solving linear and nonlinear equations, optimization methods for systems design.

Include biography of mathematicians/physicists in the course such that students go through the same critical steps which those mathematicians underwent. Example, how did Newton obtained laws of motion? Methods adopted by early mathematicians to solve quadratic and cubic equations. Invention of imaginary number and the complex plane. Various ways of obtaining the value of pi, notion of infinity etc.

These will cultivate interest in mathematics as well as generate critical mathematical thinking among students. As far as I know, what I am mentioning here is a lacking in most educational curriculum across the world.

Teach explicitly what is currently the secret syllabus. (Things like divide and conquer, Modelling 101...)

Mathematizing happens when children can create a model of the situation by using mathematical objects (such as numbers or shapes, mathematical actions (such as counting or transforming shapes), and their structural relationships to solve problems about the situation. For example, children can use blocks to build a model of a castle tower, positioning the blocks to fit with a description or relationships among features of the tower, such as a front door on the first floor, a large room on the second floor, and a lookout tower on top of the roof. Mathematizing often involves representing relationships in a situation so that the relationships can be quantified.

http://ncca.ie/en/Publications/Reports/NCCA_Research_Report_17.pdf

I would suggest that to address such a question we would need to address rather more fundamental and enduring conundrums:

What is education for?........so

What should we teach?........and so

Should we teach the same thing to all pupils?

This is why in England we have constructed a new curriculum around a body of knowledge specified by “subject experts” rather than “educational experts” in a political process where educators were marginalised.

There are two connected but different philosophical positions we could adopt in seeking a solution.

Firstly we might seek to define a curriculum of “subject domains” each consisting of a knowledge, skills and conceptual understanding which are culturally and historically defined as lying within that particular subject domain. This is “powerful knowledge”.

https://www.youtube.com/watch?v=r_S5Denaj-k

Secondly we might define an “aims based curriculum” where our approach to answering the three questions above starts not with ‘bodies of knowledge’ but with the needs of individuals and society. Here we might define what “mathematics” pupils could learn in terms of their needs. This is sometimes misunderstood as a curriculum defined only by “useful” knowledge e.g. work skills or in the case of mathematics - functional numeracy.. What an aims based curriculum is really encapsulating is “worthwhile knowledge”. Put simply - whilst a child may not become a scientist, a knowledge of the universe of atoms and galaxies is not “useful” but is a part of being human. From such a position knowledge can have a worth which is not transactional e.g. an appreciation of art or indeed of mathematical beauty. Here aesthetics is seen as a “need” as well as “functional learning” the former only being rejected in the 19th C and then only for a section of society.

https://www.amazon.co.uk/Aims-based-Curriculum-significance-flourishing-schools/dp/085473998X

I suspect that this is not an answer to your question but I do hope it might provide a framework within which it might be considered.

Thank you very much for the answers ...

I just curious about the future of school mathematics

What is school math for? 

In my opinion at least there are two things. First, school mathematics is intended to prepare students to learn pure or applied mathematics in the college level. Second, school mathematics is intended to prepare students to acquire mathematical concepts, principles, and maybe also mathematical thinking relevant to their future life in the society.

Mathematics teachers should consider both of them. They have to prepare their students for study in the higher level, which requires strong mathematical power, and to live peacefully which requires  mathematical literacy. 

I do believe that considering students capabilities, and the presence of advanced digital technology, not all of the existing mathematics topics should be introduced to them anymore. There is a need to limit the coverage of the mathematical content , and at the same time there is a request to encourage students to learn more deeply and develop higher order thinking skills. 

This means that we should reduce and reorganize the school mathematics curriculum. 

For example... do we still need to teach our students on how to find the root of quadratic equation? if so, to what entent? We know that there are some mathematical software that can be used to solve that equation. 

Similar thing are also working for other school mathematics topics such as matrices, linear programming, differential, integral erc.

So.. what school mathematics should we provide to our students now?

Should we use the existing school mathematics but change the method of teaching only?

Please inform me

Thanks

Emphasis should focus on the conceptual knowledge rather than procedural knowledge.

The discussion about the content of the curriculum is interesting (and in the UK there has been a radical overhaul of the GCSE maths content), however it is the students ability to apply maths that is my concern. In 2016 the UK A* to C grade GCSE pass rate was 61% (http://www.bstubbs.co.uk/gcse.htm ), in practice this means that I meet students at age 16 and above who can divide (when using a calculator and not without) but they don't understand when to use a division process and what the answer means; they are not sure what fractions are or what 10% off a price means. 

The content of the syllabus may be an area to discuss, but the students progression through the subject is critical to any lasting maths achievement.

