Universal Children’s Day

It was the 30th of July 2017 when I woke up to the beautiful sun streaming through the huge corner window – 6 a.m. in Lahore Pakistan. I gazed through the window. On the other side of the adjacent 7-foot brick wall, topped with barbed wire, were children playing in an empty lot. There was a boy and a girl who looked like they were 8 years old. They walked out onto the rough land, hand in hand laughing, when another little girl showed up – maybe their sister? She looked about 3 years old. No shoes. They all wanted to play – they found some garbage – some plastic – then together started playing with it. They put it onto the ground, crouched over it – oops! It flew away! They then found another piece of material. As I sat mesmerized with this scene I thought about the children who are “blessed” with being born into a wealthy family, who are constantly being stimulated with TV, the computer, or some other technology – would they be able to keep themselves occupied like these children? Would they be able to find happiness in such seemingly poor conditions?

Children want to play with one another, to appreciate a friendship and to share their experiences. Children living in similar conditions described above reside throughout the world; not just in the emerging or underdeveloped worlds. In Canada, children of all economic levels sit side-by-side in their classes. They quickly learn that all peers help to improve their own understanding of the world, and that they need each other for growth and enjoyment. Often the greater the disparity, the more the children learn from each other. Children who have very little material objects in their lives, can be stimulated in a different way, which could develop a uniquely different ingenuity.  When children are given a chance to work as a diverse team to solve a question that initially looks like it is impossible to solve, humility is learned, and so is the acceptance of others.

In December 1954, the General Assembly of the United Nations declared that all countries observe Universal Children’s Day on November 20th to remember the adoption of the Declaration of the Rights of the Child. This declaration sets out children’s rights to life, health, education, to play, to family life, and to be protected from violence, not to be discriminated and to have their views heard. The UN asked all countries to observe this as a day of worldwide fraternity and understanding between children.

We often work in “bubbles” of similar people and expose our children to the same “bubbles”. During the month of November, I encourage everyone to take one day to observe as Universal Children’s day, and to share it with their children by venturing outside the “bubbles”

Let’s work together to increase the understanding between children globally.

Reaching Out

The last time I went to Pakistan I brought a special flask that could keep cold liquids cold and hot liquids hot for many hours. I thought this would be the perfect gift for our driver, Afzal, who would, at times, sit for several hours waiting for us to finish our work. Afzal accepted the gift with a great big smile and grateful bow – he couldn’t speak English, and I couldn’t speak Urdu, so this communication style worked for both of us. A couple of days after the gift was given to him, Afzal told Zorain, our translator, that the flask was empty. Zorain told him, “yes” that is right. “But why would someone want to give me something that is empty?” replied our driver. Zorain explained to him that it wasn’t what was in the flask that was the gift: it was the flask itself. This discussion ensued for a while, with Zorain trying to link the ideas and values of one group of people with another. This is an example of the extreme dichotomy of our world. What one person may believe is the right way to think, and therefore how to understand this world, is often dramatically different than another. The reality is that all of us grew up differently and therefore understand the world differently. To understand from another perspective is very complex and critically important.

The focus of our most recent conference for all those who work with Spirit of Math was “Reaching Out”, with special attention to how one understands people from different cultures. In Spirit of Math, our cultural diversity exists within employees and students. Understanding others, and learning to understand how those from very different worlds understand “how to understand this world”, is extremely important: not only to work well with one another, but also to teach others.

As a parent who is looking to ensure that their child is educated as well as possible for the future, I believe that learning how to work with others is one of the most important skills for a child to learn. For our children to be able to be excellent problem solvers of tomorrow, and our future leaders, they will have to understand this world in multiple ways. They will need to see this world from a variety of perspectives. Providing opportunities to learn in an environment that allows students to share their ideas, and to listen to the ideas of others, will teach them to think and reason from different viewpoints. Providing them with problems that require the input and discussion from many team members will help them understand the value of others, and the value of different perspectives regarding how to think of a problem – not just how to think through the problem. This is what we are doing in Spirit of Math with our cooperative group work, and this is why we don’t just do individualized tutoring.

Reaching out isn’t only a one-way transaction: it requires a multiplicity of understanding that can be profoundly different than the way we initially thought. Working with our students, to be open to a new understanding, is opening the world to new thinking, which will generate original solutions to the biggest problems of our world.

