But, where am I going to use this?

In the profession of teaching, one of the most common question we come across from students is that “Where am I going to use what I learnt in the class today?” And in the field of mathematics this question becomes more challenging to answer as to where students would use concepts like quadratic equation and vertical asymptote in their real life. While researching about ways one can incorporate application of mathematics in the learning goal itself, I came across a video that provides an excellent example of using application of mathematical concept to enhance the overall learning experience.

Real-life application of Prime Numbers

While researching about practical applications of everyday math concepts, one topic that really fascinated me was that of use of prime numbers. Did you know every time you use your credit card to shop online, banks use prime numbers to keep that transaction safe? I was perplexed at first as to see how a prime number can help me save my credit card information from a malicious person over the internet. Before we dive deep into how prime numbers are used to keep banking transactions safe, let’s look at some key facts about prime numbers themselves.

-          A prime number is a natural number greater than one that has no positive divisors other than one and itself. A natural number greater than one that is not a prime number is called a composite number. For example, 2 is a prime number since it can be divided by 1 and itself. 4, for example, is a composite number as it is divisible by 1, 2 and 4.

-          Any whole number bigger than 1, can be created by multiplying prime numbers together. For example, 6, is a result of multiplication of 2 and 3.

Now let us look at this video and see how banks use these properties of prime numbers to keep our credit cards safe!

 

Incredible, isn't it?

 

Categories

Categories

These are the categories explain in basic term. I like this document because it is in student friendly terms and it would be easy for students with learning difficulties to also understand what the categories are all about.

Knowledge ·       Basic facts (e.g. 5x5 =25) ·       Procedures ·       Operations (+ ,-,x, /) ·       Proper use of manipulatives ·       Vocabulary, terms , “math words”	Communication ·       Organizing ideas ·       Explain their thinking orally, visually, and in writing ·       Clear expression of ideas in many ways (pictures, numbers, words, graphs)
Thinking ·       Problem solving ·       Make a plan ·       Carry it out ·       Interpret the problem and check it ·       Explain reasons and show proof	Application ·       Solve a variety of problems in different ways ·       Use all skills learned in the year not just what we’re learning now ·       Make connections to the world

Unanswerable Questions - How Big is Infinity?

A common mis-conception about the field of mathematics, especially among students growing up, is that the field has reached its limits and there nothing more to be discovered or no problems that have gone unsolved. Math is a poured set of concrete that has settled, never to be changed over time.

It is this sort of mis-conception that allows students to enter into a math course without any excitement or anticipation. Any one with a somewhat advanced interest in math though knows that this is not the case and that mathematics is a field paralleling the sciences in terms of modern breakthroughs and ancient problems yet to be solved.

At the beginning of a course a teacher has the perfect opportunity to convey this sort of thinking to their students in order to spark their interest and let them know that this course is going to be engaging. A great way to do this is through the use of multimedia which often is over looked in the math classroom. If you can set the tone early as a teacher, then you have the opportunity to create an engaging classroom in the future.

A great concept to do this with is the concept of infinity. Most people have heard of the concept but have not taken the time to try and wrap their minds around it. It is when you do that things get interesting. Take the following video for example:

The video is an exploration on how big infinity is. This is a perfect chance to begin a discussion with students: what does infinity mean? How big is infinity? The video explains how there are infinities within infinities which means that there are larger and smaller infinities. Surely this will get some of them interested. The video then goes on to explain how "there are unanswerable questions in mathematics". Suddenly the solid cement has melted and mathematics is now a field much like the sciences where inquiry is encouraged and questions remain unsolved. This short discussion and video can prove fruitful for future inquiry based activities in the classroom.  

Fun Math

A great way to start your math class is with a game, or cool trick. Students become so engaged, and they're minds are immediately focused on math. Some, like the one above, are great for reinforcing topics (order of operations), or the game GREED for simple calculation skills, and others are just fun, like different calculator tricks (such as creating different words). 

Differentiated Mathematical Instruction

Students differ mathematically. The difference between students’ mathematical abilities is an issue that we must face as teachers. Some believe that a differentiated instruction environment can be the solution, and I agree. As teachers, we need to support all students, especially those who need additional help. We need to promote the best teaching and learning in mathematics.

In order to meet each student’s needs, we need to “provide tasks within each student’s zone of proximal development and to ensure that each student in the class has the opportunity to make a meaningful contribution to the class community of learners” (Why and How to Differentiate Math Instruction, p2). Instruction within the zone of proximal development utilizes instruction effectively because it helps students to obtain new ideas that are beyond what they already know but within their reach. We can determine what the zone is by using prior assessment information. To effectively differentiate instruction, we need to focus on the big ideas, use prior assessment to determine the instructional direction and needs of different students, as well as provide appropriate choices for students.

