Reflections from a Third-Grade Classroom

Van de Walle’s chapter on Problem Solving has been incredibly helpful in planning for my own teaching. He carefully deconstructs the methods used by effective teachers to and scaffolds beginning teachers well as they begin experimenting with lesson plans. Though the focus of the book is math, these methods could easily and effectively be applied to any subject area. Here, I am choosing to highlight three that I found particularly interesting, particularly within the context of planning my own problem solving lesson, which asks the following question:

Farmer Bob raises cows and chickens on his farm. One morning, he walks into the barnyard and counts 18 legs. How many cows and how many chickens are in the yard that day?

The first is inteded to prepare students for the pending problem-solving session; the second is to be used during the session itself; the third helps students develop metacognitive awareness and demonstrates that thinking continues on after the solution of the problem posed.

BEFORE: Begin With a Simple Version of the Task

As I read this chapter, this suggestion struck me as being particularly effective for my teaching scenario. I was a little concerned with the students ability to solve such a difficult problem. However, this presents the perfect introduction. By removing one variable, and only asking about the number of chickens in the barnyard, I can make the problem much more accessible to students. This preliminary activity will mentally prepare my students for the types of thinking they will be asked to engage in during the more intensive part of the problem-solving session. It also draws attention to the fact that there are two variables to consider when presented with the “real” problem. In addition to helping my students evaluate the problem, it allows me to evaluate the students. How are they able to grapple with this problem? What challanges does the inital, “simplified,” version present?

During this introduction, I think the teacher should be both modeling and scaffolding the simplified version of the problem so that students are prepared when “turned loose” to grapple with the more complex situation. Ideally, in my mind, ideas and suggestions should be recorded on the board, without judgement or praise. By refraining from such reactions, the instructor is able to elicit the widest possible array of answers. These varied solutions (even those that are incorrect) could be an incredibly useful reference tool as children begin to experiment and test their ideas. To be sure that all answers and theories are recorded correctly, it is imperative that the teacher ask lots of follow-up questions, rather than summarize what (s)he thinks the student is saying.

DURING: Encourage Testing of Ideas/Find a Second Method

Although these are listed as two different bullet points within the chapter, they complemented each other so well that I felt compelled to point out both, here. When conducting a problem-solving session, it is very important to be able to let go and allow children to investigate any avenue that might occur to them. Teaching them to trust and investigate their own ideas is a powerful skill that will carry them through all endevors, both academic and otherwise. However, the teacher cannot simply disappear during this investigation. Rather (s)he must be available both to encourage and challenge.

For the problem posed above, these two methods work together to do just that. Children, who are very perceptive and have picked up on the fact that there are “right” and “wrong” answers in math, often want to know if they are doing the problem right. Rather than providing encouragement or suggestions for improvement, it is important to really let go and allow their instincts run the direction of the investigation. Encouraging the testing of ideas is one way to remain neutral. If a student asks, “Is this the right answer?”, the teacher should counter with, “Well, let’s see. Can you think of a way to test that and prove if it’s right or not?” If the students are particularly independent, it can still be effective to remind students to see if their thoughts make sense. One possible way to do this without making a value judgement is simply to remind students to explain their thoughts, not merely the steps.

If the students are having trouble testing their answer (or if they solve it incredibly quickly), a great strategy is to ask that they find a second method of doing the problem. This suggestion is particularly effective as it works in two different ways, depending on the student’s level of understanding. If a group (or student) is having difficulty finding a way to test an answer to see if it is correct, finding a second way of doing the problem can help provide a proof. In this way, the suggestion works as a scaffolding tool. However, if the group/student solves the problem quickly and is at a more advanced level of understanding, this same suggestion turns into a differentiation tool to provide additional challenges.

AFTER: Identify Rules, Hypotheses, and Future Problems

I found the suggestion to “identify rules, hypotheses, and future problems” particularly interesting as a way to conclude the lesson. It’s important (for both the teacher and the students!) to remember that thinking does not cease with the solution to a problem. Any scientific investigation should end with the generation of a new set of questions, and asking the question is often the hardest part of any problem. So the goal should not be, “What was the answer to that problem?” but rather, “What implications does this have?” Additionally, the process of sharing information and participating in a whole-group approach to inductive reasoning teaches students how to become active learners in their communities. They begin to develop a critical approach to discourse, where evaluation and theorizing can occur.

The teacher must once again work to maintain a neutral stance during this process. Refraining from commentary may initially stifle discussion, but if practiced continually, students will begin to provide answers without constant affirmation. This puts the focus of the discussion on the process rather than the result of inquiry. Suggestions (incorrect or not) without any rationale behind them should be recorded as “[Student's] Hypothesis.” At this point, the floor can be opened, and further investigation can occur as classmates struggle to prove or disprove the theory. Even if actual hypotheses never occur, such discussions can still generate some fascinating questions. The teacher should carefully listen to explanations to identify any questions that my be formulated, though never explicitly stated.

I feel that as a whole, this chapter has been very helpful in providing a framework within which I can develop my own problem-solving lesson. I look forward to testing the practical application of these suggestions in the near future!