The pace of technological change is amazing. Our ability to keep current and ensure that our students are current is of paramount importance. I agree that it is necessary to ensure that students are using some technology and using it in such a way that benefits their learning. Being able to show students tech uses besides just games is important for them to be able to understand that they are tools for learning. Several of you have brought up the importance of offering open questions so that everyone can be successful. The openness of the question is determined by the difficulty of it, as well as how students choose to answer it. You also mentioned that teaching students to be open to new ideas or ways of solving a problem is important if we want them to be able to keep up with change. This will help them be flexible thinkers.
With the unknown nature of future careers, we need to instil in our students habits that will make them effective: persistence, listening, thinking flexibly, questioning, applying past knowledge, gathering data, responding with wonderment, etc. These are habits that we can encourage in our math classrooms now. We can praise students for their persistence, ask questions to encourage them to use their past knowledge, get them wondering about math through the problems we choose, and help them value the power of questions.
With so much information being outdated so quickly, I think a greater emphasis in the classroom on the processing skills involved in math is necessary. Through thoughtful planning, modelling, and questioning, a teacher can help develop a student’s ability to: problem-solve, reason, prove, reflect, select tools and computation strategies, connect, represent, and communicate.
With students being bombarded by new information daily, they need to be able to analyse it and determine if it is accurate, reliable, useful, etc. By providing our students with open-ended problems where they have to decide what information is useful for their purpose, think about if their sources are reliable, and then make inferences, we are preparing them to use critical thinking skills flexibly in their ever-changing world.
Finally, according to the video, students will have 10-14 jobs by the time they are 38. If this is true, they will need to feel comfortable working in new situations, solving new problems, and working with different people. Through the problem-solving model, students will be exposed to new problems, in new contexts, and will have the opportunity to work collaboratively with a variety of their peers. This will hopefully instil the ability to be adaptable.
Manipulatives: Why Use Them
Manipulatives that are used well are central to effective instruction and have the capacity to greatly improve and deepen student understanding.” (Report of the Expert Panel, 2004)
Manipulatives provide many benefits including:
· make students’ mathematical thinking visible, so that everyone can see it, talk about it, and learn it;
· provide a context for developing mathematical concepts;
· help students explore, think about, and talk about mathematics;
· help students construct meaning and see patterns and relationships;
· allow students more easily to test, revise, and confirm their reasoning;
· help students make connections between concepts and symbols;
· help students talk about the math, with the result that teachers have a basis for assessing students’ understanding and can make programming decisions based on their observations.(Guide to Effective Instruction, Volume 3)
Studies also suggest that the use of manipulatives helps to reduce math anxiety and improves long-term and short-term retention of math. Marilyn Burns advocates for the use of manipulatives through all the grades, including high school because they accelerate and deepen student’s understanding. The Expert Panel came out with Leading Math Success: Mathematical Literacy Grades 7-12 in 2004. In it, they advocate for the use of manipulatives for many of the reasons that we’ve been talking about.
It is very important to ensure that students are using the manipulatives properly, that they are appropriate for the concepts(s) being developed and that they are developmentally appropriate. For example, if you are using popsicle sticks to teach the concept of place value, one stick may be placed on a place value chart in the ones place; however one stick should not be placed in the tens place. Instead, a package of ten sticks bundled together with string or an elastic should be placed in the tens place so that students begin to conceptualize the idea of “tenness”. This demonstrates their proper use in this situation. We should also ensure that students have an opportunity to experience free play with any new manipulative so that they can be used appropriately when needed.
I equate the use of manipulatives to students representing their understanding of what they know. Students can choose to show their understanding by using manipulatives, by drawing a picture that represents it or by using both. I think if we make it a natural part of learning mathematics for junior and intermediate students and also an expectation, I think the perceived stigma of using them will be removed.
A Constructivist Approach to Mathematics Teaching
The Dilemma:How Much to Tell?
“Unlike a more traditional approach where teachers focus on the transmission of mathematical content, in a constructivist classroom students are recognized as the ones who are actively creating their own knowledge” (Making Math Meaningful to Canadian Students: K-8, 2009)
“The basic tenet of constructivism is simply this: Children construct their own knowledge... Mathematical ideas cannot be ‘poured into’ a passive learner. Children must be mentally active for learning to take place.” Elementary and Middle School Mathematics: Teaching Developmentally, 2004)
“The single most important principle for improving the teaching of mathematics is to allow the subject of mathematics to be problematic for students (Hiebert et al., 1996). That is, students solve problems not to apply mathematics but to learn new mathematics.” (Teaching Student-Centered Mathematics, 2006)
Taken from Elementary and Middle School Mathematics, p. 53)
When teaching through problem solving, one of the most perplexing dilemmas is how much to tell. On one hand, telling diminishes student reflection. Students who sense that the teacher has a preferred method or approach are extremely reluctant to use their own strategies. Nor will students develop self-confidence and problem-solving abilities by following teacher directions. On the other hand, to tell too little can sometimes leave students floundering and waste precious class time.
