non routine problems mathematics

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.

I added a few more thoughts about strategies:

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 Polya’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 Polya.

Here is what Polya 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 Polya’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” Polya 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, Polya 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 Polya 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 Polya recognized the likelihood of forgetting (failing to learn from what you have done), and the need to make a deliberate effort there!

What is non routine and routine 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.

What is an instructional math routine?

• Instructional routines are specific and repeatable designs for learning that support both the teacher and students in the classroom… enabling all students to engage more fully in learning opportunities while building crucial mathematical thinking habits.

What are number routines?

What are they? are a collection of quick, low-prep 5 – 15 minute activities. focus on the big ideas in mathematics through activating the curricular competencies in relation to content knowledge. serve to teach, re-teach, reinforce, and enrich.

What are routine problems in mathematics?

From the curricular point of view, routine problem solving involves using at least one of the four arithmetic operations and/or ratio to solve problems that are practical in nature. Children typically do routine problem solving as early as age 5 or 6.

What are examples of routine problems?

Routine and nonroutine problems. In a routine problem, the problem solver knows a solution method and only needs to carry it out. For example, for most adults the problem “589 × 45 = ___” is a routine problem if they know the procedure for multicolumn multiplication.

What are number sentences?

A number sentence is a combination of numbers and mathematical operations that children are often required to solve. Examples of number sentences include: 32 + 57 = ? 5 x 6 = 10 x ?

What are some instructional routines?

The elements of instructional routines include: The teacher’s words in getting the student’s attention, explaining the task (lesson, project, activity), giving directions, offering assistance, giving feedback (and correction if necessary), reviewing, summarizing, and the like.

What is number Talk?

A Number Talk is a short, ongoing daily routine that provides students with meaningful ongoing practice with computation. A Number Talk is NOT intended to replace current curriculum or take up the majority time spent on mathematics. A classroom should spend only 5-15 minutes on a Number Talk each day.

What is number sense examples?

Number sense can be thought of as flexible thinking and intuition about number. When students use friendly numbers (like numbers that end in zero, such as 10, 30, or 100) or numbers that they are familiar with (for example, 27 is almost 25), this helps them to understand how numbers relate to one another.

Do you have any math routines you use in your classroom?

This routine should around 5 – 10 minutes. It can be used as a daily opening to your math block or as a morning sponge activity. This wraps up the 3 routines to build number sense. I hope you enjoyed all of these tips to help with your kids. Do you have any number sense math routines you use in your classrooms?

Which is an example of a math language routine?

A ‘math language routine’ refers to a structured but adaptable format for amplifying, assessing, and developing students’ language. More information and examples of each of these routines can be found here.

What is the purpose of mathematical routine 2?

MATHEMATICAL LANGUAGE ROUTINE 2: COLLECT AND DISPLAY Purpose: To capture students’ oral words and phrases into a stable, collective reference. The intent of this routine is to stabilize the fleeting language that students use in order for their own output to be used as a reference in developing their mathematical language.

How does the ways to make a number routine work?

I love that the “Ways to Make a Number” routine allows students to build on what they know about a number but also offers, through discussions with one another, the opportunity to clear up any partial misunderstandings. For example, during this routine one of my students told me he used ten frames to represent the number 58.

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Routine and non routine problems

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NON-ROUTINE PROBLEMS IN PRIMARY MATHEMATICS WORKBOOKS FROM ROMANIA

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Abstarct: The aim of this paper is to present a research on Hungarian 3th grade primary school textbooks from Romania. These textbooks are analyzed using two classifications. The first classification is based on how much creativity and problem solving skills pupils need to solve a given task. In this classification problems are gouped in three categories: routine problems, gray-area problems and puzzle-like (non-routine) problems. The results show that most of the problems from textbooks are routine-problems. Only about 15% of the problems are more difficult, which can be solved in few steps, but even these problems are not challenging. The second classification divide problems based on how the operation chain they have to solve is given: by numbers, by text or in a word problem. The results show that there are big differences in the percentage of problems from these three categories in different textbooks. In one of the studied textbook half of the problems are word problems, in the other one only one quarter.

Keywords: mathematical problem solving; non-routine problems, textbook.

Introduction

International Mathematics tests focus on problem solving, thus these tests includes non-routine problems too. Romanian pupils have high scores, above international average on routine problems, but they obtain lower scores than the average on non-routine problems.

The aim of this paper is to present a research regarding Hungarian 3th grade primary school textbooks from Romania. The problems from these textbooks are analyzed based on two classifications. The first classification is based on how much creativity and problem solving skills pupils need to solve a given task. In this classification problems are gouped in three categories: routine problems, gray-area problems and puzzle-like (non-routine) problems. The second classification divides problems based on how the operation chain they have to solve is given: by numbers, by text or in a word problem.

