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.

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.

Bridging the Mathematics learning gap

The learning of Mathematics is so different from 20 years ago, that the MOE Mathematics curriculum now focuses on problem solving as its primary goal. In 2009, when the use of calculators were permitted during the PSLE Mathematics Paper Two, it became obvious that the learning of Mathematics does not just focus on computation skills any more.

Importance of Heuristics in Mathematics

Today, students no longer learn Mathematics by memorizing, rote-learning or drilling. How well the child learns Mathematics does not relate to “practice makes perfect”.

Instead, they must be able to connect, reason, model what they have learned in class and apply Heuristics to solve problems.

The teaching of Mathematics has shifted from white board learning to inquiry-based learning, whereby students discover facts, concepts and relationships by themselves.

Other than Problem Solving Skills, the following are also important thinking skills that the students must adopt:

    Observation
    Visualization
    Hand-eye co-ordination
    Analytical thinking
    Logical Reasoning
    Judgment
    Categorization
    Memory



The following is an illustration of how children can be trained with multiple thinking skills: 
Q: Which picture is correct?

                                                           


Thinking skills and processes involved in solving this problem:
a)    Correspondence skill
1.    What is the mass of a Monkey?               7 units
2.    What is the mass of a Chicken?              5 units

b)    Addition skill   
1.    What is the mass of a Monkey and a Chicken?    7 units + 5 units = 12 units
2. What is the mass of 2 Chickens?            5 units + 5 units = 10 units                                                                  
c) Judgment skill
1.    Is a Monkey and a Chicken heavier than 2 Chickens?
 If yes, then the answer is a tick.            Ans: 

Where is the learning gap in problem solving?
Students may understand certain concepts that have been taught, yet may still be
unable to do well in problem solving. In order to bridge this gap, students must apply
the following strategies when solving word problems:

Step 1 Understand the problem
    Students must comprehend the text and understand what is the question asking and what is required to solve the problem.

Step 2. Devise a plan
    Students must be able to identify the correct strategies to use in solving the problem. There are many problem solving Heuristics, and the Model method serves as a good tool to allow students to visualize a solution when solving the problem.

Step 3 Model and carry out the plan
    Students will draw a model, list a table or apply the correct heuristics to solve the problem.

Step 4 Reflection
    Students must reflect by substituting the answer into the problem to meet the conditions stated in the problem.



Quote: From a Facebook posting by Education Minister Mr. Heng: “a question requires a pupil to work through a few steps to get the final answer. And even if he doesn’t get the final answer, if he gets the first few parts right, he understands his materials and gets points for it.” Unquote.

Therefore, the focus is no longer about just getting the right answers but on the process of how the students derive at the correct answers.
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This article is contributed by Edison Innovative Pte. Ltd., the Master Franchisor for eiMaths. We are a Singapore-based Mathematics program focused on creative thinking and problem solving skills to identify and bridge your child’s learning gap in Mathematics. Visit us at www.eimaths.com.

4 Types of Heuristic Method in Maths

Traditional math places emphasis on the acquisition of knowledge. Heuristic method, on the other hand, places more importance on the manner of discovery on the part of the students. The difference of the two is the performance of the student. Heuristic method allows the kid to discover the lesson on his own. It's about the manner of acquiring the information. In the long run, the kid will be able to go about the problem to achieve the solution. He will be less dependent on prior knowledge. He will do away with memorization and fact tables. That is what traditional method failed to instill in kids - the ability to discern problems and issues independently.

4 Types of Heuristic Method in Maths

1. Representation

This is most suitable in word problems. Kids are used to equations. They tend to get lost in the problem when faced with too many words. That's where representation enters the picture. Kids do best when there is a visual representation. After reading the problem, make a list. Itemize the given numbers. It summarizes the information given and gives the kids a clearer picture of the whole problem. By singling out the numbers, kids will get to identify the pattern and they will be able to provide the needed answers systematically.

2. Making a calculated guess

It's the subsequent guessing and checking on the part of the kids. It'll help them narrow down their options. This is especially useful in problems with missing information. Instead of thinking of so many possibilities and wondering whether these will work or not, the kids get to narrow down their choices. Once his guess doesn't produce the intended effect then, he can try the next one. It helps the kids bring together a systematic list of possible choices and achieve the correct one in just a short period of time. It's all about patterns and sequences.