When life presents opportunity

At certain times in your life there are people you meet, or events that happen that change your life forever. Our visit to Pakistan in November 2015 was one where my two adult children, Nathan and Kirsti, and I attended a spectacular wedding. Not only was it breathtaking with the dresses, the food, the music, the dancing, but also the focus on people and excellence. We quickly realized that we were in a land with many very bright people with immense global experience and a perspective of the world that embraces an outstanding understanding of the West and the East. It became clear very quickly that as foreigners we stood out; because of the one-sided reporting of the country, it is a country that is understood very little by the west, and therefore, almost ignored by international tourists and business. Yet, despite this, the passion and drive of the Pakistanis is creating a massive change. It was this passion for progression and excellence that intrigued me, and has resulted in Spirit of Math launching classes in three locations in Lahore, Pakistan this past August.

During our visits over the past year and a half, we have met significant changemakers, participated in a national debate, spoke at several conferences, provided workshops for parents, and trained teachers in mathematics education. One of the highlights of one of my trips was a visit to Ala-u-Din Academy, a girls school in a very poor area of the city. The building was a family home, transformed to classrooms and administrative offices. The family donated the building, and several generations later, the females of the family still run the school. The picture on the front of this NewsMagazine shows the girls and boys in the courtyard of the school. It costs approximately $60 US dollars a year to attend this school, and yet a scholarship fund had to be created because some of these students’ parents are not able to pay for this. The boys who attend are brothers or cousins in the same home, and can only do so up to Grade 5. I spoke to a room packed with girls in Grade 10 about being a female CEO, and how believing in their inner-genius and working hard to strengthen their skills and knowledge, can change their world. It was clear that they were ready to take on the world; they have already overcome extreme hurdles just to go to school, imagine what they will be able to do in their future. This is just one example of the many incredible well executed initiatives that one family has taken.

We are excited to announce that Spirit of Math is now in Pakistan! Classes are running with the same program as in Canada and in the USA. With after-school classes in three different schools in Lahore: City School Alpha Phase VI, SICAS Kids Kampus Gulberg and Kids Kampus Johar Town. My daughter, Kirsti Langen, has taken the role of Academic Director, and Zorain Masood from Lahore, is the Operations Manager. This is just the beginning of our adventure there. The requests for Spirit of Math’s classes and teacher training is immense, the support is incredible, and the opportunities to make significant changes at all levels is exciting and enormous.

Giving back in little ways, can leave a big impact

In June, as another school year came to a close I found myself watching again with pride, as our latest group of Spirit of Math graduates and assistant teachers stepped forward during our graduation to face the next phase of their lives. Like so many SoM alumni before them, they continue to demonstrate incredible enthusiasm, eager to show the world what they can do. It is inspiring to hear what our graduates have already done in their lives, and encouraging to know that they are going to be our future leaders.

With such big aspirations, and with the huge potential of our alumni, we often expect huge successes, and those huge successes are often the only ones recognized. But, as I said during the recent graduation, it is not the huge successes, and the big rewards that make lasting effects: it is the little things along the way that make the most impact. When you add up one little successful impact after another, you will have made a big impact, and that will change the world.

This reminded me of a personal story of a small gesture made by a very old man that influenced my life in a significant way. This happened in my childhood and although it was very small, it left a lifelong effect. When I was just six-years-old, I was with my father in a small village in Uganda; as my father walked ahead to meet with the village leaders, I saw an elderly man sitting in the middle of a mud hut leaning over a small fire, cooking his meal. He gestured for me to join him, so I did. He was cooking a modest meal to feed his family. The food he was cooking would have been all his family would eat that day, nevertheless, with a smile on his face, and joy in his eyes, he gestured for me to join him and offered the food to me. We could not communicate verbally, but his face said it all: he was as proud as he could be to offer all he had to a little girl. I was a stranger, and this man happily offered me what little he had…because it was the right thing to do. This small gesture taught me the importance of giving from within oneself, no matter how little you have to offer.

Graduates, I would like you to remember that you have a huge amount to give but don’t just focus on the huge successes: it’s the little things that will add up and make a big difference. As individuals you will change the lives of others, and as a group you will change this world.

I wish all of our students and their parents, a very safe and happy summer 2017!

 

Getting into the Brightest of Brains

How do you “get into” the brains of those students who are thinking at a higher level? Once you know, how do you then transfer these thinking patterns to other students so that they can also think the same way?

