Mathematical Mysteries

Revealing the mysteries of mathematics

## How to Solve Math Problems: Non-Routine Problems

Nonroutine Problem Solving , stresses the use of heuristics [3] and often requires little to no use of algorithms. Unlike algorithms, heuristics are procedures or strategies that do not guarantee a solution to a problem but provide a more highly probable method for discovering a solution. Building a model and drawing a picture of a problem are two basic problem-solving heuristics. Studying end-of-game situations provides students with experiences in using the heuristics of reducing the problem to a similar but simpler problem and working a problem backwards, i.e. from its resolution to its initial state. Other heuristics include describing the problem situation, classifying information, and finding irrelevant information. [1]

## There Are Two Categories of Nonroutine Problem Solving: Static and Active

Static-Nonroutine  problems have a fixed, known goal and fixed, known elements that are used to resolve the problem. Solving a jigsaw puzzle is an example of a Static- Nonroutine problem. Given all pieces to a puzzle and a picture of the goal, learners are challenged to arrange the pieces to complete the picture. Various heuristics such as classifying the pieces by color, connecting the pieces that form the border, or connecting the pieces that form a salient feature to the puzzle, such as a flag pole, are typical ways in which people attempt to resolve such problems. [1]

Active-Nonroutine  problems may have a fixed goal with changing elements, a changing goal or alternative goals with fixed elements, or changing or alternative goals with changing elements. The heuristics used in this form of problem-solving are known as strategies. People who study such problems must learn to change or adapt their strategies as the problem unfolds. [1]

## What is non-routine problem-solving in math?

A non-routine problem is any complex problem that requires some degree of creativity or originality to solve. Non-routine problems typically do not have an immediately apparent strategy for solving them. Often times, these problems can be solved in multiple ways.

Incorporating non-routine problem solving into your math program is one of the most impactful steps you can take as an educator. By consistently allowing your students to grapple with these challenging problems, you are helping them acquire essential problem-solving skills and the confidence needed to successfully execute them. [2]

## Step 1: Understand

This is a time to just think! Allow yourself some time to get to know the problem. Read and reread. No pencil or paper necessary for this step. Remember, you cannot solve a problem until you know what the problem is!

• Does the problem give me enough information (or too much information)?
• What question is being asked of me?
• What do I know and what do I need to find out?
• What should my solution look like?
• What type of mathematics might be required?
• Can I restate the problem in my own words?
• Are there any terms or words that I am unfamiliar with?

## Step 2: Plan

Now it’s time to decide on a plan of action! Choose a reasonable problem-solving strategy. Several are listed below. You may only need to use one strategy or a combination of strategies.

• Draw a picture or diagram
• Make an organized list
• Make a table
• Solve a simpler related problem
• Find a pattern
• Guess and check
• Act out a problem
• Work backward
• Write an equation
• Use manipulatives
• Break it into parts
• Use logical reasoning

## Step 3: Execute

Alright! You understand the problem. You have a plan to solve the problem. Now it’s time to dig in and get to work! As you work, you may need to revise your plan. That’s okay! Your plan is not set in stone and can change anytime you see fit.

• Am I checking each step of my plan as I work?
• Am I keeping an accurate record of my work?
• Am I keeping my work organized so that I could explain my thinking to others?
• Am I going in the right direction? Is my plan working?
• Do I need to go back to Step 2 and find a new plan?
• Do I think I have the correct solution? If so, it’s time to move on to the next step!

## Step 4: Review

You’ve come so far, but you’re not finished just yet! A mathematician must always go back and check his/her work. Reviewing your work is just as important as the first 3 steps! Before asking yourself the questions below, reread the problem and review all your work.

• Can I use estimation to check if my answer is reasonable?
• Is there another way to solve this problem?
• Can this problem be extended? Can I make a change to this problem to create a new one?
• I didn’t get the correct answer. What went wrong? Where did I make a mistake?

[1] “Pentathlon Institute Active Problem-Solving”. 2023.  mathpentath.org . https://www.mathpentath.org/active-problem-solving/ .

[2] Tallman, Melissa. 2015. “Problem Solving In Math • Teacher Thrive”. Teacher Thrive. https://teacherthrive.com/non-routine-problem-solving/ .

[3] A heuristic is  a mental shortcut commonly used to simplify problems and avoid cognitive overload .

“101 Great Higher-Order Thinking Questions for Math”. 2023.  elementaryassessments.com . https://elementaryassessments.com/higher-order-thinking-questions-for-math/ .