Theoretical background

While learning Mathematics, pupils solve exercises and problems in order to deeper the acquired knowledge and to develop their mathematical skills. Kantowski (1977, p. 163) highlight the differences between exercise and problem: "an individual is faced with a problem when he encounters a question he cannot answer or a situation he is unable to resolve using the knowledge immediately available to him. [....] A problem differs from an exercise in that the problem solver does not have an algorithm that, when applied, will certainly lead to a solution." In some literature problems are named "non-routine problems" in order to highlight that when solving a problem "requires a novel idea from the student" (Milgram, 2007, p. 47). In TIMMS 2011 framework "non-routine problems are problems that are very likely to be unfamiliar to students. They make cognitive demands over and above those needed for solution of routine problems, even when the knowledge and skills required for their solution have been learned." (Mullis et al, 2009, p. 45). So if the student knows what method, algorithm, technique or formula to use for solving a task, then that task is not a problem, it is a routine exercise (Schoenfeld, 1985). Thus it is possible that the same task is a problem for one student and it is an exercise for another one (Zhu & Fan, 2006). Also, a problem is no longer considered a problem for that student, who already solved it (Selden et al, 1999).

In order to be able to solve non-routine problems, students' problem solving competence has to be developed. According to PISA evaluators "problem solving competency is an individual's capacity to engage in cognitive processing to understand and resolve problem situations where a method of solution is not immediately obvious. It includes the willingness to engage with such situations in order to achieve one's potential as a constructive and reflective citizen." (OECD, 2010) They have extended the cognitive domain underlined in definition used for the PISA 2003 evaluation (OECD, 2003) by the affective domain: the willingness of solving problems. Students' interest in mathematics, their beliefs in the utility of the mathematical knowledge in their future career or in their everyday life determine in a fundamental way their problem-solving behaviour. "Belief systems are one's mathematical world view, the perspective with which one approaches mathematics and mathematical task. One's beliefs about mathematics can determine how one chooses to approach a problem, which techniques will be used or avoided, how long and how hard one will work on it, and so on." (Schoenfeld, 1985, p. 45)

Problem solving competency involves the ability to use the acquired knowledge in a new way, the ability to learn new things which are useful for the problem and to discover new methods for the solution. So the transfer of knowledge and skills to new situation is essential. Creative thinking and critical thinking are important components of problem solving competency (Mayer, 1992).

For a successful problem solving students need to use various problem solving strategies and to be flexible. Strategy flexibility is "the behaviour of switching strategies during the solution of problem" (Elia, Van den Heuvel- Panhuizen & Kolovou, 2009, p. 607). Also self-regulation is necessary when solving non-routine problems.

Teachers rarely emphasis non-routine problem solving in their classroom (Silver et al, 2005; Leikin & Levav-Waynberg, 2007). In Romania, most of the problems given on national Mathematics tests require to apply formulas or algorithms. These problems has a mathematical formulations, they don't have any connection with real life (Marchis, 2009). Thus teachers are tempted to solve many routine problems that their pupils obtain good results at these tests. But most of the pupils who pass these tests and even they get good marks don't have a good problem solving competence, they have just learnt some techniques, methods or formulas and they know which one to use for a specific problem. Another reason, that teachers don't solve non-routine problems in the classroom is that they are not confident in their problem solving competence and they are not comfortable with handling pedagogical demands required for this type of problem solving activity (Silver et al, 2005).

A study on how primary school teachers in Romania develop their pupils' word problem solving skills shows that three quarters of the teachers guide pupils in order to understand the problem and encourage them for self-control during problem solving; only one third of the respondents encourage their students to solve the problems with more methods. Almost three quarters of the primary school teachers state that they give interesting, real-life problems in class. (Marchis, 2012)

Pupils have to learn the steps of the problem solving. Pólya (1945) has identified four main stages when solving a problem: understanding the problem, making a plan, carrying out the plan, and reviewing the solution. Similar steps are described by other researchers (among others Higgins, 1997; Leader & Middleton, 2004; Ridlon, 2004). The understanding stage includes some text comprehension techniques, for example, to identify the unknown words, to reformulate the problem, to think about a picture or diagram that might help to understand the problem context, and the relations between the given and unknown data (Pólya, 1957). Based on PISA 2012 problem solving has the following four steps: exploring and understanding, representing and formulating, planning and executing, monitoring and reflecting.

Textbooks and other materials are important factors in influencing mathematics teaching (Braslavsky & Halil, 2006; Cueto, Ramírez, & León, 2006; Nicol & Crespo, 2006). Worked examples help pupils to acquire problem solving methods. Research shows that studying worked examples it is an effective and efficient way of learning mathematics (Paas & van Gog, 2006).

Research design

We have selected two textbooks for 3rd grade (we refer to them by Textbook 1 and Textbook 2) and studied the problems given in three chapters:

- Adding and substracting natural numbers;

- Multiplying natural numbers between 0 and 10 (multiplication table);

- Dividing natural numbers (divisions which can be calculated using the multiplication table).

We classified the problems using two different classifications.

In the first one we used the classification of the tasks given by Kolovou, van den Heuvel-Panhuizen, and Bakker (2009). They have divided textbook problems in three categories:

- routine exercises, which require application of a known algorithm.

- non-routine, puzzle-like tasks, which are problems that require creative thinking and a higher level of problem solving thinking.

- gray-area tasks, which can't be included in any of the above two category. These tasks can't be solved by only applying a known algorithm.

In the second classification we grouped the problems in three categories:

- solving an operation chain, where the operations are given by numbers and operation signs, i.e. Calculate 2×6+9.

