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.

Building a Strong Mathematical Mind – Strand 2: Core Concepts

Give Bright Students Concepts they can “Sink their Teeth Into”

How often has your child come home from school with math homework involving memorizing steps and repeating them ad nauseam without really understanding what is going on? Even worse: do you have a bright child who has been turned off math because they are bored with the procedures, and find it meaningless? Or, have you ever heard your child say, “this is the way you do this kind of question” – doing what they were told, not because it was what they understood, but what they were told to do?

Math programs with a primary focus on procedures not only turns people off mathematics, but also develops people who learn how to follow without questioning.  On the other hand, a strong math program, such as Spirit of Math, that focuses on a solid understanding will produce creative, critical thinkers who are excited to learn more.

As explained in the last article in this series, the first of the Four Strands in a strong mathematics program, is the drill component. The purpose of the drill component is to develop an automaticity and fluency of number facts.

This week we take a look at the second strand: core concepts, or core material. This encompasses the different topics in math in order to develop an understanding of mathematics concepts.

Finding a program with good core material is tough, but essential – especially if you have a child who is very bright. The program must thoroughly teach the basic concepts and teach the student how to think better.

These pointers will help you evaluate the core of a math program and determine if it will meet the needs of your high-performing child.

  1. 1.     The program offers questions significantly different than the regular textbook type of questions.

Giving a high performing student a textbook a grade or two above their regular grade does not address their needs. The questions in a regular textbook, no matter what level, have been designed to meet the needs of the majority of students. The thinking required to do most of the questions is not to the depth needed for a high-performing student, especially in the younger grades.

  1. 2.     The core material goes deeper and broader in each topic.

High performing students crave something that they can “dig their teeth into.” Teaching a topic just to say that it has been covered is not sufficient. They must go deeper into topics and be given the opportunity to explore and discover the more complex ideas within them. This means taking more time on each topic. A strong math program develops concepts slowly and intently so that one idea can be linked directly to another.

  1. 3.     The program should expose students to concepts not normally seen in school.

There are so many very interesting ideas in math that just simply cannot be introduced in the regular school because of time constraints. A good program for a high performing student should be exposing them to other ideas, and getting the students to stretch their minds.

  1. 4.     The program has been developed with the intent to first establish very strong numeracy skills.

Math is based on numbers, so a proper understanding of how numbers work is crucial. This may seem obvious, but very few math programs actually focus on numbers. They will often just teach “how to do the next thing”. Most of the students I have helped over the years have problems in math simply because they don’t have a proper understanding of numeracy. In fact, some teachers don’t have this understanding either because they haven’t been taught it themselves.

Look carefully at the math your child is being taught – is it just rule based, or are they being taught first how to work properly with numbers and to understand all their properties? In Spirit of Math students are taught to work with integers in grade 1 so that they can form a concept of numbers that is not just based on manipulatives.  The grade 5 course focuses entirely on numeracy and by the end of the grade, students understand how different numbers work on their own, and in conjunction with other numbers. Their thinking and understanding is consequently well above the typical grade 5, 6, 7 or even 8 student.

  1. 5.     The program was developed over many years, in the classroom with students.

Most programs have been created from someone’s imagination… someone sitting behind a desk. New and innovative ideas do come from a good imagination; however, those ideas must be tested with students to see what works. This can only happen if the program is developed with students, in the classroom. This not only takes years and years, but very dedicated and bright people working with students.

There are so many math programs out there, and many more people who think they can teach math because they know “the steps” to answering certain types of questions. It takes a very discerning parent to look deeper into the program and to make sure that it is, in fact, going to teach their child how to think mathematically,  and not just how to do the next question. A person who can think mathematically is able to logically put ideas together and that is a skill that can be applied to every area of life.

It’s a Classroom Taboo, but Missing Out Could Hurt your Child’s Learning

In North America the term “drill” has been considered, like the unmentionable name ‘Voldemort’ in the Harry Potter series, a “taboo” word. In fact, many teachers say they aren’t allowed to give any drills in their classrooms because they could get into a lot of trouble. Whether the boycotting of drills began with a single teacher or school, it progressed to the boards, the ministries and now to a realm of being considered socially unacceptable to have students do drills. But this boycott is having a negative impact on our children’s education. The truth is; practicing math drills is critical for developing math skills.

