1. Promoting Mathematical Thinking Low Floor High Ceiling Problems
When you make the finding yourself—even if you’re the last person on Earth to see the light—you’ll never forget it….Carl Sagan

2. The only way to learn mathematics is to do mathematics

3. NPR: Struggle For Smarts
NPR: Struggle For Smarts? How Eastern And Western Cultures Tackle Learning
Ted Talk: Angela Duckworth – The Key to Success? GRIT
Dan Meyer: Math Class needs a Makeover
Encouraging Perseverance in Elementary Mathematics – A tale of two problems

4. Promoting Mathematical Thinking with Low Floor High Ceiling Tasks
A few problem resources
You Cubed:
nrich:
Math Circle Problems:
InsideMathematics:
Math Counts:

5. I tell my students, “The only thing you need to have for my program – and you must bring it every day – is ganas Jaime Escalante…Stand and Deliver
Mike said to Linda, "Bet you can't guess the number of marbles I have in this sack." "Give me a clue," she said. "I have more than 50 but fewer than If you divide them into piles of 8, there are 2 left over. If you divide them into piles of 7, there is one left over." Help Linda figure out how many marbles are in the sack.

7. Climbing the Staircase of Mathematics

8. I think we do need to spend more time on problem solving in the school system rather than have the teacher solve problems for us.  If teachers simply give us the equation used to solve the problem and don't let the students figure out how the equation came to be, they will never truly understand …
Casper College Elementary Ed Major

9. Low Floor High Ceiling Problems
Low floor High Ceiling Tasks are those that
all students can access but can be
extended to high levels ( You Cubed).
Low Threshold High Ceiling Tasks are
activities that everyone in a group can
begin, and then work on at their own level
of engagement, but which has lots of
possibilities for the participant to do much
more challenging math (nrich.maths.org).

10. “I think we should be spending much more time on problem solving
“I think we should be spending much more time on problem solving! Personally, I feel like I was sort of cheated in my math classes because it was so easy to get lazy and just go through the motions and I was never forced to do critical thinking. I wish I had been in a classroom where problem solving was a priority!” Casper College Elementary Education Major

11. Is it a Low Floor High Ceiling Problem?
Is the working on the problem more important than finding an answer?
Is it Low Floor? Can all students access the problem?
Is it High Ceiling? Is there room for students to explore?
Is there the potential for mathematical discourse?
Think about questions might you ask a student who is “stalled to get them going again.

12. How many numbers can you
find between 1 and 20 using
only four 4’s and any
operation?

14. ….before I took this class I probably would've gotten really frustrated with the problem and it would've taken me a lot longer to figure out but …[now]… I feel like I am a stronger math student because I can try to problem solve on my own and create those "AHA" moments…

17. Consider the list of numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Pick any two numbers from the list above, say a and b. Remove a and b from the list and add to the list a+b+ab.
For example, if you choose 5 and 13, cross those two numbers off and replace them with *13 = 83.
Continue until there is only one number left
What are all possible final numbers?

18. How many rectangles are there in a 8x8 chessboard?
"The best way to learn is to do; the worst way to teach is to talk."—Paul Halmos
How many rectangles are there in a 8x8 chessboard?

20. Adding them all up!

21. Total 36 100 12 32 62 102 More Checkerboard Solutions:
Rect. 1 2 3 4 5 6 7 8
1x1 9
1x2
2x1
2x2
1x3
3x1
2x3
3x2
3x3
4x4
Total 36
100 12 32 62
102
More Checkerboard Solutions:
Look at a simpler problem

22. Now, where have I seen that pattern before?

26. I think we do need to spend more time on problem solving in the school system rather than have the teacher solve problems for us.  If teachers simply give us the equation used to solve the problem and don't let the students figure out how the equation came to be, they will never truly understand … Casper College Elementary Ed Major
How many squares will need to be painted on the 50th figure

28. Teachers need to sometimes pose problems that they don’t know the answers to….It’s good for students to see teachers “struggle” with problems from time to time…create a culture of problem solving and discovery….
 22 ends in a ends with Find a square that ends with One that ends with 4444?