- solving an operation chain discribed by text, where the operations are given by text, i.e. Find the sum and the difference of 56 and 34.

- word problem, where pupils have to discover which operation to use, i.e. Ana has 45 glass ball, Peter has 4 less than Ana. How many glass balls they have together?

We also studied if there are tasks in which pupils has to create a word problem. In these task pupils have to formulate a problem:

- based on an operation chain;

- based on a graphical representation;

- based on a drawing.

Results and discussion

Dividing tasks in routine, gray-area and puzzle-like categories

Analyzing the problems in the three selected chapters from Textbook 1 and Textbook 2 we can conclude that there is no any puzzle-like problem. There are problems, which can be solved with backward method, which could have been challenging for pupils, but the idea of the solving method is given. It is the same situation with some problems which can be solved by the graphical method. There are some problems, which are more difficult, require more steps in the solution plan, but these problems can't be considered puzzle-like problems. Kolovou, van den Heuvel-Panhuizen, and Bakker (2009) have arrived to similar conclusion in case of two textbooks studied by them. In the other textbooks the maximum percentage of puzzle-like tasks is 2,43%.

It is very difficult to distinguish between routine-problems and gray-area problems, as a problem could be routine problem for somebody already solved something similar and gray-area problem for somebody first time sees that type of problem. Thus problems, which can be solved in more steps, could be considered gray-area problems for some pupils and routine-problems for other pupils. In this study we considered multiple step problems as gray-area problems. Problem 1 is an example of grayarea problem from Textbook 2 and Problem 2 is such an example from Textbook 1. We have counted the number of routine-problems and gray-area problems for each operation in Textbook 1 and Textbook 2 (see Table 1). We could observe that the number of gray-area problem is quite similar for the two textbooks, 16.8% for Textbook 1 and 14.8% for Textbook 2. In Textbook 2 the percentage of gray-area problems for addition and substraction is much higher than in Textbook 1 (30.0% in Textbook 2 and 17.5% in Textbook 1). In case of division in Textbook 1 there are more gray-area problems than in Textbook 2 (26.5% in Textbook 1 and 16.7% in Textbook 2).

These results are similar with that obtained by Kolovou, van den Heuvel-Panhuizen, and Bakker (2009). They the textbooks analyzed by them the percentage of gray-area and puzzle-like tasks together were between 5% and 13%. They also mentioned the difficulty on deciding if a task is grayarea or routine.

Dividing tasks by how the operations needed to carry out are given

The most easier is when the operations to be performed are given by an operation chain with numbers and operation signs (i.e. 3×4+24:6).

A bit more difficult is when the operations to be carried out are hidden in a text (i.e. calculate the sum of 345 and 234). In this case pupils have to know specific mathematical terms, as sum, difference, product, etc.

The most difficult is when a text problem formulation is given because solving these problems requires text comprehension, problem representation, selection of the adequate operations, solving these operations (Kintsch & Greeno, 1985; Swanson, 2004). When solving these problems the most important difficulties are related with forming an operation based on the text of the problem (Carey, 1991; English, 1998). Usually pupils try to use the operations which they last learnt (i.e. if they learn multiplication they tend to use multiplication when solving text problems). Thus it is important that in a chapter related with some operation to include also text problems which have to be solved using other operations. In Textbook 1 we have found few problems hidden in other chapters than the operation needed in those problems. A sequence of problems included in the division chapter highlights the differences between "3 more", "3 times more" and "3 times less" (see Problem 3, 4 and 5). In Textbook 2 there are also problems which need other operations than discussed in the current chapter. Problem 6 is in multiplication chapter, but it highlighs the difference between increasing by 2 (so adding 2 to the given number) and inceasing by 2 times (so multiplying by 2 the give number). Problem 7 is included in division chapter, but it contains also addition not only to calculate the total, but also to calculate the numbers of green balls, as the problem states that "András has 3 more green balls, than red balls".

To see the difference between the problems where the operations are given in text and word problems, see Problem 8 and 9 from Textbook 1). In Problem 8 the terms difference and sum indicates the operations needed. In Problem 9 the operation is also indicated by the word total, but it is not so obvious than in case of the term sum.

It is important that in different word problems the operation which pupils should use to be expressed differently. For example, addition could be suggested by the words total, together, more, etc. Multiplication also could be expressed in different ways. Problems 10 and 11 from Textbook 1 show two different problems in which multiplication should be used.

We have counted the numbers of each type of problems in above described three categories in each chapter. The results are included in Table 2.

In the studied three chapters we have analyzed 185 problems from Texbook 1 and 155 problems from Textbook 2. We can observe, that in Textbook 1 the percetage of word problems is double than in Textbook 2 (44.9% in Textbook 1 and 21.9% in Textbook 2). In Textbook 2 almost half of the problems (46.5%) are operation chains which have to be calculated, while in Textbook 1 only one third (30.3%) of the problems are of this type.

Composing word problems

Composing their own word problems also helps students in changing their attitudes regarding these problems and becoming familiar with the mathematical terminology (Edwards et al., 2002). In Textbook 1 there are problems in which pupils have to formulate the question(s) of the problem or to formulate the problem based on given arithmetic operations (see Problem 12) or graphical representation (see Problem 13). In Textbook 2 there are problems where pupils have to formulate a word problem based on a picture (see Problem 14, in the original problem three pairs of snowmen where represented, we have simplified the drawing using hexagons instead of snowmen).