 

When developing the initial Spirit of Math program, Charles wished to expose the students to as many different problems as possible. He then proceeded to provide answers according to his background and his adult understanding. This is when he realized that the students did not think like he did and did not have the knowledge base to access so that they could use his solutions. Having recognized that this was a rare opportunity to look at how very bright and successful students think, Charles closely watched and listened to those students who were solving questions with ease. He documented their methods, and with Fraser Simpson developed the Problem of the Day book. The solutions to any of the problem-solving assignments have since been developed by watching, listening and questioning bright students, using approaches that encourage their thinking processes. The material in the primary grades has also been developed in the same manner, so that younger students will know how reason mathematically when they get older.

 

When trying to develop new material, better thinking processes and a strong skill set, we have found that listening to kids as they talk and explain their thoughts in a peer-to-peer group setting is the best way to come up with effective material. Students will often think of looking at a problem in a very unique way. Often their explanations will be more efficient than our “adult” methods. Rather than interpreting problems through our mature eyes, we often will find the “jewels” when looking at the problems through the students’ eyes.

 

As parents and teachers, we often think that we have the best way to solve a problem. However, students often relate best to others their own age because their understanding is at the same level. A very effective way to change the way students will approach problems, therefore, is to listen to the brighter students, and to teach others to solve problems in the same way. All students can learn how to think differently; how to think like the brightest.

 

When developing our program it was essential that we paid attention to the brightest students, rather than thinking that we knew all the answers. As a parent, make sure that you take time to listen to your child. Listen to their talk. Listen to their explanations. Look at the pictures they are drawing to explain themselves. Work with these and try to understand them rather than insisting that they first listen to you.

Do you know what is available for your child?

Searching for the right educational program for your child can be the most complicated decision you will make with some of the most far-reaching consequences. These consequences can be extremely positive if your decision is the right one for your child.

Understanding how your child learns, solves problems and makes decisions is your responsibility and an educator’s job. It is also a parent’s job to ensure that their child is provided with proper education that will train a brain to work efficiently and effectively. You want your child to learn the skills needed to open doors to their future. A parent is thus responsible for finding the right program and effective educators. A strong marriage of the two – a solid program with good teaching, will give the greatest benefit to your son or daughter.

Let’s first examine the teaching. This starts with the teachers. One of the key ingredients to a good teacher is their expectations of students. If they expect students to do well, the teacher’s perceptions of the students’ abilities rises and the children’s own self-image is enhanced. You want to find a teacher or set of teachers who work together to build a positive environment that does not limit a child’s intellectual growth. Perceptions influence the way a brain works. If a child is told that he/she is very smart and can learn easily then the child will more likely respond appropriately. If a person has a negative perception of themselves then they will also act it out. Look for a teaching environment where students are encouraged to perform at peak efficiency, and expect positive results.

Content and how it encourages different ways of learning is a key to a good program. In Spirit of Math the content was chosen to expose students to different ways of learning. Charles Ledger (2004) stated this as follows: “… content must exemplify ways of learning, for these are more important than knowledge about any particular subject of investigation. Knowledge of methods makes it possible for a person to continue learning, and to undertake inquiries on their own. Every discipline has its own methods that allow for the simplification of learning. A key element in the Spirit of Math approach is the emphasis on group, or team, work – developing an ability to work with others, an understanding of the great advantages available through a team effort, building up respect for others’ ability and a realistic understanding of one’s own strengths and weaknesses.”

Finally, materials must help a teacher impart a sense of the extraordinary and surprising so that learning becomes a continuous adventure. An imaginative use of materials generates habits of thought that will enable a student to respond to changes in knowledge and belief with zest instead of dismay. Many young people have lost their fear of mathematics, and have found that they approach the learning of mathematics with excitement simply because they have mastered the drills! Students develop a sense of meaning and value simply from being challenged in appealing ways.

When looking for the best program for your son or daughter, do your research:

1. Listen to others.

2. Find out what other concerned parents are doing.

3. Look at the progress of those kids in the program. Watch the children who go to various programs. Are those kids succeeding? How do they conduct themselves around adults and around other children? How are they doing at school? Have their parents noticed differences in them since beginning the program? What types of differences? Is it that their child is getting better marks, or is the child both getting better marks and also able to apply their knowledge in other areas too? Do they have a better understanding? Are they self-confident, motivated, etc.?

4. Look at the kids in the program. Your child is going to be spending time with these children in the classroom and associating themselves with them. They learn not only the material taught in class, but how to act, what is important in life, and what values others hold.

5. Examine the people who will be working with your child. Do they exemplify a person who can be a good role model?

6. Don’t just get caught up with the “in thing” as it might not be right for your son or daughter.

7. Read educational columns and get to know those journalists who aren’t frightened of stating their fundamental beliefs.