3. Going through a process

Kids learn best through experience and this type of heuristic maths method does just that. Kids will be asked to do the problems using actual objects. He can use balls or sticks to represent the items in the problem. This allows the kids to explore their options in an actual situation.

It's not all about the problems or words swimming in their heads. They can bring those concepts to the real world. What happens when you put this and this together? What happens when you put it this way instead of the other way around? They could try out their options. Find answers to their inquiries themselves. They get to understand the changes happening. What is the scenario before you added this to the pile? What happens after you placed it on the pile? By analyzing their actual objects, they could understand the before and after concepts.

4. Restating the problem

Not all kids think alike. Some can grasp the facts faster than the rest. By changing how the problems are presented, kids will achieve a certain level of understanding that is not present before. They can view the problem in a different perspective. Use items that they can recognize or tap into their interests. Once they started viewing it in a different perspective, they'll be able to know how to work it out. If the problem is too broad, breaking the problem down into simpler parts can help the kids understand better.

Joining Math Enrichment Classes Singapore – Why?

Interesting concept maths for Singapore preschool,visit here

Mathematics is one of those subjects which have rapidly become one of the best and the most interesting as well. The subject is quite challenging and those who study it also get to acquire many high end benefits in both the short as well as the long run. Learning maths can be an arduous process but with the right kind of guidance, it can be rather easy. Math enrichment classes Singapore are the best way for individuals to enhance their mathematical skills and that too, in a very short period of time. Joining such classes enables people to gain knowledge that they never be able to acquire otherwise.

Acquiring Mathematical Knowledge

The math enrichment classes Singapore are responsible for helping people understand the fundamentals of maths. This is something that all people should consider as being an expert at maths is really not ever possible without knowing all the fundamentals of the subject first. Such classes have experienced staff and faculty which provide people with all the important information they need in the process of getting their fundamentals straight. It is exceptionally important for people to be aware of all the basics of maths in order to score well in examinations as well as tests in the near future.

Concepts of Maths 

On the other hand, the enrichment classes are also the best place for people who are interested in knowing all there is about mathematical concepts. While understanding them is most likely to be difficult at first, all that people need is some attention as well as focus in order to know these for good. These classes offer the utmost assistance as well as faculty which is very well are of everything there is about maths in general. Once an individual is aware of all mathematical concepts, they can understand maths and even learn to apply it within real life.

Perfect Affordability

One of the most prominent benefits of joining math enrichment classes Singapore is the fact that such classes are provided at reasonable everyone these days. From students to teachers and anyone who has a desire for learning mathematics on a large scale, the classes are undoubtedly the best place for them in both the short as well as the long run. While a lot of other such classes and entering institutes in general may cost a lot of money to people initially, joining these classes is really not costly and hardly makes people spend truckloads of money in the procedure.

Certified & Qualified Teachers

Since these classes are dedicated in providing the best possible knowledge of maths, they always have the best faculty members. With a lot of experience on hands, the coaching provided to individuals by the people in these classes is definitely going to be top notch and the best of its kind. These classes are the most exclusive place for everyone who has an interest in maths and wishes to take that interest further by actually enrolling there to learn it in great depth later on. Such classes also offer high end and reliable services in terms of the learning of mathematics.

The Bottom Line

Joining these classes for the purpose of knowing mathematics in great detail has been highly recommended to everyone who has a keen love for maths in the first place. However, those who merely want to brush up on their existing skills can also join them in order to get to know more about the subject and how it can help them in the long run. Finding the best kind of places for learning maths may be complex at first but after searching online, individuals are most likely to come across a wide range of different top notch choices that they can easily choose from.

Concept Maths – The Top Kinds

A lot can be said about maths, which is one of the most exclusive subjects in the present times. Even decades ago when it was created, people found it to be the most helpful subjects which taught them to survive in the world. Till date, the subject has managed to prove its significance and how everyone is the world needs in order to acquire personal growth in both the short and the long run. Concept maths is amongst the best ways for people to learn everything there is about the subject and all that it has to offer initially and afterwards; therefore, gaining an insight about the main types of the subject is something that all people must do before studying it.