We need to introduce Vedic Mathematics in school, it's a supreme system that develops mathematical thinking of a child. Based on an ancient system from India, the child can calculate using 16 sutra( word formula) mentally any known mathematical problems through pattern recognition. This will drives them to towards sense of self dicovery and they will develop logical mathematical intelligence.

The only purpose of teaching the solving of quadratic equations is to teach that there is not necessarily always one solution to a problem. There may be another equally valid solution lurking away somewhere!

Learning mathematics helps people learn logic. It also provides skills to understand how mathematics works in the real world.  

For example, many surveys you read about in the news are plain convenience samples and thus have no validity.  Even worse, online surveys often do not prevent people from voting more than once.  

Understanding the fundamentals of math can help us function in the real world.

Thanks Michael W Marek. 

I absolutely agree. Mathematics help our students to having better logical skills. 

But... let me tell you...

In grade 7, students in our country have to learn many mathematical topics such as: integers and fraction, algebraic expression, set, proportion, social arithmetic, parallel lines, area and perimeter of triangles and quadrilaterals, data presentation. 

In addition to understand it, they have to be able to explain/communicate, and apply it the problematic setting. 

Do you think they need to learn all of these? Does not it too much?

Dear Henk Smid,

I am not thinking of teaching students to use electronic during their learning period at school level (although sometime they need to realize that there are a lot of electronic tools available to solve mathematical problems). I want to teach them mathematical contents. But... to what extent? 

If they are not mathematicians, I stongly believe that,for their daily lifes, rather than using traditional mathematics procedures, they will use electronic tools. If this is so, it means that what is given to them during their school years are useless. They do not use and apply it for their life.

So... what school mathematical contents should we provide to our students?  

I think,  Basic Mathematics which is aplied in normal life should be done in lower levels,like arithmetics and  simple algebra.   Advanced Mathematics should be done for those who want to be  mathematicians in advanced and higher education level.

Using electronic devices in lower levels lead to habit  of dependancy even for simple mathematics, so it is better to use these devices in higher learnings.

Frank

The answer to your question depends on many aspects of your educational system. Does your country have nationally prescribed mathematics curriculum? Is it a  farmework mapping the structure within the outer boundaries of what is to be learned at a certain level or is it specific in each and every detail? Are there different tracks for pupils depending on their assesed abilities or do you have inclusive education for all children and the teacher in the classroom is responsible for accomodating for the learning needs of each child individually? Pupils need to be exposed to the big ideas in mathematics and the clever ways of approacing problems which have been developed during the long history of mathematics but to understand their internal logic and to experience the beauty ceratin levels of procedural knowledge have to be acquired along with conceptual understanding. Needless to say, practicing mathematics is very much like playing an instrument: the more you do it the better you are and the more pleasure you get from it. We should not teach children tedious techniques for the sheer purpose of mastering them (the new technology makes it unnecessary), yet downsizing the content to be learned may lead to crippling effects on children's prospective opportunities to learn science, technology, and even social sciences.

Slagjana

I think a good answer is given by Conrad Wolfram in this TED Conference: 

http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers

Hi, organizaría el  urriculum a tarvés de los procesos de construcción de conocimiento matemático: defining, conjecturing, proving, modeling.

I think school mathematics should learn about Mathematics content integrated with mathematical skill and relate to real life. The Mathematics content still important because the calculator or any program can calculate but can not doing mathematics. 

Business mathematics, statistics and probability, and basic algebra will do fine. A good extent of geometry, trigonometry,  calculus, etc. are very important for those who plan to take STEM related careers.

Thank you very much

All of you answers are good. But, I think I need a more detailed answer

I would like to say that in the 20th century, two main part of mathematics education are core content and assessment. Meanwhile, in the 21st century, beside those matters, we have additional things to teach to our students, namely: thinking skills, ICT literacy, and others. It means that  since the allocated time to study is remain the same (no change on the time allocated for study), we need to reduce, eliminate or even delete many mathematical content for students to learn. 

Therefore, my point now is: What school mathematical topics should we maintain in the curriculum and what others should we eliminate? I am thinking of which mathematical topics is the best tool for preparing students to face the problems in the 21st century which require 4C's, and/or effective thinking. acting, relating, and also effective accomplishing. 

Thank you very much for your kind attention.