8. Ask good educational authorities. Apparently someone at OISE (Ontario Institute for Studies in Education) has been recommending Spirit of Math. You might want to ask an educator there for further information regarding other programs.

Become knowledgeable about what educational programs are available and how they have been developed. Ask the following questions:

Do they have a proven record?

Who created it and how was it created?

Who is teaching it? What is the training required for the teachers?

Persistence, Attention to Detail, “Sticking with It”, and The Discipline to Follow-Through

Bright kids who are used to getting answers quickly, and not having to work very hard at questions can develop habits that encourage speed, and the “easy way out”. Somehow, students must change those habits so that they realize that speed is not necessarily the critical route to success. Bright students must be given questions that force them to slow down, and to look more closely at their work. If they were ever in a situation, in their future, where they had a large amount of data to analyze, such as financial statements, scientific research, medical results, or even legal matters, they must have the skills necessary to be able to look at the details. These skills include having the patience to painstakingly go through what appear to be “non-consequential” numbers.

When sitting with a student a couple of weeks ago, going over the grade 6 material, she told me that she understood how to do the work on rationals, but kept making “little” mistakes, such as missing the negative sign. That negative sign changed the whole answer. She was feeling a little frustrated because it was those “little” mistakes that brought her mark down, even though she felt that she understood the concept. What she was missing was some more basic skills required for this course.

The skills needed for the grade 6 course are very similar to those needed in much of the Spirit of Math program. For example, as students progress through the grade 5 Order of Operations unit, one question, which is at least 10 steps long, is only worth 1 mark, and to get that mark, EVERYTHING in the question must be perfect – not just the answer. The procedures in the process are crucial, and are just as crucial as is the ability to find their mistake, if they made one. Students are held accountable to find the mistake by comparing their complex answer with others in their group. If they decided to do the question their own way, then they will very likely have got it wrong. They quickly learn that having a consistent method makes a task easier and much more efficient in the long run – that sometimes just following procedures in a certain way just works much better than being creative. There is a time to follow procedures and a time to be creative.

Students these days are inundated with technologies that have been specifically created to satisfy an immediate demand. Technology is fast, simple to use, stimulates a quick “good feeling”, and can be turned off when it gets frustrating. I feel that media has gone overboard with their promotion of quick fixes, and that marketing is so sophisticated now, that it is tough not to get “sucked in”. This sincerely concerns me because it is much harder now for parents to justify to their kids that hard work, patience, discipline and persistence are necessary skills. I strongly urge you to look closely at our curriculum, and look beyond the surface. Students who are able to succeed well in our program are those who work hard, and have the patience, discipline and persistence to do the questions. Even if your child doesn’t master the concepts behind all the math, they will have developed very valuable skills that are tough to instil. And, yes, it does take some work.

What are some of the skills we want students to learn to value?

 

1. Patience.

 

2. Discipline to stick with a problem until it is properly completed.

 

3. Learning how to follow procedures.

 

4. The discipline to work hard to meet a standard; paying attention to the details and procedures.

 

5. The discipline to double-check your work, and to take it one more step: to go back and correct your work. It is OK to have a wrong answer. It is not OK to let that wrong answer stay there, if you know it is there, and if you have a chance to correct it.

 

6. Ability to talk to others in a meaningful way, to help others with their work.

 

7. Ability to find out where you went wrong, if you did.

Challenge is Not a Four Letter Word

“The harder your brain is obliged to work, the greater will be its capacity for work.” Lewis, D., Greene, J., Thinking Better

Explore how much of a challenge your child is able to accept. We all have the innate desire to be challenged and conquering a good challenge delights us.

Everyone likes to do something easy, but only for a short period of time. Think of the last time you had something challenging and you succeeded in conquering it. You probably felt a great sense of satisfaction. It may have been very frustrating going through the process, but once you did it, WOW! You likely felt good about yourself and realized that you could do something you didn’t think you could do. Your self-confidence skyrocketed, even if only for a short period of time. You felt that you were now armed and equipped to do more.

Why is it then that people believe that giving a child something easy to do is the key to improving the child’s self-confidence? An easy task may be a start, but the effects short-lived. This “easy” idea has permeated the education system so much so now, that educators are frightened to challenge students. Programs have been developed based on this premise, and in the US alone, billions of dollars have been spent focusing on the problems of the weakest kids, rather than on programs designed to challenge them.

