05. Mathematical Understanding: An Introduction

        Many people associate math as something very negative.  Very often the principles of How People Learn are overrided in math instruction.  Instead of building on and refining mathematical understanding, it is ofter replaced by sets of rules disconnected from problem solving.  These rules are often the center of instruction, instead of core mathematical ideas.  For these same reasons, metacognitive skills are rarely engaged.  An Adding It Up report by the National Research Council identified five strands for mathematical proficiency in elementary mathematics.  They include conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition.  


        Even at very early ages, children show understanding of numbers.  Also, studies have showed that lay people who work with numbers successfully have trouble completing the same concepts in abstract, school-like settings.  This all suggests that people have mathematical reasoning and strategies available but the bridge between informal and formal instruction is not automatic.  The first principle suggests that this bridge be built.  

Some Common Preconceptions About Mathematics

Preconception #1 - Math is about learning to compute: While computation is certainly important, it needs sense-making at the very least for guidance.  When asked to add 7/8 plus 12/13, using sense making one could say it is a little less than 2.  If just computing and a little mistake occurs, a totally different answer can appear.  

Preconception #2 - Math is about "following rules" to guarantee correct answers: Rather, math is a dynamic field that is constantly evolving and designed to solve the problems of the day.  For this reason, the abacus was used, but later replaced by paper and pencil and now by calculator.  Each procedure has advantages and disadvantages depending on the problem to be solved.   Comparisons of these, for example, can deepen students' understanding and skill.

Preconception #3 - Some people have the ability to "do math" and some don't: This is not only commonly held, but also becomes a self-fulfilling prophesy.  Furthermore, there is an interplay between effort and natural ability as being the key variable in success.  Some teachers want students to struggle to solve problems, others try to simplify for the students.  Not having math experience grounded in nature leads to rules without conceptual frameworks which are easily forgotten.  

Engaging in Student's Preconceptions and Building on Existing Knowledge

How do we teach that math is not just a set of rules, but clever human inventions to solve problems?  How can we link math training with students' informal knowledge?  While there is no single best instructional approach, there are certain features that support these questions:
  • Starting with informal reasoning and guiding students towards more effective strategies
  • Encouraging math talk so students can clarify their strategies to themselves and others
  • Designing activities that bridge preconceptions and targeted math understandings 
        One approach is to allow multiple strategies.  Give a problem and allow students to find their own ways to solve the problem.  Then, open up the classroom with discussion on each of the strategies, evaluating which may be correct or incorrect and why.  
        Math talk is a technique to make students' thinking visible.  While it seems obvious, it is rarely employed in traditional classrooms.  Community-centered aspects of instruction are important, so that students feel comfortable sharing their preconceptions, even if they are wrong.  They can even use pictures.  
        Research has uncovered many misconceptions that can be addressed preemptively.  Activities such as having students represent situations in different ways - words, notation, pictures, etc. - help students make connections with their existing understanding.  


        The debate over procedures and conceptual understanding is most hotly debated in mathematics.  Both are equally important, however, as without conceptual understanding students do not become problem solvers, and without procedures the problem is not solved, just changed.  Rather than seeing teaching as either-or, both must be closely linked.  This is often accomplished by beginning with informal concept development and slowly abstracting and generalizing the procedures, at the same time allowing for multiple methods.  

Developing Mathematical Proficiency

        Multiple methods does not need to be taught for every domain, but certainly allowing for alternative methods which arise in the classroom can facilitate flexibility and understanding.  Some countries purposely give problems which are conducive to multiple solutions.  Less advanced classroom students must be considered as well, and for them the most accessible methods may be preferred.  
        To teach in a way to truly develop proficiency, primary concepts must be clear to the teacher.  Adults who were taught procedurally may have difficulty identifying the core conceptual understandings.   Research has begun identifying the key conceptual hurdles of students in the various domains  (i.e. rational numbers, functions), and these should be used.   


        The idea that people are "non-mathematical" is pervasive.  It is a weak intervention to tell students that "though this may be difficult now, it will get better in the future."  Rather, students need to be successful in finding solutions and truly understanding.  Research on metacognition suggests that even more must be done: students need to reflect on their experiences and see their ideas as larger categories of ideas.  

Instruction that Supports Metacognition
        Facilitating classroom discussion on students' thinking facilitates teacher and student assessment.  Also, students can learn specific strategies, such as making a drawing or ask yourself questions that can help with self-monitoring.  
        Metacognition is also facilitated by moving beyond just what is right and wrong to a more detailed focus on "debugging" wrong answers.  Their is plenty of research showing that getting students to talk about mathematics is very important.  This can help them compare methods, ask questions, coach each other, etc.  This works among all age and cultural groups.  Teachers must, however, facilitate this working together process.  Students must also feel comfortable seeking help when they are stuck. 

The Framework of How People Learn: Seeking a Balanced Classroom Environment

        The recent TIMSS study showed that the US is overwhelmingly traditional.  Traditional teaching has tended to emphasize knowledge networks and pays insufficient attention to conceptual supports and the need to build upon learner knowledge.  On the other hand, an overemphasis on learner-centered teaching results in insufficient attention to connections with valued knowledge networks.  Students can't invent all their methods, but can get focuses on sense making in a balanced way, whether the student invented it or not.  Not all methods are equal as well, and therefore must not be treated as such.  A discussion of advantages and disadvantages must be shown.


       Even without curricular supports, teachers can drastically improve their practice by following some of these principles.  There are several community-based methods, such as lesson-study, that facilitate such teacher development.  3 case studies are shown to demonstrate some of the principles in action in the chapters that follow.    


As you are reading/listening/watching, make note of things that you find interesting, important, or insightful.  When you finish, choose 3 of these items that most stand out for you.

  • For one of these, choose a colour that you feel best represents or captures the essence of that idea.
  • For another one, choose a symbol that you feel best represents or captures the essence of that idea.
  • For the other one, choose an image that you feel best represents or captures the essence of that idea.

With a partner or group first share your colour and then share the item from your reading that it represents.  Tell why you choose that colour as a representation of that idea.  Repeat the sharing process until every member of the group has shared his or her Colour, Symbol, and Image.

1.  Color - White

White is the presence of all colors, and this captures the idea that in math it is important to value the thinking and conceptions of students, then to work from there and build their understanding.  As such, it is important to encourage various methodologies to solve problems, or, to absorb all the colors.  

2.  Symbol - Chaos

This is the symbol for chaos-convergence.  The chapter is suggesting that for math understanding to happen, it can start with problems that students need to solve informally, without tools (such as with chaos).  Then, the teacher's job is to help students create generalizeable, abstract rules or procedures that make sense of the chaos (convergence).  

3.  Image - 

This image shows the idea of metacognition - that in math development it is important for students to think about the moves they are making by talking about them and having a community-centered environment that encourages such conversations