30. PATTERN STRUCTURES: How many blocks are needed for the 100th term?

31. Math Magic
Here is a math magic trick you can use to amaze your students. Turn your back to the blackboard and have a student pick two positive integers and write them one below the other on the board. (You might suggest small numbers because there is a lot of addition to be done after this!). You of course cannot see the numbers nor are you told what they are. Below this have the student write the sum of the two numbers, so there is now a column of 3 numbers on the board. Then add the bottom two numbers in the column and put the result of this addition below the three. Continue in this way, adding the two most recent numbers in the column and writing the sum below, until there are 10 numbers in the column. You then declare that you will turn and almost instantly find the sum of the ten numbers. How is it done?

32. Three Cent Coin Problem
Suppose that a certain country has a 3-cent coin and a 4-cent coin. Then it is not possible to obtain certain amounts of money using only these coins, such as five cents. Determine which amounts can be obtained with these coins.
Now repeat the previous problem with a 3-cent and 5-cent coin. Then try it once more with a 3-cent and 7-cent coin.
Based on the results of the previous two problems, make a conjecture as to the largest amount that cannot be obtained using only 3-cent and b-cent coins for any value of b not divisible by 3. Also conjecture the number of amounts that cannot be obtained. Explain how the situation changes when b is divisible by 3
Show that all amounts greater than or equal to 3b can be obtained using 3 cent and b cent coins
From Circle in a Box.

33. On Fostering a Good Mathematical Disposition
"I suppose it is because nearly all children go to school nowadays, and have things arranged for them, that they seem so forlornly unable to produce their own ideas. "
- Agatha Christie

34. What the teacher says in the classroom is not unimportant, but what the students are thinking is a thousand times more important.
-George Polya
A family grows oranges in a desert oasis. They have 3000 oranges and market is 1000 miles away. They only have one camel to transport oranges, but there are two problems:     i) The camel can only carry at most 1000 oranges at a time     ii) The camel will only walk if munching on an orange. He eats one orange for every mile he walks. What is the maximum number of oranges the family can get to market using ONLY the camel to transport them?

35. Math Magic Take any 3 digit number Reverse the Digits
Subtract the smaller from the larger
Tell me what the units digit in the difference is
I’ll tell you the difference
Generalize

36. Proof and Reasoning Exploration Inductive and Deductive Reasoning
1. Write down any “non-palindromic” 3 digit number
2. Reverse the number and then subtract the smaller from the larger
3. Tell me the last digit of the number in the difference and I AMAZE you by telling you the entire difference.

37. For example

39. 99
198
297
396
495
594
693
792
891

40. Deductive Reasoning

41. I don’t know what’s the matter with people…
I don’t know what’s the matter with people….They do not learn by understanding…They learn by some other way – by rote, or something....Their knowledge is so FRAGILE!
Richard Feynmen

42. The Game of Take Away – How do you win?
“I think we should be spending much more time on problem solving! Personally, I feel like I was sort of cheated in my math classes because it was so easy to get lazy and just go through the motions and I was never forced to do critical thinking. I wish I had been in a classroom where problem solving was a priority!” Casper College Elementary Education Major
The Game of Take Away – How do you win?
To play, begin with a set of pennies or other small objects. Two players take turns removing pennies. At each turn a player must remove anywhere from 1 to 4 pennies (inclusive). The winner will be the last player who makes a legal move.

43. Now here is a question. Suppose you took 7 to the 9999thpower
Now here is a question. Suppose you took 7 to the 9999thpower. That would be a number with 8450 digits. In 12 point type, that is a number about 70 feet long. What are the last three digits of that number? Why would we ask such a question? No reason, that’s just what we do.

44. Engage them in Problem Solving

45. Find 4 ways of finding the difference 769 - 203

46. Multiply 6 different numbers by 11
Multiply 6 different numbers by 11. What is an easy way to multiply by 11?

47. George Polya’s TEN COMMANDMENTS FOR TEACHERS
Be interested in your subject.
Know your subject.
Know about the ways of learning: The best way to learn anything is to discover it by yourself. 
Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place. 
Give them not only information, but "know-how," attitudes of mind, the habit of methodical work. 
Let them learn guessing.
Let them learn proving.
Look out for such features of the problem at hand as may be useful in solving the problems to come—try to disclose the general pattern that lies behind the present concrete situation.
Do not give away your whole secret at once—let the students guess before you tell it—let them find out for themselves as much as feasible.
Suggest it, do not force it down their throats.