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 #8 Revised 12/1/20
- to normalise 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. Experiencing
that "ah-ha" moment is a gift and once bitten problem solving
becomes established. Problem solving is creative human in
action. Develop this skill and students will naturally make
their own meaning and transfer the learning to other areas. Hence
to "wicked problems" (see below). All of the mathematics and
statistics curriculum is set in the context of "problem solving".
Problem solving is not an extra... it is an expectation.
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.
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, Book Depository 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".
STOP READING! Go on, Define problem solving. What does it mean to
To me, a problem is any situation that does not have an
obvious solution and is new to the solver. 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 "I have 2 sheep in
this paddock and 9 in this one, how many sheep will I have if I
put them all together?" 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 noisy 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
confused but 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
At you peril, rell answers, solve problems for
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, getting a
drink for them, making their bed, making their lunch 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
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 these 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. Sequence
1,2,3,4,5,... Series 1+2+3+4+5+...
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 "5" 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 in 5 different ways. Toss three dice and see if you can
form a triangle. This can be simulated and calculated. I did not
figure a probabilistic/algebraic way of doing this problem but
"all possibilities and simulation are quite accessible.
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
something new? Have a SOLUTIONS display in the class room.
My 7 year old grand-daughter when asked "How many blocks in the
staircase?" See picture below. She immediately asked for a
calculator and proceeded to add 1 + 2 + 3 + 4 + ...20. Her
strategy was sound and her method modern and after a few
minutes of persevering 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
I rearranged the 1 to the 19, the 2 to 18 and quite quickly she
finished the pattern. "Ah-Ha" she said "it is 10 lots of 20 and 10
more but I am not sure what that is. The calculator again showed
10x20 +10 confirmed the "210" answer. So here we see a
counting based solution and not a lot of base ten ideas yet. Those
more advanced ideas will happen and very soon. I did like the
checking and two solutions.
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. Brilliant.org
is a goodie.
Resources abound but the best problems happen in a daily
lesson. The questions that students ask; genuine problem
solving. Ignore these oppportunities at your peril and treat every
problem exposed 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
someone's understanding conflicts with what they already know
(hence cognitive disonance). "Can fractions be bigger than one
Sir?" "Can 4/2 be a fraction?" "Why did you say there are just as
many numbers between 1 and 1.1 as there are on the 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 the size of a sample needs to be about size
20 to 50?" Is there another answer?
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 new generations
will have to solve. Problems such as global climate warming, over
population, obesity, future energy sources, freshwater supplies,
clean oceans, food supply, rising oceans, antibiotic resistance,
asteroid impacts, equitable wealth and little poverty, robots,
environmental sustainability, plastic contamination, recycling,
trigger happy people in control of missiles adn on and on. This is
a growing list. When I was 18 I thought by 2020 we would have as a
world solved most of the problems and be concerned more with
sustaining all we have.
Yeah because I think we should all be problem solvers and
be able to contribute and help solve these big issues.
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 are developed. I could easily write a book about the
future of the human race but I do not think there will be anyone
to read it!
Team Problem Solving
A great way to involve and have students collaborate is to put
them in groups of 3 (or 4 or 5) 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 usually multichoice and each targets a skill,
knowledge or curiosity. I developed a FileMaker Pro app to make up
the sets of questions for a new topic rather efficiently.
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 patterning and sums of
squares. It deserves to be investigated and 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 conext
for finding out the perfect place to sit should you wish to be
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 a little over 23 people which is
much less than what I expected. Now prove it!
4. Breaking the Stick Problem
Break a stick in two random places. What is the probability of
making a triangle with the three pieces?
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! 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 solvable 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.
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
When 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?
I made a power point and have about 12 solutions to this one.
How far away is town?
What is his wife's name?
What colour is his tractor?
What does he sell?
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 surprising outcomes. This Wikipedia
Link has some cool simulations and exposes Brownian Motion!Maths
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
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.