New questions that now exist from our research on constructivism

Who thought this would best suite the educational field?

With regards to the traditional way of teaching and constructivist teaching is there anymore evidence that there is no difference?

What are some other proposed strategies that have not been implemented into the classroom, but are being worked on and looked at now?

How long does it take for ideas such as constructivism to be implemented into the educational system?

What are some constructivist ways to assess students in the classroom, beside the traditional methods?

Should the teacher always act as a facilitator, or is there any evidence that the teacher should be actively involved by other means within the classroom?

What are children's views on constructivism? Do they enjoy it?

Will constructivism work in classes of really large or really small numbers of students?

How does culture play into the success of establishing a constructivist classroom?

Does constructivism in the primary and elementary grades lead to higher academic achievement in middle and high-school?

A Final Thought...

Trying to cover the topic of Constructivism in Mathematics Classrooms was quite an undertaking. The amount of activities available for teachers to use in their classroom is endless. As you might have noticed, we only included a small sample in our blog, and instead focused on the foundations and conditions needed to establish and maintain constructivism principles in the classroom. However, we have not left you high and dry! On the left hand side of this blog are websites that provide countless activities for any grade level. After much internet surfing we have selected these and stand behind them.

We hope you have enjoyed our blog and that you leave ready to bring constructivism to your current, or future classroom!

Thank you,
Tara Tobin & Amanda-Rae Warren

Ideas for the Classroom

Some Ideas for classroom teachers to keep in mind:

• Teachers could use is arranging the classroom so that students can and feel free to work with other studentswhen having difficulty or trying to complete a task. Sometimes that is one of the most beneficial things a teacher can do to help his or her students.
• The comfort level that exists in the class is key for students taking risks, if students feel that they can express themselves then they will enjoy risk taking and exploring different methods of trying mathematical problems.
• Another thing that teachers can do is to have a variety of materials available for students to experiment with and work with. This opens students minds to a variety of possibilities.
  
• Allow for games and activities to be placed around the classroom at all times so students can work with them on their own time, as well ensure that these activities incorporate a variety of learning styles, such as kinesthetic, visual, and auditory.
  
• One key point that should be encouraged, is making sure that students questions are answered. Because students are curious and need help in understanding certain ideas, concepts, or just want to make sense of things. This can be done to help students. For example if students are having trouble with integers, then relate them to the weather (Class notes, mathematics winter 2007). There are a wide variety of ways to do this, this is only one example.

Possible Activities for Primary and Elementary Students

As Clements (1997) maintained, constructivism is more than just teaching, it's a philosophy of learning. Here are some activities that are excellent examples to use for a unit on geometry, area, shape or space in a constructivist classroom:

• Triangle areas
• Shape-construction game
• Magic Bugs and Mobius Strips (strategy/problem solving)
  
• 3-D tic-tac-toe (spatial skills and strategy/problem solving)
  
• 3-D Shapes
• Geometric Shapes Colouring
• Tessellations (Mobius Strip)
• Polygon and tower construction with wooden pieces (stimulating geometry discussion)
  

For activity details and instructions consult Ann E. Kajander's article (see Resource section on bottom)

Can we do it? Yes we can!

So, now that you have the literature, you have the background and you know the facts - can you bring it?? That is, can you bring constructivism to your classroom? We know, it's nerve wrecking. But no one said it would be easy right? As we figure it, there are basically two ways to start thinking constructivily. You could:

A) Dive right in with the information you have and see what happens, or
B) Start gradually, and immerse yourself and your students into the world of constructivism slowly.

Althought both options have their merits, our research leads us to conclude that option B is probably the better route. We've come to this conlcusion for different reasons.

• better equips new-constructivist teachers to maintain classroom management
• teachers can optimize success for themselves and their students when they start with a few good lessons compared to many poor ones.
• teachers can observe students reactions to this "new" type of learning and change their instruction accordingly
• Starting slowly can also mean teachers have a chance to build up resources

Copeland (1970) proposed for beginning teachers that they "allow only one group a day to do the practical "discovery" work while the others continue their regular routine. This will allow a pilot approach to the new procedure to see how it works, and later the program can broadened as the tacher feels more secure with it" (p.331). For particularly weary teachers, or those who are more comfortable in highly organized classroom atmospheres, this may be the best option. The bottom line is that experimentation with constructivism is far more beneficial for students than none at all.

Constructivism in Action!

