Cling to the main vine, not the
Kei hopu tōu ringa kei te aka tāepa, engari kia
mau te aka matua
Thoughts on Teaching and Learning of Mathematics
Problem Solving and Lesson #7
- to value problem solving as a purpose of mathematics and
causation of learning
- to open the pathway to being a lifelong problem solver
Problem solving is a complex skill involving curiosity,
creativity, connecting to prior learning, collaborating, critical
thinking and having insight, that "ah-ha" moment, to mention a few
traits. Develop this skill and students will naturally transfer
the learning to other areas and hence to "wicked problems" (see
below). Being a problem solver reveals all the attributes of the
main vine of mathematics. Success in the senior Y11 to 13
NCEA Achievement Standards, all of which involve the term "in
solving problems", naturally follows. Well, that is how I reason
it all fitting together. Being able to problem solver is also a
darn good skill to possess.
The patron of Problem Solving is George Polya, https://en.wikipedia.org/wiki/George_Pólya,
and he wrote a book called "How to
Solve a Problem". Read it! There are new copies available
from online book sellers like Fishpond and Amazon. Berkeley have
summarised his ideas https://math.berkeley.edu/~gmelvin/polya.pdf.
What is a problem?
Before reading on try and define "a problem".
To me, problem is any situation that does not have an
obvious solution and is new to the student. No "telling" or
"suggesting" ways to solve are to be provided. A problem in a
school context is somewhat artificial as many students and
teachers have previously solved most problems presented. That
said, there is much to learn and experience in solving past
fascinations. Problems in maths can be as simple as "What is the
sum of the numbers 2 and 3" and as complex as "How long is the
parabolic curve formed by y=x^2 on the interval x = [-1,1].
Relevant problems depend on age, maths knowledge and previous
experience. Problem solving almost always raises the word
An Old Problem to Ponder.
A prince once decreed upon his death that 1/2 of his
stable be given to his eldest son, 1/3 to his next eldest and
1/9 to his youngest son. On his death he owned 17 horses. Much
argument and confusion was noticed by a woman walking a donkey
nearby and she asked of the dilemma. "You may borrow my donkey"
she said thoughtfully. "You now have 18 animals. 1/2 is 9 so
give the eldest 9 horses, 1/3 is 6 so give the second son 6
horses, 1/9 is 2 so two horses are for the youngest son. The
total 9 + 6 + 2 is 17 so you may now give me my donkey back and
I will leave you all in peace." With that she left and everyone
How does this all work out so sweetly?
This problem is said to have been recorded in ancient Babylon some
5000BC. Wow! That is some problem and today it continues to expose
fundamental understanding of fractions.
THE EXPECTATION OF BEING A PROBLEM SOLVER
Tell answers, solve problems for others, at your peril.
Problem solving should be normalised as an everyday event and
expectation. It is fun to be in the world of "I have no idea!" and
to collaborate and use creative thought for a path ahead. By
solving problems for others you create dependence. This
may well be what you want but it does not create independent
future focused citizens in my mind.
Most problems in school mathematics are related to counting
numbers, square numbers, the powers of 2 or are exponential, and
all in the form of sums or products. This also means we should
know about sets of numbers and sums of sequences.
A fundamental and important sum is 1+2+3+4+5+... . The sum of the
whole numbers. The general form of sum to n is 1/2*n*(n+1) and
appears in all sorts of circumstance. If 8 people in a room all
shake hands with each other, just once, we see this formula. The
number of diagonals in a polygon again shows this formula. Two
Some common problem solving skills are "simplifying the problem",
"finding patterns", "connecting to prior knowledge or a similar
problem", "listing all possibilities". I have a sheet of these I
sourced using Lighting Mathematical Fires by Prof Derek Holton.
Any successful method is OK when it comes to problem solving.
It is better to solve
one problem 5 ways than 5 problems one way. - Polya.
Encourage other solutions. Share solutions between class groups. I
have often seen a better solution after solving it one way.
My 7 year old grand-daughter when asked "How many blocks in the
staircase we had made immediately asked for a calculator and
proceeded to add 1 + 2 + 3 + 4 + ...20. Her strategy was sound and
her method modern and persevering she announced 289. We asked
Siri...who said "210". Repeating the quest on a calculator rather
than the iPhone app gave 210. She suggested the iPhone was not a
good calculator! I rearranged the 1 to the 19, the 2 to 18 and
quite quickly she finished the pattern. "Ahh" she said "it is 10
lots of 20 and 10 more. 10x20 +10 which she did with a calculator
and of course confirmed the 210 answer. So here we see a counting
based solution and not a lot of base ten ideas yet. They will
happen and very soon.
Ex Prof of Mathematics at Otago University, Derek Holton, has an
excellent site of useful problems. He was involved heavily with
the development of the nzmaths web site, education and is a great
problem solver and promoter of such things. There is http://www.cuttheknot.org,
math100, NLVM, nRICH and a many sites in UK and Aussie. The
Australian math Associations such as MAV have sources of problems.
Resources abound but the best problems happen in daily lesson. The
questions that students ask; genuine problem solving. Ignore at
your peril and treat everyone as genuine and nontrivial.
"What is the next number after 99?"
is as valid as...
"What is half way between 1:1 and 1:2?
