETALLE
7 + 5 ...
6 4 ...
7 – 5 ...
Laat kinders hul intuïsies gebruik en formaliseer!
Activity 1: Working with negative numbers
1. You may have heard on the weather forecast thatthey 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