Mathematical problem solving will continue to be (as it is today) an essential skill in facing the peculiar challenges of the 21st century, especially with the potential developments in science & technology.

In more specific terms, I think that "procedural knowledge" or "knowing/creating/using strategies" along with the "creativity" and "affective factors" like perseverance will play a large role in facing technological challenges (such as creating new technologies and using them).

Thank you Henry!

But in some countries, mathematical problem solving is being left out of the classroom practices. In some cases, since the curriculum doesn't address it properly, teachers only use "teaching to the problem", that is, they teach specific contents to be used in specific problems later on.

How do you think a teacher can intertwine "knowledge", "strategies", "creativity" and "emotions" in the mathematics classroom activity?

I think the intertwining can be done in many different ways...but perhaps the real challenge lies on designing "problem-based activities" that would expose to students to a diverse set of strategies and opportunities to use them. In addition, we have also to consider how teachers can effectively encourage & support creative ideas and emotions attached to problem solving.

That sounds to me as a very complex curricular experience. =)

Hi Helia Jacinto & Henry Encabo,

I think one important step (or I could say a small step) is to integrate different fields into one field while attempting to TEACH problem solving skills in the later field.

i.e. while teaching problem solving skills in Physics & Mathematics, rather than bluntly asking a school student to find out the maximum vertical distance traveled by a PARTICLE in a Projectile motion, they could ask the Height the BALL travels that is kicked by Beckham, given the angle of the Kick.

Questions like these could be modified to fit into different culture and ONE IMPORTANT factor that must be inculcated in school students is the MULTI DISCIPLINARY Nature of today's problem solving requirements.

From my own experience, it was only during my engineering that I came to know that the main MOTIVATING factor for inventing CALCULUS is the HUMAN NEED to find out the AREA & VOLUME occupied by irregular shapes !

I hope that many schools, in many places, start a class in Calculus with some formula, without considering what Motivation would it give to students to develop a Self-Interest in learning them !!

In my view, the schools must make the Students feel the NECESSITY for each topic that is in a particular field, which made some one to invent/discover it.

So, UNLESS THESE MOTIVATING FACTORS behind each topic is experienced by the Students, it is very difficult to achieve required standard.

Hi Jacinto,

One aspect of learning that has very little research given to it is in the area of developing question asking skills for the LEARNER. The most important problem solving skill is asking the right question. Because no attention or value is given to this core learning skill right through schooling, students often are in a sate of high stress when they get to college and have to start asking their own questions and have not been helped to ask in all the ten or so years before. The average number of questions asked by school pupils is one a month. How skilful would one be at tennis for example if one only got the chance to hit a ball just once a month! One of the great blocks to student questions is the old one of "I did not want to look stupid!"

When a student asks a question, doesn't that mean they have engaged with the subject at their level of understanding? It should therefore be possible to assess a student on the questions they ask rather than their answers.

What do you think?

Laurence Smith, Independent Educator

I would offer three processes to consider. First, John Boyd developed a decision making theory called OODA Loops. OODA stands for observe orient decide and act. Browsing the web will yield several diagrams of this process. It is a wonderful way to solve problems. I have used it well in my career. However, teaching the skill is a daunting task in which I have been less successful. Second, The socratic method or the Harkness Table (Phillips Exeter Academy), predicated on student initiative, this process works well when led by a committed and practiced teacher. And finally, Blooms Taxonomy is a clear 6 position method to teaching better questioning. The last two depend on the student asking questions that develop a dialog that leads to a theory and the testing of theories.

By developing inquiry techniques, students are able to map problems in their mind or on paper and engage in discussion that achieves perspectives beyond the original problem. This should yield better problem solving skills.

Of these three suggestions, OODA loops resembles mathematical processes most noticeably, the other two apply the scientific method more so than a mathematical model.

I do agree with Henry Encabo's statement above. I too find perseverance a characteristic lacking in many high school students today. His suggestion ""procedural knowledge" or "knowing/creating/using strategies" " clinically define the three methods I propose here.

Helia, we are barely into the second decade of the 21st century. I sometimes think that while general phrases like "21 century skills" might be helpful in some contexts they are not always useful. The challenges we are facing now are likely to be very different within even the next decade let alone later in the century. It's also interesting to look at the variety of responses you already have to this question.