⭐ “Developing Mathematics Thinking with HOTS (Higher Order Thinking Skills) Questions”. 2023. saydel.k12.ia.us . https://www.saydel.k12.ia.us/cms_files/resources/Developing%20Mathematics%20Thinking%20with%20HOTS%20Questions%20(from%20classroom%20observations)PDF.pdf .

“Higher Order Thinking Skills in Maths”. 2017.  education.gov.Scot . https://education.gov.scot/resources/higher-order-thinking-skills-in-maths/ .

“How to Increase Higher Order Thinking”. 2023.  Reading Rockets . https://www.readingrockets.org/topics/comprehension/articles/how-increase-higher-order-thinking .

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

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## What is non-routine problem-solving in math?

A non-routine problem is any complex problem that requires some degree of creativity or originality to solve.  Non-routine problems typically do not have an immediately apparent strategy for solving them.  Often times, these problems can be solved in multiple ways.

Incorporating non-routine problem solving into your math program is one of the most impactful steps you can take as an educator. By consistently allowing your students to grapple with these challenging problems, you are helping them acquire essential problem-solving skills and the confidence needed to successfully execute them.

One of the best ways to prepare students for solving non-routine problems is by familiarizing them with the four steps of problem-solving. I have a set of questions and/or guides for each step, that students can use to engage in an inner-dialogue as they progress through the steps.  You can download this free Steps to Non-Routine Problem Solving Flip-Book {HERE}.

## 1. Understand:

This is a time to just think! Allow yourself some time to get to know the problem.  Read and reread. No pencil or paper necessary for this step.  Remember, you cannot solve a problem until you know what the problem is!

• Does the problem give me enough information (or too much information)?
• What question is being asked of me?
• What do I know and what do I need to find out?
• What should my solution look like?
• What type of mathematics might be required?
• Can I restate the problem in my own words?
• Are there any terms or words that I am unfamiliar with?

Now it’s time to decide on a plan of action! Choose a reasonable problem-solving strategy. Several are listed below.  You may only need to use one strategy or a combination of strategies.

• draw a picture or diagram
• make an organized list
• make a table
• solve a simpler related problem
• find a pattern
• guess and check
• act out a problem
• work backward
• write an equation
• use manipulatives
• break it into parts
• use logical reasoning

## 3. Execute:

Alright! You understand the problem.  You have a plan to solve the problem.  Now it’s time to dig in and get to work! As you work, you may need to revise your plan. That’s okay! Your plan is not set in stone and can change anytime you see fit.

• Am I checking each step of my plan as I work?
• Am I keeping an accurate record of my work?
• Am I keeping my work organized so that I could explain my thinking to others?
• Am I going in the right direction? Is my plan working?
• Do I need to go back to Step 2 and find a new plan?
• Do I think I have the correct solution? If so, it’s time to move on to the next step!

## 4. Review:

You’ve come so far, but you’re not finished just yet!  A mathematician must always go back and check his/her work. Reviewing your work is just as important as the first 3 steps! Before asking yourself the questions below, reread the problem and review all your work.

• Can I use estimation to check if my answer is reasonable?
• Is there another way to solve this problem?
• Can this problem be extended? Can I make a change to this problem to create a new one?
• I didn’t get the correct answer.  What went wrong? Where did I make a mistake?

My Brain Power Math resources are the perfect compliment to this free flip-book.  Each book has a collection of non-routine math problems in a variety of formats.

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Pentathlon Institute

## Active Problem-Solving

Learning to resolve problems that are continually undergoing change.

The Mathematics Pentathlon ® Program provides experiences in thought processes necessary for Active Problem Solving. The series of 20 Mathematics Pentathlon games provide K-7 students with experiences in deductive and inductive reasoning through the repeated use of sequential thought as well as nonlinear, intuitive thinking. Exposure to such forms of thought helps students relate to real-life problem-solving situations and learn to “think on their feet.”

The Four Sections below explain the following: Active Problem Solving Defined, What is Mathematics, Three Types of Mathematical Thought, and Conceptual Understanding Using Concrete and Pictorial Models.

## Active-Problem Solving Defined

Problem solving can be divided into two categories:, routine & non-routine.

Routine Problem Solving , stresses the use of sets of known or prescribed procedures (algorithms) to solve problems. The strength of this approach is that it is easily assessed by paper-pencil tests. Since today’s computers and calculators can quickly and accurately perform the most complex arrangements of algorithms for multi-step routine problems, the typical workplace does not require a high level of proficiency in Routine Problem Solving. However, today’s workplace does require many employees to be proficient in Nonroutine Problem Solving.

