1. Developing Mathematical Thinkers
Marian Small
Sydney
August, 2015
#LLCAus
#LoveLearning
@LLConference

2. What is math?
Is it doing things?
Is it thinking about things?

3. For example
If you are teaching students to deal with problems involving addition and subtraction of numbers to 18 using mental strategies…

4. Is your goal…
Getting the answers?

5. Or is your goal… That students know when to add and when to subtract?
That they can predict answer sizes?
That they recognize what strategies make calculations simpler to do?

6. Or suppose
You are working with problems requiring estimation and calculation of perimeters and areas of rectangles.

7. Is the goal?
To simply solve those problems?
Do the calculations?

8. Or is the goal?
Realizing when area numbers are bigger and when perimeter numbers are bigger?
That perimeter and area are somewhat independent?
Realizing that knowing any two of length, width, perimeter, and area forces the other two?

9. If you are …
Working on problems using the Pythagorean theorem…

10. Is the goal…
To just solve the problems?
To apply the theorem?

11. Or is the goal…
To recognize the uniqueness of a situation where knowing two sides of a triangle automatically gives you the other side
Knowing when the theorem can be applied
Knowing why the theorem makes sense?

12. If the topic is..
Solve systems of two linear equations involving two variables, using the algebraic method of substitution or elimination

13. Is the goal…
To solve the systems?

14. Or is the goal… To know what it means when you solve a system
To know when or why you would do it algebraically vs when you would not
To know why there is only one solution (most of the time)

15. In general Math can be about “doing” or can be about “thinking”.
In our current society, thinking probably matters more.

16. Thinking vs Doing Consider these alternative questions.
How many dots are there? OR
How would you arrange 8 dots to make it easy to tell it’s 8?

17. Thinking vs Doing Consider these alternative questions.
What is the name of this shape? OR
What shapes are a lot like rectangles, but not quite rectangles?

18. Thinking vs Doing Consider these alternative questions.
What is this number? OR
A number is represented with some hundred flats, twice as many ten rods and three times as many ones as rods.
What could the ones digit be?

19. Thinking vs Doing Consider these alternative questions.
How long is an activity that begins at 11:00 and ends at 11:45? OR
You complete three activities between 10:00 a.m. and 11. Activity 2 takes twice as long as Activity 1. Activity 3 takes twice as long as Activity 2. Make a schedule.

20. Thinking vs doing Consider these alternative questions.
How many days are in 8 weeks? OR
Choose either minutes, hours, days, weeks or years to fill in the blank.
Then write your amount of time using other units.
500 ____ is the same as about ….

21. Thinking vs Doing Consider these alternative questions.
Read this number: 4023 OR
What numbers take exactly four words to say?

22. Thinking vs Doing Consider these alternative questions.
Measure this angle. OR
An angle’s measure is really easy to estimate. What might it look like?

23. Thinking vs Doing Consider these alternative questions.
Divide 483 by 3. OR
Without actually dividing, tell why ÷ 3 has to be more than 484 ÷ 4.

24. Thinking vs Doing Consider these alternative questions.
What is the perimeter of this rectangle? OR
A rectangle has a perimeter triple its length. What could its dimensions be?
4 cm
10 cm

25. Thinking vs Doing Consider these alternative questions.
Determine the perimeter of this rectangle. OR
Which number is usually greater- the number of centimetres in the perimeter of a rectangle or the number of square centimetres in the area? Explain.
5 cm
9 cm

26. Thinking vs Doing Consider these alternative questions.
4/22/2017
Thinking vs Doing
Consider these alternative questions.
What is the ratio of green to blue counters? OR
Two arrangements of counters both show the ratio 3:4, but they use different numbers of counters. How could that happen?
3 green and 4 blue OR 3 green and 1 blue OR 6 green and 8 blue OR…

27. Thinking vs Doing Consider these alternative questions.
What is the 20th term in 3, 5, 7, 9, …? OR
The 20th term in a growing pattern is 41. What are possibilities for the pattern?

28. Thinking vs Doing Consider these alternative questions.
What is 10% of 300? OR
Is 10% a lot or not?

29. Thinking vs Doing Consider these alternative questions.
4/22/2017
Thinking vs Doing
Consider these alternative questions.
How does the mean of the numbers 5, 10, 10, 20, 100 change if the number 100 is eliminated? OR
You calculate the mean of a set of 5 numbers. It decreases to almost 1/3 of its size by removing one number. What could the numbers have been?
Goes from 29 TO OR , 10, 10, 10, (30) to 10

30. Thinking vs Doing Consider these alternative questions.
4/22/2017
Thinking vs Doing
Consider these alternative questions.
What is the solution to
3/4 x – 2 = 5/8 x + 9? OR
Do equations with fractions in them usually have whole number solutions or fraction solutions?
Goes from 29 TO OR , 10, 10, 10, (30) to 10

31. Suppose You have to cover: Demonstrate an understanding of place value
You might ask students to write a number with a 7 in the ones place, 3 in the tens place and 4 in the hundreds place. OR