Concept Mathematics

By studying concept maths, individuals go on to also learn all the concepts that lay behind it. It is these concepts that maths is based on solely and with their proper knowledge, people can understand the subject a great deal better in the near future. Concepts are mostly rather complex and require a lot of time as well as understanding. However, when individuals give a good amount of time in finding the meaning behind these very concepts, they are likely to find all the answers they have been looking for since the beginning. This type of maths has several types and one of the most prominent ones include the concepts that surely must not be missed out.

Theoretical Mathematics

Moreover, theories are also a major part of this sort of maths and for it has to offer, individuals should most definitely give it a good amount of attention. Theories are an essential part of the subject and while some may be difficult, others may not be. Some are considered to be rather lengthy whereas others are much more easy and small in comparison. Those who understand the concepts as well as the theories are definitely on the right path and are likely to get a chance of becoming experts in the subject later on in their lives.

Basic Concept Mathematics

Mathematics starts off with the basics, which are also known as fundamentals of the subject. These are nearly not as lengthy or difficult as what has to come later on; hence, these are fairly easy for everyone to learn in a short period of time. On the other hand, these also allow people to comprehend what the subject is actually about in the first place. Those who wish to score well in the subject should definitely clear away their knowledge of the basics as with their help, they can surely understand the more detailed branches of mathematics.

Advanced Concept Mathematics

Apart from concepts, another important type of this sort of maths includes the more detailed parts of it. Advanced maths is quite complicated and often, many people fail to understand all that it has to offer in both the short as well as the long run but it is through the study of this kind of maths that people can actually go on to score immensely well in the subject without having to struggle too much in the matter. With concept maths, individuals get the ability to learn all sorts of parts and branches of the subject which are necessary for everyone to know in order to gain a better understanding of the subject.

The Bottom Line

All these types of this sort of mathematics have been considered to be exceptionally important and without the knowledge of one, acquiring the knowledge of the other is simply not as easy as one may expect it to be. Therefore, individuals should study all of these types in order to understand what mathematics is all about in the first place.  

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The Need for Learning Concept Maths

Learning mathematics is one of the most important things that people must pay attention to these days since it surely is worth it. The subject is a major part of studies in the loves of students who are attending schools, colleges as well as universities. Therefore, it is something that not a lot of people can miss out on as the choice is simply theirs, which includes going on with it for the purpose of joining prestigious universities in the long run. Concept maths has been considered to be the best way for gaining many high end benefits through the subject. It is this type of maths which tends to allow people to understand the main subject in a short period of time and in great detail.

Real Life Applications

The best part is that the type of maths also has real life applications, which means that people can use it for the purpose of solving problems that exist in real life. On the other hand, it also enables people to solve issues within different topics, debates as well as discussions as maths tends to allow them with all the logic and reasoning they need in order to prove something. Since it has real life implications, it has been labeled as one of the most interesting and unique subjects till date. 

Analytical Skills

Through concept maths, individuals can go on and acquire a high level of analytical skills. These skills are important for anybody who wishes to do well in other aspect of their lives. It is due to the fact that analysis is required in every other part of life, which is precisely why studying this type of maths has been highly recommended to all those who wish to enhance their analytical skills at the earliest convenience. Through increased analytical skills, people can achieve high end jobs as well as entry in top notch universities that most definitely ensure a bright future in the long run. 

Critical Thinking Skills

Similarly, learning this kind of maths is also the best way for acquiring critical thinking. Mathematics has been labeled as the best subject that people should consult in the matter of enhancing their critical thinking skills. According to many surveys as well as education experts, critical thinking is something that requires a lot of time to develop and only through maths is this possible in the first place. Therefore, individuals should spend a good amount of time in the process of learning the subject as it ultimately allows them to have impressive critical thinking skills, which are going to help them in many aspects of life later on.

Mathematical Concepts

Another thing that makes concept maths a must these days is the fact that it enables students to enhance their mathematical concepts on a large scale. Concepts are something that holds a lot of importance in the subject and as it is completely based on these concepts, studying them in great depth has been required by anyone who wishes to know everything there is about it in the first place. People who wish to do well in the subject have been recommended to study it right from the beginning as that surely enhances their knowledge as well as their ability to think better. 

