Teaching Problem Solving Heuristics in Mathematics

What are heuristics?

Heuristics are techniques students can use to tackle a problem when the solution to the problem is not obvious. Some examples of heuristics are listed below and grouped into four categories according to how they are used: To give a representation e.g. draw a diagram, make a list, use equations To make a calculated guess e.g. guess and check, look for patterns, make suppositions To go through the process e.g. act it out, work backwards, before-after To change the problem e.g. restate the problem, simplify the problem, solve part of the problem Here is an example of one that involves guess and check: The perimeter of a rectangle is 42 cm. Its area is 108 cm2. Find its length and breadth. Since the perimeter of the rectangle is 42 cm, the sum of its length and breadth will be 21 cm. Make a list of the possible lengths and breadths.



Singapore mathematics syllabuses

The Singapore mathematics syllabuses, developed by Curriculum Planning and Developing Division (CPDD), Ministry of Education Singapore (MOE), have identified thirteen heuristics that are applicable to mathematical problem solving.
1. Act it out
2. Use a diagram/model
3. Use guess-and-check
4. Make a systematic list
5. Look for patterns
6. Work backwards
7. Use before-after concept
8. Make suppositions
9. Restate the problem in another way
10. Simplify the problem
11. Solve part of the problem
12. Think of a related problem
13. Use equations (Heuristics 12 and 13 are not in the primary syllabus.)

We can also treat these four ideas as four general heuristics, “use different representations”, “simplify your problem”, “approach your problem from different directions”, and “bring in solutions”.


Figure 1: Model for problem solving in mathematics (Tiong, Hedberg, & Lioe, 2005)

By definition, all heuristics have the following two characteristics: 1. Heuristics do not guarantee a solution. All heuristics do is pointing us towards possible ways in which we might be able to find our solution. 2. Heuristics do not come with specific procedures. When we use heuristics, we are required to make some judgments of our own regarding what we should do. Heuristics help us to deal with difficult problems or problems that we are not familiar with. Other than that, heuristics usually enables us to find solutions with less time and effort as compared to when we use algorithms to find solutions. Not all heuristics are the same in term of specificity. There are ones that give very general and ambiguous instruction, while the others have more specific procedures that we as problem solvers would like to follow. For example, heuristic “represent you problem differently” only gives us a very general direction to what we should do, where as heuristic “draw a diagram” tells how we can represent our problem, visually. Heuristic “draw a histogram” on the other hand gives a much more specific instruction than the previous two. Representation Simplification Pathway Bring in solution P R O B L E M S O L U T I O N To follow this heuristic “draw a histogram”, we will need to know first the procedures in which we can draw a histogram, whereas to follow the heuristic “draw a diagram” we can invent our own diagrams and the rules or procedures in which we can manipulate them; here we have the freedom to choose and be creative. Of course, we can still end using the histogram, since it might be the best representation for our problems. Here we can construct hierarchy of heuristics according to their specificity. We propose to put the four “general heuristics” from the model above on top of the hierarchy. Using the example in the previous paragraph, we have Figure 2. We should note that Figure 2 is not an exhaustive hierarchy for “representations”, we can still add in more representation into the hierarchy, such as “manipulative” into the second level, “pie chart” into the third level, and so on. Here we are just trying to get a rough idea of how a hierarchy of heuristics might look like. Since the number of heuristics are only limited by our creativity and imagination, it is not possible to construct a hierarchy that contain all possible heuristics.



Figure 2: An example of hierarchy of heuristics

Here we need to clarify first that diagram, symbol or histogram, matrix themselves are not heuristics, but “use diagram”, “use equation” and “draw table” are heuristics. Representations themselves are not heuristics, however suggestions to use representations are. Of course Figure 2 is only part of the whole hierarchy of heuristics for representations, and we can have similar hierarchy for “simplification”, “pathway”, and “bring in solution”. Heuristics on the top of the hierarchy are just very basic and general ideas that can be applied to most problems. These ideas can be further broken downward to make them more specific, making them easier to follow and apply to problems. However by doing so, we have restricted the applicability of the heuristics to only a specific few types of problems.

Heuristics Maths with specific procedures are usually less applicable than those with fewer procedures. Besides that, specific heuristics require less of problem solvers’ interpretation and intuition or creativity. As we can see in Figure 2, the “heuristics” at the bottom of the General Specific Representation Diagram Symbol Histogram Bar chart Matrix Equation Table hierarchy are pointing us to topics that we learn in mathematics lessons that come with a lot of procedures and rules governing how to create and manipulate them. After all, we can see mathematics as a big collection of tools or “representations” in which we use to solve problems. 

Problem-solving Heuristics

Apply logic to solve puzzles – that’s all it is. And there’s a systematic science to it.

Most children love activity books. This proves their innate need to be challenged and mentally stimulated. Maths puzzles tap into this instinctive desire to stretch children’s minds. 

Challenging Maths questions challenge them to find and apply faster, more creative solutions. This trains them in higher-order thinking and decision-making skills on top of subject knowledge.

Now, don’t we all apply these skills at home, at work and at play? Therefore, heuristics practically take children beyond school, right through their lives.

Heuristics in PSLE Maths 

More than half of PSLE Maths marks go to heuristics-related questions.

Approximately 55% of PSLE Maths marks are dedicated to long, open-ended questions that necessitate problem-solving. Several are routine questions, easily solved using the four basic operations of addition, subtraction, multiplication and division. Many are non-routine questions, requiring the four basic operations AND problem-solving heuristics.

The Ministry of Education has identified 11 heuristics for primary-level Maths, and two more for secondary-level Maths.