Nonroutine Problem Solving , stresses the use of heuristics and often requires little to no use of algorithms. Unlike algorithms, heuristics are procedures or strategies that do not guarantee a solution to a problem but provide a more highly probable method for discovering a solution. Building a model and drawing a picture of a problem are two basic problem-solving heuristics. Studying end-of-game situations provides students with experiences in using the heuristics of reducing the problem to a similar but simpler problem and working a problem backwards, i.e. from its resolution to its initial state. Other heuristics include describing the problem situation, classifying information, and finding irrelevant information.

## There Are Two Categories of Nonroutine Problem Solving:

Static and active.

Static-Nonroutine problems have a fixed, known goal and fixed, known elements that are used to resolve the problem. Solving a jigsaw puzzle is an example of a Static- Nonroutine problem. Given all pieces to a puzzle and a picture of the goal, learners are challenged to arrange the pieces to complete the picture. Various heuristics such as classifying the pieces by color, connecting the pieces that form the border, or connecting the pieces that form a salient feature to the puzzle, such as a flag pole, are typical ways in which people attempt to resolve such problems.

Active-Nonroutine problems may have a fixed goal with changing elements, a changing goal or alternative goals with fixed elements, or changing or alternative goals with changing elements. The heuristics used in this form of problem-solving are known as strategies. People who study such problems must learn to change or adapt their strategies as the problem unfolds.

## What is Mathematics?

There is a growing belief in the mathematics community, as well as society in general, that the study of mathematics must develop in all students an understanding of mathematics that continues throughout one’s lifetime and evolves to meet ever-changing situations and variables. From our perspective, mathematics is an area of investigation that develops the ability to critically observe, classify, describe, and analyze data in a logical manner using both inductive and deductive methods. In contrast to the sterilized and unrelated manner in which school mathematics has often been taught, mathematics is a creative and aesthetic study of patterns and geometric and numerical relationships. It is dynamic rather than passive in nature and should involve students in strategic thinking by exploring multiple possibilities and variables that continually change, much like life.

The Mathematics Pentathlon® Program, which integrates Adventures in Problem Solving, Activity Books I & II, the Mathematics Pentathlon® Games and Investigation Exercises, Books I & II was designed to implement the definition of mathematics described above. The games are organized into four division levels by grade, K-1, 2-3, 4- 5, and 6-7 with five games at each level. The name of the Program, Mathematics Pentathlon®, was coined to liken it to a worldwide series of athletic events, the Decathlon component of the Olympics. In the world of athletics the Decathlon is appreciated for valuing and rewarding individuals who have developed a diverse range of athletic abilities. In contrast, the mathematics community as a whole has rarely valued or rewarded individuals with a diverse range of mathematics abilities. The Mathematics Pentathlon® Games promote diversity in mathematical thinking by integrating spatial/ geometric, arithmetic/computational, and logical/scientific reasoning at each division level. Since each of the 5 games requires students to broaden their thought processes, it attracts students from a wide range of ability levels, from those considered “gifted and talented” to “average” to “at-risk.”

The format of games was chosen for two reasons. First, games that are of a strategic nature require students to consider multiple options and formulate strategies based on expected countermoves from the other player. The Mathematics Pentathlon® further promotes this type of thought by organizing students into groups of four and teams of two. Teams alternate taking turns and team partners alternate making decisions about particular plays by discussing aloud the various options and possibilities. In this manner, all group members grow in their understanding of multiple options and strategies. As students play these games over the course of time, they learn to make a plan based on better available options as well as to reassess and adjust this plan based on what the other team acted upon to change their prior ideas. Through this interactive process of sharing ideas and possibilities, students learn to think many steps ahead, blending both inductive and deductive thinking. Second, games were chosen as a format since they are a powerful motivational tool that attracts students from a diverse range of abilities and interest levels to spend more time on task developing basic skills as well as problem-solving skills. While race-type games based on chance are commonly used in classrooms, they do not typically capture students’ curiosity for long periods of time. Students may play such games once or twice, but then lose interest since they are not seriously challenged. The Mathematics Pentathlon® Games have seriously challenged students to mature in their ability to think strategically and resolve problems that are continually undergoing change. As a result, we view active-problem solving and strategic thinking as described above as a critical focal point of the mathematics curriculum.

