Many math lessons follow a very traditional format that is not conducive to learning. Here’s how I see one unfolding.
STAGE 1
The teacher goes over assigned homework while students are supposed to make corrections.
Reality
The majority of students are not involved in this process because:
a. If they did the homework correctly they see this as a waste of time,
b. If they did not complete or even attempt the homework they are not interested in highlighting their lack of understanding or disinterest in the work.
Result
Little or no learning takes place at Stage 1.
STAGE 2
The teacher “teaches” the new content by talking at the students, writing notes on the board or on the overhead.
Reality
Students either:
a. blindly copy the notes without thinking about what is being taught or what they are writing down. They believe that when it is time for a test they can study their notes and all will be clear.
b. become bored and do not bother to copy notes or listen to the explanation.
Result
Little or no learning takes place at Stage 2.
STAGE 3
The teacher works through an example on the board or on the overhead.
Reality
Students:
a. copy the solution without thinking about what they are doing,
b. tune out and do not pay attention or copy the solution,
c. work through a similar question. This is done by referring to the steps of the previous example so thinking is unnecessary.
Result
Still, little or no learning takes place at Stage 3.
STAGE 4
The teacher assigns questions for students to complete on their own and assists when requested; usually by once again explaining how to complete the question.
Reality
Students will:
a. work through the questions by repeating the steps shown in the worked example,
b. attempt to look busy while avoiding doing the questions as they do not understand what to do,
c. ignore the entire process as it seems to have little or nothing to do with them.
Result
Still, little or no learning takes place at Stage 4.
STAGE 5
Teacher assigns homework. Often students have worked on “set A” of the questions in class. The teacher, therefore, assigns “set B” for homework. Set B questions are usually more difficult and often do not follow the pattern in the worked example.
Reality
Students will:
a. note the homework and stop work for the remainder of the class,
b. think they will be able to complete the work based on the worked example, which, unfortunately, will not help them complete the more difficult questions,
c. ignore the homework—if the teacher goes over the questions during the next lesson, they can copy then (See Stage 1).
Result
Little or no learning has taken place during the lesson.
As a way to learn mathematics, this lesson format is inefficient and unsuccessful for many, if not most, students. They learn little, but soon become convinced that mathematics is difficult, boring and they are not good at it.
Over 25 years of research on how students learn has made it clear that students learn best by solving problems and being actively involved in the learning process. Active involvement does not just mean working with geoboards, pattern blocks, algebra tiles, etc. (which are excellent tools to introduce concepts) but also requires active mental involvement.
So, why is there so little active involvement in math classes?
New methods of teaching which include the learner as an active participant in the learning process may be more demanding on some of us as we can no longer follow a prepared script from the textbook resources. Encouraging students to investigate problems and ask questions can be threatening, as it requires us to have a sound knowledge of the math content. In addition, supporting student learning by asking questions rather than simply explaining the concept or algorithm is challenging.
Here is one possible lesson format to encourage active involvement.
STAGE 1
• Teacher assigns an interesting problem related to the lesson.
• Students work on the problem singly or in pairs.
• Teacher circulates keeping students focused on the task by asking leading questions.
Reality
Students are actively involved in trying to answer the question while those who are off task are quickly redirected to the problem at hand.
Result
Students are engaged in solving the problem and are involved in the learning process from the start of the lesson.
STAGE 2
• Students share their solutions and insights.
• The teacher leads a discussion on the solutions, summarizes the math and brings a focus to the upcoming lesson.
Reality
Students share their thinking processes and discuss alternate strategies and solutions. The teacher involves all students and focuses attention on important aspects of the upcoming lesson.
Result
Students are engaged and involved in clarifying and reinforcing their knowledge.
STAGE 3
• Teacher reviews homework as it relates to the lesson. This goes beyond going over the answers. The teacher involves different students in explaining their methods and solutions.
• Students discuss the concepts involved in the homework assignment and ask questions for clarification.
Reality
Students review various ways to solve the assigned questions.
Result
Students are involved in reinforcing and clarifying their prior knowledge and understanding.
STAGE 4
Teacher builds on the work done on the problem and the homework, develops new concepts and skills and involves the students by asking in-depth questions and seeking explanations.
Reality
Students are able to relate the concepts and solutions addressed during the problem solving and the discussions during the homework review to the to the concepts being developed by the teacher.
Result
Students are not simply learning how to solve one type of question, but rather a multitude of different types by building on prior knowledge and through involvement in the learning process.
STAGE 5
• Teacher assigns questions ensuring they cover skill reinforcement and higher level thinking skills. She/he assists students by asking questions rather than providing answers.
• Students work on questions, discussing difficulties with peers and teacher.
Reality
Students are actively involved in solving problems. If they experience difficulties, the teacher supports learning by asking questions to focus attention on specific concepts and prior learning.
Result
• Students understand and practise the concepts and skills developed, and work to reinforce and integrate new learning with prior knowledge.
• Learning is taking place.
STAGE 6
Teacher summarizes the main points of the lesson by asking questions and assigns home study rather than homework.
Reality
Students are empowered. They feel they are grasping the concepts and understand different strategies for solving a wide variety of problems.
Result
When students are responsible for and involved in their own learning, they gain confidence and enjoy the subject and the learning process.
The most valuable aspect of using this method or a variation is that lessons become more enjoyable for the teacher and a much more exciting and rewarding learning experience for the students. The ensuing enthusiasm for learning is infectious!
ABOUT THE AUTHOR
Jim Mennie
Jim Mennie is the driving force behind Root 7 Educational Services (root7.ca), a company dedicated to improving mathematics education by providing high-quality math resources that are easy to use and easy to incorporate into lessons and unit plans. Root 7 also offers workshops, information and assistance to teachers, students and parents. Jim has nearly fifty years experience in mathematics education having taught math at all levels from grade five through AP Calculus to university courses; presented workshops, in-service courses and at conferences; written and edited textbooks and resources; and worked as a consultant in Canada and overseas. Jim can be contacted through his website or directly at jimmennie@gmail.com.
This article is from Canadian Teacher Magazine’s Nov/Dec 2013 issue.