1. Use Diagrams / Models 2. Act it Out 3. Use Before & After 4. Use Systematic Listing 5. Look for Patterns 6. Work Backwards 7. Use Guess & Check 8. Simplify the Problem 9. Make Supposition 10. Solve Part of the Problem 11. Paraphrase the Problem
12. Think of a Related Problem 13. Use Equation
Source: Ministry of Education of Singapore (2007). Mathematical Syllabus Primary. Singapore: Curriculum Planning and Development Division.

Opening Hearts & Minds

• Accertation
  We teach students problem-solving heuristics. Students simply add this new information (heuristics) into their existing mindset.

• Tunning
  We guide students in heuristics application. Students begin to realise the limitations of their existing mindset. They begin to modify their existing mindset to incorporate heuristics.

• Restructuring
  We expose students to the variety of challenging Maths problems that necessitates heuristics application. Students begin to address the inconsistencies between their existing mindset and heuristics. They begin to recreate their existing mindset to feature heuristics.

Habituating the Process

• Understanding the problem
  Students will be trained to look for, visualise, organise and connect information. They will also develop Maths language proficiency.

• Choose an appropriate heuristics
  Students will learn how to select, and combine where necessary, the most appropriate heuristics for different problems.

• Perform the chosen heuristics
  Students will develop computational skills, geometrical skills and logical reasoning.

• Reflect
  Finally, students will be trained to check their solutions, to improve on the methods used, to seek alternative solutions, and to extend the methods to other problems.

Click here to know more about Heuristics Maths.

Basic Concept Maths

For students to understand and work with formal mathematical concepts successfully, they must understand the concepts of classification, conservation, seriation, ordering and one-to-one correspondence. Students must first work with and understand these concepts on the basis of quality (e.g., attributes such as shape, size, weight) before moving on to their application to general quantity (e.g., attributes such as many, few, none) and then on to number (e.g., attributes such as "fiveness", 100=10x10, 4+1=1+4.

In order for students to develop their innate number sense, and a working knowledge of the above concepts, they must have a great variety of interactions with their environment, exploring and manipulating, comparing, arranging and rearranging real objects and sets of objects. Many of these types of interactions and experiences occur incidentally for sighted children, but the blind child is at great risk for missing valuable and relevant incidental information. Therefore, it is critical that teachers and parents provide both structured and informal opportunities to handle and explore, note likenesses and differences, match, group and classify, order, and experience other relationships with real objects to prepare them for understanding the same relationships with numbers.

One of the earliest concepts to be developed is that of classification.



Classification involves discrimination, matching, and grouping or categorizing according to attributes and attribute values. A sampling of these attributes and attribute values at the quality level follows:

• Shape (square, circle, triangle, rectangle)
• Size (large, small, big, little)
• Weight (heavy, light)
• Length (short, long)
• Width (wide, narrow, thick, thin)
• Height (tall, short)

At the quantity level, these attributes would involve general number concepts (e.g., many, few, more, less, none), and later, more specific number values (e.g., sets of 2, sets of 10, sets of values greater than 2).

The development of classification concepts involves several sequential stages:

1. discriminating between same and different (note: if a child has difficulty with the dichotomy of same/different, the dichotomy of same/not same may be more effective to begin with); attention should be called to the critical features of objects and their attributes;
2. matching, grouping and categorizing according to specific criteria; and
3. classifying according to a variety of dimensions.

To promote the development of classification concepts, the teachers can:

• Begin working on simple discrimination and matching with objects that are familiar to the child and that occur naturally in his or her world (e.g., shoes, toothbrush, squeeze toys, blocks, etc.), then move on to noting and analyzing specific attributes (e.g., shape, size); later, those specific attributes can be applied to naturally occurring objects in the environment (e.g., circle shape of a plate).
• Provide numerous opportunities for the child to handle and explore objects, note their critical features or attributes of shape, size, position in space, length, etc.
• Provide many opportunities for the child to match objects, and build groupings or sets of objects on the basis of specific attributes.
• Follow a logical or Piagetian sequence with regard to matching, grouping or categorizing, and later classifying: start with a single criteria or attribute by which to discriminate or group (e.g., shape/circle), change to a different criteria (e.g., small/large), progress to two attributes simultaneously (e.g., small circle), add additional attributes (e.g., small thin circle), and finally discriminate according to attributes NOT present (e.g., item that is not round, not small).

Another basic concept maths that children must understand is that of seriation, or ordering objects, then quantities, and eventually numbers, according to specific given criteria. As with the concept of classification, the child must begin working in this area with real objects on the basis of quality (e.g., ordering family members' shoes or belts according to attributes such as length). Only then will the child be able to apply the concept to quantity (e.g., ordering jars of coins or chains of keys–one having many, one having several, one having few and one having one or none), and later to number (e.g., ordering the numerals 2,10, 3, 5). The concepts of classification and seriation can be taught in conjunction with each other very effectively. For example, after the child can match and sort according to size, he or she can work on ordering from largest to smallest.

In addition to the understanding of the concepts of classification and seriation, the child must develop an understanding of conservation-knowing that a given amount remains the same though its appearance may change. Also, as with classification and seriation, the concept of conservation must be developed first with real objects (e.g., a bowl of cake mix is the same amount as when it is divided into 12 cup cakes). This must be understood before a child can be expected to understand the "partners" that make up numbers (10=5+5, 10=7+3, 10=6+4), units of measurement and money (a nickel is the same amount as five pennies), fractions (one whole is the same amount as two halves or four quarters) or the associative principle (7x3 equals the same as 3x7).