I know your secondary question focused on mathematical problem solving skills & I agree with Henry's response about the role of mathematics. But there's so many problems to be solved & so many ways to tackle them. Scientific & systems approaches will remain important but they are not the only tools to draw upon. Conundrums are important for learning & they can take many forms -- solutions to problems can be intellectualized & systematized but in the case of "zen buddhist koans" (seemingly irrational statements like "what is the sound of one hand clapping?") there are other cognitive pathways that "solve" such problems. And then there's thinking laterally, around or outside the problem.

Dear Helia,

Mathematical problem solving means solving problems using mathematical knowledge. So mathematical knowledge, especially well developed mathematical concepts, is the main factor in solving problems in which mathematics are applicable. How to develop such knowledge, is a different question. Using mathematical concepts one can perceive the situation in mathematical way, formulate a question in a mathematical way so the situation can be solved in a mathematical way.

However, mathematics is not all you need to solve different problems. For example one principal of an elementary school thinks that problem solving is about solving interpersonal conflicts between students. So this skill should be in the 21 century curriculum.

Elena

Yes, James, The Socratic method is good. One of the most exciting developments in education I saw in the last ten years was Philosophy with Children ( See also SAPERE or P4C). It was one of the few times while I was researching this subject of learner's questions, that I heard students asking questions that were really valued and developed.

I remember doing an OU course in Biology some years ago and we were told that we would be taught to think "scientifically". But of course, what we were taught was simply how to précis text in our own words. Several times I challenged the organisers of the course to say that I thought science was about asking questions and challenging authority and nowhere in that course did they allow for this. At the end of an experiment or investigation we should be encouraged to say what we think the next questions we ought to be looking at will be. Knowledge would then be seen as a ever opening subject and not a set of closed boxes.

Learning about question asking, I believe, is the most neglected subject in our education system. I have looked at hundreds of Education books and almost none of them have anything to say about developing question asking by students; if there is a chapter on questions it is always about how teachers can ask better questions.

The skill of asking is taken from our youngest pupils and they rarely recover it!

Hello Helia

this is a very interesting question but as Mr Mason noted below individuals nowadays do not only need mathematical problem solving skills to face constant challenges and problems. I suppose they need a large variety of problem solving skills in order to gain better ways of understanding and improving their lives. Due to the mass of complicated information we receive everyday from other people and the media, developing information processing skills should play a crucial role at the moment and in near future as well. Systems theories, chaos theory and happenstance theory could offer us a vast background to deal with uncertainty and multiplicity in the postmodern world in various fields such as science, human relationships, career and education among others of course. People have more and more to be capable of identifying and generating meaning from the Interrelationships among subsystems such as family, school, workplace and society in order to resolve problems which stem from these subsystems and affect the quality of their lives.

Furthermore, in my opinion decision making skills development play a vital role in the "21st century world" because everyday we have to take important decisions in relation to numerous kinds of problems which are most of the times complex and demanding. Mathematical problem solving skills are necessary but not adequate. Curiosity, persistence, emotional intelligence, robust self-efficacy beliefs, broad experience and several competencies such as communication skills and flexibility could help us to face effectively every challenge.

The graphic representation of data is a field that blends mathematics, language arts and fine arts, and lends itself to ICT implementation. Envisioning curricula in early education, adolescent, early adult, undergrad, postgrad -- the development of the literacies of interpreting and communicating the enormous amount of numerical and non-numerical data should be considered problem solving tools as essential as basic reading, writing, etc. Static charts, cartooning, computer graphics, representational visuals -- providing training in the skills of producing these objects as well as interpreting them are key problem-solving skills for 21st century challenges.

Hello everyone! Thank you so much for your contributions!

Let me clarify some issues that some of you have mentioned previously. I will try to condense my comments.

– I have not stated that mathematical problem solving (MPS) is the ONLY skill we need in order to face the 21st challenges (whatever they are). My questions are: is problem solving an important skill? If so, how is it that mathematical problem solving activity can contribute to the development of the general skill of solving problems of the 21st century?

– I can relate to the issue of “what are the 21st challenges? Are they real? Or are they any different from the 20th century challenges?” I believe they are real and different. We are now witnessing some changes around us… Check out these 10 new jobs that didn´t exist in the 20th century: http://www.forbes.com/sites/meghancasserly/2012/05/11/10-jobs-that-didnt-exist-10-years-ago/. As teachers, are we preparing our students to enrol in one of them? Or to create new jobs? How is it that “teaching mathematical concepts to solve classroom mathematical problems”, help?