Researchers connected with four different constructivist programs, while noting that there will never be a simple solution to this dilemma, offer the following guidance: Teachers should feel free to share relevant information as long as the mathematics remains problematic for the students (Hiebert et al., 1997). That is, “information can and should be shared as long as it does not solve the problem [and] does not take away the need for students to reflect on the situation and develop solution methods they understand.” (p.36)
They go on to suggest three types of information that teachers should provide to their students:
The social conventions of symbolism and terminology that are important in mathematics will never be developed through reflective thought. For example, representing “three and five equals eight” as “3 + 5 = 8” is a convention. Definitions and labels are also conventions. What is important is to offer these symbols and words should only be introduced after concepts have been developed and then specifically as a means of expressing or labeling ideas. They should rarely be presented solely as things to be memorized.
You can, with care, suggest to students an alternative method or approach for consideration. You may also suggest more efficient recording procedures for student-invented computational methods. A teacher must be cautious in not conveying to students that their ideas are second best. Nor should students ever be forced to adopt a teachers’ suggestion over their own approach. The rule of thumb here is that the value of a procedure should always be the judgment of the students, not a dictate of the teacher. In this spirit, students can learn to appraise a teachers’ suggestion without feeling obligated to use it.
Clarification of Students’ Methods
You should help students clarify or interpret their ideas and perhaps point out related ideas. A student may add 38 and 5 by noting that 38 and 2 more is 40 with 3 more making 43. This strategy can be related to the make-ten strategy used to add 8 + 3. The selection of 40 as a midpoint in this procedure is an important place-value concept. Such clarifications reinforce the students who have the ideas. Discussion or clarification of students’ ideas focuses attention on ideas you want the class to learn. Care must be taken that attention to one student’s ideas does not diminish those of other students. Nor should teacher attention to one method suggest that it is the preferred approach.
Problem Solving Models
The word Bansho comes from the Japanese word for blackboard. It is a method that focuses on teaching math through problem solving and encourages students to observe and learn about different ways in which a problem can be solved. This strategy encourages students to discuss their strategies and ideas related to the problem in a collaborative manner.
There are many benefits to students and teachers. The list below highlights a few of those benefits:
- students see many different ways of solving a problem
-helps them see connections among concrete, symbolic and numeric representations
-see the thinking of others
-helps students become more comfortable sharing their ideas
-helps students reason, reflect and communicate their ideas to others
-teachers are able to continuously assess
-instruction can be differentiated
-conceptual understanding and success criteria can be built collaboratively and a permanent record of this information is kept for future reference
-promotes deeper mathematical understanding
For a demonstration of this method, please use the following link:
The Guide to Effective Instruction in Mathematics defines a balanced math program as being one that involves solving, reflecting on, reasoning, connecting, communicating and representing when problem solving. These can be accomplished by incorporating the following three components: Guided, Shared and Independent Mathematics. These names are familiar to us as they have been used in our language programs; however, in mathematics these approaches are different.
In a traditional guided language lesson, we model for students what we want from them. There is a specific way for them to answer the question. In Guided Math, the goal of the lesson might be to reinforce a specific skill or concept; introduce the new skills or concepts required to solve a problem; introduce a specific process (a problem solving strategy or algorithm); teach specific conventions, and/or model mathematical language, thinking and problem solving. It can involve the whole class, a few students while they are working on a problem in groups or in a small group with the teacher. Reflection, discussion and sharing are still key components. We can model a problem of a mathematical idea using appropriate materials and tools and use the think aloud strategy while solving a problem (see page 70 at http://www.eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_1.pdf). In guided math, the lessons are short, target specific, have clear goals and target a specific concept or skill. Guided mathematics should not be the primary focus of a mathematics program (Guide to Effective Instruction, Vol. 1, page 69).
In Shared mathematics, the teacher guides the lesson and supports students in sharing, discussing and exploring mathematical concepts. Students can work at centres, work collaboratively on a problem, demonstrate a skill to their peers and communicate their ideas.
During Independent mathematics, students work on their own on a particular concept to further develop their mathematical understanding. If needed, students are able to access support from their peers or teachers. Sufficient time to complete the task is important to ensure that students do not become anxious and as a result, prevent them from demonstrating their full potential.
Access this slideshare for a comprehensive explanation of this instructional strategy that was developed by Fosnot and Dolk.
Parent Resources List
Faber, Adele, and Elaine Mazlish. How to Talk so Kids Can Learn: What Every Parent and Teacher Needs to Know at Home and in School. New York: Simon & Schuster, 1996.