In addition to the concepts of classification, seriation, and conservation, children need to understand basic spatial and positional concepts. For example, the concepts of top, bottom, around, middle, center, corner, line, straight, curved, next to, beside, are very relevant to basic mathematical understanding. Later, concepts such as diagonal, parallel, perpendicular, intersecting, angles, and rotating will be relevant. Positional ordering concepts are also critical for sorting, for seriation, and for working with sets; these include concepts such as first, second, third, next, last, before, and after. However, these concepts require basic counting ability.

When teaching any of these basic concepts, it is important to start with real three dimensional objects, progressing to two dimensional shapes or diagrams and finally to more symbolic representations. It is also advantageous to have students develop the ability to express their discriminations in complete sentences (e.g., "These are the same because they are both square," or "This is the longest belt.") because doing so helps them to focus their attention on the concept rather than simply naming a descriptor.

Activities for teaching basic concepts

• Involve children in daily living activities around the home or classroom. For example, helping to put silverware away in a divided tray with a sample in each section provides practice in matching, sorting and categorizing; helping to sort different sizes of towels or different items of clothing provides additional practice with these concepts.
• Give children numerous opportunities to use everyday items for matching and categorizing: eating utensils, grooming tools, foods, and toys for function; shoes and shoelaces for matching by size or length.
• To work on seriation, have children arrange boots belonging to family or class members from smallest to largest size; boots could also be arranged by height.
• The same type of activity could be carried out with other personal items such as belts of different lengths, books of different thicknesses, milk cartons of different sizes, or later with Unifix towers or Cuisenaire blocks. Students should not only identify the "extremes" of a series (e.g., longest or shortest), but also the "next shorter".
• Having family members or class members line up according to height can also help to facilitate understanding of seriation.
• Provide chances for children to work with the concept of conservation: give them a ball of clay and let them divide it into smaller amounts as they wish, and then combine the smaller shapes to demonstrate the constancy of amount.
• Using a sorting tray, place a variety of small items (buttons, paper clips, keys) in the larger section; to categorize, place one of each type of item in each of the smaller sections of the tray and have the child match and sort the remaining items; to classify, have the child form his or her own groups without providing a model. This activity could also be done using attribute blocks.
• Have children fold stiff fabric and paper to make different shapes. Squares can be folded to make triangles or smaller squares. Later, origami can be used to facilitate understanding of geometry.
• Children can explore shapes and size by building with Legos and Unifix blocks; they can also work with conservation by making a variety of different groupings from a given number of blocks.
• Have children copy simple shapes on geoboards; later they can make their own shapes based on names or clues such as "four corners", etc.
• Provide children with opportunities to explore and compare the three-dimensional shapes from Essential Geometric Forms which can be gotten from the American Printing House for the Blind.
• Have children walk, hop, run, jump through an obstacle course made from large shapes on frames, available from several children's catalogs, or arranged from items in the natural environment (e.g., jump 3 times in the circle, hop through the square, step in and out of the triangle).
• Use shapes, sizes, orders, patterns, planes, and eventually numbers in the real life environment (classroom, home) to teach concepts (e.g., compare the size of books to each other and to the size of tables, use corners of rooms to demonstrate angles, etc.).
• To practice positional ordering, have a student line up the rest of the children in a group, and then identify each as first, second, third, . . . last. Also have the student identify which child is before or after a particular individual, which one is next, etc. Children can also do the same activity by arranging toy cars or other manipulatives.
• Make a mathematical "pattern block" to enable students to build shapes and patterns with manipulatives that stay in place. To make the pattern block, drill ten or twelve evenly spaced holes into a long block (22 inches x 3 inches) such as the ones found in kindergarten block centers. Hammer thin wooden dowels or glue pieces of thick stranded wire into the holes, leaving about 2 inches; protruding up out of the block.
Assemble a collection of small objects that slide easily over the dowels or wires (e.g., beads of various sizes/shapes, washers, straws, plastic Unifix cubes, large paperclips, uncooked pasta, small pretzels). Students slide objects over the dowels in a left to right sequence to make a pattern (cube, cube, pretzel, cube, cube, pretzel, etc). The teacher can also start a pattern and have the student finish it. This device can also be used to teach ordinal number positions such as first, second, next, last.

• Use magnet boards or felt boards for children to match shapes, size, position, order, and patterns; later, children can match numbers or form simple number statements to accompany the arrangement of manipulatives.

The above described activities can be used to good advantage in helping young severely visually disabled children to lay the groundwork for understanding the fundamental concepts underlying the study of mathematics.

Don’t Teach Math, Coach It

PEOPLE ask me all the time how they can get their kids excited about math. That ought to be a softball for me, because I teach math for a living. I wake up excited about math.

But it’s not that simple. With the college students I teach, it’s a straightforward transaction. They’re paying me to teach them math, and my job is to cajole or incentivize them into doing the work that’s necessary to learn the subject, whether they feel like it or not.

It’s a different story with your own children. None of us want to be Leo Wiener. Yes, Wiener helped shape his son, Norbert, into a child prodigy who got a Ph.D. at Harvard at 18, and who later became a groundbreaking mathematician. But this was how Norbert recalled the process:

“He would begin the discussion in an easy, conversational tone. This lasted exactly until I made the first mathematical mistake. Then the gentle and loving father was replaced by the avenger of the blood. ... Father was raging, I was weeping, and my mother did her best to defend me, although hers was a losing battle.”



No parents want this story told in their child’s memoirs. But how can we encourage kids in a difficult task like math without doing so in a way they’ll come to resent?

