afemo - One, Two, Infinity…
Transcription
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 • • • • 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… • • • • • 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… • • • • 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. Download • www.onetwoinfinity.ca • Recent presentations • AFEMO