What might constructivism look like in a real classroom? Check out this video of a "Math Night" at West Mercer Elementary School in Mercer Island, Washington. Notice how the children are engaged and seemingly enjoying participating in, and demonstrating to their parents, various math activities. What a great idea!

Constructivism and Teaching Mathematics in the Classroom

There is much debate today as to when constructivism first began, what exactly is included in a constructivist classroom and how to run these classes smoothly. In a constructivist classroom there needs to be an open environment, one in which children feel free to express themselves in new and creative ways. This is important and one of the main reasons why play and experimentation is so important. We all learn through our environments; therefore, what better way to learn than through direct interaction with the environment.

The same goes for the math classroom. Exposing children to manipulatives and other items which will help them during math class (although sometimes frowned upon) is an excellent, and pretty standard, constructivist practice - but there has to be a distinction as to why one is using or doing so. In the article “Middle School Mathematics Reform: Form Versus Spirit,” Gary Tsuruda (1990) states that in the traditional way of teaching, students that are exposed to manipulatives only once or so and this is not enough to help students. Manipulatives can be used to help with almost every topic in mathematics, but their effectivness will be determined by the way in which the teacher incorporates them into his or her lesson.

Although on the otherhand, one might ask is constructivism really as beneficial as some people think? According to one study done by Insook Chung at Saint Mary's College, Notre Dame, Indiana, the difference between two schools of instruction and learning (constructivist vs. Traditonalist) was non-existant. In fact, it showed that students really had little difference while being taught multiplication in a traditional or constructivist method. For us, this raised the question of does constructivism really make a difference or was the constructivist classroom created properly? As, Gary Tsuruda states, a classroom can have the look of a constructivist classroom, but does not have to run like so. For example, the desk can be put into groups our, yet no students have to work together ( Tsuruda, Pg. 6). Clements (1997) raises similar issues when he discusses the teachers who practice "constructivism on fridays."

The fact of the matter is that constructivism is a philosophy, a practice and an atmosphere. Students have to be shown that they are permitted to help each other and are able to work together to solve problems ( Tsuruda, Pg. 6). In our experience through our observation days, we have come into contact with all kinds of new and motivating ways to promote independent learning for students. One example that comes to mind is allowing students the opportunity to express their answers in different ways and not be condemned for doing so. This is constructivist learning at its finest. Pentominoes are a great way for children to discover new ways to express their answers. This involved students creating different shapes out of blocks, which taught the concepts of shapes and patterns, while not restricting children to simply read a text book.

There is more support for constructivist learning now, more than ever before. This is especially true since there has been more studies done in the past decade that study and promote the effects and benefits it has for students and all learners. One thing that comes to mind is Gardner’s Theory of Multiple Intelligences. This has had a big impact on the way some people teach there class. This concept also ties in with the idea of constructing knowledge. Because if students are permitted to complete tasks in ways that help them learn and process information more efficiently then are we not doing our job as teachers? This is the challenge, but as educators we must try and overcome to the best of our abilities.

Myths about Constructivism

With the great debate on the relevance of constructivism there have been many questions raised and many myths formulated. The following are some of the myths about constructivism according to Douglas H. Clements in his Article “Constructing Constructivism”:

1.) Students should always be actively and reflectively constructing.
Clements states:
-"Our minds actively construct ideas without our “working at it” or even being conscious of it.
-There is a time for many different types of constructing:
- Time for “experiencing”
- For “intuitive” learning.
- For learning by listening.
- For practice.
- And for conscious, reflective thinking.
*We need to Balance these times to meet our students goals."*
2.) Manipulatives make learners active.
"Clements states that manipulatives are helpful unless teachers use manipulatives to impose prescribed procedures for routine problem types, causing them to learn to use manipulatiaves only in a rote manner."
3.) Constructivist are lonely learners.
Clements notes that learners do not build their ideas in “isolation.”
4.) Cooperative learning is constructivist.
“ The way students think and interact is more important than the size of the group in which they work. Just using cooperative groups does not necessarily make teaching more “constructivist” (Clements, Pg. 2).
5.) Everybody’s right.
“Everybody’s effort can be respected without abandoning the notion that some solutions are better than others and that some just do not make sense” (Clements, pg.2).

All information has been retrieved from Douglas H. Clements article “Constructing Constructivism”
http://investigations.terc.edu/relevant/MisConstructing.html

Here's a funny perspective on just how "right" some students may think they are. It's just a reminder that sometimes students' work may be logical, but still incorrect!

