afemo - One, Two, Infinity…

Comment développer la
pensée critique en
mathématiques à
l’intermédiaire?
Marian Small
October 2016
A REQUEST
•  Please do not videotape this presentation.
•  Please do not tweet out more than one or two slides, if
any.
Critical thinking involves
•  Reflection on your own and others’ thinking and
reasoning; this involves criteria setting
•  Recognizing that there are alternative points of view
•  Recognizing that there are always assumptions made
•  Even though we as teachers might initiate ideas about
critical thinking, it is only when the student does it
without being told that we really have a critical thinker.
For example…
•  Which bank account grew the most last year?
Ben’s
Lea’s
January 1
$1000
$100
Dec 31
$1200
$250
The point of view issue
•  Did you mean additive or did you mean multiplicative?
Or…
A car goes 280 km in 3 hours.
Which would be easiest for you to figure out?
How far it goes in
•  9 hours?
1 hour?
1.5 hours?
•  Why would it be easiest?
Criteria are used
In deciding whether the case for ease is well-made.
Or…
•  A shape does not have much area but it has lots of
perimeter.
•  Predict what features it would have to have.
•  How did you predict that?
Criteria come into play
•  In terms of how well the argument for prediction is made
This problem requires stated assumptions
•  Estimate the number of square centimetres of pizza that
all of the kids in Ontario eat in one week.
Assumptions
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How many kids are there in Ontario?
What proportion eat pizza in a week?
How much pizza do they eat?
How does that translate into square centimetres?
Critical thinking might occur if..
•  I asked you to make an argument as to why you can
divide by 5 by dividing by 10 and then multiplying by 2.
•  Just saying because 10 = 2 x 5 won’t be good enough
since it doesn’t tell me why it’s ok.
•  What would you expect as criteria for an explanation?
Maybe
•  Either something symbolic (e.g. n ÷ 5 = q means that
n = 5 x q, which is the same as 10 x q ÷ 2.
If n= 10 x q ÷ 2, then 2 x n = 10 x q, so 2 x n ÷ 10 = q,
so n ÷ 10 x 2 = q OR
• Something that relates to the meaning of division, e.g.
creating 5 equal groups happens if you create 10 equal
groups and then double up the groups.
• More than just a single example that works
• More than just the relationship between 2, 5 and 10
Asking the right questions
•  This is the heart of the issue.
•  We need to ask questions that encourage or even
demand critical thinking behaviours.
•  You could make it the “normal” way you teach.
For any topic, the potential exists
•  Grade 7:
•  établir les liens entre la multiplication, la division, le
raisonnement proportionnel et les concepts de rapport et
de taux
You might ask
•  Why does knowing how to multiply and divide allow you
to solve this problem?
•  Problem: You know that 10 boxes of cookies cost $18.
How much do 45 cost?
•  What assumptions are you making?
Criteria
•  You don’t just tell how to do the calculation, but tell why
the calculations are relevant.
•  I am assuming the same unit price for the 10 boxes as
for the 45 boxes.
For any topic, the potential exists
•  Grade 7:
•  établir et expliquer à l’aide de matériel concret, la
relation entre les fractions, les nombres décimaux, les
pourcentages et les rapports
I might ask…
•  How and why would using a hundredths grid help you
figure out the decimal and percent for 2/5?
I might ask…
•  How and why would using a hundredths grid help you
figure out the decimal and percent for 2/5?
I might ask…
•  Why might someone say that using money would be a
better model?
•  Criteria would focus on strong reasons and not just doing
it for the first part.
•  Focus on seeing an alternate point of view for the
second part
I might ask…
•  Think of a couple of problems where you would think of a
percent as a fraction to solve a problem.
•  Tell why using a fraction would be a good idea.
•  Criteria: again focuses on rationale.
For any topic, the potential exists
•  Grade 7:
•  additionner et soustraire dans divers contextes des
fractions positives en utilisant une variété de stratégies
I might ask
•  Amélie says that to figure out 1 2/3 – 3/4, you should add
¼ to 2/3, but she didn’t tell why.
•  How should she have explained it?
•  Criteria:
•  Focus on reasoning
•  Explanation of where ¼ and 2/3 came from AND why
they are added and not subtracted.
•  Explanation of what subtraction means in relation to this
problem
For any topic, the potential exists
•  Grade 7:
•  estimer et calculer des pourcentages
You might ask
•  To estimate 45% of 54 using mental math, what options
make sense?
•  Which do you think is best? Why?
•  Maybe 40% of 60
•  Maybe 50% of 50
•  Maybe 50% of 54
For any topic, the potential exists
•  Grade 7:
•  établir, à l’aide de matériel concret ou illustré, les
relations entre l’aire du trapèze et l’aire du
parallélogramme et entre l’aire du trapèze et l’aire du
triangle
You might ask
•  Someone says that you don’t need to learn the formula
for the area of a trapezoid. You can always just break it
up into triangles.