There are two core strategies for differentiating mathematical instruction: open questions and parallel tasks. Asking open questions can encourage students to have variety of responses or approaches. An open question should be mathematically meaningful. It can also enrich a mathematical conversation, because every student will be able to contribute and gain from the discussion. Open questions can also provide opportunity for teachers to help students see that mathematics is multifaceted. Parallel tasks are sets of tasks, “which are designed to meet the needs of students at different developmental levels” (Why and How to Differentiate Math Instruction, p11). When developing open questions and parallel tasks, we need to keep in mind that they should be created in such a way that all students can participate in follow-up discussions. 

Mathematics in Daily Life

“Mathematical literacy involves more than executing procedures. It implies a knowledge base and the competence and confidence to apply this knowledge in the practical world.” When I pay tuition fees, buy textbooks, do grocery shopping, pay rent, and plan any budget, I’m using simple mathematical principles in my daily life. In fact, everything in our life involves simple or complex math principles. “A mathematically literate person can estimate; interpret data; solve day-to-day problems; reason in numerical, graphical, and geometric situations; and communicate using mathematics” (Leading Math Success: Mathematical Literacy, Grade 7-12, pp10). Math can help us understand and solve daily problems. Whether I’m taking a flight back to China or financing a car in Canada, math literate is everywhere. As difficult as aircraft navigation; as useful as compound interest, Mathematics underlines everything that occurs around us. Even cooking and baking involve math, look at the proportion of different ingredients that I need in a recipe to make a tasty cheesecake.

1-1/2 cups  graham cracker crumbs

3 Tbsp.  sugar

1/3 cup  butter or margarine, melted

4 pkg.  (8 oz. each) PHILADELPHIA Cream Cheese, softened

1 cup  sugar

1 tsp.  vanilla

4  eggs

Math is a language that can be understood everywhere, it is universal.

Student Involvement

Although I did not have a practicum placement with a math teacher, I was able to spend my prep periods observing and assisting a grade nine math teacher at my school. He had been recommended to me from other members of the faculty and even some of his students.

One of their main points of praise was his ability to get the entire class involved in learning. He managed to achieve this by having students continually out of their desks and performing math questions on the boards, which completely lined his classroom. This activity worked well for his classes because it allowed students to move around and stay active, which is helpful for a subject like math. It also made students pay attention to the material because they knew that they would be putting their answers on the board.

This method helped the teacher understand where his students were at in their learning because he could easily see how quickly they answered the questions. He could also tailor his teachings to the students individual difficulties if he noticed that more than one were getting stuck on the same step or making similar mistakes. This is an easy way for a teacher to check understanding without marking individual papers or asking if everyone "gets it."

It's important to note that the teacher did not mark or grade any of these student answers. He went around the classroom and observed and made suggestions when necessary. The students were comfortable enough to ask their peers next to them for help if they were stuck. They also used each other to check the validity of their answers. All students attempted the problems in some ways, possibly because they enjoyed writing on the board and did not want to have a blank space where they were standing. Since students were standing up, and the noise levels were a little louder than most classrooms, students felt less anxiety about raising their hand and asking the teacher a question.

In conclusion, students seemed to really enjoy this method of teaching and I would encourage other math teachers to try and have their students get over the fear of having other people look at their math by collective board-writing.

The Dividing Factors Between Classrooms and the Real World

https://www.youtube.com/watch?v=xYONRn3EbYY

 

The following is a video of Conrad Wolfram, a fantastic mathematician and speaker who argues that computers are the dividing factor between the classrooms and the real world. By this, he means that the pedagogy and practices currently used in classrooms today, are unrelated to the real life examples of where these concepts can be applied.  Mathematics is process that moves people towards logical reasoning, something we need in our day to day activities. As teachers, we need to teach students to

1.       ask the right questions

2.       set up their data

3.       calculate

4.       take their answer and apply it to the real world.

Currently, Wolfram believes that we spend up to 80% of math classes focused on step 3, calculating when we should be teaching students all steps evenly. His biggest issue with the current math system is that because the emphasis is so great on getting students to calculate answers, students are not learning 'big picture' math in the context of real world examples and implications. So much time is spent on computations in a classroom whereas in the real world solutions are often solved by computers. I believe that to change the current math system would be a massive undergoing, where the system bottom-up would have to re-educate teachers to be more tech savvy  far passing their current knowledge of technology and programming. Top-down, government ministries are just as responsible for restructuring curriculum to move away from computational strategies and move toward comprehension strategies. However, this change could be revolutionary in terms of the leadership and innovation our next generation of citizens could offer.