The number of those task were pupils have to formulate a word problem is very low, 2 in Textbook 1 and 4 in Textbook 2.

As regarding the first classification, dividing problems in routine, gray-area and puzzle-like problems, the results show that most of the problems from textbooks are routine-problems. Only about 15% of the problems are more difficult, which can be solved in few steps, but even these problems are not challenging, pupils don't need to develop new solving methods. None of the problems can't be considered puzzle-like problem.

As regarding the second classification, dividing problems based on how the operation chain they have to solve is given (by numbers, by text or in a word problem), the result show that there are big differences in the percentage of problems from these three categories in different textbooks. In one of the studied textbook half of the problems are word problems, in the other one only one quarter. Thus teacher can choose the most appropriate textbook for their classroom.

The number of tasks where pupils have to formulate word problems is very limited.

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Iuliana Marchis, Babes-Bolyai University, Cluj-Napoca (Romania).

E-mail: [email protected]

Acknowledgment

This research was founded by Domus Hungarica research grant, contract number DSZ/34/2012.

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The aim of this paper is to present a research on Hungarian 3th grade primary school textbooks from Romania. These textbooks are analyzed using two classifications. The first classification is based on how much creativity and problem solving skills pupils need to solve a given task. In this classification problems are gouped in three categories: routine problems, gray-area problems and puzzle-like (non-routine) problems. The results show that most of the problems from textbooks are routine-problems. Only about 15% of the problems are more difficult, which can be solved in few steps, but even these problems are not challenging. The second classification divide problems based on how the operation chain they have to solve is given: by numbers, by text or in a word problem. The results show that there are big differences in the percentage of problems from these three categories in different textbooks. In one of the studied textbook half of the problems are word problems, in the other one only one quarter. [PUBLICATION ABSTRACT]

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Professor of Mathematics, University of North Georgia

3. Non-routine Problem Solving

Almost none of the work kids do in K12 math classrooms makes them better problem solvers. The vast majority of kindergartners enter school with the ability to solve problems they haven’t seen solved before. The vast majority teenagers in the US graduate high school without that ability. That’s almost criminal negligence since we know how to improve problem-solving abilities.

What’s the problem?

To get better at problem solving, you have to actually solve a problem. Not mimic a solution. Not practice a bunch of slightly altered versions of an example. Not drill and practice someone else’s solution. You have to dig in yourself, get messy and figure it out.

The idea that you learn lots of problem-solving tricks in math class that you whip out when needed is almost a joke. A few kids with a certain flair for mathematics do manage to transfer their knowledge to other domains, but most great problem solvers didn’t learn it in math class.

Non-routine problems are those for which most of us have no idea what the solution is before we start. Scientists, mathematicians, doctors, lawyers, teachers – no one has a head start on these problems.

The best non-routine problems are:

If you want some good non-rountine problems to use yourself or with your kids, check out Stella’s Stunners which has some good ones or these excerpts from Polya’s How to Solve It .

Need an example?

You have an eccentric uncle who is leaving you his collection of twelve valuable pirate coins on one condition. One of the coins is fake and weighs either slightly less or slightly more than the other. It is otherwise identical to the others. Your challenge: find the fake and verify the 11 that are real in only three weighings with a standard balance scale. If you can do it, you uncle will give you the coins which are worth millions.

Give it a try!

When I did this problem as a part of a problem-solving journal activity, I logged over 12 hours of active problem-solving time before I finally got it. (You can probably do it faster, because you won’t have to journal about every step in your thought process.) I spent four or five more hours optimizing my solution to account for the number of coins that could be sorted with 4, 5, 6 or even N weighings. That’s the kind of revisions and extensions Polya meant.

If you’re going to make the effort to solve a problem, why not spend some additional time to leverage all that learning into lots more learning? Problem solving without reflection is like eating all your vegetables and then skipping dessert anyway.

International Conference on Artificial Intelligence in Education

AIED 2020: Artificial Intelligence in Education pp 409–413 Cite as

Improving Students’ Problem-Solving Flexibility in Non-routine Mathematics

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 12164)

A key issue in mathematics education is supporting students in developing general problem-solving skills that can be applied to novel, non-routine situations. However, typical mathematics instruction in the U.S. too often is dominated by rote learning, without exposing students to the underlying reasoning or alternate ways to solve problems. As a first step in addressing this problem, we present a cognitive task analysis study that investigates how students without a mathematics-related background solve novel non-routine problems. We found that most students were able to identify the underlying pattern that yields the final solution in each problem. Furthermore, they tended to use various forms of visualization in their draft work, but occasionally made computational mistakes. Based on these results, we propose our plan for developing an instructional platform that leverages learning science principles to train students in problem-solving abilities.