Components of Constructivism

There are many components to constructivism and the constructivist classroom. As Clements and Battista state in their article “Research into Practice: Constructivist Learning and Teaching.” These five items are:

1.) “Knowledge is actively created or invented by the child, not passively received from the environment.

2.) Children create new mathematical knowledge by reflecting on their physical and mental actions.

3.) No one true reality exists, only individual interpretations of the world. These interpretations are shaped by experience and social interactions.

4.) Learning is a social process in which children grow into the intellectual life of those around them (Bruner, 1986).

5.) When a teacher demands that students use set mathematical methods, the sense-making activity of students is seriously curtailed.”


All the above information can be found at http://investigations.terc.edu/relevant/ConstructivistLearning.html


Two Major Goals of Constructivism:

According to Douglas H. Clements and Michael T. Battista there are two main goals of constructivism and they are:

1.) “ Students should develop mathematical structures that are more complex, abstract, and powerful than the ones that currently possess so that they are increasingly capable of solving a wide variety of meaningful problems”.

2.) “Students should become autonomous and self motivated in their mathematical
activity. Such Students believe that mathematics is a way of thinking about problems.”

Clements and Battista’s article "Research into Practice: Constructivist Learning and Teaching" can be found at http://investigations.terc.edu/relevant/ConstructivistLearning.html

We've all been there...

One of the advantages of the internet is that we now have access to videos. This one, found on youtube.com perfectly exemplifies how the majority of us were taught mathematics. No constructivist teaching - we were all simply sponges absorbing information.

The History of Constructivism

Although prior to their formal education, many pre-service teachers have probably never heard of the theory, constructivism quickly becomes a staple concept during their university careers. In fact, pre-service teachers have probably, at some point during their formal education, had to write a paper or conduct a mini-lesson on the topic. But do they really know what it is? Ann E. Kajander (1999) doesn’t think so. Her work with pre-service teachers has shown her that teachers still believe there should be a reliance on technical skills in mathematics, and they fail to see how math can be made enjoyable (“Creating Opportunities for Children to Think Mathematically”). Well, let’s change that and get some things straight!

So, what is this “new fangled” idea of constructivism? Well, to start, constructivism and constructivist principles are not new at all. In fact, some proponents trace its roots all the way back to eighteenth century philosophers Emmanuel Kant and Giambattista Vico (Jaworski, 1994, p. 14). Granted, a lot has changed in 400 years, but constructivism has stayed on the forefront largely due to the efforts of one psychologist in particular: Jean Piaget. Piaget’s work with genetic epistemology is credited as the conceptual root for modern constructivism; Jaworski (1994) adds that Piaget, alone, has been perhaps “the greatest single influence on education generally, and mathematics education in particular” (p. 15). However, this should come as no surprise to any university student who has taken a psychology course.

Don’t let “genetic epistemology” slip you up; to see the connection between modern day constructivism and Piaget one need only look in their educational psychology books. Piaget’s “Constructivist Approach” is a staple in every text. Therefore, terms such as schema, accommodation and assimilation should be well known by the majority – if not all – pre-service teachers. Despite the big words, the theory is simple and should not be forgotten. If we break it down to the brass tacks, Piaget believed that “much of our logic comes not from without but from within by the force of our own logic…” (Copeland, 1970, p.12). Currently, constructivism is viewed as “learning [that] rejects the notion that children are blank slates who absorb ideas as teachers present them. Rather, the belief is that children are creators of their own knowledge” (Van de Walle & Folk, 2005, p.28).

One might ask, but has this been enacted in schools? Surely, the majority of people can remember their own childhood education and remark that, no, they most certainly did not participate in any form of constructivist teaching. And sadly, that is usually the case. Constructivism, as a teaching “method” did not become popular until the 1960’s in the United States, and was certainly not legitimized until the NCTM (National Council of Teachers of Mathematics) grounded their mathematical standards in the principles of constructivism in the late 1980’s (Jaworski, 1994, p. 14). Nearly 3 decades later, this has held true and even crossed borders.

The Foundation for the Atlantic Canada Mathematics Curriculum maintains a vision that “fosters the development of mathematically literate students who can extend and apply their learning and who are effective participants in an increasingly technological society.” To accomplish this, the foundation relies on the standards of the NCTM as a “guiding beacon for pursuing this vision.” (Atlantic Canada Mathematics Curriculum – Grade 4, 2003, p. 1). And so it is, that constructivism, with roots in the 1700’s has propelled itself into the twenty-first century so that a small province off the coast of mainland Canada holds it as a “guiding beacon” in their aim to teach mathematics effectively.