HOW TO MAKE MATHEMATICS CLASS INTERESTING

Make it meaningful

Many math courses suffer from the following issues:

1. The teachers don't know why they are teaching particular math topics, and they often don't know what else the students are learning in other subjects.
2. As a result, the students don't know why they learn those math topics, either. The common question, "Why do we have to learn this?", is a reasonable one. Do you have a good answer, beyond "It's in the exam" or worse, "Because it's good for you"?

Some possible ways to fix this:

• Find out where the students will use each math topic you teach (it may be in their science class, or some engineering subject). It's great when you can use actual examples from those other subjects and let the students know that's where they'll use each math topic.
• Help students make connections between the math topic and the "real world". If you're not sure how it's used in the real world - do a search!

Start with concrete examples - leave the abstract concepts to later

Math is largely about abstraction. Mathematicians for centuries have thought about real problems and come up with practical ways to solve those problems. Later, they have generalized the process, usually presenting the solution using algebraic formulas.
When students have no idea what the original practical problems actually mean, how can they be expected to understand the abstractions of those problems (using the formulas)?

Use watermelons to teach calculus

Instead of starting each topic with a formula, start with concrete examples of the problems that were originally solved using that math. Then, help the students see how the math theory can help to solve such problems by showing them the thinking behind the solution.

Start with an interesting, real-world problem (preferably localized)

Most math lectures start with "Here's the new formula for today, here's how you plug in values, here's the correct answer."
Problem is, there's no attempt to motivate the learners.
It is good to pique curiosity with a photograph, a short video, a diagram, a joke, or perhaps a graph. This trigger should outline an interesting problem in your local area (so students can relate to it better and feel more ownership).

Where you can, use computers to do the drudge work

Many math courses seem to be more about calculation rather than concepts. These days, it doesn't make sense for humans to spend hours learning how to calculate using complicated algebra.
To quote John Allen Paulos:
Mathematics is no more computation than typing is literature.
And then this one, usually attributed to Albert Einstein:
Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination.
For the vast majority of your students (who will not eventually become mathematicians), it's more important they understand the concepts and which process to use when confronted with different real problems. They then should learn how to use computer algebra systems (or graphics calculators) to solve such problems.
Some examples:

1. I observed a lesson recently where the math teacher got his students to calculate the standard deviation of a set of scores. He started with the formula then allowed the students to use Excel. He wanted them to apply the formula (rather than let Excel do it directly using STDEV, which is what many students actually did).The problem was, the students didn't even know what "standard deviation" meant. They were just plugging in numbers with little idea of what it meant. Finally someone remembered the standard normal curve and could explain what they were all doing. (See Probability Distributions - Concepts)
   The teacher should have made sure the students had the general idea first, including the general meaning of "distance from the mean".
2. When finding the length of a curve using calculus, it's actually not possible to find the answer using ordinary integration (because the integral does not exist). But using a computer algebra system, we can easily find the answer for such problems and spend more time understanding the problem and the solution given. (See Arc Length of a Curve.)
3. Creativity and ownership

Many math students feel very little ownership for what's going on.
They have little say in what the topics are (that's usual for most formal education) and the exact same assignments are given to everyone. It's not surprising there is little enthusiasm for such "one size fits all" approaches.

Computer-generated flower

We are all creative, and we all enjoy being creative, but in most school systems creativity is discouraged. 
There are many ways we can encourage creativity in math. Technology is one avenue - get students to use creative means to describe a mathematical concept (it could be a video, an animation, a diagram or perhaps a concept map).
Such individualized assignments get them thinking about the bigger picture, encourages creativity, and is more likely to generate feelings of ownership than the normal mass-produced assignment.

TEN CREATIVE WAYS TO TEACH MATHEMATICS

Here are some activities for your classroom to add a bit of sparkle and creativity. As children work, ask critical questions such as "Did you try this?" "What would have happened if?" "Do you think you could?" to enhance children's understanding of mathematical ideas and vocabulary.

1. Use dramatizations . Invite children pretend to be in a ball (sphere) or box (rectangular prism), feeling the faces, edges, and corners and to dramatize simple arithmetic problems such as: Three frogs jumped in the pond, then one more, how many are there in all?
    
