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.