I found an answer in something my 8-year-old son, C. J., likes even better than math: baseball. Let me be clear here. My level of skill at baseball — actually, with every kind of ball — is pretty much the opposite of my mastery of math. I’ve reached 40 and I still throw in the way that we used to call, before they started showing college softball on TV, “like a girl.”

But C. J. is a baseball fanatic. He lives and dies with the Milwaukee Brewers and he’s pretty set on being one of them when he grows up. He plays Little League with a fierce concentration I seldom see at home. And I’ve learned a lot about what kind of math parent I want to be from an unexpected source — his coaches.

Baseball is a game. And math, for kids, is a game, too. Everything for them is a game. That’s the great thing about being a kid. In Little League, you play hard and you play to win, but it doesn’t actually matter who wins. And good coaches get this. They don’t get mad and they don’t throw you off the team. They don’t tell you that you stink at baseball, even if you do — they tell you what you need to do to get better, which everybody can do.

What does it mean to coach math instead of teaching it? For C. J., it means I give him a “mystery number” to think about before bed. “I’m thinking of a mystery number, and when I multiply it by 2 and add 7, I get 29; what’s the mystery number?” And already you’re doing not just arithmetic but algebra.

For his little sister, who’s 4, that’s too formal. But say we’re at the grocery store and we need four cans of soup and she brings me two, and I say, “So we need three more, right?” and she says, “No, Daddy!” That’s really funny when you’re 4. It’s a game, and it’s math.

Lots of games are math. There are the classics you know about: chess, which builds the ability to follow a series of logical steps; Monopoly, which demands basic arithmetic and probabilistic reasoning; and Rubik’s Cube, which is fundamentally an exercise in geometry and group theory.

I have fond memories my 4th and 6th grade elementary school teacher, Miss Marks, talking our class out to the school yard one day to play...

But there are new classics, too, that weren’t around when you were a kid: Rush Hour, a board game about search algorithms; Set, a study in higher-dimensional geometry in the form of a viciously competitive card game; and DragonBox, an app for phone or tablet that teaches the formalisms of algebra. Every one of these games shows kids mathematical ideas in a spirit of play, which is a big and often hidden part of the true spirit of math.

These games are difficult, but also, for many kids, kind of addictive. Which means they also teach sitzfleisch, the ability to focus on a complicated skill for the length of time it takes to master it. Math needs that. (Baseball does, too.) It fits with the research of the psychologist Carol Dweck, which suggests that mentors should emphasize effort over native ability. We can’t really teach kids to do things; we can only teach them to practice things.

There are many things we’d like to coach our kids to do. And we can’t help playing favorites to some extent. I’ll admit, I’d rather C. J. aimed to be a mathematician than a shortstop. I tried to open his eyes to some more realistic careers that could still satisfy his hunger for the major leagues. “You know,” I told him, “you really like math, and all the teams now have people who work for them analyzing the players’ statistics. You’d probably enjoy that!”

At this suggestion he became agreeably eager. “Daddy, that’s a really good idea,” he said. “Because almost all major league players have to retire by the time they’re 40 — so then I could get a job analyzing the statistics!”

Well, I tried.

You can find more information about maths coaching at our site: http://eimaths.com/

Solve math problem using free online tutors

Friends as we all know that for most of the students, math is one of the most complex subject to score maximum marks. That's why everyone looks for a better tutor who could help them to excel their skills in mathematics. So there is an excellent solution for those students which is online math tutor. You do not need to go anywhere else to study and wash out your weakness in mathematics, because math online tutor is there for you anytime through World Wide Web or we can say Internet, most friendly environment for present generation.

The 24 hrs availability of online math tutors for user makes this facility extraordinary than normal private tuition. Online math tutor helps you to solve your mathematics problems in a way that you will definitely feel confident while solving your other math questions., because any problem gets more complex when you solve it in a meshed way, so this disqualification of yours is improved by online math tutor to make you learn that how to solve any problem in an appropriate way.
We know that in present time every field have its number of competitors, so that many online tutoring services are also there to provide you online math tutors . All of these online tutoring services provide an easy going platform on Internet for user and online tutor, because of which there is fluency maintained in communication while taking lessons. There are various practice session sheets organized for all lessons you learned to make you more comfortable, because Practice makes a man perfect.



Every online tutoring services have individual math online tutor for each branch of mathematics like algebra solver for Algebra, Calculus math tutor, Statistics math tutor etc. Each of them helps user to solve their mathematics problem related to particular branch of math.

Let us discuss about Statistics tutor that what he actually do for user?€ Statistics online tutor helps user, when user's problems are related to term €statistics€. Statistics is a branch of mathematics in which problems are related to probability and linear algebra analysis. In statistics problems the value of data is not so fixed, it has uncertainty in it. Statistics aims to elaborate given data for the particular condition or we can say it is used for data analysis. Online math tutor is a required facility of present syllabus, to help students anytime they want, so that they can be flexible in their study schedule.

For more information about Problem Solving Maths, click here.

Math Learning Made Easy With The Following Tips!

Math is a subject that raises your eyebrows with its strengths and challenges. It could be fun or bore- the way you look at Math doing. Let us see how Math skills could improve your life style as such and what measures you have to incorporate to improve the Math skills of your kids or others.
Math- the lifeline for life activities

Math is an important aspect of life. Without the fundamental skills of Math, you are nowhere in the world. Basic addition, subtraction, multiplication and deduction are essential for everyday activity in your life. You need Math to improve your accounting skills and manage your personal finance with efficiency. Without Math, you have no place for Engineering, Finance, Marketing or any skill related job in your life. Since Math skills are elementary for Art and Architecture, you could make your home more beautiful with Math precision. Be it cooking or any real time activity, Math plays a major role in making your personal life function with perfection and accuracy.