## 3 Types of Mathematical Thought

Integration of spatial/geometric, computational, and logical/scientific reasoning.

Most mathematics instruction stresses students’ knowledge of basic arithmetic facts. While the Mathematics Pentathlon Program provides a great deal of practice with mastery of the basic facts, it goes far beyond learning arithmetic skills. The Mathematics Content and Standards Chart for the 20 Mathematics Pentathlon Games shows how each game addresses several mathematical content and process objectives (see inside back cover of manual). These objectives have been clustered into logical/scientific reasoning, computational reasoning, and spatial/geometric reasoning. Each of these categories is described below.

## Spatial/Geometric Reasoning

Spatial visualization involves the ability to image objects and pictures in the mind’s eye and to be able to mentally transform the positions and examine the properties of these objects/pictures. A large body of mathematics research concludes that spatial reasoning ability is highly related to higher-level mathematical problem-solving and geometric skills as well as students’ overall achievement in mathematics. Many of the Mathematics Pentathlon Games stress spatial reasoning and several integrate this form of thinking with logical and computational reasoning.

## Computational Reasoning

Many of the Mathematics Pentathlon Games incorporate computation into the game structure. More time-on-task practicing arithmetic skills does indeed result in students’ increased performance in the classroom as well as on standardized tests. But in the Mathematics Pentathlon Games that stress computation, it is not sufficient to rely on arithmetic skills alone. To be successful in these games, students must also use their logical reasoning abilities to consider several options and to decide which ones will maximize their ability to reach the game’s goal(s).

## Logical/Scientific Reasoning

One of the most important life skills, not to mention mathematical skills, is the ability to think logically. The process of observation, classification, hypothesizing, experimentation, and inductive and deductive thought are required for logical reasoning. Yet where do children learn these fundamental life skills? Strategic games provide students the opportunity to develop this form of thinking. Each of the Mathematics Pentathlon Games is a strategy game that develops students’ logical reasoning skills through the process of investigating a variety of options and choosing better options.  At the same time students develop scientific reasoning skills by learning how to be better observers of game-playing variables and options. Playing the games over the course of time allows for hypothetical reasoning to evolve since students analyze sequences of “if-then” situations and make choices based on linking inductive and deductive thought.

While each of the 20 games may stress one form of the mathematical thinking over another, each game integrates at least two categories of mathematical thinking. Furthermore, the five games at each Divisional level balance the three types of reasoning.

## Conceptual Understanding Using Concrete and Pictorial Models Understanding

Since its inception, the National Council of Teachers of Mathematics (NCTM) has called for a conceptually-based curriculum in schools throughout the country. The most recent psychological and educational research has shown that conceptual understanding is a key attribute of individuals who are proficient in mathematics. Furthermore, a large body of research over the last four decades suggests that effective use of physical and pictorial models of mathematics concepts improves students’ conceptual understanding, problem-solving skills, and overall achievement in mathematics. Research also indicates that the use of concrete and pictorial models improves spatial visualization and geometric thinking.

The  Mathematics Pentathlon ®   Program  incorporates a variety of concrete and pictorial models to develop students’ conceptual understanding of many important mathematics concepts that involve computational, spatial, and logical reasoning. In addition, by playing these games in cooperative groups, as suggested in this publication, students also improve their oral and written communication skills through their discussion of mathematical ideas and relationships.

MATHEMATICAL THINKING RESEARCH GROUP

mathematics, thinking, teaching, learning critical thinking, problem solving, non routine

## Routine and non-routine examples

Routine problems.

A routine problem requires a sequence of actions (usually with some formulaic structures) regularly followed in order to arrive at the correct prescribe answer. These problems often have an immediately apparent strategy for solving them.

## Non-Routine Problems

A non-routine problem is any complex problem (within Zone Potential Construction) that requires some degree of creativity or originality to solve.  These problems typically do not have an immediately apparent strategy for solving them.  Often times, these problems can be solved in multiple ways.

## Non-routine Algebra Problems

(a new problem of the week).

Last week I mentioned “non-routine problems” in connection with the idea of “guessing” at a method. Let’s look at a recent discussion in which the same issues came up. How do you approach a problem when you have no idea where to start? We’ll consider some interesting implications for problem solving in general, with an emphasis on George Pólya’s outline.