•  Do you agree or disagree? Why?
Or you might ask
•  What would be a good strategy for creating a trapezoid,
a triangle and a parallelogram with the same area?
•  Why is it a good strategy?
•  Focus is on criteria for good.
For any topic, the potential exists
•  Grade 7:
•  estimer et calculer le volume de prismes droits dans
divers contextes.
I might ask…
•  What is the volume of a minivan?
•  This requires assumption making.
For any topic, the potential exists
•  Grade 7:
•  identifier un solide à partir de ses vues de face, de côté
et de dessus.
I might ask…
•  Why do I need three views to pin down what a structure
looks like?
•  Does it have to be top, front and right side?
For any topic, the potential exists
•  Grade 7:
•  additionner et soustraire des monômes à l’aide de
matériel concret (p. ex., tuiles algébriques) dans le cadre
d’une résolution d’équation simple.
I might ask
•  You add a monomial you can represent with 8 tiles to
one you can represent with 4 tiles.
•  How many tiles MIGHT you need to represent the sum?
•  How many would you never use? Why?
For any topic, the potential exists
•  Grade 8:
•  décomposer des nombres naturels inférieurs à 144 en
produits de facteurs premiers
I might ask
•  Describe some different strategies you can use to
decompose a number into primes.
•  Which do you think is the better strategy?
•  When is it better? Why is it better?
For any topic, the potential exists
•  Grade 8:
•  déterminer le plus petit commun multiple de nombres
naturels à l’aide de facteurs premiers.
I might ask…
• 
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You have some counters.
When you make groups of 4, there are 3 left over.
When you make groups of 3, there is 1 left over.
Find a pattern in listing the possible numbers.
Explain the pattern.
Could be
•  7, 19, 31, 43, 55, 67,…
•  Why?
•  If you had 7, there would be 1 group of 4 and 3 left over.
•  There would be 2 groups of 3 and 1 left over.
Could be
•  If you add groups of 12, you have more groups of four or
more groups of 3, but no changes in leftovers..
For any topic, the potential exists
•  Grade 8:
•  multiplier et diviser des fractions positives, à l’aide ou
non de matériel concret ou semi-concret dans divers
contextes.
I might ask.
•  You multiply two numbers.
•  The result is a tiny bit less than one of the numbers and
just a little more than the other.
•  What could the numbers have been? Why?
Assumptions: What is tiny ? What is a little bit?
Criteria: Addresses why the answer has to be what it is,
including giving meaning to multiplication.
Maybe: 9/10 x 10/9
For any topic, the potential exists
•  Grade 8:
•  estimer et calculer l’aire de cercles.
I might ask
•  How could you convince someone, without just stating a
formula, why it makes sense that the area of a circle is
just a little more than 3 x r2?
Maybe
Or I might ask…
•  Since A = πr2 for a circle, Sarah thinks it is not possible
for the area to be a whole number of square units. What
do you think? Why?
For any topic, the potential exists
•  Grade 8:
•  expliquer l’effet d’une rotation (multiples de 90º) de
centre à l’origine sur les coordonnées d’un point dans le
plan cartésien
I might ask
•  The point on the image of a shape after a rotation with
the centre at the origin is in Quadrant III.
•  In which quadrant could the coordinates of the point on
the original shape have been located?
For any topic, the potential exists
•  Grade 8:
•  évaluer des expressions algébriques et des équations
simples en substituant des nombres entiers, des
fractions positives et des nombres décimaux
I might ask
•  I evaluated an algebraic expression.
•  Whenever I substituted in a fraction with a denominator
of 2, the output was an even whole number.
•  What could my expression NOT have been? Why?
•  What could it have been?
•  E.g. It was not x/2.
•  It could have been 2x +4
Or I might ask
•  I evaluated two algebraic expressions. When x increased
by 4, y increased by 8.
•  What could the expression have been? Why?
•  Different points of view: Do you mean whenever or
when?
•  It could be 2x+5 (then it always happens), but it could be
X (x + 5) –2 when you go from x = 2 to x = 6.
For any topic, the potential exists
•  Grade 8:
•  justifier la pertinence de conclusions basées sur le calcul
de la moyenne, de la médiane ou du mode.
I might ask..
•  I calculated the mean income at two different tech
companies for the employees.
•  At Company A, the mean was $95,000 a year.
•  At Company B, the mean was $104,000 a year.
•  Do you agree that Company B is a better place to work?
So…
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We need to teach students to consider:
Assumptions they make
Criteria for good solutions/explanations
That there are sometimes different points of view
•  And then we can ask questions where these can be
practised.
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