2. Use children's bodies. Suggest that children show how many feet, mouths, and so on they have. When asked to show their "three arms," they respond loudly in protest, and then tell the adult how many they do have and show ("prove") it. Then invite children to show numbers with fingers, starting with the familiar, "How old are you?" to showing numbers you say, to showing numbers in different ways (for example, five as three on one hand and two on the other).
    
3. Use children's play. Engage children in block play that allows them to do mathematics in numerous ways, including sorting, serializing, creating symmetric designs and buildings, making patterns, and so forth. Then introduce a game of Dinosaur Shop. Suggest that children pretend to buy and sell toy dinosaurs or other small objects, learning counting, arithmetic, and money concepts.
    
4. Use children's toys. Encourage children to use "scenes" and toys to act out situations such as three cars on the road, or, later in the year, two monkeys in the trees and two on the ground.
    
5. Use children's stories. Share books with children that address mathematics but are also good stories. Later, help children see mathematics in any book. In Blueberries for Sal, by Robert McCloskey (Penguin, 1993), children can copy "kuplink, kuplank, kuplunk!" and later tell you the number as you slowly drop up to four counters into a coffee can.
6. Use children's natural creativity. Children's ideas about mathematics should be discussed with all children. Here's a "mathematical conversation" between two boys, each 6 years of age: "Think of the biggest number you can. Now add five. Then, imagine if you had that many cupcakes." " Wow, that's five more than the biggest number you could come up with!"
    
7. Use children's problem-solving abilities. Ask children to describe how they would figure out problems such as getting just enough scissors for their table or how many snacks they would need if a guest were joining the group. Encourage them to use their own fingers or manipulatives or whatever else might be handy for problem solving.
    
8. Use a variety of strategies. Bring mathematics everywhere you go in your classroom, from counting children at morning meeting to setting the table, to asking children to clean up a given number or shape of items. Also, use a research-based curriculum to incorporate a sequenced series of learning activities into your program.
    
9. Use technology. Try digital cameras to record children's mathematical work, in their play and in planned activities, and then use the photographs to aid discussions and reflections with children, curriculum planning, and communication with parents. Use computers wisely to mathematics situations and provide individualized instruction.
    
10. Use assessments to measure children's mathematics learning. Use observations, discussions with children, and small-group activities to learn about children's mathematical thinking and to make informed decisions about what each child might be able to learn from future experiences. Also try computer assessments. Use programs that assess children automatically.

ALGEBRA TEACHING TIPS

Solving Equations

After explaining the basic concepts in solving equations, I show students a simple equation to solve such as 2x + 4 = 8.  I tell students that the goal is to get x by itself on one side of the equation. So the first step is to subtract 4 from both sides, and we are left with 2x = 4. Most are able to follow this procedure. Then when I ask them what should be the next step, some students want to subtract the 2 from the 2x. So I explain that 2x means 2 multiplied by x, so to get x by itself we need to divide by 2. This satisfies some students but others remain confused.

I have achieved good results by turning this equation into a word problem: Two cantaloupes = $4. I ask “How much is one cantaloupe?” and they all answer $2. Then I ask “How did you figure this out?” This helps them realize that they had to divide both sides by 2 in order to get the answer.

Collecting Like Terms
I use a similar strategy when I am explaining how to add like terms. For example, if we have an expression such as 6x² + 2x⁴ - 2x² + 7 + 3x² - 3 and the goal is to combine like terms, I first ask three students to name their favorite fruit. If I get answers such as apples, oranges and bananas, I first rewrite this expression as 6 apples + 2 oranges – 2 apples + 7 bananas + 3 oranges - 3 bananas. Then I ask them to evaluate how many fruits of each kind are left. Most can figure this out as 4 apples + 5 oranges + 4 bananas.

Then I tell students that in math we also have different kinds of fruits but they have mathematical names such as x⁴, x² and just plain numbers. They are then usually able to make the connection and get the right answer.

Factoring Polynomials
Most algebra textbooks introduce factoring of polynomials before describing the purpose of factoring. I prefer to start by showing my students a graph of a quadratic equation which is a parabola, and explaining that the solutions of this quadratic are the x intercepts (they are already familiar with the concept of intercepts from linear equations). Then I indicate that there are three ways of solving this type of equation: 1) The quadratic formula, 2) completing the square and 3) factoring (when possible). Then the reason for factoring becomes more understandable in view of the ultimate goal.