## First problem: mindless manipulation?

This came to us in March, from a student who identified him/herself as “J”:

Hi, Recently I had to solve a problem If (a + md) / (a + nd) = (a + nd) / (a + rd) and (1 / n) – (1 / m) = (1 / r) – (1/n) , then (d / a) = -(2 / n) i.e. Given the two expressions above I need to prove the last equality. I don’t understand problems like these. Basic Algebra books talk about problems like equation solving or word problems, but those are easy because there’s always some method you can use . For example regarding equation solving you move x’s to the left, numbers to the right; word problems can be solved using equalities like distance = rate * time. But a problem like the one above it seems has no method; it seems like you’re supposed to just manipulate the symbols until you get the answer . For example I tried to solve it like this: Regarding the first expression, after multiplying numerator of the first fraction by the denominator of the second I get (d / a) = ((m + r – 2n) / (n^2 – mr)) and 2mr = nm – nr then substitute for mr in the first expression. I reached the solution by luck; I just manipulated the symbols and it took me a lot of time . So is there a more efficient way to solve problems like these? How to think about these problems? Am I supposed to just mindlessly manipulate the symbols until I get lucky? Finally are there any books that deal with problems like these ? Because like I mentioned it seems like most precalculus books talk about equation solving etc., problems which have a clear method. Thanks.

## The solution

Before we deal with the question, let’s look more closely at his solution.

We are given two equations:

$$\displaystyle\frac{a + md}{a + nd} = \frac{a + nd}{a + rd}$$

$$\displaystyle\frac{1}{n} – \frac{1}{m} = \frac{1}{r} – \frac{1}{n}$$

We need to conclude that

$$\displaystyle\frac{d}{a} = -\frac{2}{n}.$$

J gave only a brief outline of what he did; can we fill in the gaps?

My version is to first “cross-multiply” in each equation to eliminate fractions, and do a little simplification:

The first becomes $$(a + md)(a + rd) = (a + nd) (a + nd),$$ which expands to $$a^2 + rda + mda + mrd^2 = a^2 + 2nda + n^2d^2,$$ then $$rda + mda – 2nda = n^2d^2 – mrd^2,$$ which factors to yield $$(r + m – 2n)da = (n^2 – mr)d^2.$$ Dividing, we get $$\displaystyle\frac{d}{a} = \frac{r + m – 2n}{n^2 – mr}.$$

(You may notice here that in dividing both sides by d , we obscured the fact that the line before is true whenever d = 0. I’ll be mentioning this below.)

The second equation, multiplied by $$mnr$$, becomes $$mr – nr = nm – mr,$$ which easily becomes $$2mr = nm + nr.$$ (J had a sign error here.)

Now, replacing $$mr$$ with $$\displaystyle\frac{nm + nr}{2}$$, we get $$\displaystyle\frac{d}{a} = \frac{r + m – 2n}{n^2 – \frac{nm + nr}{2}} = \frac{2(r + m – 2n)}{2n^2 – nm – nr} = \frac{2(r + m – 2n)}{-n(r + m – 2n)} = -\frac{2}{n}.$$

## How to solve it

Taking the question myself, I replied:

I tried the problem without looking at your work, and ended up doing almost exactly the same things. That took me just a few minutes. So probably it is not your method itself, but your way of finding it , that needs improvement. In my case, I did the “obvious” things (clearing fractions, expanding, factoring) to both given equations, keeping my eyes open for points at which they might be linked together , and found one. It may be mostly experience that allowed me to find it quickly. That is, I didn’t “mind lessly manipulate”, but “mind fully  manipulated”. And the more ideas there are in your mind, the more easily that can happen. So maybe just doing a lot of (different) problems is the main key.

There may be a better method for solving this, but finding it would take me a longer time than what I did. So perseverance at trying things is necessary , regardless. Solutions to hard problems don’t just jump out at you (unless they are already in your mind from past experience); you have to explore . The ideas I describe for working out a proof apply here as well: Building a Geometric Proof I like to think of a proof as a bridge, or maybe a path through a forest: you have to start with some facts you are given, and find a way to your destination. You have to start out by looking over the territory, getting a feel for where you are and where you have to go – what direction you have to head, what landmarks you might find on the way, how you’ll know when you’re getting close. (By the way, in my work I also found that d/a = 0 gives a solution, so that if d=0 (and a ≠ 0), the conclusion is not necessarily true. Did you omit a condition that all variables are nonzero?) You are probably right that too many textbooks and courses focus on routine methods, and don’t give enough training in non-routine problem solving . They may include some “challenge problems” or “critical thinking exercises”, but don’t really teach that. One source of this sort of training is in books or websites (such as artofproblemsolving.com ) that are aimed at preparation for contests. Books like Pólya’s   How to Solve It  (and newer books with similar titles) are also helpful. Here are a few pages I found in our archives that have at least some relevance: Defining “Problem Solving” Giving Myself a Challenge Preparing for a Math Olympiad Learning Proofs What Is Mathematical Thinking? Others of us may have ideas to add.