"How can you show that a sample only needs to be about size 20 to
This needs a special mention. It is the notion of the authors of
the NZCER publication http://www.nzcer.org.nz/nzcerpress/key-competencies-future
that building student competencies and problem solving ability
will prepare them for future monster problems that they will have
to solve. Problems such as global climate warming, future energy
sources, freshwater supplies, clean oceans, food supply, rising
oceans, antibiotic resistance, asteroid impacts, equitable wealth
and little poverty, robots, environmental sustainability...
Yeah because I think we should all be problem solvers and be able
to contribute and help solve these big issues. Taking an action.
Nah because who, realistically, has the resources to do so. We can
all be aware of these issues, be concerned and acknowledge them
but that does not solve them.
Team Problem Solving
A great way to involve and have students collaborate is to put
them in groups of 3 and have them compete for 20 or so questions
as a summary or review of a topic just taught. I reserve these for
"last period Fridays" when students are often a bit over school
for that week. My style of questions for these sessions is
multichoice and each targets a skill.
1. What is 2 thirds plus 3 quarters?
(a) 4/7 (b) 1and
5/12 (c) 17/12 (d)
12/17 (e) 1/2
2. What is multiplied by 1/2 to make 3/4?
(a) 6/4 (b) 3/8 (c) 1 and a
half (d) 2 (e) 1/4
3. Which are equivalent to 3
(a) 9/3 (b) 1/10 of 30 (c) 1/7
of 42 (d) 3/7 of 7 (e) One and a half divided
by a half
and so on.
More on Teams and Problem based Learning coming!
Here are a few problems that might last more than a day and
illustrate different groups of problem solving and techniques for
1. Chessboard Problem
How many squares on a chessboard?
I will concede that there are 64 1x1 squares, but what about the
2x2 and 3x3 and so on squares as well?
This is a great problem involving sums of squares and can be
generalised to any sized grid, 3d and beyond.
2. King Arthur Problem
Sit the class in a circle and selecting a student as start point
call"In", "Out", "In", "Out","In", "Out","In", "Out", ...as you
proceed around the circle until you end up with one student. The
problem is about the way King Arthur selected the man around the
Round Table to marry his daughter. All nonsense of course, he
never did that as far as I know, but it is an interesting problem
finding out the perfect place to sit should you wish to be chosen.
The problem is about patterning and powers of 2, odd numbers
as well surprisingly enough.
3. Birthday Problem
How many people do you need in a room to have a 50% chance that 2
will have birthdays on the same day?
Surprisingly the answer is something like 23 people
which is much less than expected. Now prove it! 1- ?
4. Breaking the Stick Problem
What is the probability that you can form a triangle with a stick
broken in two random places?
The solution connects to geometry and some intuition or "ahha"
moment. It is solvable in several ways all accessible to
5. What is 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... to infinity?
We do not usually have these infinite problems presented at early
years but we should do!
Stand 2m away from a wall and then step halving the distance
to the wall each time you do so. Does this help?
6. Square Pegs and Round Holes.
Which is better use of space? Putting a round peg in a square hole
or a square peg in a round hole?
Just a bit of calculation here.
7. The Runes of Hippocrites
A famous problem accessible to Y10 and Y11 students. See the LOGO
fro NZAMT at http://www.nzamt.org.nz
More calculation but a surprising result.
8. Clock Problem
The hands of a clock are exactly together at noon. When, exactly
are they next together?
Understanding the problem is key here.
9. Farmer Brown
If Farmer Brown travels to town, to sell his chicken eggs ,at
20km/hr he arrives an hour late. When he travels at 30km/hr he
arrives an hour early.
What speed should he travel at to arrive on time, eggsactly?
How long does it take him to get there?
How far away is town?
What is his wife's name?
What colour is his tractor?
What does he sell?
I made a power point and have about 12 solutions to this one.
10. Random Walks
Using a grid and a coin, find a start point, move one grid step,
toss the coin and turn Left for Head and Right for Tails.
How many steps do you typically take before returning to the start
This can be simulated using a visual programming language like
Hence Lesson #7
Normalize problem solving as a daily event. Normalise the
expectation that "Thou shalt problem solve!" Find and collect
great problems of all types. Try cryptic crossword clues! Refuse
to answer another's problem. Do not tell answers. Telling is like
doing things for another... it creates dependence.
How often should problem solving be offered to students?
Find 10 sources of problems suitable for your classes.
Find ten problems.
How can you build problem solving into every lesson.
What is the one key action a teacher should never do when
promoting problem solving.
Intro and Relationships, L#1
This is to help look around my
pages. I have tried to make it consistent in all chapters.
The Planned chapters are only ideas at the moment.
2. The Main
3. Beginning a
4. Number and
Geometry and Measurement L#5
Probability and Statistics L#6
Problem Solving L#7
Visual Mathematics L#9
Assessment and Learning L#10
Team and Problem Based Learning L#11
12. Engagement L#12
13. The Classroom L#13
14. Being a Teacher L#14
15. Being a Leader L#15
16. Managing the Principal L#16
17. The Importance of Whanau (Family)L#17
18. The Importance of the Student L#18
19. Math Topic A - Squares
20. Teacher Tools
22. Math Phobia