Learning Philosophy

“Practice grows a mathematical mind”

by Barry Stanley

It is my belief that practice is the central core of all learning, particularly for mathematics. Practice it to learn it, practice it to develop and expand it, and then practice it in life. Logically thinking, a lesson is taught first, the students learn some principal, they might practice it, and then they get tested. However, learning doesn’t necessarily come from a taught lesson --- often doesn’t. It is my belief that it is during practice that learning takes place, the type of learning that will stay with the child.

Consider that without a reasonable time in practice, the brain will discard even the most relevant of information, as it keeps only what is being used. Practice is where understandings are permanently transferred to long term memory (the word permanently is used somewhat lightly here, because if these skills are not used on a regular basis, they will be discarded, or automatically pushed to the back of long term memory to the point that they are barely reachable). Systematic practice (presenting a concept over a number of practice sessions and in a variety of problems and contexts (½ an hour, ½ the cake, ½ the group, ½ the gym, ½ plus ½ is a whole, etc.), will connect a concept to many other concepts, broadening the understanding of the concept (and mathematics as a whole), and will evade the possible discarding of the concept by the brain. “Use it or loose it” is perhaps the simplest way to understand the need for practice as presented in these worksheets (or as some call, black line masters). Through practice students will find and experience the many subtle patterns in mathematics and how these fit into the more complex patterns of numbers, shapes and algorithms which make up mathematics.

“Practice Makes Perfect” or “Drill and Kill”, take your pick.

The slogan, “Drill and Kill” only has one fact and that is that two words rhyme. There’s nothing more to it than that. It is not factual. Students like to practice when they can enjoy some fair level of achievement. Practice can be very satisfying, while it broadens understanding. Practice may not make perfect but it will make the person because a person becomes what they practice. I get some criticism about paper worksheets (there has been and is an expressed bias against practice). Fortunately I am not handicapped by that misconception.

Most of these worksheets try to offer a few levels of ability and achievement, such that there is always some room for practice as students of varying levels solidify their knowledge and understanding of known concepts, but also face the opportunity to face some new challenges. These sheets aim to have a mix of easy, medium and difficult problems. In some sense, the experience is one of “Differentiated Instruction”. Weak students may only master the easiest, and are challenged by some of the medium level problems. Average students, will solidify easy level problems and broaden their ability at the medium level, while they are challenged by the highest level. Strongest students may breeze through the easy, solidify their knowledge of the medium level, but attain some satisfaction by achieving the most challenging (often problems and concepts that have not even been taught). Teachers, knowing the ability of their class as a whole, should pick worksheets that are at the threshold of their students’ learning; not too easy, and not too hard.

Differentiated Instruction refers to the concept of Readiness, while I prefer the expression “threshold”. The threshold is that border in a student’s mind between what they now know and what they should learn next. Understanding is very incremental, and it is up to the teacher to strategically select the next increment. Skipping increments leaves gaps in learning that not only will slow learning down later, but can often lead to misunderstandings. These worksheets can and should be modified by the teacher so that they meet the needs of his or her students, as individuals and as a class. Teachers should be trying to address the weaknesses of their lowest students, offering them some level of achievement and satisfaction, while they also address the needs of the average and strongest students, providing appropriate practice and challenges for them as well.

This practice of meeting the needs of most of the students can be met using multi-level practice. This objective often can’t be met during class instruction or group work, as this style of teaching generally targets an average ability, rather than individual ability. An important advantage of the use of these practice sheets to the progress of all students is that while students work on their sheet, the teacher is able to spend individual time with students at their level. By the way, these sheets should be completed, or at least started, during class time --- they are not meant to be just homework sheets. But circulating through the class the teacher is able to learn where students are struggling, and is better able to address those concerns immediately. In a way these worksheets act as a continuous diagnostic and formative assessment, while the teacher daily experiences the needs of the students, and adjusts to it in the practice for the next day.