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1 Introduction

The ability to tackle non-routine problems – those that cannot be solved with a known method or formula and require analysis and synthesis as well as creativity [ 9 ] – is becoming increasingly important in the 21st century [ 5 ]. However, when faced with a non-routine problem, U.S. students tend to apply memorized procedures incorrectly rather than modify them or develop new solutions [ 8 ]. One possible source for this difficulty is the typical instructional focus in U.S. schools on memorization and application of routine procedures [ 2 , 6 , 7 ]. Such an approach makes students proficient at executing rote procedures, but it does little to help them understand the conceptual basis for the procedures or to think creatively about novel problems - both of which are essential for developing problem-solving flexibility.

An important first step in addressing this issue is to assess how students currently approach non-routine problem solving, so that we can design the appropriate learning interventions. In this work, we present an empirical cognitive task analysis where participants were asked to think aloud while solving a series of non-routine problems from discrete mathematics. We chose this domain because discrete math problems can often be tackled from multiple perspectives while not requiring any advanced background beyond the high school curriculum [ 3 ]. Based on the findings from this study, we propose our plan for developing a tutoring system for non-routine problem-solving ability. Then, we discuss the system’s broader implications and the challenges we need to address in deploying this system at scale.

2 Assessing Students’ Problem-Solving Skills

We conducted interview sessions with three students at a private university in a midwest US city. None of the students had a mathematics-related background. The participants were asked to solve three non-routine mathematics problems on paper in one hour. They were also encouraged to think aloud and write down their draft work. The three problems in our study, taken from [ 3 ], and a brief summary of their sample solutions, are as follows.

In an air show there are twenty rows. The first row contains one seat, the second three seats, the third five seats, the fourth seventh seats, and so on. How many seats are there in total ?

Sample solution: In the first row there is 1 seat. In the first two rows there are 1 + 3 = 4 seats. In the first three rows there are 1 + 3 + 5 = 9 seats. In the first four rows there are 1 + 3 + 5 + 7 = 16 seats. In the first five rows there are 1 + 3 + 5 + 7 + 9 = 25 seats. Based on this pattern, in the first k rows there are k 2 seats. In our case, there are 20 rows and therefore 400 seats in total.

Find all integers between 1 and 99 (inclusive) with all distinct digits.

Sample solution: there are 99 integers between 1 and 99 in total, and 9 of them have non-distinct digits, namely 11, 22, 33, …, 88, 99. Hence, the remaining 90 integers have distinct digits.

What is the digit in the ones place of 2 57 ?

Sample solution: Looking at the sequence of powers of 2–2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, … – we see that the corresponding sequence of digits in the ones places is 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, … In other words, this sequence is a cycle of length 4. Therefore the last digit of 2 57 is that of 2 53 , which is that of 2 49 , …, which is that of 2 1 , which is 2.

We then analyzed recordings of the participants’ think-aloud and their draftwork, from which we derived the following insights:

Pattern Identification.

Participants were aware that they had to find a pattern or formula to solve the problems, because it was not feasible to directly compute the final answer. All participants were able to identify the expected pattern for each problem as outlined above, except for one student who failed to do so for Problem 1 . While this participant realized that the number of seats on row k is the k-th positive odd number, this pattern alone was insufficient to solve the problem.

Visualization.

Participants tended to visualize the problem by drawing examples and making lists or tables (Fig. 1 ). They expressed that these visualizations were crucial in helping them identify the correct pattern and solve the problem.

Participants’ attempts at visualizing the problem in their draftworks.

Computation.

Participants occasionally made computational mistakes while calculating the initial sequence values, especially in Problem 3 . As a consequence, they could not identify any pattern based on the wrong values, and took some time to realize the mistake. All students who corrected their mistakes were able to subsequently solve the problem.

In summary, we found that participants were aware of the idea behind identifying patterns, and they all did so via some kind of visualization. On the other hand, computational mistakes, while not directly related to our learning objectives, can be detrimental to the overall problem-solving process. From these insights, we propose the following next steps.

3 Developing a Tutoring System for Flexible Problem-Solving

Moving forward, our plan is to iteratively conduct more cognitive task analysis interviews and develop a prototype of the system. Our initial conceptualization of how the system will work is as follows. A single round of exercise in the system incorporates four learning stages, all of which are built on established learning principles: 1) Reviewing a worked example of a non-routine mathematics problem, 2) Explaining the worked example to a partner, 3) Solving a new problem which is isomorphic to the worked example problem, and 4) Explaining the isomorphic solution to a partner. Between rounds, the student can review previous solutions, look at materials related to the problem space, or practice basic math skills. This design is intended to (1) formally introduce students to a complete solution through worked examples, (2) reinforce their understanding of the worked example through self-explanation, and (3) assess students’ learning through an isomorphic problem. Our hypothesis is that through the learning system, students will get a better sense of how to approach a novel non-routine problem, so that in case they have not yet found the solution – for example, like the participant in our study who did not identify the true pattern in Problem 1 – they can still adopt a different viewpoint and explore other strategies.