For several years, I helped a local community initiative that was assisting low-income students with their academics. With these free programs, you never know who is going to attend from week to week and you have to be very flexible. I was given 30 minutes each Saturday to provide a math program to students ranging from grades 1 through 10. Many of the students who came had very poor math skills. I thought carefully about what to provide them so that they would establish a skill set that would empower them for the rest of their lives, and that would help them in their future mathematics. I decided to give them the Spirit of Math drills. For the first six months, it was all we focused on.

At the 5-month mark, I asked some of the weaker students how their math at school was progressing. A grade 6 girl, who had really struggled at the beginning of the year piped up. “Great!” she responded enthusiastically. So I asked her how the drills were helping. She said that she was doing geometry and measurement at school and the teacher said that the width of the rectangle was 7 and the length was 8, therefore the area was 56. Whether she realized it or not, this girl told a very revealing and insightful story. She used to be failing, not because she wasn’t able to understand the concepts, but because she didn’t know where the numbers came from, so she would have been lost at the “56”, wondering how the teacher “magically” came up with that number. Now she knows. It isn’t a mystery anymore. The numbers make sense, and she is able to follow the teacher.

Research has shown that drills do work, but be careful, because many are ineffective. It was these ineffective drills that gave drills a bad name. Find the right drill system and your child will develop a fluency in numeracy that will set them up for a life-long understanding of numbers.

To determine if your child’s drill system is good, ask yourself these important questions:

1.      What is the purpose of the drill?

The main purpose of a drill is to develop an automaticity and fluency of facts. In math, it can include addition, subtraction, multiplication, and division. At Spirit of Math Schools, students also do decimal equivalence drills (from ½ to 11/12), fraction addition drills, perfect square drills (from 12 to 602), percent drills, radical drills and radian to degree drills.  We believe that a student should not just know the facts, but know them so well that their minds are calculating without realizing it.

2.      Are the drills themselves actually developing the skills?

The old fashioned drills, giving students a sheet of questions to see how many questions the student can do, is an ineffective and outdated way of doing mathematics. Provide drills that will consistently reinforce the facts and simultaneously push the student to retrieve them faster.

3.      Can you clearly see students’ progress?

There should be a way to measure a student’s progress each and every time they do a drill. In Spirit of Math Schools, students plot their results on individual graphs, after every drill. In addition, they have a class average graph so that they can see their own progress and compare to other classes. Teachers, students and parents can immediately see if progress is being made.

4.      Is there a goal set that is well defined and challenging to achieve?

Goals help to hold students accountable. Students should know what level they must achieve before passing that drill and going on to another one. Very strong drills will also hold the students accountable to each other: their learning affects others too.

5.      Does the drill set a high standard?

It is not good enough to just know the number facts. Students should be able to retrieve the facts and use them at almost lightning speed. The drills should be timed.

In Spirit of Math, grade 5 students are given 10 minutes to complete 80, three digit x one digit multiplication questions in a grid. When the class average reaches 70 out of 80, the students move on to the next drill.

There is so much involved in mathematics. Drills are just one strand, but a very critical strand. I always say that asking a student who doesn’t have an automaticity or fluency of number facts to do math, is like putting a student with crutches into a running race. There is no way that the student with crutches will be able to properly keep up, never mind, compete with the other racers, and it is the same with math.

What Happens if Algebra is Taught too Early?

Quick, solve this grade 6 question… and do it without algebra:

Two bugs are running back and forth along a straight branch at constant speeds without stopping. They start from opposite ends of the branch at the same time and meet for the first time 40cm from one end of the branch. They continue to the ends and return, meeting for the second time 20cm from the other end of the branch. How long is the branch?

This is just one of the many questions that a teacher must answer during their screening process when applying to Spirit of Math Schools. Most teachers are able to answer it with algebra, but have difficulties without it.

Because Spirit of Math was developed for high performing students, many applicants entering our program have taken other math programs outside of day school with the intent of getting ahead. Sometimes parents of younger children will proudly tell us that their child has learned algebra, so therefore, according to the parent, the child is ahead of other students. Unfortunately, this is not true. Those students may be very bright, and are looking to do more than what is given to them in their day school, but by learning algebra at a very young age, they have gone in a direction that reinforces a convergent, procedurally based type of thinking.

Most people don’t know how to properly challenge these bright students. Instead of going deeper into topics, challenging the child’s thinking, and getting them to think divergently, they teach new ideas based on procedures and linear thinking – thinking that many adults believe is the way to teach because it is easy to teach, and very nicely gets to a solution to question. This type of thinking and working is prevalent in topics such as algebra. It is very important that a student learns it, but when they learn it is just as important.

