Van De Walle, Ch. 3 on Reflective Thinking

I am using a process called "writing to learn" which is the way Dr. Van De Walle intended Elementary School Mathematics to be read.  Since each question asked by Van De Walle for this chapter has  been so thought provoking, it hasn't made sense for me to simply move on from chapter 3 without a fuller exploration of the essential questions.  After three days, I'm ready to elaborate on the six ways in this chapter Dr. Van De Walle suggested "reflective thought" could be promoted.


Here is how Dr. Van De Walle defines reflective thought:

Active thinking about or mentally working on an idea can be called reflective thought.  (Van De Walle, 29)

Here are the six ways suggested to promote reflective thought: (Van De Walle, 30)

1. Create a problem-solving environment.
2. Use models:  manipulatives, drawings, calculators.
3. Encourage interaction and discussion.
4. Use cooperative learning groups.
5. Require self-validation of responses.
6. Listen actively.

Create a problem-solving environment

Creating a problem-solving environment is the first thing Dr. Van De Walle suggests that is needed to promote reflective thought.  Dr. Van De Walle is not proposing that by chaining students to their desks and forcing them to "solve" lots of problems they will become better mathematicians.  In fact, fewer problems may be better.

In The Culture of Education, Jerome Bruner writes,

One of the great triumphs of learning (and of teaching) is to get things organized in your head in a way that permits you to know more than you “ought” to.  And this takes reflection, brooding about what it is that you know.  The enemy of reflection is the breakneck pace – the thousand pictures.  (Bruner, 129)

The key differences Van De Walle has with traditional math instruction in the United States involves the relationship of the learner to new information, the relationship of the teacher to the learner, plus a focus on developing relational understandings versus instrumental understandings.  Here, we avoid problems associated with teaching knowledge in a way that is "a mile long and an inch deep," or teaching math procedures in an unconnected way.  How would Dr. Van De Walle or Bruner evaluate the pacing guide / data-driven classroom?  

The 5 other ways of promoting reflective thought are components of a problem-solving environment.  My question is where in pacing guides is there time for instructing math the way Dr. Van De Walle intended?

Each of the 5 other ways of promoting reflective thought are components of a problem-solving environment.

Use Models:  manipulatives, drawings, calculators

In Jean-Jacques Rousseau’s Emile, Rousseau writes:

In any study whatsoever, unless one has the ideas of the things represented, the representative signs are nothing.  However, one always limits the child to these signs without ever being able to make him understand any of the things which they represent.  (Rousseau, Book II, p. 109)

Models are math tools which represent math concepts physically; they enable students to internally grasp relationships embodied in the tool. (Van De Walle, 30) A flat, for example, can represent one hundred or one, depending on the concept being modeled. 

Here are a few exemplars Dr. Van De Walle uses to demonstrate how models can be used to help a learner construct relational knowledge in his or her own mind. (Van De Walle, 31)

Model	Relationship
Counters	The concept of six can be demonstrated with counters and number mats: it relates to counting words one, two, three, etc.; also, it relates to the idea of one more than five.
Base-10 blocks, sticks and bundles	The concept of hundred relates to the relationship of the flat to the unit, not to the flat in isolation.  Sticks and bundles can also show place-value relationships.
Rods	The concept of length can be shown as a comparison between two objects.  Rods can be used to measure the spaces within objects.
Spinners	The concept of probability can be shown as a relationship of the spinner to comparative outcomes
Dot grid paper	The concept rectangle can be illustrated with dot paper to show it as a relationship between length and space.
Calculator	Concepts of pattern can be shown using calculators:  we can test relationships between starting and jumping numbers using repetitive entries.
Number line and arrows	The concept of integers can be shown on a number line with arrows to direction and length.

Learners need opportunities to handle these tools and “change, move, count, compare, draw, measure” with them to build mental schemas. (Van De Walle, 30)  Thus, Dr. Van De Walle suggests they be used in three parts of a lesson: (Van De Walle, 32)

1.   To introduce unit concepts.

2.   To help children connect concepts to symbols as they record their observations and defend their answers.

3.   To assess understanding – if math tools were used in instruction, student should be able to use them when demonstrating understanding.

Encourage Interaction and Discussion

When a child explains an idea, several things happen.  First, a child organizes her thoughts, thinking about what she knows.  Next, she considers how she will explain her understanding to others:  Will she draw a picture or some other model?  What words will she use? What does her listener understand?  What questions remain to be answered?  In writing, people are forced to clarify their thinking, which is why “writing to learn” is so important.   (Van De Walle, 30) Likewise, student talk is an essential part of a problem-solving environment, because it enables students to build connections.  A classroom management plan that enables these kinds of discussions to regularly take place  is critical.

Cooperative Learning Groups

Dr. Van De Walle suggests that students learn best in groups of 3 or 4 because it maximizes the level of interaction and discussion that can occur in a classroom.  Similarly, my experience with guided reading was that the program worked best when the class was organized in groups of 4.  Dividing the class into groups enabled me to differentiate instruction, and provided regular scheduled opportunities for small group instructional opportunities.  When I taught a 4^th grade class in Prince William County, my Literacy Coach and I used Guided Reading Levels as a starting place, but groupings were flexible.

Math students need similar kinds of opportunities for structured small group learning.  Within a Guided math format, students could rotate through different “centers”, discussing the nature of math problems, using math tools to help develop understandings, and explain their answers while I listen in and facilitate discoveries one group at a time.  Another specialist or Instructional Assistant could set up shop in a different part of the room.  For this to happen, routines would need to be established over the first 6 weeks.

Require Self-validation of Responses

A good explanation requires evidence.  A requirement that students use representations and logical arguments to defend their answers in class should help eliminate bad habits such as guesswork and mechanical answer finding.  The reading strategy QAR (question answer relationships) provides a framework for defending an argument.  Is the answer right there in the text (a simple addition problem)?  Do I need to think and search for clues (find key ideas in a story problem, collect data, and draw a picture)?  Is it a relationship that I just know in my head (already mastered math facts)?  Is it a relationship that involves the author and me (do I need to think about what the author does and does not tell me and need to make inferences)?  Self-validation is a skill that can be taught and reinforced in small group instruction.

Listen Actively

In a problem-solving environment with lots of reflective thinking going on, the relationship of students to their learning and to their teacher must necessarily change.

When children respond to questions or make an observation in class, an interested but very nonevaluative response is a way to ask for an elaboration.  “Tell me more about that, Karen,” or “I see.  Why do you think that?”  Even a simple “Um-Hmm, followed by silence is very effective, permitting the child and others to continue their thinking.  (Van De Walle, 31)