Key Instructional Strategies

Authentic math questions-ones that encourage students to think about all aspects of the problem-are not solved in five minutes. The typical "word problems" given to students merely require routine applications of the concepts currently being studied-whether they involve multiplication or algebra-in which students must simply choose the correct algorithm in order to solve the problem. They rarely make students think about the issues surrounding the problem. For this reason, we strongly suggest that students engage in extended problem-solving activities. This will enable students to see how mathematical concepts are applied in natural contexts in which the mathematics of the problem are not readily evident.

The key instructional strategies for extended problem solving can be broken down into three steps:

 

Step 1: Problem Identification and Clarification

Unless you are an expert in a specific area, authentic math problems generally do not have immediate answers because the nature of the problem is not clear (e.g., there are too many loose ends, not enough is known at the beginning, some parts of the problem are just fuzzy) or data need to be collected. The teacher's first job is to help students re-frame and clarify the problem through classroom discussion. This is perhaps the most difficult step in the problem-solving process: Students will tend to adopt a passive role and let teachers do the work for them, so the teacher must find a balance between keeping the discussion going and not giving students the answers.

We do not have enough space (nor is it the purpose of this strand) to provide a complete "how to" for conducting classroom discussions. Rather, we will provide a range of strategy options that a teacher may consider. Throughout this discussion, we will refer to complete problems that are presented in this strand.

In presenting the problem, the teacher needs to help students see that a common activity such as donut sales before school, selling t-shirts so a club can go on a field trip, or, in the case of social studies, examining data about countries involves a range of issues that are not readily apparent. To get the discussion started and to keep it going, the teacher should consider the following techniques:

 
Elicit Background Knowledge Ask students what they know about the activity, problem, or subject under discussion. In the case of donut sales, you might ask if anyone knows who is selling the donuts, why they are being sold, or what the goal is. In the case of comparing different countries and their standard of living, ask students if they have traveled or done reports on different countries.
 
Carefully Examine Specific Dimensions of the Problem The teacher often needs to formalize different dimensions of a problem and show how there are mathematical issues involved. In the case of donut sales, students need to think carefully about the cost of the donuts to the vendors and the relation of this cost to the sales price. Be careful not to rush to immediate solutions. In the donut problem, as students think about profit they also need to think about the loss associated with leftover donuts. Thus, the teacher needs to continue developing dimensions of the problem so that students see it clearly.
 
Model Questioning and Reasoning It is crucial that the teacher model how to ask questions about the problem or the available data. In some cases, this is merely a matter of showing how something in the problem requires clarification. For example, in the donut problem we need to ask, "What is the goal of these Wednesday morning sales?" or "Would it be useful to know how much the group makes on a weekly basis?" One of the key purposes of modeling questions is to show how one question either clarifies the problem or leads to another important question.
 
Other questioning and reasoning techniques involve: 1) reasoning by analogy (e.g., "Maybe it would be easier to understand this problem if we thought about it this way.") 2) presenting an easier version of the problem first (e.g., "Let's just say the group can buy the t-shirts for $6 each and they can sell them for $10. That would be a $4 difference, and we could call that profit.")
 
Representing the Problem Be sure to use the blackboard or overhead to document key ideas or relationships. This running record will help students visualize the problem, and it is an important reference point for the discussion. For example, in the countries problem, we can first look at life expectancy. Once we look at the data, it leads us to ask questions about average number of family members, infant mortality, and other related concepts. By writing these concepts on the board and linking them, the teacher can develop important relationships that students can eventually explore in the data. Text, diagrams, or what are often called "semantic webs" are important aspects of problem solving.
 
Scaffolding Rather than telling the students the important aspects of a problem, it is critical that the students actively engage in articulating the issues. Students must be able to talk freely about their ideas and the teacher can help by clarifying them. Therefore, the teacher needs to create and maintain a classroom environment in which students feel safe in stating their ideas (e.g., they won't be put down by others). You will also want to guide students in their thinking, paraphrase their statements, and encourage them to elaborate.
 
One useful technique is to have other (perhaps quieter) students restate what someone in the class has just said or summarize the discussion to this point. Teachers can also selectively summarize as a way of making explicit the important ideas developed in the discussion. We call this "adding closure" to the discussion.
 
Brainstorming Next Steps Once students have fully discussed the problem, it is important that they brainstorm the next steps. In some cases, they will need to collect data. This would be true in either the donut or t-shirt problem. Students will need to develop a complete set of questions that they will ask the vendors or the club about their activities. This could entail a formal questionnaire. In the case of the social studies problem, they will need to note what categories of information are important as well as how they will be represented using the spreadsheet (e.g., numerical averages, bar charts). The listing of what to do next as they collect and analyze the data is a logical link to Step 2 in our process.