It took over 20 years to develop the Spirit of Math program. A significant part of that time was spent determining what skill sets and thinking a student required to compete with the top students in the nation.

Initially, it was thought that the students would answer the tough problem-solving questions at the grade 7 to 9 level by using algebra. But, when watching the brightest students and observing their solutions, we realized that they were not thinking in terms of algebra, (the adult thinking) –  they had a much more practical type of thinking that did not involve algebra. Their thinking required seeing the bigger picture, and putting several ideas together – ideas that weren’t necessarily introduced in the question itself, but inferred. This is what many call divergent thinking.

It has been our experience that excellent problem solvers are those who have first developed the ability to think divergently before going deeply into the linear and procedural type of thinking. It is more difficult to teach students how to think divergently, but once they have it, they will be able to use it in all areas of their lives. If a student is taught algebra before problem solving, then they will want to use the easier, linear type of thinking required in algebra, and avoid the practical, divergent thinking.

High performing students tend to learn procedures very quickly, and then find ways to apply the procedures. Knowing and developing a proficiency with procedures is essential in math, but it does not necessarily mean that the concepts were understood.

Understanding algebra is extremely important, and the foundation for a huge amount of mathematics. There is a time to teach it, and it is after a student has developed a solid conceptual foundation of numeracy and the ability to think on their own. You don’t want them to learn to just blindly follow procedures because “that is what you do”. Teaching your child to think first, and to have creative independent ideas is a gift that will last them a lifetime. Teaching algebra is relatively easy in comparison.

The answer to the bug question above? 100cm. How do you do it without algebra? Ask your grade 6 or 7 child to help you out!

The intent of this section is to give you a peek into some of the ideas behind the Spirit of Math program, so that you will understand why it works and how these methods can work for your high performing child.

Competition – No, It’s Not Just for Sports Enthusiasts

Spirit of Math makes the case for healthy math competition.

“…and pencils down.”

The contest is over. Fidgeting legs and hands make an effort to wait patiently until all the contest papers have been collected.  Anxious parents are waiting outside the doors.

“Okay, now you can talk.”

The room erupts with excited chatter. “How did you do this one?” “I got 2 for question number 4, what did you get?” In the large hall, over 120 students from grades 3 to 12 have just completed writing one of the nation’s largest series of mathematics contests – one that tens of thousands of students write. These Spirit of Math students have opted to miss a morning of school just so that they could have the opportunity to compete in something they get excited about – math.

Yes, math. Even in North America, many students think of it as another sport – one in which they too can enjoy competitively. They love it because they know they’re good at it, and want to show what they can do.

There are great benefits to exposing students to contests, as long as the focus is on learning. So often teachers, parents and students get caught up in the testing frenzy, studying and teaching to the contest. There is so much more value to these contests, then just “teaching to the test,” and this value is often lost when the focus is exclusively on how to do the next question.

Many people can give past questions and show students how to answer the questions – the solutions are there. But how many people can actually look at all the problems, determine the skill sets that are needed, and teach those skill sets so that multitudes of mathematical ideas and questions can be tackled? This method is hard to come by, but it has been perfected  – over 25 years – in the Spirit of Math program. And the results of this method of teaching skill sets are outstanding. With over 700 placements on the national mathematics honour rolls this year – an outstanding feat – Spirit of Math makes a great case for not just teaching to the test. Spirit of Math students are excelling in mathematics because they have the skill sets needed to think mathematically.

With this focus, the educational value in doing the contests shifts from looking at how many questions are answered, to an opportunity to evaluate how well the math skill sets have been developed, and the concepts understood. In addition, students learn how to take a risk, and are pushed to do more than they thought they could.

Putting students into contests gives them an opportunity that many students don’t have. If you want your teenager to be able to compete against others for university math scholarships, they must achieve at least an honours standing on a contest.

Spirit of Math students have an impressive list of contest achievements to add to their resumes. For over 25 years now, students studying with the Spirit of Math program, have succeeded in many national mathematics contests. In fact, for a period of 13 years, when students had the opportunity to study this program in their day school, there were more students on the national honour roll from that one school, than all other Canadian schools combined. Did their teacher teach to the test? Absolutely not, but they did have an incredible program that not only developed extraordinary math skills, but many other interpersonal skills. These students developed minds able to think in ways that propelled them to the top in all areas of life.

From doctors to physicists to engineers, these Spirit of Math alumni are now leaders in their communities, changing the world.