Some these have been mentioned in previous posts such as How to Write a Proof: The Big Picture and  Studying Math: Want a Challenge? .

## Another problem: following Pólya

The next day, J wrote in with another problem, having already followed up on my suggestions:

Hi. I posted here recently asking about problem solving and algebra and I was recommended a book called “How to solve it” by Pólya . I bought that book and now I am trying to solve some algebra exercises using it. Today I came across this problem If bz + cy = cx + az = ay + bx and (x + y +z)^2 = 0 , then a +/- b +/- c. (The sign +/- was a bit confusing to me since it’s not brought up anywhere in the book besides this problem, but Wikipedia says that a +/- b = 0 is a + b =0 or a – b = 0.) In the book “How to solve it” Pólya says that first it’s important to understand the problem and restate it . So my interpretation of a problem is this: If numbers x, y, z are such that (x + y + z)^2 = 0 and bz + cy = cx + az and bz + cy = ay + bx, then the numbers a, b, c are such that a + b + c = 0 or a – b – c =0 Next Pólya says to devise a plan . To do that he says you need to look at a hypothesis and conclusion and think of a similar problem or a theorem. The best I could think of is an elimination problem, i.e. when you’re given a certain set of equations and you can find a relationship between constants. Can you think of any other similar problems which could help me solve this problem?

I first responded to the last question:

Hi again, J. I would say that the last question you asked was “similar” to this, so the same general approach will help. That’s essentially what you said in your last paragraph, I think. I know that isn’t very helpful, but it’s all I can think of myself. You’d like to have seen a problem that is more specifically like this one, such as having (x + y + z) 2  = 0 in it, perhaps, so you could get more specific ideas. I only know that I have seen a lot of problems like this involving symmetrical equations (where each variable is used in the same ways), and I suspect those problems can be solved by similar methods. But I don’t know one method that would work for this one.

I’ll get back to that question. But let’s focus first on Pólya.

Here is what Pólya says (p. 5) when he introduces his famous four steps of problem solving:

In order to group conveniently the questions and suggestions of our list, we shall distinguish four phases of the work. First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan . third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it.

This process is then explained in more detail, and used as an organizing principle in the rest of the book. It can be amazing to see how many students jump into a problem before they understand what it is asking, or do calculations without having made any plans . On the other hand, it would be wrong to think of these four steps as a routine to be followed exactly; often you don’t fully understand a problem until you have started doing something , perhaps carrying out a half-formed plan and then realizing that you had a wrong impression of some part.

## Understanding the problem

And J has here a good example of a misunderstanding. This problem uses the plus-or-minus symbol (±) in a rare way, which in this case requires asking (not explicitly one of Pólya’s recommendations, but valuable!).

The problem says this:

$$\text{If } bz + cy = cx + az = ay + bx \text{ and } (x + y + z)^2 = 0 \text{, then } a \pm b \pm c.$$

(No, that doesn’t quite make sense! We’ll be fixing that shortly.)

What does it mean when there are two of the same symbol? The Wikipedia page J found says, “In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the + or − symbols, allowing the formula to represent two values or two equations.” They give an example (the quadratic formula), where either sign yields a valid answer; then an example with two of the same sign (the addition/subtraction identity for sines) in which both must be replaced with the same sign ; and third example (a Taylor series) where the reader has to determine which sign is appropriate for a given term. Later they introduce the minus-or-plus sign ($$\mp$$), which explicitly indicates the opposite sign from an already-used ±.

But here, we have two ±’s with no clear reason why they should be the same, or should be different. Is this a special case? J has assumed they are the same, so that it means “$$a + b + c = 0$$ or $$a – b – c = 0$$“. This is the first issue I had to deal with:

First, though, did you mean to say that the conclusion is a ± b ± c  = 0 ? That wouldn’t quite mean what you said about it, because the two signs need not be the same. Rather, it means that  either  a + b + c = 0, or a + b – c = 0, or a – b + c = 0, or a – b – c = 0: any possible combination  of the signs.

