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
Lesson #7 • Revised 5/1/19
- to value problem solving
as a purpose of mathematics
- to develop problem solving as an engaging cause of
- 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 all just to start describing this amazing approach. Develop
this skill and students will naturally make their own meaning and
transfer the learning to other areas. Hence to "wicked problems"
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 whole mathematics curriculum is now applied
mathematics and each year level has descriptions that are all
embedded "in solving problems". We must involve and embed the
learning in problem solving situations.
The patron saint 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, a 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
Problems in mathematics 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, mathematical knowledge, skills and
previous problem solving experience. Problem solving almost always
raises the word "creative" as new solutions are devised, current
knowledge is applied in new ways, new connections are discovered,
new learning is made.
There are some wonderful old and tradition problems that have been
used by masters for many years to expose deep understanding,
develop curiosity and create learning.
An Old Problem to Ponder from Babylonian times 5000BC.
A prince once decreed
upon his death that 1/2 of his stable of horses 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. After an explanation she said
"You may borrow my donkey. Now you 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 all the horses have been
shared. Now give me my donkey back and I will leave you all
in peace." With that she left and everyone was happy.
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
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
By solving problems for others you will create dependence.
This may well be what you want but it does not create
independent future focused citizens in my mind. This
statement connects to what I call the "Mummy Syndrome"
where I see teachers (and Mummies) "doing things for students
(children) that students(children) should be doing". Doing their
thinking for them, telling them how to behave, telling them what
to look for, taking their pencil to draw a diagram and so on, all
illustrating the "Mummy Syndrome". A teacher (and a Mummy) will
say "It is just easier if I do it!". Maybe, but it is just
creating dependence. It is not creating that resourceful and self
managed problem solver that you should be creating! Easy is
not always the best pathway. Expedience is not always best for
everyone. Why be in a hurry when it comes to learning?
Problems in Maths
Almost all problems in 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 (or series) is 1+2+3+4+5+...
or 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 appear. The number of diagonals in a polygon again shows
this formula. Two good examples. This sequence of triangle
numbers is the most important first step for all students. Add
them and the sequence becomes a series. Some literacy
Methods of Problem Solving
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.
George Polya stated this in 1945 in his book called "How to Solve
a Problem". He focused on some very cool math problems in his book
and like Paul Erdōs exposed how important it is to be a problem
solver. I think this statement is one of the best he made! I try
and solve all problems in many ways. Breaking a stick in two
random places and making a triangle sometimes works and sometimes
does not work. What is the chance of it working? This problem I
solved about 5 or 6 different ways. Toss three dice and see if you
can form a triangle. This can be simulated and calculated. I did
not figure an algebraic way of doing this problem.
Encourage other solutions. Share solutions between class groups. I
have often seen a better solution after solving it one way. I
often get ideas for new solutions from seeing other ideas. How
many times do you look at someone using a computer and learn
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 many sites in UK, BBC, and Aussie,
AMC. The Australian Math Associations such as MAV have sources of
problems. Just Google these sites.
Resources abound but the best problems happen in a daily
lesson. The questions that students ask; genuine problem
solving. Ignore them at your peril and treat everyone as genuine
and nontrivial. Encourage curiosity and innovation. Answer "Why
did you do that Sir?" with a question like "How would you do it?".
I loved creating "cognitive dissonance". These are the times when
someones understanding conflicts with what they already know. "Can
fractions be bigger than one Sir?" "Can you have 4/2 as a
fraction?" "Why did you say there are just as many numbers between
1 and 1.1 as there are on teh number line? Surely that can not be
correct! The number line is way longer!"
"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.
The solution to "wicked problems" like these is a civilization
responsibility and will require the redistribution of lot of
wealth so 1000x the number of people capable of solving these
problems happens. I could write a book about the future of the
human race as well but there may well not be anyone around to read
it, at least not on this planet!
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 or perhaps to find
out what they know about a planned topic. Make learning fun! 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, knowledge or
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! See a L8R
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 deserves
to be investigated and generalised to any sized grid, 3d and
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. Can you know where to sit for any
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!
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! Infinity is fun.
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. A very cool problem solveable
by all ages of students.
7. The Lunes 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. You do need some
knowledge of Pythagorus in some solutions.
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. How many times does the
minute hand overtake the hour hand in 12 hrs.
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.
His wife's name must be Mrs Brown!
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
LOGO. Try and write a program for the Edison (programmable robot)
to do this. Seeing this random walk in action is quite
mesmerizing and has a surprising outcome.
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.