– Now, mathematical problem solving isn’t only about using “mathematical concepts” to solve “scholar mathematical problems”. In fact, there is a tremendous amount of research on MPS. Many researchers have stated that it is possible (and desirable) to teach mathematics THROUGH problem solving, that is, the mathematical concepts or procedures are an outcome of the PS activity itself (look for the Models and Modeling Perspective, for instance). On the other hand, there is no doubt that many people in many different jobs, besides mathematicians, solve problems that require some sort of “mathematical thinking”. See, for instance, the studies from R. Noss, C. Hoyles and colleagues, focused on the mathematics used in several workplace contexts.

– Finally, I agree with Laurence: posing questions is an interesting activity that can lead to more inventive and innovative activities! And that can be done in the mathematics classroom! In fact, there are maths education researchers claiming the importance of developing the ability of “posing problems”. So, I have a new question: how can we teach our students to look at the 21st century society and pose interesting problems (mathematical or not), which are worth solving?

A cornerstone to the Socratic method is to definitions. " which problems are worth solving" may not be the first question. "what criterion will define our "problem"?" might be asked earlier on. Each classroom or micro community may have a different approach to defining this question.

Ask questions like what is our gaol? Is it to define a global problem in which MPS can be used? Or is it to tailor a problem asking for help to solve needs in an emergency? The earth quake at Haiti was the target that produced a student designed portable shelter with water filtration, a light and was compressible enough to ship by air several shelters on one pallet. A high school student MIT project. It required many disciplines to define aspects from weight verse strength verse durability rations.

Provide a definition and the next discussion will offer both questions and answers until your problem is fleshed out.

@ Professor Latchanna - Contrary in certain countries where I have exposed myself to teaching and learning, my experience leads me to interpret that the youth and adults in schools now (21st century) have difficulty facing challenges in their lives. Many students whom I have interacted think simplistically and, as my friend often comments, are ONE LINERS, because they have been trained to solve problems through multiple-choice questions, true or false, and simple ONE-LINE answer to questions. Many of them are followers and tend to follow the joneses. For instance, when the nursing occupation was a boom, they jumped in to be trained as nurse. I have known of even doctors and accountants who submit themselves to this unrealistic transformation to be nurse without realizing that bird (e.g., accountant) cannot swim and fish (e.g., nurse) cannot fly. What I am emphasizing here is CONTEXT.

I agree with you that there is a dire need to change the educational systems and likely the need for radical change is urgent in all countries. Thus, this question is timely because the 21st century challenge demands integrated 21st century skills.

Thank you for your contribution to this forum and for the opportunity to think and to share my opinion to dissent. :-)

Interesting discussion. I think that all of us learn better when we get to ask questions and seek the answer. Seeking the new is what makes learning universally enjoyable according to Aristotle, and nothing has changed since then except that we know about serotonin. There are ways to create a canned experience that is still meaningful and interesting though, and teachers have to know these tricks. Like presenting a seeming impossibility, so that we can stimulate student questioning along the lines we desire, which allows us to structure the curriculum to the ends we want or need to, yet maintain the student voice and experience.

I think we should call it the Aspasian method though.

Though not directly related, I find persistence to be a significant issue with Critical Thinking as an aspect of 21 Century Skills. Students tend to migrate from one idea to another without fully developing the original concept. This directly affects asking follow up questions and solving for solutions requiring additional calculations. John Dewey suggested we create skepticism to keep the learner asking questions and searching for answers. In modern digitally oriented learners, I think persistence is inhibiting the process of "drilling" into a concept with quality student originated inquiry.

Hi David, your last sentence seems to contradict what you said before . Can you amplify it further perhaps with example?

Laurence,

I consider persistence an issue because it is absent. No persistence equals surface analysis only hence poor interrogatives. I think students need to develop a self regulation level that is more persistent than is the present case.

Yes, that is a good point. The next thing then is how to encourage , develop and give value to PERSISTENCE from an early age so that it becomes part of the habit of study. Many of the great discoveries have been made by people who have had "super stickability."

Learning to think logically, to recognize patterns and relationships in content. Learning to set goals. Learning to finish something, even when the execution of the study or project seems to be more 'boring' than it's beginning. Learning to plan and manage one's own projects. This suggests that doing homework and finishing group and independent projects on time is something that should be learned in the primary school years.