 

Step 2: Analysis of the Information or Data

Step 2 has three major activities associated with it. First, the teacher needs to work with the students on:

All of this will involve ongoing discussions. Together with the students, you can summarize findings, think through steps for further analysis, and organize information in a way that will lead to a clear explanation or way of presenting findings. It is important that you follow the advice offered in Step 1. That is, the students play a critical role in summarizing information or determining what kind of analysis needs to be done. If the teacher is overly directive in these activities, students are likely to adopt a passive role and wait to be told what to do next.

It is also important that the teacher prod students on a regular basis as to whether or not they have a complete picture of what they are studying. Students need to be asked, "Have we collected enough information?" "Have we analyzed all of the information?" "Is there another way in which we can think about this problem?" "Is there a better way to summarize or explain this information?"

How Data Will Be Collected and Organized We have provided a number of guides or "think sheets" to help students collect and organize their findings. All of these guides can be found in guides.doc as part of the files to download in each of the sample lessons. The initial guides direct students to questions to be answered and any relevant data associated with those questions. This can be done in a variety of ways, as the think sheets indicate. The sheets are a running record of important findings or statements about the problem.
 
We also recommend that you use the grid worksheets (gridport.doc or gridland.doc) that we have used in the Math Concepts strand because students are actually going to organize and enter the actual data into the spreadsheet. Like the guides.doc, you will be able to download these files from each of our sample lessons or you can find them under Spreadsheet Basics in the Math Concepts Strand.
 
How Data Will Be Analyzed We assume that the students' time in the computer lab will be limited. Therefore, it is important that they know what the data are, how the data are organized, and have a plan for analyzing this. All of this can be done in class prior to the lab. Again, the guides are an important vehicle for helping students make their time in the lab as productive as possible. The most important part of the guide for this phase of the lesson-the Data Analysis Guide-is the last step. Students need to think about the data and determine what kind of mathematical operations need to be applied. Will they just total the data? Find averages? Calculate percents? Convert decimals to percents? These are all questions that you can explore with students using this guide before they go to the lab. This will reduce the amount of "What do I do now?" questions that typically arise in the lab.
 
A Framework for Explaining or Presenting the Findings The last activity in this step is to organize the findings so that students can write them up for a specific audience. This activity is a bridge between Step 2 (Analysis of the Information or Data) and Step 3 (Clear Communication of the Results).
 
To complete this activity correctly, it is important that you work with students in developing an organizational or "text structure." There are a variety of text structures that are used in writing up expository or non-fictional material. For a long time, teachers have had students use a "Comparison-Contrast" or "Cause-Effect" structure to write about a topic.
 
The text structure that we feel works the best with the data collection activities found in Examples 1 and 2 is the "Findings - Recommendation" structure. In essence, students translate their findings into statements that tell what they found and what they recommend.
 
To do this, use the guide "Finding-Recommendation Pattern Guide" found in guides.doc. Have them first restate the problem they are trying to analyze. Next, point to the key words at the top of the guide. They can be used in the statements that follow as a way to reinforce the findings or recommendations. These are often called "key words" in text structure writing. The findings section should be relatively straightforward at this point, but you may want to list findings on the chalkboard first before students write them on their guides.
 
Next, conduct a discussion about recommendations. This will be important because many recommendations are possible. Again, use the board to record possible recommendations. At some point, you may want to analyze each one for its feasibility, likelihood of being adopted, etc. Once you have achieved some consensus on the recommendations, students can use the last half of the "Finding-Recommendation Pattern Guide" to record their recommendations.

 

Step 3: Clear Communication of the Results

Step 3 focuses on clear written communication. It can be adapted if students are making an oral presentation using note cards or a presentation program such as Powerpoint™. In this strand we will focus on written communication in the form of memos and short essays.

If you choose to have students write a memo or essay to explain their results, there are 2 key activities involved. First, they transfer their findings and recommendations (or different writing structures such as cause-effect, comparison-contrast, problem-solution) to a form that facilitates writing. We have provided a "Summary Sheet" in guides.doc for this purpose. This sheet is optional, and you may want to move directly to a first draft where you emphasize key aspects of writing such as topic sentence, supporting information, and use of key words.

The second and final activity is a checklist to make sure the final draft is of high quality. In the case of a memo, it may be helpful to collect or create copies of professional memos to serve as models for this activity. A checklist for the memo is included in guides.doc. Various parts of this final checklist can be developed for oral presentations, particularly if you are having the students make Powerpoint presentations.