Cling to the main vine, not the loose one.

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

Problem Solving

- to value 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 skillinvolving 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" (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 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.

Task 1

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 fascinations.

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.

NORMALISE THE EXPECTATION OF BEING A PROBLEM SOLVER

Tell answers, solve problems for others, at your peril.

Problem solving should benormalisedas 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 createdependence. This may well be what you want but itdoes not create independent future focused citizensin 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. Thissequenceof triangle numbers is the most important first step for all students. Add them and the sequence becomes aseries. Some literacy appears.

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 usingLighting 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 something new?

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 butthe 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?

and

"How can you show that a sample only needs to be about size 20 to 50?"

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 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...Nah.

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 curiosity.

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 ProblemHow 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 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. 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 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 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 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. 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 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.

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.