32. It could be A question like:
4/22/2017
It could be
A question like:
Create a number that has two 7s in it and two 4s in it, BUT
One 7 has to be worth 100 times as much as the other and
One 4 has to be worth 10 times as much as the other
7447 Or or 744.7

33. Suppose You have to cover: Adding and subtracting money amounts.
You might provide some prices and ask for total cost and change from $10. OR

34. 4/22/2017
It could be
A task like:
You have to buy 3 items so that your change if you gave the clerk two $20-bills and a $10-bill would be one bill and 8 coins.
What could the prices be?
How much is your change and what bill and coins make it up?
Change of $5 bill and 8 quarters– total $43, so $20 and $20 and $3
Change of $5 bill and 8 dimes- total $44.20, so $20, 20 and 4.20

35. Suppose
You are focusing on creating and analyzing symmetrical designs through reflections.
You could simply have students do a reflection and look at the symmetry OR

36. 4/22/2017
It could be
A task like:
You reflected a design that involved a square and a triangle.
One of the vertices of the square moved a total of 16 cm.
One of the vertices of the triangle moved a total of 8 cm.
Draw several possible designs.

37. Possible designs
8 cm
4 cm

38. Possible designs
8 cm
4 cm

39. Suppose You are focusing on equivalent fractions.
You could ask students to create equivalent fractions for a given one or test whether two fractions are equivalent OR

40. It could be A question like:
4/22/2017
It could be
A question like:
You found an equivalent fraction to 2/5 but the numerator and denominator were 36 apart.
What could that fraction have been?
Why does your answer make sense?
24/60

41. Suppose
You are focusing on multiplicative relationships involving fractions and decimals.
You could ask questions like: I have 20 books. My brother has 1 ½ times as many. How many does he have? OR

42. 4/22/2017
It could be
A task like:
Create a pattern block design where the area of the red section is 2 ½ times the area of the yellow section and the area of the blue section is 2/3 the area of the green section.
7447 Or or 744.7

43. Possibility

44. Suppose Suppose you are teaching about unit rates.
You could ask the price for one item if you give the price for 5 OR

45. It could be A question like:
4/22/2017
It could be
A question like:
You know that you can buy a 1.77 L container of laundry detergent for $5.95. It does 38 loads of laundry.
Calculate each of these unit rates. Decide which you think is more useful.
7447 Or or 744.7

46. It could be How many loads/1 L? How many loads/$1? How many litres/$1?
4/22/2017
It could be
How many loads/1 L?
How many loads/$1?
How many litres/$1?
Cost/1L?
Recall: 1.77L for 38 loads for $5.95.
21.5 loads/1L loads per $ L for each $ $3.37 per litre

47. Suppose
You are teaching about the estimation and calculation of the areas of triangles and the areas of parallelograms.
You could simply give dimensions and ask for areas OR

48. It could be A question like:
4/22/2017
It could be
A question like:
Create a parallelogram and a triangle so that the parallelogram’s area is ¼ of the triangle’s.
How do you know there are lots more answers?
7447 Or or 744.7

49. Maybe

50. Suppose You are exploring addition and subtraction of integers.
You could ask straightforward addition and subtraction questions OR

51. It could be A question like:
4/22/2017
It could be
A question like:
I added some positive and negative integers using two-colour counters.
There were five times as many negative counters as positive ones.
What sums could I end up with?
What sums could I not end up with?
-4, -8, -12, …

52. Suppose Students are learning about evaluating
algebraic expressions by substituting
natural numbers for the variables.
You could simply ask students to do substitutions OR

53. It could be Questions like:
4/22/2017
It could be
Questions like:
An algebraic expression is worth 20 more when x = 4 than when x = 2. What could it be?
An algebraic expression is worth 30 less when x = 4 than when x = 2. What could it be?
10x – – 15x

54. Suppose
Students are exploring theoretical probability of two independent events.
You could ask for the probability of rolling a 6 and a 2 on two dice OR

55. 4/22/2017
It could be
A task like:
You are going to create two different spinners.
Create them so that the probability of spinning two blues is around 16%.
7447 Or or 744.7

56. Suppose Students are exploring equations of lines.
You could simply provide a graph or table of values and ask for an equation OR

57. 4/22/2017
It could be
A task like:
Provide some reasonable and some unreasonable equations for the line connecting these two points. Explain.
7447 Or or 744.7

58. Mathletics task
Leftovers

59. Task

60. Interactive

61. Interactive

62. Interactive

63. Mathletics task (to come)
Hypotenuse
The hypotenuse of a right triangle is about 2.6 times as long as one of the legs. What could the three side lengths be? Think of lots of possibilities, including some involving whole number centimeter lengths and some involving rational number of centimeter side lengths.
Verify that each combination is correct using the Pythagorean theorem.

64. Your turn Choose three different topics you teach.
Think about what “doing” questions would look like.
Now create “thinking” questions to replace them.
Then we’ll share.

65. So what are you thinking?
Is this possible for every topic?
How do you get yourself in this mode?
How do you get your students to go along too?

66. Download
You can download this at
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