So, Math learning is an important aspect in education, career and life. If so, how to make it easy, fun and interesting? Here are:

Some tips to teach Math to kids

€ Use Math flash cards to make the kids follow basic Arithmetic skills in Addition and Subtraction
€ Cut out paper figures in various colors to teach basic pattern of Geometry
€ Search for online Math fun games to make Math learning a mind-blowing activity. You can take recourse to Math tutoring online services for this practice as well
€ Ask the kids to help you in cooking and give them a chance to work with real measurements
€ Same way make them work with tools to find out actual measurements

Tips for elder students

€ Do not neglect your failures in Math. Go through them and check out your weak areas like where you go wrong, which step you falter, which concept you are not able to concentrate and so on
€ Don't develop a negative attitude towards Math with the feeling that you are not born for it- not so. Everything comes by practice and you are no exception
€ Try to understand the fact that you cannot skip concepts to do Math. You have to learn Concept Maths in a particular order, since the concepts are built upon one another. Algebra skills are essential for Calculus and Pre Algebra skills are needed for Algebra 1 and 2. Basic Arithmetic is a definite helping factor to enter Pre Algebra- the chain goes like this and you are to follow the chain to reach the endpoint. Hence the importance of seeking every basic concept clarity in Math to attain perfect skills.

€ Write down the problems repeatedly to get into the essence of the problem. If Algebra poses problems, get proper assistance through Algebra tutoring to make your skills strong in the area.
€ Math homework could be a head ache for a lot of students who negate Math doing on this ground. Get good online homework help to wipe out your fear in doing Math homework.

Math Anxiety Caused By Teacher's Teaching Style

Math anxiety can be caused by many factors. But one of the major contributors is that of the math teacher's teaching style. Have you met a teacher's teaching style that does not suit our learning style? It puts us off straight away. If not that, the going will be tough and learning math becomes taxing. It therefore appears that the role of the math teacher very much influence the outcome of our math learning. It makes us like or dislike math. The character or personality of the teacher also affects the interest towards the math subject.

However,looking back, how is the learning experience of our math teachers during their students' day studying for math themselves? Was it exciting then? Or has their math teacher taught them the way they are teaching us now? These are the very questions that we, the students, ponder also. I believe the math teachers, themselves, sometimes do model after their teachers as the image flashes back when they deal with math. The experience formed from the past do affects their teaching style, as a normal human should. If the current math teacher was taught by a dedicated and motivated one, the math learning experience would be a beautiful and enjoyable experience. However, if it was the opposite, then the impression received will goes to their teaching style and attitude towards the math subject.

If the teaching style is positive, math anxiety for the students will be reduced as the teacher are able to express their interest through the way math lessons are conducted. The math lessons and concept maths will be delivered and expressed with excitement and enthusiasm. This teaching style will indirectly affects the learning emotion of the students and the extent of their math anxiety. However,having said that, the math teacher has still to constantly monitor the students' progress. Being confidence with math does not guarantee good math lesson delivery, it only serves as a good starting point where students of the math teacher are less resistant to learning math given a good role model by the math teacher. The learning math journey of the students will not have this favourable headstart given a math teacher who is not confidence and has math anxiety himself.

Therefore, to summarise, math teacher plays an important role for our students and selecting the appropriate teacher becomes more crucial as it decides between a good start and better math learning journey. For math teacher who demonstrates math anxiety, be aware and constantly self-remind that your teaching style has an impact on the outcome of the students' learning. For student, be aware that math teachers are also human with their own feeling and personality. Adjust your learning style to match your teacher's teaching style to reap the best possible result in your math learning and aim to reduce math anxiety.

Heuristics Methods Used in Mathematics - Part 2

Part 2

Heuristics can be divided into 4 main types, which another 2 types would be discussed in this article.

·         The heuristic ‘act it out’ requires pupils to use physical objects or manipulatives to represent information. This skill is used to introduce new concepts and to allow pupils to explore the concepts using manipulatives.

Example: The figure is made of 17 sticks. Move 4 sticks to form 8 squares. Solution:

Pupils can use matchsticks to represent the information and use the matchsticks to find the solution to the problem.

Example: There were some chocolates in a basket. Michael and three of his friends took 8 chocolates each. 25 chocolates were given to Shirley. There were then 11 chocolates left in the basket. How many chocolates were there in the basket at first? Solution: Using the ‘work backwards’ method, first find the number of chocolates taken away by Michael and his three friends. 4 × 8 = 32 chocolates Then, find the total number of chocolates taken away from the basket. 32 + 25 = 57 chocolates Number of chocolates in the basket at first = 57 + 11 = 68 chocolates

·        Pupils can use the ‘before-after concept’ for problems that provide information given before and after the event to find the unknown. This skill allows pupils to compare the information and relate different events together to solve the problems.

Example: Jacky had $56 more than Jill. When he spent $108, Jill had thrice as much as what he had left. How much did Jacky have at first? Solution:   2 units → $108 - $56 = $52   1 unit → $52 ÷ 2 = $26  $26 + $108 = $134 Jacky had $134 at first.

Four: Changing the problem

Example:Find the sum of 10 + 12 + 14 + … + 146 + 148 + 150. Solution:           There are 75 numbers in the sum 2 + 4 + 6 + … + 146 + 148 + 150.   There are 4 numbers in the sum 2 + 4 + 6 + 8. 75 – 4 = 71 Hence, there are 71 numbers in the sum 10 + 12 + 14 + … + 146 + 148 + 150.    There are 35 pairs of 160 and a ‘80’. 10 + 12 + 14 + … + 146 + 148 + 150 = 35 × 160 + 80  = 5680

Thus, do try to keep in mind the various heuristics and apply them when you are doing problem sums. With regular practice, you will be able to handle problem sums with ease.