Best regards

I think we should maintain the essential topic like number and operation, Measurement, Geometry, Algebra, Probability and the most important is Mathematical skill.

we should eliminate the difficult and complex situation for students in some question that just show the memory or show how to use quick formula to solve. in some question have to solve by 2 - 4 step so some student feel boring. for one hour can do just 1 or 2 questions so that not suit for school math class. some topic suit for university level like calculus or complex trigonometry.

one thing is the national test that not relate with level of student and the government focus on the test not the feeling of student. I think the number of student in the class is very important to organize the happy thing in class if we would like teach the hard or complex question can not just show on the blackboard.

Thank you.

Thanks Supannika

I am going to go to a bit more detail now.

For example, a teacher asks his/her students to solve a question.

Determine the solution set of x^2 - 3x - 4 = 0

The students than answer the following

x^2 - 3 x - 4 = (x - 4) (x + 1) = 0

therefore x - 4 = 0 or x + 1 = 0

Hence x = 4 or x = -1\

So, its solution set is {-1,4}

Well... the students are able to follow the required procedures to solve the equation

He/She can produce the correct answer

But...  what does it mean for general person who don't want to be a mathematician? who don't want to work at STEM sectorr? What is the important for them?\

This is part of algebra, isn't it? 

I don't think this should not the focus of algebra learning for 21st century students.

The 21st century students have all the capabilities to find to solution of this type of equation, er even the more complicated problems, directly using digital tools. They are borned in digital era. They live with digital environment. They use digital tools most of their time. They will never try to use this mechanical way to solve it when they are already working. So, why should we keep train them to do this mechanical practices. 

However. I still believe that learning algebra is important. But, which algebra then? 

Thank you very much.

Regards

You cannot just remove topics from the Maths syllabus because the problem can now be solved by calculator.

For example, the purpose of teaching the solving of quadratic equations is to remind the student that the graph cuts the axis in up to two places.

That is, the hidden curriculum is for the student to realize that there in not necessarily a single one optimum answer to every problem.

I know that this is not written into the syllabus. It should be, otherwise too-hasty scissors may cut it out of the syllabus!

I repeat what I said earlier:

"The only purpose of teaching the solving of quadratic equations is to teach that there is not necessarily always one solution to a problem. There may be another equally valid solution lurking away somewhere!"

I totally agree with Dr. Ian Kennedy. In addition, basic skills should be taught and maintained but we should develop and improve the way of thinking and conclusion even with using advance technology to solve problems. Please refer to the link below.

Regards

SM Najim

http://www.sciepub.com/portal/downloads?doi=10.12691/education-4-2-1&filename=education-4-2-1.pdf

Dear Dr. Kennedy,

You are right ... we should not easily delete the mathematical contents just because of the digital tools availability to simplify any mathematical work. I also agree that we should encourage our students to find the meaning behind any mathematical works. I strongly agree with the Mr. Najim' message. We need to improve our way of using mathematical content to teach thinking. Teaching mathematics for thinking is one among the cores of any 21st century curriculum. We should use math content to teach thinking or other important skills.

But, since there will be no specific additional time provided in a week for teaching the thinking, we need to reduce or sorry to say, delete, several aspects of math topics. We won't have a chance to teach thinking if there are so many math topics to teach still to cover in the curriculum. 

I raise this issue since there are so many topics to learn in my country. Coverage of school mathematics contents are so broad. It seems that reduction or even elimination some materials is really a need. 

If so, then which ones. Shall we reduce the teaching of math procedures?

In my country, most of teachers explain about how to use procedures. They are happy if their students are fluent in implementing the procedures. Following the existing math procedures is mostly the focus of our teachers' teaching preferences. 

If we are going to teach math for thinking, actually we can still teach math procedures. But, we should not expose directly the procedures to students. Even, we should encourage students to discover their own procedures. But, it will take a very long time.

In addition

Not so many teachers consider teaching thinking as the main thing in mathematics teaching. Most of them assume that they just have to deliver what is written in the textbook and help the students to understand it. They usually assume that teaching thinking is only a theoretical concern. It they think that these theoretical matter will make them in trouble or having difficulty in implementing it, they will newer do it.  

From my experiences, the limited amount of time is frequently used as the argument for not teaching thinking. They say that the allocated time is inadequate for them to teach thinking. Even, they need additional time to cover all the materials.

So. If we want to teach our students to live in the 21st century successfully, we need to reorient the school mathematics curriculum. Isn't it? But, how?

Thanks so much for your kind attention. 

Regards

Dear Kolita

I strongly agree with your opinion that calculator or computer can ever replace the mathematics. But... to what extent we still teach the school math. According to Gagne (?), there are 4 mathematical objects, i.e.: (1) mathematical facts, (2) mathematical concepts, (3) mathematical principles/theorems, and (4) mathematical procedures. Which should be the focus for students learning now?