We have already begun mapping the problem space by developing a non-routine problem-solving flowchart and identifying sets of potential non-routine problem solutions. Once we have tested our solution space, we will develop and pilot a low fidelity paper prototype version of the system with college students to further refine the mathematical content and identify areas for revision to the design. We are also looking at which technological features could be useful for students learning in this domain. As a first step, our system will include a canvas for students to perform their draftwork on, as well as a simple calculator interface with basic arithmetic operations to help students avoid computational mistakes. An important follow-up question is whether students’ draftwork can be analyzed to infer their thinking process, which could in turn guide the design of appropriate feedback mechanics. While this task has previously been performed manually by domain experts [ 1 ], employing a machine learning technique to automate it to some extent would greatly enhance the system’s adaptive support functionality and scalability.

4 Conclusion

This research will provide concrete, generalizable evidence about the utility and implementation of worked examples, multiple solutions, and self-explanation to promote skills in non-routine problem solving. Results will inform future tutoring system design by identifying how and when the instructional features are most beneficial for developing problem-solving skills. We also intend to have a practical impact by distributing a tutoring system that is accessible to a wide range of students, including lower-performing students who would typically not be exposed to these types of problems and strategies [ 1 , 4 ]. In addition, we will provide a teacher’s guide to support educators in using the system adaptively to support their instructional goals.

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Nguyen, H.A., Guo, Y., Stamper, J., McLaren, B.M. (2020). Improving Students’ Problem-Solving Flexibility in Non-routine Mathematics. In: Bittencourt, I., Cukurova, M., Muldner, K., Luckin, R., Millán, E. (eds) Artificial Intelligence in Education. AIED 2020. Lecture Notes in Computer Science(), vol 12164. Springer, Cham. https://doi.org/10.1007/978-3-030-52240-7_74

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José G. Vargas-hernández

Inside Problem Solving

The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity. The problems were developed by the Silicon Valley Mathematics Initiative and are aligned to the Common Core standards.

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Courtney’s Collection Cut It Out Cutting a Cube Digging Dinosaurs Diminishing Return First Rate Friends You Can Count On Game Show Got Your Number Growing Staircases Measuring Mammals Measuring Up Miles of Tiles Movin ‘n Groovin On Balance Once Upon A Time Part and Whole Party Time Piece it Together Polly Gone Rod Trains Surrounded and Covered Squirreling It Away The Shape of Things The Wheel Shop Through the Grapevine Tri-Triangles What’s Your Angle?

Cutting a Cube (K.G.B.4) Digging Dinosaurs (K.OA.A.2) First Rate (K.CC.B.5, K.CC.C.6) Growing Staircases (K.CC.B.5) On Balance (K.MD.A.2)

Cutting a Cube (1.G.A.1) Growing Staircases (1.OA.A.1) Rod Trains (1.MD.A.2, 1.OA.C.6) Measuring Mammals (1.MD.A.1) Miles of Tiles (1.OA.A.1) Movin ‘n Groovin (1.OA.A.1) Piece it Together (1.G.A.2)

Courtney’s Collection (2.MD.C.8) Digging Dinosaurs (2.MD.C.8) Got Your Number (2.OA.B.2, 2.NBT.A.1, 2.NBT.A.4, 2.NBT.B.5) Miles of Tiles (2.NBT.B.5) Part and Whole (2.G.A.3) Piece it Together (2.G.A.1) Squirreling It Away (2.OA.1) The Shape of Things (2.G.A.1) Through the Grapevine (2.MD.D.9, 2.MD.D.10) What’s Your Angle? (2.G.A.1)

Measuring Up (3.OA.A.3) Once Upon A Time (3.MD.A.1) Part and Whole (3.G.A.2, 3.NF.A.1, 3.MD.C.6) Party Time (3.OA.A.3) Piece it Together (3.MD.C.5, 3.MD.D.8) Polly Gone (3.MD.D.8) Surrounded and Covered (3.MD.C.6, 3.MD.D.8) The Wheel Shop (3.OA.A.1, 3.OA.A.2) Tri-Triangles (3.OA.A.3)

Courtney’s Collection (4.MD.A.2) Digging Dinosaurs (4.MD.A.2) Diminishing Return (4.OA.A.3, 4.MD.A.2) Friends You Can Count On (4.OA.A.3) Game Show (4.OA.C.5) Growing Staircases (4.OA.C.5) Measuring Mammals (4.OA.A.2) Measuring Up (4.OA.A.3) Once Upon A Time (4.OA.A.3) Part and Whole (4.G.A.3) Party Time (4.NF.B.4c) Piece it Together (4.G.A.2, 4.MD.C.6) Squirreling It Away (4.OA.3) The Shape of Things (4.G.A.3) The Wheel Shop (4.OA.A.3) Tri-Triangles (4.OA.C.5)

Digging Dinosaurs (5.NBT.B.7) Movin ‘n Groovin (5.NF.B.4)

Courtney’s Collection (6.NS.B.4) Cutting a Cube (6.G.A.4, 6.RP.A.3c) Diminishing Return (6.RP.A.3a, 6.RP.A.3b) First Rate (6.RP.A.3b, 6.RP.A.2) Measuring Up (6.RP.A.3c, 6.EE.A.1, 6.EE.B.7) On Balance (6.EE.B.5, 6.EE.B.6, 6.EE.B.8) Once Upon A Time (6.NS.B.2, 6.NS.B.4) Movin ‘n Groovin (6.RP.A.3d) Part and Whole (6.G.A.1) Piece it Together (6.G.A.4) Polly Gone (6.G.A.1) Surrounded and Covered (6.RP.A.2, 6.RP.A.3b) Tri-Triangles (6.EE.A.1, 6.EE.B.6, 6.EE.C.9)