Developing question asking according to Lawrence above emerges from home. Some homes teach children to be silent as in one little town in my country. People are "good" workers because they do their jobs no questions asked. Just the town next door is totally different. People and students show their discontent and ask questions if they need to. I thus believe that asking the right question, at the right time, to the right person, in the right manner (Mey, 1996) is a skill that emerges from home just as emergent literacy helps children become better equipped learners.

@ Jose Lobo - You are right. Developing students to ask the right question is important. Knowing to ask the right question is a essential skill.

I would consider hypothesis testing an increasingly important skill (systematically explicating one's assumptions on how things work and why they don't; as well as adapting them to informative feedback) in the 21st century.

I agree with the above comments, but I also think that being able to effectively search for and evaluate information is a key skill for the 21st Century. In my view, the largest change in the last 20 years is the shift from knowing where to find information, to knowing how to filter information. Since the explosion of the Internet, there is more information available on any subject that we could ever digest. The skill is in evaluating the quality, relevance and validity of that information.

Some considerations are:

1- Creative capacity, self-employment, entrepreneurship and conditions

to adapt to emerging situations

2- Developing the potential to be constantly updated

3- Skills to work in interdisciplinary groups, including communicational potential of handling tools computer and the domain of both the home language and other languages universal

4- Skills to identify, access and use relevant information in the moment.

I am wondering how much the concept of problem solving and active questioning becomes a broader issue in terms of control as well as expectation within the culture of a learner.

In many environments the ability to ask a question is limited by the freedom to do it. It strikes me that a Freirean approach, linked to communities of enquiry , Lave and Wenger and perhaps SOLES and SOMES might afford a process through which the language and freedom to question might evolve.

Continuing contributions, it is important to note that some critics, such as the group Common Core (http://www.commoncore.org/), are in favor of greater importance of content and a more comprehensive curriculum that letters of teaching critical thinking skills and learning to learn. The main argument of this approach to teaching is that, although such skills are very important, can not be taught independently and can not be outside of a specific field of knowledge those designed as traditional academic subjects, nor students are able to apply these skills if they lack knowledge appropriate factual about a particular field of study.

From another perspective quite different also been argued that, although the term competence is valuable to guide how it should be teaching and learning in the classroom, this is an expression that comes from the world of businesses and companies. The rhetoric of the XXI century skills is conceived in many cases as another side of an approach to education economistic according to their main goal: to prepare workers for highly qualified knowledge economies or in some cases, even toindividual companies.

Instead of putting the emphasis on the harmonious development of all abilities human discourse on the powers exaggerate the significance of the skills related to the workplace. In addition, some argue that such and how they are defined, not all students will achieve skills century, because first of all not all young people of today will be workers tomorrow's highly skilled even in developed countries, and in Second, this rhetoric forget the needs of the vast majority of countries of the world.

STEM ...! Science, Technology, Engeneering, MATHEMATICS !

No less, No more !

I would say, to be a good problem solver, he/she requires (i) the knowledge, (ii) the process, and (iii) the confidence. With that he/she can face any challenges.

Great comments. As Samantha says, there is information overload and instantaneous access to information. Students are aware of the latest in any domain.

They are more inquisitive than those students who studied in 1980-90s. They are experimenters. They are technology handlers and application focused. In case we match their imagination and questioning the class room experience would be great.

You will never know the economy if you do not know mathematics.

Dear Helia Jacinto, In my opinion the problem solving skills required to face the challenges of 21st century are:

- Critical thinking,

- Problem solving,

- Communicating,

- Collaborating,

- innovating and

- Self-direction & orientation

To successfully integrate problem solving into teacher education programs, it is vital to first seek to categorize these problems and determine the cognitive skills and conditions needed to effectively solve them.

Is this a 21st century skilled kid?

;)

https://www.youtube.com/watch?v=Uc4CD1aEFwE

Dear Ms Helia Jacinto,

Thanks for sharing nice and informative link.

Hola Helia :)

This is an excellent site! I hope you will find it helpful :)

http://galileo.org/teachers/designing-learning/resources/inquiry-and-assessment/

http://www.galileo.org/tips/rubrics/math_ps_rubric.pdf

Thanks Lorraine! 

 At a first glance it looks quite interesting and useful, especially the maths PS rubric! 

Regards,

Hélia