What do we teach math for if it is not to have students use their mathematical knowledge and understanding to solve problems? I used to tell my students that their employer was not going to ask them to do a straight computational problem such as, 346,328 - 132,004. No, they were going to ask them how many widgets were available to ship after the Acme Company’s order of 132,004 widgets was filled.
Now some of you will argue (and I would argue) that the problem above is not problem solving for let’s say a fifth grader, it is really more of an example of a contextual math problem. But, I was making the point that we should be teaching math in the way that students will use the math so that they will have a context within which to remember what to do with their math knowledge and skill, now and in the future. I would also say that at some levels of ability, the widget problem could be considered problem solving.
So let’s look at what the research says about problem solving. There is a ton of research on problem solving but I am going to really focus on cognitively demanding tasks and how they may differ from story or word problems. Margaret Smith and Mary Kay Stein are researchers from Pennsylvania who study the professional development of teachers and teaching and learning as they occur in classroom settings. Smith and Stein wrote an article for Mathematics Teaching in the Middle School (NCTM, 1998) called, Selecting and Creating Mathematical Tasks: From Research to Practice. There are some really key points from this article that I want to highlight. I would also encourage all of you out there who are members of NCTM or who have access to a professional data base to get this article and keep it for reference.
In this article, Smith and Stein talk about the most important elements for meaningful mathematics teaching/learning. They refer to the QUASAR project and the data that was collected from this five-year study that looked at teaching practices that included the use of small groups, the tools (manipulatives) that were available for student use, and the nature of the mathematical tasks the students solved. They found, “...that the highest learning gains on a mathematics-performance assessment were related to the extent to which tasks were set up and implemented in ways that engaged students in high levels of cognitive thinking and reasoning.” So basically what this means is that the level of difficulty of the tasks to which students are exposed determines what they learn. They go on to say (and this is really a key point) how important it is to begin with a high-level, cognitively complex task if the ultimate goal is to have the students develop the capacity to think, reason, and problem solve. Isn’t this what we are trying to achieve with our students??? Smith and Stein also say that while selecting and setting up a high-level task well does not guarantee students’ engagement at a high level, starting with a good task does since low-level tasks almost never result in high-level engagement.
What does a high-level task look like? How does it compare to a low-level task? Will we know the difference between high and low-level tasks? First we have to consider the students’ age, grade level, prior knowledge, etc. Smith and Stein used the example of a task that asks students to add five two-digit numbers and explain the process. For a fifth or sixth grader who knows and understands the traditional algorithm, “explain the process” would be considered routine and low-level. But for a second grader who has just started to work with two-digit numbers, and who is still grappling with re-grouping and place value, and who is asked to use base-ten pieces to “explain the process” this could be considered a high-level task.
Smith and Stein use four categories of cognitive demand when considering tasks:
Memorization (lower-level demands) - regurgitating facts, rules, formulas, or definitions, or committing those to memory; tasks that involve exact reproduction of previously seen material and what the student is reproducing is clearly and directly stated (no ambiguity); or there is no connection to the concepts or meaning that underlie the facts, rules, formulas, or definitions being learned or reproduced (think cross-multiplying fractions to determine equivalence without learning the underlying math).
Procedures without connections (lower-level demands) - use of an algorithm when it is specifically called for or its use is evident from prior instruction, experience or the nature of the task in relation to the concept being taught such as only providing addition word problems when teaching addition; requires limited demand for a successful completion and there is little doubt about what needs to be done and how to do it; there is no connection to the concept or meaning tied to the procedure being used; there is more emphasis on producing a correct answer than developing mathematical understanding; no explanation is required or the explanation focuses solely on the procedure that was used.
Procedures with connections (higher-level demands) - student attention is focused on the use of procedures for the purpose of developing deeper levels of understanding of math concepts and ideas; broad, general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that may not have clear connections to the underlying concepts; the use of multiple representations such as visual diagrams, manipulatives, symbols and problem situations (making connections among multiple representations helps develop meaning); some degree of effort is required and procedures can be used but not followed mindlessly and students need to engage with the conceptual ideas that relate to the procedures to complete the task successfully and that develop understanding.
gDoing mathematics (higher-level mathematics) - requires complex, non-algorithmic thinking, a predictable and clear approach is not apparent and there is no example to follow; requires exploration and understanding of the nature of mathematical concepts, processes or relationships; metacognition (self-monitoring of one’s own thinking) is required; students must access relevant knowledge and experiences and use them appropriately through the task; requires students to analyze the task and examine constraints that may limit possible strategies and solutions; it requires considerable effort and may involve some anxiety for students because of the unpredictable nature of the solution process required.
I know that this post has been a long one, but the information in it is so important that I didn’t know what I could leave out. I wanted you to have something to measure how cognitively demanding the tasks are that you are asking students to complete. While there may be a place for all four levels of tasks in our classrooms, I think you will agree that we want most of the tasks we give to our students to be higher-level. More on this later!