Systems of Equations
Before showing the two methods of solving a system of two equations (substitution and elimination), I first write this equation on the board: x + y = 4.

Then I ask my students “What is x and what is y?” After the common answers of 2 and 2, 3 and 1 and 4 and 0, most of them soon realize that an equation in two variables does not have a unique solution, because there are an infinite number of possible answers. Then I write two equations like this:

x + y = 4
x - y = 2

I ask “What is x and what is y?” Most are able to figure out that x is 3 and y is 1. This makes the explanation that we need two equations to solve for x and y more understandable.

Finding the LCM (Least Common Multiple) of Rational Expressions
This topic is often quite challenging for students. What I try to do is clearly demonstrate the analogy with finding the LCM of numbers, which they have studied in an earlier chapter and know fairly well by the time we get to rational expressions.

In order to find the LCM of  12, 15 and 18, break each number down into its prime factors:

12:  3,2,2
15:  3,5
18:  3,3,2

So the LCM is 3 x 3 x 2 x 2 x 5 = 180

To find the LCM of x² - 4, x² + 4x + 4, 2x - 4, break each expression into its prime factors (after defining a prime factor for an algebraic expression):

x² - 4:  (x + 2) (x - 2)
x² + 4x + 4:  (x + 2) (x + 2)
2x - 4:  2(x - 2)

Using the same procedure we used for numbers, we pick each prime factor to the maximum number of times it appears in any of the three expressions:

So the LCM is 2(x + 2) (x +2) (x - 2)

10 POINTS FOR EXPLORING MATHS CREATIVELY

1. Empowers pupils to take ownership of their learning as active learners

It is no longer about the teaching but rather about the learning. The distinction reflects the important notion that pupils should be actively engaged in the learning process. Pupils are given the opportunity to enquire, investigate and choose from a variety of resources. They can direct the focus of their learning according to their interests and prior knowledge. Pupils’ motivation and expectations increase and so does their confidence in engaging with maths skills and concepts. 

2. Promotes investigative and problem-solving skills

Creative mathematics is all about developing problem-solving skills which enables pupils to solve unfamiliar mathematical problems creatively. Pupils realise that there might be more than one possible solution to solving a given situation and learn how to adopt diverse strategies towards problem-solving which best suit their learning styles, capabilities and situation. Pupils are also given the time, space and resources to explore mathematical skills and concepts and can devise their own path to a solution. 

3. Establishes connections to real life making learning more relevant

The notion of a classroom has been subject to strong competition with the real world beyond its walls, as well as the instantly accessible virtual world. In today’s information-based and highly globalised society, it is simply absurd to teach without acknowledging real data that is surrounding and bombarding us every second. Teaching and learning should be ever more connected and contextualised in real life circumstances. We cannot have pupils ask; “Why are we learning this?”. The more we establish links between learning and real life, the better can pupils apply their knowledge and skills, and regard the learning as valuable and relevant. Learning tasks should be more based on real life situations, enabling learners to tap into their prior knowledge whilst becoming more engaged with the task at hand. Such examples of realia include; menus, TV schedules, informative websites, transport information, published newsletters, promotional leaflets, sports websites, etc. 

4. Presents opportunities for collaborative learning and communication

Creative learning tasks entail the exploration of diverse learning modes which include collaborative group work. Pupils learn to work with other learners who have different learning abilities and together attempt to find a strategy on how to produce something or solve a given task. Throughout this process, the pupils are actively engaged in dialogue. They learn to verbalise their mathematical thinking and to consolidate their use of mathematical vocabulary. Pupils learn to enage in self-assessment and they evaluate their best capabilities, and assign different tasks of the project to specific members of the group in order to reach their final goal. Such tasks allow pupils to develop their social and communication skills, which prepares them for the future. It is the teacher’s responsibility to form functional group clusters by having diverse learners grouped together. Groups should be kept small, so that every student remains engaged and feels important to the rest of the team. One important tip is to assign specific roles within the student’s abilities, and whose duty is necessary in order for the group to reach their final objective or produce their desired outcome. 