Concept Maths – The Bottom Line

These days, mathematics is behind all the technological changes that are being faced by people in the present times. Every other thing requires it, which is why having an adequate knowledge of the subject is something that is important for all those who wish to do well in the subject later on. Similarly, every other field these days requires the subject and those who know maths can always go on and choose any field that they love or prefer more than the other.  

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Daily Life and Primary School Maths – Its More Important than One Thinks

How many things one has to learn throughout his or her academic life that he or she thinks is of no or least importance and use ahead in their life? A handful really, name a subject and half of it if not more appears to be dustbin worthy when it comes to its practical usage in the real world. All philosophical mumbo jumbo turns to dust every time a student gets promoted to the next class. All what he or she had studies in the previous class begins looking like rubbish. Same is the case with math, the parts that are the easiest also are the ones that are more practical. The propositions like addition and product, percentages and probabilities etc are the thing that one can bet to use one day. These problems of math are covered in the very early stage of a student’s academics. Primary schools make the base of a student in all the subjects.

The level of math taught in high school level and beyond, is the one that seems the most useless. Students just study to pass the subject and to get promoted and the teachers simply teach to end the course in time. But the fact is that it needs to be given respect that is due. Math is never out of someone’s life, not even if they are done studying it, not even if they are becoming medical doctors or lawyers. Ancient philosophers stated math as not a subject of study but as means to understand and if used right, to spend the life.

Managing one’s finance is amongst the most common tasks that anyone encounters. At a young age, it is pocket money and at the working stage it is the salary of a person that matters the most for him. Managing it right brings leisure to life but then the question is that who uses it right? Primary school maths can help in saving so much money that one cannot even imagine. Finance is a scarce resource for everyone and it cannot be allocated in all the places all at once. If a person spends more on one thing, needs to offset and trade off another thing for it. But who is to tell how much should be spent on what. All of this comes to knowledge by learning a little advanced math than just the basic primary school maths.

Almost every person invests money at some stage of their life sooner or later. They may buy bonds, may deposit money in a savings account with a bank, or may even buy stocks of a company. Doing this requires calculation as well as analyses skills. Failure in doing even one of these two things can result in loss of whole life savings. Future projections of values can be easily calculated if one has studies a graduate level math course. Doing this helps in making big decisions related to finances like buying a house or not.

Math also comes in great use when it comes to taking care of health and fitness. Simple calculations tell a person about their body mass relation. It also tells them about the amount of dosage they need to take in relation to the degree of their ailment. In body building, they help in making sure that the weight being lifted is increased at a constant level. With all this being said, math is arguably the most widely and commonly used subject of all and has the most to do in routine daily life. Regardless of what one’s field is or what profession they are pursuing, math remains an unmovable necessity for them at all levels. 

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Why do People Require Maths Enrichment?

Maths is one of those subjects without which, the survival of people is not possible in the present times. Due to this fact, the subject has been made a necessary part of the curriculum within schools and all other educational institutions that exist in the present times. People need to have a good amount of knowledge of the subject for a variety of different reasons, all of which are unique in their own special way. Maths enrichment gives individuals the ability to do a lot more than they usually can in their lives as they can easily go on to achieve a massive amount of success later on.

Solving Real Life Issues

It is important for people to know all there is about the subject due to the fact that it has real life applications and this means that individuals can always go on to apply it within their actual lives for the purpose of solving almost any kind of problems in both the short as well as the long run. The subject is amongst one of the most practical ones, infact there is not another subject that teaches more practicality than mathematics in the first place and that is precisely why studying it completely has been highly recommended to everyone.

Enhanced Career Options

On the other hand, maths enrichment enables people to pursue top notch careers that involve the subject. Almost every other field in the present times consists of maths, which means that people cannot avoid it even if they wish to. Other fields such as those of science as well as IT also require a good amount of knowledge of maths since without it, learning and understanding a wide range of different things is simply not possible. With enhanced knowledge of the subject, individuals get the amazing opportunity of being able to enter any field they want or prefer. It gives them an ability to choose from a wide range of different career options; something that people who don’t learn maths cannot attain.

Successful Careers

Furthermore, due to learning the subject from an early stage, individuals can always go on to become experts in a short period of time. As every other field requires great knowledge of maths, it is best to learn it right from the beginning in order to do well in it later on in life. However, through maths enrichment, individuals can go on to have the careers of their dreams as the subject has a lot to offer in terms of learning and practical problem solving, which allows everyone these days to gain success in record time.