The word ‘heuristic’ is taken directly from the Greek verb, heuriskein which means ‘to discover’. In Mathematics, there are usually different ways to go about solving problem sums. These ways or methods are known as heuristics maths.

Three: Going through the process

·        Pupils can apply the ‘work backwards’ method when given a problem that provides the final result and that requires them to find the initial quantit

·        By restating a problem in another way, pupils can view the problem in another perspective.

·        When facing a complex problem, pupils can split the problem into smaller parts and start by solving the simpler parts. After doing so, the problem is simplified and solving the problem is much easier.

Heuristics methods used in Mathematics - Part 1

Part 1

The word ‘heuristic’ is taken directly from the Greek verb, heuriskein which means ‘to discover’. In Mathematics, there are usually different ways to go about solving problem sums. These ways or methods are known as heuristics maths.

Heuristics can be divided into 4 main types, which will be covered in this 2-part article.

One: Giving a representation

·         Pupils can transform word problems into pictorial representations and represent information with a diagram/model. This skill helps pupils to understand the question better when they see the visual representation of the word problems.
·         A systematic list should be made for word problems that require pupils to identify patterns such as repeated numbers or a series of events that repeat. This skill helps pupils in identifying patterns easily as the list organises all possible answers systematically.

Example: Michele saved $150 on the first month. On the second month, she saved $60 more than the first month. On the third month, she saved $70 more than the second month. On the fourth month, she saved $55 more than the third month. How much did she save in four months? Solution: Making a list: 1st month → $150 2nd month → $150 + $60 = $210 3rd month → $210 + $70 = $280 4th month → $280 + $55 = $335  Total amount saved = $150 + $210 + $280 + $335                               = $975  She saved $975 in four month

Two: Making a calculated guess

·         The ‘guess and check’ method is used for word problems when certain information is lacking. It requires them to make a guess first and check it, and making subsequent guesses and checks until the correct answer is derived. It is often used together with a systematic list as it helps pupils to narrow down the possibilities within a short time frame.

Example: Jenny has a total of 7 dogs and parrots. The animals have 20 legs altogether.                 How many dogs does she have?  Solution: Using the ‘guess and check’ method, Number of dogs Number of legs Number of parrots Number of legs Total number of legs Check 1 1 x 4 = 4 6 6 x 2 = 12 4 + 12 = 16 X 2 2 x 4 = 8  5 5 x 2 = 10 8 + 10 = 18  X 3 3 x 4 = 12 4 4 x 2 = 8 12 + 8 = 20  √ She has 3 dogs.

·         The ‘look for patterns’ method is usually used by pupils when they have to identify a certain pattern in a number sequence.

Example: 12 4 5 13 11 9 16 18 16 12 3  X Solution: Making a list of possibilities:  12 - 4 + 5 = 13 11 - 9 + 16 = 18 16 - 12 + 3 = 7 The value of X is 7.

Hence by using the systematic list, it is more effective to find the underlying pattern.

HEURISTICS AND WORD PROBLEMS

Heuristics and its application What is heuristics?+ How do we use heuristics in problem solving?+ Steps to solving word problems Step 1: Understanding the problem Step 2: Deciding on an approach Step 3: Solving the problem Step 4: Checking the solution Worked Example with the "Guess and Check" strategy

Content Heuristics and its application What Is Heuristics?+ Heuristics Maths refers to the different strategies that we can adopt to solve unfamiliar or non-routine Maths problems. How Do We Use Heuristics In Problem Solving?+ There are different types of heuristics and they can be grouped into four categories, based on how they are being used:

Steps to solving word problems With word problems as an introduction to problem sums for Primary 1 pupils, does your child know the different steps to approaching a word problem? There are 4 basic steps that pupils can take to solving word problems: Step 1: Understanding The Problem It is important for children to understand the word problem that they receive, before they begin solving it. A good way to help children grasp a question's requirement is to break the question up into smaller parts. Alternatively, parents can also provide guiding questions for the children to help them pick out the important information inside the question. Step 2: Deciding On An Approach There are many approaches (different heuristics) to solving word problems. Children will need to decide on the approach they wish to use based on the question's requirement. Step 3: Solving The Problem After deciding on the approach, children will proceed on to solve the word problem with the selected approach. Over here, children will need to have a good knowledge of the approach selected in order to solve the problem. Step 4: Checking The Solution The last step for children to take when solving word problems will be to check their worked solution to ensure that they have solved the problem correctly. For this step, children will need to make reference to the question provided and do the necessary comparison.

Worked example Here we have a worked example using the 4 steps introduced above: Question: At a café, there are 3-legged stools and 4-legged stools for customers to sit on. Vincent counts 15 stools with a total of 50 legs altogether. How many 3-legged stools are there in the café? Solution: Step 1: Understanding the problem (through asking guiding questions). Step 2: Deciding on an approach. To use the “Guess and Check” strategy. Step 3: Solving the problem with the "Guess and Check" strategy. The total number of stools must always be the given number, 15. Step 4: Check the solution. Number of legs for ten 3-legged stools: 3 x 10 = 30 Number of legs for 4-legged stools: 50 – 30 = 20 Number of 4-legged stools: 20 4 = 20

Can Learning Heuristically Join Forces With Education?

There are now waiting lines for pre-school. Yes, two years before my boy steps foot into his pre-school, he gets put on a waiting list. This must be the ivy league of pre-schools.