I believe that as a society as a whole, we CANNOT be scared to challenge others. If it is our intention to help a person improve his intelligence, or to prepare him/her for their future challenges, then isn’t it our responsibility to challenge them when they are young?

How can you, as a parent, help your child meet a challenge with “zest”?

First, a child needs to learn how to approach a challenging situation or problem. They need to know how to mentally deal with something that they think cannot be done. By going through a difficult process and experiencing success, they will develop better self-esteem. They won’t be as frightened of a tough situation or problem the next time it comes to them. Teach them that there are always solutions to problems, even if the problem doesn’t look possible, or the challenge appears to be way beyond what they think is possible. This is when your attitude “kicks” in. If you as a parent approach your own challenges with a positive attitude, believing in yourself and believing that you can do it, then your kids will see this and react in the same way when they are in similar situations. They also need to see that you can’t always solve all your problems on your own, and that often others are needed to solve the problem too. Let them hear how you ask questions and work with others in a positive way.

Second:  a student needs to learn the basic “question solving” skills. First teach them to look for the right question(s) to ask. They need to learn techniques on how to approach a challenge and be armed with a set of skills they can refer back to, so that they can find the right question. Once the question is known then their other skills will help them. A strong skill set in this area is transferable to many types of situations. If kids are given opportunities to use their problem-solving skills in a variety of combinations, then they will learn how to approach any type of new challenge without fear. Another fact that becomes apparent to them very quickly is that they won’t be able to solve all their problems/questions on their own, and that is OK. They will learn that they will need to work with others, and that working with others, as a team member, is crucial to success. Learning to depend on others at times, and learning that others need to depend on you too is important. We are not alone in this world and it isn’t just about “me”.

Third:  the fear of a challenge. If the challenge seems too great, many of us will stop in our tracks and freeze, or turn around and go back; or worse, just ignore or avoid the situation. You need to set realistic expectations for your child, but make them tough enough that your child will be stretched just a bit. I believe that God gives us challenges in life so that we are the best person that we can be. In the classroom, and at home, it is important that we continue to challenge children.

Our main focus with math problems at Spirit of Math, is not just to improve math skills. We want kids to develop the skills necessary to deal with difficult situations. Using math as the means to do this is very effective. Learning how to get kids to work on their own and with their peers without telling them how to do everything is our teachers’ challenge.

Your expectations and your attitude are crucial. Challenging kids takes courage, self-confidence, and confidence in your kids’ abilities.

Surrounded by Brilliance – Just Imagine the Possibilities

Building a Strong Mathematical Mind – Strand 4: Co-operative Group Work

Imagine having the opportunity to be in a room of brilliant people, not just for one day, but week after week. Imagine that, while in that room, you were able to listen to those brilliant people: to their logic; to their thoughts; and to their thinking. Also, imagine that you were able to share your own thoughts, getting immediate feedback from those people. In addition, imagine that you were able to ask them to help you with problems that you just couldn’t seem to get through. Wouldn’t that be exciting?

Now, imagine your children having the same opportunity, once a week for an entire year, and that even more is happening in that room: they are being taken care of and guided through the process while there. This is the magic of a Spirit of Math class.

When a group of high performing students who enjoy work like it is candy to their minds, and who thrive on challenging questions, get together they just can’t help but to get excited when they discuss intriguing problems. Their minds are stimulated by the interaction. The challenges they pose each other not only feeds their enthusiasm, but change the way they think –  for the better. Who wouldn’t want their bright child to be in such an interactive environment?

Mathematics has often been taught as an independent subject, and unfortunately, one often thought of  for people who want to keep to themselves. Because many very good math problems can take time to solve, math can take a person to a place where they will tend to isolate themselves. However, that’s not where it has to end. Working through difficult problems starts first by struggling through them on your own; but after a little while, you will need to ask someone else for some help. Learning to use the ideas of other bright people will generate an explosion of new ways to think, and consolidate well thought-out, logical and realistic ideas.

When a student writes their ideas on a piece of paper, they are able to view their ideas from afar; they have the opportunity to look at their thoughts in a more objective manner. The same happens when they have the opportunity to talk out loud: they are able to sort out their ideas in their minds even faster than writing it down. This allows others to then accept or challenge their ideas. By having their thoughts questioned by others, students are then able to change or consolidate their logic and their thinking.

Cooperative group work and presentation skills are stressed in the Spirit of Math class, not only for their essential life-skills benefits, but because teamwork is often essential for a student to get the solution to a problem. Through teamwork, students develop the ability to work effectively with others, learning from their peers, and learning how to make helpful contributions to group learning.