Quick Thinking

Another reason to study the subject is the fact that through it, individuals get the ability to think better in the first place. Enhanced thinking is something that not everyone possesses but it can be achieved by studying maths for a long period of time. According to many reports as well as studies, proper learning of the subject is the reason behind increased thinking in both the short as well as the long run. With high end thinking skills, individuals can do better in many aspects as well as different situations they face in their day to day life.

The Conclusion

As far as the benefits of learning maths are concerned, they are countless in number. Ever since the subject was created decades ago, it has been immensely helpful in many things and till date has proven how important it is to survive in the world. Therefore, individuals who want to gain success in this life and that too, in a short period of time, learning the subject in all ways possible is something that they must put their mind to immediately. 

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The Main Importance of Primary School Maths

Over the past few years, more and more children can be seen to be learning primary school maths since the subject has a lot to offer in many ways later on. While children should always give an equal amount of attention to all subjects, they have been suggested to send more time in the process of learning maths as it can be challenging and often quite difficult to comprehend. However, qualified and experienced teachers from all over the world can be seen to be teaching maths to children in order to enhance their knowledge as well as information about the unique subject in the near future. 

Leaning the Basics & Concepts of Maths

Studying the subject at this level allows children to be able to learn the basics of the subject. The basics are exceptionally important in order to have a good know-how of the subject within the long run. Therefore, teachers have been highly recommended to teach mathematical basics as well as concepts to children who can later on use these for the purpose of acquiring good grades in the subject without having to face any issues in the process.

Saving Time

Children should be taught maths at this stage also due to the fact that it teaches them to save a lot of time. Children are new to the concept of time and how it has to be managed when they are so young. Therefore, by teaching them mathematics, they are able to see how time works and what it actually is in the first place. With proper information about time, children can learn to use it to their advantage and go on to learn saving time for many important events and things, such as tests, in their lives.

The Development of Interest

Studying mathematics at primary school maths, the subject also teaches children to be able to acquire an interest in the subject since they get to learn it for the very first time. When children are introduced to something, they may love and they may not, but with a passage of time, they can learn to adapt to it and in this case, it is highly important considering the significance of this subject for their futures. Therefore, as soon as they have been introduced to the subject, they are likely to acquire a growing fondness for it, which will help them study the subject in a much better way later on.

Out of the Box Thinking in General

A major reason for studying primary school maths level is the fact that it allows students to achieve out of the box thinking. While many children are creative right from the beginning, many others are not, which is why teaching them mathematics has been highly recommended since it allows them to think deeply on a day to day basis and that results in increased creativity and out of the box thinking. While there are many other subject that can be taught in order to achieve this, maths has been labeled to be the best one when it comes to enhancing creative and deep thinking process.

The Conclusion

With all that mathematics has to offer to children, learning it right from the beginning tends to give them an opportunity to understand the subject efficiently and that too, without having to struggle too much later on in their lives. Maths is an exceptionally diverse subject and enables people to join a wide range of different fields in the near future. Therefore, giving the subject importance since this stage of school is something that all parents and teachers should take into thorough consideration as without it, surviving in today’s world has become rather impossible. Moreover, the subject at this level is also much more fun and interesting. 

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Should you Study Heuristics Maths?

There are many other important subjects that people should study these days but maths is certainly the best one amongst them. It is because of the fact that it tends to help people in becoming successful in every possible career there is. Therefore, those who have not yet been acquainted fully with it should now do so since it really is mandatory. The study of heuristics maths is important for everyone to learn since it basically allows everyone to understand everything there is about maths in the first place. It may be complex at first but with time, individuals can easily understand all the concepts behind it.

Wide Career Choices

Individuals should be fully aware of the fact that heuristics maths comes in handy when choosing any kind of field. Be it maths, or science or information technology, it tends to help them in all ways possible. This is due to the fact that maths is required on a large scale in all of these fields and in order to understand them fully in the long run and to be successful, most importantly, individuals need to acquire high end knowledge of heuristics mathematics and see how everything in different careers in linked to it thoroughly.

