Show and tell activity (aka lecture method) may work for some but never in a mathematics class. Getting students to explain and ask questions are nice but only when the explanation and the questions are mathematical. Reasoning and justifying are good habits of mind but they are only productive if they are based on mathematical principles. Explaining, asking questions, and substantiating one’s conjecture or generalization make a productive class discussion but they are only productive for learning mathematics if the mathematics is kept in focus. Orchestrating a productive class discussion is by far the most challenging work of mathematics teaching. Stein, Engle, Smith, and Hughes* (https://doi.org/10.1080/10986060802229675) proposed five practices for moving beyond show and tell in teaching mathematics. I have always practiced them in my own teaching whether with students or with teachers and I find them effective especially when the lesson involve cognitively demanding tasks and with multiple solutions. Continue reading “How to orchestrate a mathematically productive class discussion”
Tag: teaching mathematics
Bedrock principles of math and what it means to understand math
In one of my LinkedIn group, someone started a discussion with this question What is the bedrock principle of mathematics? May I share some of the answers.
1. “The bedrock principle of mathematics is the axiomatic system. The realization that there are propositions that must be taken for granted in order to have something to build upon.”
2. “Speaking about foundations, in my opinion the bedrock should be enlarged at least as follows:
- discerning that two things are different;
- identifying two things which share same property;
- discovering relations among properties.”
3. “I would say the bedrock principles of mathematics are:
- The ability to differentiate two things
- The ability to rank two things (as to most value, shortest route, least danger, etc.)
- The ability to expand the above to more than two things”
Bedrock principles of any discipline of course can’t tell us how one can know if he or she understands a piece of that discipline. So I asked How can one tell if he/she understands a piece of mathematics?
According to Peter Alfeld you understand a piece of mathematics if you can do all of the following:
- Explain mathematical concepts and facts in terms of simpler concepts and facts.
- Easily make logical connections between different facts and concepts.
- Recognize the connection when you encounter something new (inside or outside of mathematics) that’s close to the mathematics you understand.
- Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)
You may also want to read my post To understand is to Make Connection.
The challenge of course will still be this question: What does all these imply about teaching mathematics?
The Four Freedoms in the Classroom
You will find that by providing the following freedoms in your classroom an improved learning environment will be created.
The Freedom to Make Mistakes
Help your students to approach the acquisition of knowledge with confidence. We all learn through our mistakes. Listen to and observe your students and encourage them to explain or demonstrate why they THINK what they do. Support them whenever they genuinely participate in the learning process. If your class is afraid to make mistakes they will never reach their potential.
The Freedom to Ask Questions
Remember that the questions students ask not only help us to assess where they are, but assist us to evaluate our own ability to foster learning. A student, having made an honest effort, must be encouraged to seek help. (There is no value in each of us re-inventing the wheel!). The strategy we adopt then should depend upon the student and the question but should never make the child feel that the question should never have been asked.
The Freedom to Think for Oneself
Encourage your class to reach their own solutions. Do not stifle thought by providing polished algorithms before allowing each student the opportunity of experiencing the rewarding satisfaction of achieving a solution, unaided. Once, we know that we can achieve, we may also appreciate seeing how others reached the same goal. SET THE CHILDREN FREE TO THINK.
The Freedom to Choose their Own Method of Solution
Allow each student to select his own path and you will be helping her to realize the importance of thinking about the subject rather than trying to remember.
These freedoms help develop students skills and habits of mind.
Should the historical evolution of math concepts inform teaching?
Should the history of a math concept inform the way we should teach it? Some camps, especially those that strongly object to the usual axiomatic-deductive style of teaching, advocates the use of a “genetic” teaching model that takes seriously into account the historical roots of mathematical knowledge. Here are some studies that support this approach.
Harper (2007) compared the historical analysis with students’ empirical data and found a parallelism between the evolution of algebraic symbolism and the way students understand the use of letters in school algebra, concluding that “… the sequencing of conceptual acquisition appears to parallel that which is to be detected through the study of the history of mathematics.”
Moreno and Waldegg (1991) found that “… in situations involving the concept of infinity, the student response schemes are similar to the different response schemes given by mathematicians throughout the history of mathematics,…, when faced with the same kind of questions”
However, there are also those who contradicts this conclusions: For example, on solving linear equations, Arcavi argues that,
….solution methods generated throughout history are quite different from the usual methods generated by students. Consequently, we cannot assert that a reason for the study of linear equations is based on or inspired by parallels between history and psychology – these parallels do not seem to exist (Arcavi 2004, p. 26).
Herscovics acknowledges that while obstacles in the nature and evolution of knowledge are in parallel with some of those met by the learner and are associated with his/her cognitive evolution, she also warns that this parallelism should not be taken too literally, since learning environments in the past are significantly different from those of our learners now (Herscovics 1989, p. 82).
In their investigation of the parallelism between historical evolution and students’ conceptions of order in the number line, Thomaidis & Tzanakis (2007) has this to say:
If room is left for genuine problems to help the emergence of the new concepts, motivate students to appreciate their necessity, or formulate their own alternative ideas (as it happened historically), teaching will not be restricted to the presentation of formal constructs in their polished final form, as it is often the case under the additional pressure of factors peculiar to the modern educational system itself, but will help students conceive mathematics as a creative, adventurous human activity.
Like in most issues related to teaching and learning, there is no clear cut answer here, but it will always pay to know for teachers to have a sense of how specific math concepts evolved in history. It could provide valuable information both in the design of instruction, in anticipating cognitive obstacles and, for making sense of students difficulties in learning the concept. Teachers must also always remember that the evolution of a math concept is always tending towards abstraction. And because definitions of math concepts are already abstractions of those concepts, starting with definition in teaching is a no-no. Read why I think it is bad practice to teach a math concept via its definition.