Courtney’s Collection (7.SP.C.8b) First Rate (7.RP.A.2b, 7.RP.A.3, 7.EE.B.4a) Friends You Can Count On (7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b) Game Show (7.SP.C.8a, 7.SP.C.8b) Got Your Number (7.NS.A.3) Measuring Mammals (7.RP.A.2a, 7.RP.A.2b, 7.RP.A.2c 7.RP.A.1) Measuring Up (7.RP.A.2b, 7.RP.A.2c, 7.RP.A.3, 7.EE.B.4) Movin ‘n Groovin (7.RP.A.2c, 7.RP.A.3) Part and Whole (7.NS.A.1D) Piece it Together (7.G.B.6) Polly Gone (7.G.B.6, 7.G.B.4) Rod Trains (7.SP.C.8b) Squirreling It Away (7.SP.8b) Surrounded and Covered (7.G.B.4, 7.G.B.6) Through the Grapevine (7.SP.A.2)

Cutting a Cube (8.G.A.1a) Digging Dinosaurs (8.EE.C.7b, 8.F.B.4) Diminishing Return (8.EE.C.7.b) Miles of Tiles (8.EE.C.8b, 8.EE.C.8c) Movin ‘n Groovin (8.EE.B5) On Balance (8.EE.C.8b, 8.EE.C.8c) Once Upon A Time (8.EE.C.8b) Squirreling It Away (8-F.1) Through the Grapevine (8.SP.A.1, 8.SP.A.2) The Wheel Shop (8.EE.C.8b, 8.EE.C.8c)

Courtney’s Collection (A-CED.A.2) Digging Dinosaurs (A-CED.A.2) Diminishing Return (A-CED.A.1) Growing Staircases (A-CED.A.2) Measuring Mammals (A-CED.A.2, A-REI.B.3, A-REI.C.6) Measuring Up (A-CED.2) Miles of Tiles (A-APR.A.1, A-SSE.A.1a, A-SSE.A.2) On Balance (A-CED.A.2, A-REI.C.6) Once Upon A Time (A-CED.A.1) Part and Whole (A-APR.D.6) Polly Gone (A-REI.C.6) Squirreling It Away (A-CED.2, A-CED.3, A-REI.6, A-REI.8, A-REI.10) The Wheel Shop (A-REI.C.6, A-REI.D.12) Tri-Triangles (A-CED.A.1, A-REI.B.4b, A-SSE.A.2)

Cut It Out (F-BF.A.1a) Digging Dinosaurs (F-IF.C.7b, F-IF.C.7e) Diminishing Return (F-BF.A.1a) First Rate (F-IF.B.6, F-BF.A.1a) Growing Staircases (F-LE.A.2, F-BF.A.2, F-BF.A.1a) Movin ‘n Groovin (F.BF. A.1a) Rod Trains (F-BF.A.1a) Squirreling It Away (F.LE.2, F-BF.1a, F-BF.2) Surrounded and Covered (F-BF.A.1a) Tri-Triangles (F-BF.A.1a) What’s Your Angle? (F-BF.A.1a)

Cut It Out (G-CO.B.6) Growing Staircases (G-MG.1) First Rate (G-SRT.C.8) Measuring Mammals (G-SRT.B.5) Miles of Tiles (G-MG.A.3) Once Upon A Time (G-C.A.2) Piece it Together (G.MG.A.1, G-MG.A.3, G.GMD.A.1, G.SRT.C.8) Polly Gone (G-CO.B.7, G-GPE.B.7, G-MG.A.3, G-GPE.B.4) The Shape of Things (G-C.A.2, G-CO.C.10, G-CO.C.11, G-SRT.B.5, G-MG.A.1) What’s Your Angle? (G-MG.A.3, G-C.A.2)

Digging Dinosaurs (S-ID.6.a) Diminishing Return (S-CP.A.2, S-CP.B.8) Friends You Can Count On (S-CP.A.4, S-CP.A.5, S-CP.B.6) Game Show (S-MD.A.1, S-MD.A.2, S-MD.A.3) Growing Staircases (S-ID.6a) Party Time (S-CP.A.1, S-CP.B.9, S-CP.B.8) Squirreling It Away (S-ID.6a) Through the Grapevine (S-IC.B.4, S-ID.A.1, S-ID.A.2, S-ID.A.3, S-ID.B.5, S-ID.B.6c) The Wheel Shop (S-CP.A.1)

Why Problem Solving?

Problem solving is the cornerstone of doing mathematics. George Polya, a famous mathematician from Stanford, once said, "A problem is not a problem if you can solve it in 24 hours." His point was that a problem that you can solve in less than a day is usually a problem that is similar to one that you have solved before, or at least is one where you recognize that a certain approach would lead to the solution. Bu t in real life, a problem is a situation that confronts you and you don’t have an idea of where to even start. Mathematics is the toolbox that solves so many problems. Whether it is calculating an estimate measure, modeling a complex situation, determining the probability of a chance event, transforming a graphical image or proving a case using deductive reasoning, mathematics is used. If we want our student s to be problem solvers and mathematically powerful, we must model perseverance and challenge students with non-routine problems.