5. Fosters initiative, innovation and creative thinking

Much focus is being placed upon the terms initiative and creativity, as part of the list of transversal skills required in the 21st century. Society needs citizens who are able to take initiative, who are good decision-makers, problem-solvers and who are able to be creative and think outside the box. Exploring mathematics creatively involves providing open-ended opportunities for our pupils to work collaboratively and to design innovative strategies and solutions to a given situation. This practise allows pupils to foster such important skills which allow them to thrive and to be better equipped for tomorrow’s world. 

6. Explores maths through technology

Our children are constantly surrounded by technology especially mobile touchscreen devices. From a very young age they seem to hold an instinctive disposition to interact with screens and to respond to visual cues. This exposure is enforced both in households as well as in other locations outside the home, such as shops, restaurants, shopping centers, etc. Technology has revolutionized the concept of education and has shifted the learning process to one which is more self-directed, creative and also game-based. Exploring maths creatively acknowledges and values the potential and vast resources which technology can provide us. Through technology, pupils learn various skills such as language, creativity, social skills, mathematical thinking, and problem-solving. Technology also provides models and opportunities for pupils to explore their learning through an appealing and relevant medium. One great example of creative maths through technology is the soaring use of coding, which can be carried out either using apps (such as online, mobile or tablet apps) or else through floor robots such as the Pro-Bot. Pupils have the possibility to learn how to programme, design, engage in content creation and problem-solve whilst exploring different skills related to maths such as shape, measure, fractions, angles and position. Coding skills are considered as a language in itself and it forms the basis of logical reasoning. Pupils can be given the opportunity to create their own problem-solving tasks which they can then share with their peers or other pupils from other schools. 

7. Supports pupils with diverse abilities

Adopting this approach towards exploring maths in real life, also serves to cater for the different students who are diverse in terms of learning abilities and preferences. Such an approach can be considered inclusive and through the continuous representations of mathematical situations as drawn from real life, pupils will have the chance to explore maths from different perspectives while the learning becomes more appealing. The variety of learning modes, enables most students to participate and to remain engaged on the task. 

8. Nurtures mathematical thinking and reasoning

Fundamental to mathematical learning is the ability to think and reason mathematically. It is important to present opportunities whereby the students are able to explore the process of problem-solving through mathematical thinking and reasoning. These situations also enable the learner to become better at communicating their thinking and in finding the appropriate vocabulary to explain their reasoning. Students are also able to observe that there might be more than one possible and reasonable solution to a given problem. Mathematical reasoning can be exemplied verbally, visually or through models.

9. Blurs the boundaries among different curricula areas

One very positive aspect about exploring maths creatively is that it does not only establish a more dynamic relationship among the teacher and the students, but it also establishes links with other learning areas, creating a multi-disciplinary approach to learning. Teachers can collaborate into providing project-based learning scenarios whereby students work in groups and explore a given situation or location and require a myriad number of skills in order to reach their goal or final product. Students learn to establish connections between maths and language, art, history, science, technology, physical education and other aspects of the curricula. 

10. Heightens understanding and retention

Hands-on experiences allow students to apply their learnt skills and concepts in practice. Such opportunities provide pupils with a repertoire of experiences which they can recall and allows them to become more confident at applying their knowledge in the future. Through hands-on practice, learners are able to self-assess where they require further support. These activities provide a meaningful context to learning and promotes retention of learnt maths skills and concepts.

Throughout my past years as a maths support teacher, I have had the possibility to visit many primary schools in my country. These visits implied opportunities to collaborate with class teachers and together explore innovative mathematics pedagogies, which appeal to different students and which attracts them towards participating in engaging maths activities. I have gained a lot of insight into the type of quality education which our pupils require not only during their young age but also to increase their participation in tomorrow’s world. We as teachers realize, that we are preparing our learners for a world which is rapidly changing, and for future jobs which are yet to be created. Our teaching has to remain abreast with what technology presents to us and valid to today’s rising generations. It remains, however, our duty to instill in our pupils a sense of love for learning and creating, and to recognize the relevance, inter relatedness and beauty of education in the real world beyond the classroom.