Now, how did I know that, when it goes against what Wikipedia seems to be saying? I’m not sure! There is actually some ambiguity; really, we just shouldn’t rule out this possibility . But I saw from the start that if the two signs are the same, then the problem has an odd asymmetry , requiring b and c to have the same sign in this equation, but not a . That simply seems unlikely, considering the symmetry elsewhere.

Sometimes we discover, as we proceed through the solving process, that we have to interpret the statement one way or another in order for it to be true – an example of my comment that understanding can come after doing some work. (That was actually the case here. But the problem really should have been written to make this clear!)

## Hints toward a solution

What this means is that we don’t know the signs of the numbers. One thing that suggests is that we might be able to show some fact about a 2 ,  b 2 , and c 2 , so that we would have to take  square roots , requiring us to use ± before each of a, b, and c. It’s also interesting that they said that (x + y + z) 2  = 0, which means nothing more than x + y + z = 0. That also makes me curious, and at the least puts squares into my mind for a second reason.

Here I am just letting my mind wander around the problem, pondering what the givens suggest. This is part of both the understanding phase, and the “looking for connections” Pólya talked about.

Not even being sure of the conclusion, I just tried manipulating the equations any way I could, just to make their meanings more visible; and then I solved x + y + z = 0 for z and put that into my derived equations, eliminating z. That took me eventually to a very simple equation that involved a, b, x 2 , and y 2 . And that gave a route to the ± I’d had in mind.

We could say that my initial plan is, as I suggested at the top, to explore ! We can refine the plan as we see more connections. (As I said, Pólya has to be followed flexibly.)

There’s a lot of detail I’ve omitted, in part because much of my work was undirected, so you may well find a better way. But the key was to have some thoughts in mind before I did a lot of work, in hope of recognizing a useful form when I ran across it . The other key was perseverance , because things got very complicated before they became simple again! (I suspect that as I go through this again, I’ll see some better choices to make, knowing better where I’m headed.) I don’t think you told us where these problems came from; they seem like contest-type problems, which you can expect to be highly non-routine. As I said last time, until you’ve done a lot of these, you just need to keep your eyes open so that you are learning things that will be useful in future problems! I am not a contest expert, as a couple of us are, so I hope they will add some input.

Since we never got back to the details of this problem, let’s finish it now. Frankly, I had to look in my stack of scrap paper to find what I did in March, because I wasn’t making any progress when I tried it again just now. Clearly I could have given a better hint! I was hoping that just the encouragement that it could be done would lead to J finding a nicer approach than mine.

But here’s what I find in my incomplete notes from then. First, I rewrote the equality of three expressions as two equations, and eliminated c; I’ll use a different pair of equations than I did then, with that goal in mind: $$cx + az = ay + bx\; \rightarrow\; c = \frac{ay-az+bx}{x}$$ $$ay + bx = bz + cy\; \rightarrow\; c = \frac{ay-bz+bx}{y}$$

Setting these equal to eliminate c, $$\frac{ay-az+bx}{x} = \frac{ay-bz+bx}{y}$$

Cross-multiplying, $$ay^2-ayz+bxy = axy-bxz+bx^2$$

Solving $$x + y + z = 0$$ for z and substituting, $$ay^2-ay(-x-y)+bxy = axy-bx(-x-y)+bx^2$$

Expanding, $$ay^2 + axy + ay^2 + bxy = axy + bx^2 + bxy + bx^2$$

Canceling like terms on both sides, $$2ay^2 = 2bx^2$$

Therefore, $$\frac{x^2}{a} = \frac{y^2}{b}$$

We could do the same thing with different variables and find that this is also equal to $$\frac{z^2}{c}$$. So we have $$\frac{x^2}{a} = \frac{y^2}{b} = \frac{z^2}{c} = k$$

Now we’re at the place I foresaw, where we can take square roots: $$x = \pm\sqrt{ak}$$ $$y = \pm\sqrt{bk}$$ $$z = \pm\sqrt{ck}$$

Therefore, since $$x+y+z=0$$, we know that $$\pm\sqrt{ak}+\pm\sqrt{bk}+\pm\sqrt{ck}=0$$

and, dividing by $$\sqrt{k}$$, we have $$\pm\sqrt{a}\pm\sqrt{b}\pm\sqrt{c}=0$$

In March, it turns out, I stopped short of the answer, thinking I saw it coming. But in fact, I didn’t attain the goal! I hoped that a , b , and c would be squared before we have to take the roots. We seem, however, to have proved that they must all be positive , which makes the conclusion impossible!