IMAGES

Solve Non-Routine Real World Mathematics Problem Workbook 5
Non Routine Mathematics Problem
Math Enrichment: Math Challenge Problems for Younger Students -Distance Learning
Math Problem Solving Examples With Solutions For Grade 4
Solve Non-Routine Real World Mathematics Problem Workbook 6
Image result for non routine problems in mathematics worksheets

VIDEO

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Can you solve this? #maths #problem #math #study #useful #youtubeshorts #shorts #ytshorts #viral

COMMENTS

Non-Routine Mathematics
Creative problem solving or Non-Routine Mathematics involves finding solutions for unseen problems or situations that are different from structured Maths problems. There are no set formulae or strategies to solve them , and it takes creativity, flexibility and originality to do so.
Non-Routine Problems
Non-routine problem fRoutine problem defined as a problem in mathematic lesson that involves easy and simple problem solving. It present a question to be answered with out need certain strategies. It means, the routine problem can be solved by direct application of previously learned algorithms. fExample for routine problem:
Routine and Non-Routine Problems in Mathematics
• 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. Oftentimes, these problems can be solved in multiple ways. -Melissa from teacherthrive.com
Non-routine Algebra Problems
(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?
What is non routine and routine 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. What is an instructional math routine?
Routine and non routine problems
Solving routine and non routine problems involving multiplication without or ... EMELITAFERNANDO1 4.4k views • 24 slides Math 7 Curriculum Guide rev.2016 Chuckry Maunes 18.9k views • 26 slides K to 12 math curriculum Melvin Rey Alim 28.2k views • 50 slides Grade 7 Learning Module in MATH Geneses Abarcar 585.3k views • 261 slides
The Difference of Routine and Non-routine Problems in Mathematics
This video shows the difference between routine and Non-routine Problems in Mathematics with some examples for each type of problem. Changing Improper Fraction into Mixed Number and Vice Versa...
Routine and Non-Routine Problems
Articulate the importance of Mathematics in one; Harnessing Inclusive Opportunities of Globalization around the world in making things happen; ... Non - Routine Problem Solving - 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 ...
NON-ROUTINE PROBLEMS IN PRIMARY MATHEMATICS
My Research and Language Selection Sign into My Research Create My Research Account English; Help and support. Support Center Find answers to questions about products, access, use, setup, and administration.; Contact Us Have a question, idea, or some feedback? We want to hear from you.
PDF Non-routine problem solving and strategy flexibility: A quasi
a non-routine problem. Leading math educators argue that non-routine problems are indispensable for the development of students' problem-solving and reasoning skills (e.g. Polya, 1957; Schoenfeld, 1992). Non-routine problem-solving strategies can be defined as procedures used to explore, analyze and examine aspects of non-routine problems to ...
Routine and Non-Routine Problem.pdf
Non-Routine Problems 1. Jean thinks of a number. If he multiplies it by 30, then divide by 5 then subtract by 250, he will have an answer of 62. What was Jean's number? 2. Mr. Thomas raise chickens and cows in a farm for a total of 32 heads in all. How many of each does he have if there are 98 feet?Lesson 2
PDF Non-routine problem solving performances of mathematics teacher ...
the mathematics teacher candidates in non-routine problems at different difficulty levels. The descriptive survey model was preferred because it can be used to summarize the characteristics (capabilities, preferences, behaviors, etc.) of the study group, as it is intended to describe a situation that exists in the past or still
Teaching problem solving using non-routine tasks
This study examines several activity tasks incorporated with non-routine problems through the use of an emerging mathematics framework, at two junior colleges in Brunei Darussalam.
3. Non-routine Problem Solving
Non-routine problems are those for which most of us have no idea what the solution is before we start. Scientists, mathematicians, doctors, lawyers, teachers - no one has a head start on these problems. The best non-routine problems are: Open-ended, Allow multiple, correct solution paths, and Are completely new to the person trying to solve it.
Improving Students' Problem-Solving Flexibility in Non-routine Mathematics
The ability to tackle non-routine problems - those that cannot be solved with a known method or formula and require analysis and synthesis as well as creativity [] - is becoming increasingly important in the 21st century [].However, when faced with a non-routine problem, U.S. students tend to apply memorized procedures incorrectly rather than modify them or develop new solutions [].
Learning To Solve Non-routine Mathematical Problems
The findings of this study underscore the importance of implementing writing as an integral part of the mathematics curriculum and emphasize the need for additional research on writing in mathematics. The present study is related to seventh and eighth grade students' solutions to non-routine problems.
Non-routine Mathematical Problems: Phenomenological Analysis of
Specifically, non-routine problems were given emphasis in this study to understand education students' dispositions, appreciations, and experiences in applying mathematics in practical life situations and, consequently, will employ a more personalized teaching technique which suits their mathematical learning styles especially in doing problem …
Inside Problem Solving
The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.
Differences between routine and non-routine problems
In an effort to inculcate higher order thinking skills (HOTS) among students, the form of mathematical problems given to them should be changed from routine to nonroutine problems. To achieve...