Problem Solving in Real Life

Heuristics maths enables people to solve real life problems as well. This is due to the fact that the subject has real-life applications which can completely benefit people in a lot of ways. This is why it is currently being taught in schools, colleges as well as universities worldwide. By learning this, individuals can find the solutions for many real life problems that can be pretty difficult otherwise. It is amongst one of the many reasons why this type of maths has managed to progress immensely over the past couple of years into various locations of the globe.

Acing Entry Tests

It also tends to help students do well on entry tests and all other tests as such. This is due to the fact that it allows them to think out of the box and to apply critical thinking into various parts of the test which are complicated in general. Critical thinking as well as proper analysis can make students do great within tests and it is only through the proper learning of this type of maths that they can actually go on the achieve their goals in the near future. All in all, test scores can be drastically increased by learning heuristics mathematics.

High End Intelligence

On the other hand, it has a lot to do with intelligence. According to many surveys as well as studies, this type of maths is the answer to the questions of all those individuals who wish to find a way through which they can enhance their thinking. The subject requires a lot of analysis as well as thinking, which makes it rather complex at first but it can eventually be understood when people push their mind further for the purpose of acquiring the answers they need in the first place. Basically, through it, the thinking process can be increased, which in turn tends to eventually enhance the intelligence of an individual.

Heuristics Maths – Is it worth Studying?

One can easily say that this kind of maths is very essential for anyone who wishes to choose a mathematics related field. It is also exceptionally important for anyone who wants to do well at maths in general and needs to acquire enhanced intelligence as well as critical thinking. For all that it offers, heuristics mathematics has been fully introduced into the curriculums of many schools in the US, UK and many other countries in the present times. Moreover, it can now be seen to be spreading all over the world as well.

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Ways to Boost Memory – Enhanced Math Skills

Children of younger age are more prone to getting disheartened after not being able to catch up with the number related subjects. This is a very big problem because at that stage, the career path of students gets selected and impacts them throughout their lives. They get wore down by not being able to solve equations and as a result they simply lose interest in the subjects and may even go rebellious against studies. 

Memory problem is the root cause of children not being good at mathematics. It has been revealed by research that majority of children that have a sharp memory, are the ones that actually do good at arithmetic subjects. Faulty working memory causes faults in calculations and also accounts for wrong formula usage. 

Heuristics Maths is a difficult subject and before the numbers get written o the paper, naturally, it gets solved in the mind. This mental math needs a very good and reliable working memory, in the absence of which, the learner starts feeling sick and uninterested in the subject altogether. Here are however 5 ways through which one can boost the memory that results in better performance at calculations.

Help Working Memory:

Using tools can overcome this disadvantage. One can use an audio recorder to record the teacher’s audio during the class that can be heard afterwards. This is good in case the child has poor concentration level and complains about not being able to listen to the lecture carefully. Simply sitting in the front row does not remedy this problem because it has nothing to do with the distance from teacher and has everything to do with the mental strength of the student. Calculator in use needs to be suitable with the user’s age. A scientific calculator should not be used till the child turns at least 15 since there is no use of its operations at least till high school. The child needs to repeat each calculation on the calculator that the teacher does on the board.

Calculation Games:

It is really surprising how games like the “minesweeper” could teach children math and improve their calculation skills. Games are interesting and can be helpful too. There is next to infinite number of games available that can improve the math skills of children. Engaging games that come in good graphics teach students instead of spoiling them and damaging their eyesight. Geometry, trigonometry and calculus are the areas that specially benefit from the right usage of such games.

Brain Training:

Sometimes it helps to let children research on their own a bit. This trains and warms their brains up making it easier for them to accept ideas and grasp concepts. The topic to research not necessarily has to be math, the basic purpose is to have them sit and study for a few hours developing this habit in them. Once this gets developed, then they can easily learn even the most difficult of concepts in the toughest of subjects.

Over Learning Method:

One can always get their children to over learn things because doing that makes them lesser dependent on their working memory. When children over learn, they tend to grow confident in learning the next levels of math. The knowledge turns into reflexes through this method. Solving heuristics maths, eventually becomes not harder than solving a fun puzzle.  

Multimodal Approach:

For very young aged children, pictorial learning is what appeals to them the most. Visual representations of calculation based questions makes children ready to comprehend with word problems in the next classes to come. This also helps in aiding the working memory of the child.    

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