Cooperative learning is an integral part of the Spirit of Math approach. Our experience has shown that when we present mathematics at an appropriately complex and thought-provoking level, students naturally want to analyze their work and discuss ideas with one another. New understandings emerge and new ideas are generated more effectively.

Tutoring is good for students who have specific issues that need to be addressed. Students who come to Spirit of Math are looking to improve and stretch in all areas, and that is why a classroom situation is much more appropriate and exciting for them.

Spirit of Math is a school designed to meet the needs of high performing students. Students who want to excel past the capabilities of the regular day-school need to be in an environment where they are given material that necessitates discussion. Our students are given some very challenging material, especially in grades 5 and up. To be successful in this program they must learn the skills involved with working as a team. To the extent that they succeed, they join the top students in their nation.

Problem Solving – Strand 3 of Spirit of Math

“Problem Solving” Oh, those words! So many people shy away from me when I suggest that they do some math problems. Many other people are encouraged that there is a program that actually incorporates proper, intense problem solving, but they can often be afraid of doing the problems themselves. What is it that they are not comfortable with?

It is important to note that problem solving in math is not just putting words to a basic math question. A good math problem will test the conceptual understanding of an idea or topic, or encourage a student to think differently and read carefully.

Here are a couple of problems.

Question #1:  Grade 1

There are 2 shapes: a circle and rectangle. One of them is not a rectangle, what is the other one?

Common wrong answer: circle.         Correct answer: rectangle.

A large part of being able to do math problems is in the language. One must determine what the question is asking, then to logically organize the material to create a strategy to answer the question. This “other one” question does just that. Students, (and many of their parents), will want to automatically say the other one is the circle, without thinking about the problem. Students are given a chance to come up with a strategy to solve this, and most of them realize fairly quickly that if you point to the shape that is not the rectangle, (the circle), then the other one is a rectangle.

Question #2:  Grade 3

Brian Bunny is reading a book called “How to Grow Your Own Carrots”. If he starts reading at the top of page 8 and he reads to the bottom of page 20, how many pages will he have read in all?

Common wrong answer:  20 – 8 = 12.  Correct answer:  20 – 7 = 13.

The key to this question is what is not stated in the question. What was missing in the question, but implied, is the number of pages NOT read. If Brian read all 20 pages, except the first 7, then he read 13 pages. In many cases, it is not what is written in the question, but rather that information which is not written that is the key to the question. Good problem solvers are able to see this. You want your child to look for a complete picture of the problem, and not just to always look for a solution that includes only that information given in the question.

Question #3:  Grade 7

How many numbers are there in the following arithmetic series?

8 + 9 + 10 + … + 19 + 20

Notice the similarity to question #2. Instead of pages in a book (a concrete concept), this question has numbers. Again, the answer is (20 numbers) – (7 numbers that are not in the series) = 13. This uses the same concept as in question #2, but it just looks different.

Question #4: Grade 4

It takes 30 seconds for a clock to strike 6 o’clock. How long does it take the same clock to strike 11 o’clock? (Assume the strikes are instantaneous)

Common wrong answer:  55 seconds. Correct answer:  60 seconds.

Again, we want students to think about what is really happening. If the strikes are instantaneous, then the time period, or interval, between each strike is what is important. If there are 6 strikes, then there will be 5 intervals. The complete time for the intervals is 30 seconds. Each interval therefore takes 30 ÷ 5 = 6 seconds. When the clock strikes 11 o’clock, there are 10 intervals, each taking 6 seconds, for a total of 60 seconds. There are several processes that have to happen in a student’s mind to come up with this answer. Often we encourage students to draw out a picture illustrating what is happening so that there is a proper understanding.

One last note:  stay away from introducing algebra until students learn how to think divergently with these types of problems! Many parents have come to me, very proudly announcing that their child has learned algebra at a very young age. I strongly urge you to be careful:  once students learn the linear thinking required for algebra, it is very difficult to get them to think divergently. Algebra is relatively easy to learn, compared to the thinking required for problem solving.

In every grade in Spirit of Math there are at least 400 problems that students must answer each year. The problems are presented to students as the Problem of the Day (POW) for the younger students, and progress to larger assignments. These assignments provide for research and experimentation with numbers, but are largely aimed at developing both problem solving and relationship skills. In various ways, students are encouraged to share their insights and understandings without just giving away answers.