I had a fluke event during kindergarten reading hour (which morphed into nap time). My friend Davy was lying down next to me, on my left. I don't remember the story being read, but I distinctly remember what happened next.

The fluorescent light above us began to flicker. As I stared at the light in awe, the light popped and broke on one end. Something inside me forced me to push Davy--into a row of desks, bloodying his nose. The weight of the light dangling from one end proved too heavy, and the light crashed directly where we had been lying.

Davy was crying. He failed to see that I'd saved his frickin' life. His nose was bleeding, the teacher was screaming, little girls peed their pants. Nap time was over.

This was the beginning of my education. And make no mistake, I liked learning. Still, the very act of sitting in rows staring at a teacher never made sense. How is a test going to really going to prove you've learned something? Answer: It doesn't.

My son will enter this school system soon, a system that closely resembles the one I trudged through. Classrooms with ~30 kids regurgitating what the Education Department believes is important.

I have already drawn up my plan towards my sons' classes. I plan to know his teacher better than most. I want to know where he/she has been and where they're going. I want to know where they stand and how they grew up. I want to know if they've travelled. I want to hear them tell a story and tell a joke. I plan on holding them to a higher standard. When you give a kid responsibility and trust, they shine. So will my teacher.

Our educational system shows signs of being inflationary. The degrees needed when I went to university are only the beginning of what is needed today. Tuitions are astronomical and are not matched by quality. Textbooks are obsolete before they're finished printing. I must face the reality that this is not a world I've seen before. We must gaze at our environment with new lenses.

How can we (an obese nation, by the way) move our classrooms out of an actual 'room'? Biology in a nearby field, language at a public speech, Heuristics Maths...well, anywhere but behind a chalk board.

Everyone learns in their own special way. I won't get into it, but I know that you can read 10 books on starfish, but until you see one in its own habitat, you don't know starfish. We live in a world where everything is digital. There will come a day when "I've actually seen a starfish!" means more than anything.

This isn't about entertaining kids. It's about knowing the world around them, writing about what they see, and learning actively.

Just look at them--sitting in their seats in neat little rows, all day-dreaming...reminds me of the unemployment line.

Are You Looking For More On Math Online

One in every of the fantastic perks with the on-line is usually that schooling is now accessible for any person and everyone. Those who have missed out on school whenever they were younger can initiate downloading programs internet based and get a certification and even a graduate or post-graduate diploma.

For a lot of the internet based maths courses you need to sign-up and fork out a tiny payment while there is a number of far more courses which might be without cost. There are actually numerous choices in maths itself you can nearly pick and opt for the subjects that you simply want like calculus, differentiation, trigonometry, geometry and so on. As soon as you wish to get Maths activity class therefore you never have the time to go out and meet with anyone, acquiring the knowledge over the internet stands out as the finest solution. This kind of online maths is practical for teenage pupils and all those in superior school. They will get all of the particulars and allow that they will need without needing to go away place and can get extra done in a smaller amount time. It's also possible to get this math tutoring online any time in the day or evening and this makes it a whole lot more effective for everyone to perform clearly. A good amount of college students have study techniques and many people can only examine underneath a number of issues and at several instances of day. As you appearance in the way that this can be performed via the internet, you're able to see how advantageous it happens to be for most persons. With mathletics there to assist you every action within the way with your maths, you will see that you just improve your grades tons swifter and quicker. This can be viewed as the subsequent generation in math tutoring and getting 100 % on the internet dependent it truly is most suitable for all students.

While you make use of the mathematics strategy you will find that it performs so much more beneficial than the previous conventional procedures of gaining a math tutor from a class. It makes use of entertaining video games and remarkable approaches that can aid you to remember the math and basically know it additional. Most internet based mathematics lessons begin the process of with concentrating on most of the general procedures that will involve addition, subtraction, coupled with order of operations. Aside from the basic principles, these affordable web based programs concentration within the totally different types of figures like decimals, fractions, percentage, absolute worth, radical signs, exponents and ratios. Online maths software programs are one of the hottest instructional coursewares between all topics remaining provided above the net. You can find countless educational establishments, some non-public and some will be the most desirable globally who are offering zero cost downloads of different maths assignments, tutorials as well as other coursewares. Almost all of the compact tutorial centered courses are for free but there are actually selected certification programs too for which you ought to spend a minimum charge depending on the topics. Learning math is a little diverse than studying for other subjects. Math can be a vital thinking study course and is realized by actively doing dilemmas. It's essential for students to complete the assigned research and even more math complications for practice. Doing research and practice function aids college students understand formulas and strategies they need to know. This also facilitates boost trouble fixing and crucial imagining abilities.

Learning for math begins from the classroom. It truly is essential for students to get duty for learning and becoming conscious of strengths and weaknesses. Attending just about every class is crucial. Not only attending, but becoming an energetic participant with the classroom, combined with taking notes. Examination material is typically coated in class, so it's always crucial that pupils pay awareness.

Making Math More Fun Review

For the greater majority, mathematics is one subject that doesn't interest quite a few individuals. It is meant to be hard, stressful and requires plenty of logical thinking. Well, this is the most popular misconception of individuals about mathematics. Unless they've heard of Making Math More Fun ebooks, then they will have a different view of this subject area, seeing maths is fun.

1. Each Child's Ultimate Games Book.

Making Math More Fun is not your ordinary textbook wherein the basics are taught. That way, kids are given a various view of what mathematics is all about. Having a fun concept of the subject will make them appreciate it a lot more. In fact, in performing the activities, they'll not even understand that they are learning the fundamental mathematical abilities at the exact same time. Since the e-books come in four different parts and of distinctive sets of games, kids can have their pick from board games to puzzles and other fun-filled activities which are meant for the subject to be as easy as ABC.