I’m wondering if the problem, which was never quite actually stated, might have been different from what I assumed. In fact, armed with this suspicion, I tried to find an example or a counterexample, and found that if $$\begin{pmatrix}a & b & c\\ x & y & z\end{pmatrix}= \begin{pmatrix}1 & 4 & 1\\ 1 & -2 & 1\end{pmatrix}$$ satisfies the conditions, with $$bz + cy = cx + az = ay + bx = 2,$$ but no combination of signed a , b , and c add up to 0. So the real problem must have been something else …

## Remembering how to solve a problem

At this point J abandoned that path, and closed with a side issue:

Hi Doctor. I have one more question about problem solving. I spent some more time on the problem we discussed then I skipped it and decided to focus on other problems instead. I managed to solve a few of them but then I took a long break when I came back I couldn’t remember the solutions without looking at my work . I don’t know if you read How to solve it by Pólya. I ask since at the beginning of that book Pólya gives an example of a mathematical problem. The problem in question is this: Find the diagonal of a rectangular parallelepiped if the length, width, and height are known. He asks the reader to consider the auxiliary problem of finding the diagonal of the right triangle using Pythagoras theorem. I am telling you this because the solution to this problem is very clear; I can recall it even long after I finished reading. I do not feel the same about algebra problems. I solve them, do the obvious things, and then I almost immediately forget. Does that happen to you? If not how do you remember the solution? I just want to know if you find these algebra problems as unintuitive as I do.

My memory is as bad as anyone’s! I replied,

I wouldn’t say that I remember every solution I’ve done, or every solution I’ve read. The example you give is a classic that stands out, particularly the overall strategy. Others are more ad-hoc and don’t feel universal (in the sense of being applicable to a large class of problems), so they don’t stick in the memory. I don’t have my copy of Pólya with me (I’ve been meaning to look for it), but I recall that one of his principles is to take time after solving a problem to focus on what you did and think about how it might be of use for other problems. This is something like looking around before I leave my car in a parking lot to be sure I will recognize where I left it when I come back from another direction. I want to fix the good idea in my mind and be able to recognize future times when it will fit. But even though I do have that habit, there are some problem types that I recognize over and over, but keep forgetting what the trick is. (Maybe sometimes it’s because I’ve seen two different tricks, and they get mixed up in my mind.) So you’re not alone. For me, though, it’s not such much being unintuitive , as just not being memorable , or being too complex for me to have focused on them enough to remember.

So Pólya recognized the likelihood of forgetting (failing to learn from what you have done), and the need to make a deliberate effort there!

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• DOI: 10.1080/03055698.2019.1701993
• Corpus ID: 213509456

## The relationship between routine and non-routine problem solving and learning styles

• Z. Kablan , S. Uğur
• Published 19 January 2020
• Education, Psychology
• Educational Studies

## 11 Citations

The link between teacher-student relationships and non-routine problem solving in ninth-grade mathematics: a moderated mediation model, the relationship between college freshmen's self-understanding and problem-solving ability: a structural equation modeling analysis, profile of argumentation ability of undergraduate students in chemistry education based on non-routine problems, school-type difference among rural grade four malaysian students’ performance in solving mathematics word problems involving higher order thinking skills, prospective primary teachers’ initial mathematical problem-solving knowledge, top achievers in mathematics in the end of upper secondary school, an examination of the relationship between secondary school students’ abstract thinking skills, self-efficacy perceptions and attitudes towards mathematics, the principles of four basic steps of scientific stage: problem, hypothesis, trial, report, gaya belajar kolb dan kemampuan pemecahan masalah matematis pada mahasiswa pendidikan matematika universitas la tansa mashiro, sınıf öğretmenlerinin rutin olmayan problemlere i̇lişkin görüşleri, 97 references, turkish primary school students' strategies in solving a non-routine mathematical problem and some implications for the curriculum design and implementation., the investigation of teacher candidates’ skills of solving exercises and non-routine problems related to the topic of series, the primary school teacher trainees’ skills and opinions on solving non-routine mathematical problems, common and flexible use of mathematical non routine problem solving strategies, non-routine mathematical problems among in-service and pre-service mathematics teachers, the relationship between secondary school pre-service mathematics teachers' skills in problem solving dimensions and their learning style characteristics, making it count: strategies for improving problem-solving skills in mathematics for students and teachers' classroom management, a metacognitive-based instruction for primary four students to approach non-routine mathematical word problems, exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics, fourth graders and non-routine problems: are strategies decisive for success, related papers.

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