Cling to the main vine, not the loose one.
Kei hopu tōu ringa kei te aka tāepa, engari kia mau te aka matua

123Thoughts on Teaching and Learning of Mathematics

Lesson #8 • Revised 12/1/20
Problem Solving
- to normalise problem solving as a purpose of mathematics
- to develop problem solving as an engaging cause 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. 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

Polya PS
Task 1
What is a problem?
Before reading on try and define "a problem".

STOP READING! Go on, Define problem solving. What does it mean to you?

 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 fascinations.

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.

definition

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.

NORMALISE THE EXPECTATION OF BEING A PROBLEM SOLVER

At you peril, rell answers, solve problems for others.

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 pathway forward.

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 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 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 calculator!

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.

Staircase of blocks

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?
and
"How can you show that the size of a sample needs to be about size 20 to 50?" Is there another answer?

Wicked Problems
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...Nah.

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.

Eg Fractions
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 chapter.

Here are a few problems that might last more than a day and illustrate different groups of problem solving and techniques for solving them.
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 chosen.
The problem is about patterning and powers of 2, odd numbers as well surprisingly enough. Can you know where to sit for any sized circle?

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 students.

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?
• 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. Continue.
How many steps do you typically take before returning to the start point?
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 is useful!

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.

Teacher TASK
• 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.

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