2. A Teacher's Effective Teaching Companion.

Just as kids have difficulty in understanding mathematics, teachers too have a difficult time on tips on how to teach the subject effectively to students. With Making Math More Fun [http://hubpages.com/hub/Making-Math-More-Fun-Review], a teacher's resources are unlimited. 1 can pick from the quite a few board games and puzzles that can be discovered inside the four distinctive books. Because of this, keeping the kids' attention will never be challenging once more.

This book may be purchased on the web and can be downloaded and thereafter printed. Imagine acquiring something in just a couple of clicks of your mouse. It'll also save you from a great deal of effort and hassles.

3. A Parent's Welcome Relief.

Making Math More Fun is each and every parent's tool in teaching their youngsters the way to do well in mathematics. Since most of the activities are some thing the children will surely take pleasure in, teaching them becomes less stressful. Since it's inexpensive, parents can also save additional money from buying unique resource materials. Very best of all, the activities are something that everybody in the family will surely appreciate. It promotes family bonding by providing 1 activity that the entire family will definitely love performing.

Discuss With Your Tutor and Solve Math Problems Instantly

The Fact Behind Math Problems

Math is a big hurdle for students as many of them try to solve tricky math problems without knowing the relevant concepts properly. It is a different subject and it demands concentration and step-by-step understanding. It sharpens students' logical reasoning skill. But studying Math is not a good experience for everybody. Most of the students go from one class to another without getting a concrete idea about different concepts and hence, they face difficulties in solving tough mathematical problems. They lose interest in Math and end up with disappointment.

4 Steps to Learn Math

Math seems a tough subject to many students, so every student needs a proper guidance to learn Math step-by-step. Some useful tips to improve problem solving maths skills are discussed below.

• Understanding the problem is required to solve any tricky sum. Students need to focus on the problems to get solutions.

• Each Math question gives some known and unknown information. Students are required to identify these information to get the right solution.

• Practice can make a student better in Math, so students should practice each topic as much as they can. Several websites offer free worksheets. Students can download and practice these whenever they need help.

• After completion of the entire syllabus, students need to revise it thoroughly. Without revision, students may forget the concepts and formulas and can face the same learning difficulty again.

Discussion With Tutors Is a Great Way to Solve Math Problems

Students can discuss their learning problems with the tutors. It is the easiest way to get comfortable with Math. Students can clear their doubts in a step-by-step manner as subject experts can give a fair idea on each topic. They help in enhancing student's confidence and give several worksheets on each topic. Thus, students can do more practice and get a good command over each topic. Additionally, discussions with tutors can reduce exam anxiety.



Online Math Help Improves Student's Mathematical Skill

Online tutoring makes learning any subject easy and comfortable. Students can opt for online assistance for any subject including Math. Several positive features of online learning keep students' stress free. With this service, students can schedule their tutoring session anytime from home. Thus, they can save their time and energy and additionally, the virtual environment of online session allows students to clear their doubts unhesitatingly. So students can enhance their mathematical skill by scheduling required numbers of sessions with their preferred tutor.

The Effect of Good Solved Math Problems in Education

There are probably an infinite number of math problems. When you are studying math in elementary or high school, you have no idea of the huge world of math that exists at the college and post-college levels. Additionally, when you're studying elementary-level math, its sometimes hard to make the connection between seemingly-insignificant math problems and the ultimate power that math has to solve problems in real life.



Think of medicine, for example. Students who started out the same as you and I, learning about square roots and fractions in elementary and junior high schools, have ended up using math to solve major health problems such as polio and tetanus. By turning health problems into math problems, collecting data and turning it into numbers, public health workers and epidemiologists figured out what was causing these diseases. Then, they found the problem solving maths and figured out how to get rid of the diseases.

Without the beginning elements of addition, subtraction, algebra, geometry, calculus, and statistics, this could not have happened. Mastery of finding solutions to math problems allowed scientists to solve health problems and relieve human suffering. By performing statistical analysis of the numbers, they developed vaccines for these problems. All of this would be impossible without math.

At the college level, students usually see those seemingly meaningless math problems, like how much bread Joe can carry if his bicycle has a basket that is 1 foot by 1 foot, turn into real-life issues. If you study social science, you'll do research using math. When they get to graduate school, the statistical part of the math problems is often done with SPSS. However, the student has to understand what the data is telling her/him and know how to input it into the program in order for it to work.

Improving your house is also another area where you will encounter math problems. If you want to repaint one or several walls, you have to figure out many issues. Though this may seem pretty simple, you still have to know how to add, multiply, divide, subtract, and do basic algebra. Its for this reason that everyone in the United States is required to achieve at least a basic competency in math. People who study education are aware that all aspects of our daily lives involve math in many ways.

For a lot of people the simple words "math problems" always go together and seem like a negative thing. We think of the word "problem" as something we want to get rid of. It would be better if we called them "math puzzles" instead. Isn't that more inviting? Math puzzles would be something fun, playful, or exciting.Problem has a negative connotation, whereas puzzle sounds mysterious and exciting, something you just want to delve into to figure out how to put it together. And that's what math is about.

When we do math problems, we take various parts of the puzzle, various concepts, and we put them together. That is the mystery part and provides the frame for the puzzle. In real life we always have some elements of the puzzle, like the speed of a vehicle and how far the vehicle is going to go. That's the information that we have to put together with the concepts. We can put the concepts together with the information and that's how we complete the puzzle. This is what math problems really are.