06. Fostering the Development of Whole-Number Sense

        There are 3 important components to teach math [in the perspective of the author of this chapter, a 15-year math teacher veteran -Ed].  They are knowing where you are now, where you want to go, and what is the best way to get there.  These are similar to the principles outlined in HPL.  The resources chosen are an important variable in fostering these principles.  For example, to know where students are, a Number Knowledge test is used to determine it.  A specific framework is used to know where you want to go, and there are specific learning tools used to get the students to the right level of understanding.  Answering these three questions every time a lesson is developed is a useful way to organize an instructional unit. 


        Deciding what to teach is difficult because there are various forces pulling at you - national standards, state standards, school curriculum, etc.  Many elementary teachers choose number sense, though often fail to define it well.  As exemplified simply by asking students to solve the following problem, "If you had four chocolates and someone gave you three more, how many would you have altogether?" there is a lot that goes into number sense.  This includes knowing the counting sequence, knowing that "four" refers to a particular size, that "more" means to add, etc.  This network of understanding for this question makes up what is called a central conceptual structure for whole number.  This happens to be fundamental to children's mathematical learning.  Helping teachers understand the procedural, factual, and conceptual understandings of this can help teachers build clear learning paths.  The principles underlying these learning paths are building on children's current knowledge, choosing new knowledge to teach that is the natural next step, making sure students understand before moving on, and giving students many opportunities to use the concepts in a broad range of contexts. 

        It is important for teachers to know the range of understandings their students have.  The Number Knowledge test can be administered orally and individually formatively for this purpose.  The test is organized into levels, and for each level there is a set of understandings.  Using the test, testers can figure out that by the age of 4, most children can count chips, even when mixed with colors and asked to count certain colored chips.  Children that make mistakes typically forget which chip they started with (they often don't pull them out) or have trouble corresponding the next number with the next chip they count.  Such specific demonstrations have led researchers to post hypotheses about children's understanding.  These include counting and quantity schemas.  By the age of 5, a major change takes place.  While many children still need to use fingers or physical objects to continue counting, many can count mentally as if they have a mental counting line.  Such examples are given through 8-year old understandings.  


        Teachers can similarly use these tests to assess themselves, as they frequently also have partial understandings that are important to rectify.  There are 3 in particular in this case:
        Insight #1 - Math is not about numbers, but about quantity - as such, children should learn in the context of quantity representations
        Insight #2 - Counting words is the crucial link between world of quantity and formal symbols - as children begin with counting words before symbols, their understanding should gradually become formalized
        Insight #3 - Acquiring an understanding of number is a lengthy, step-by-step process - A concept isn't just understood with a single, "aha" experience.  Rather, it takes time and many connections to form. 


        The Number Knowledge test is excellent for the individual students, but at the class level there are 2 clear objectives.  First, to teach the central conceptual structure for whole number, and second, to make clear the way number and quantity are represented.  Using the framework based on the Number Knowledge test, clear learning goals can be set for each grade. 


        A program called Number Worlds is described here, which was designed with the principles of HPL.  Over 200 activities were designed under this framework and includes 6 design principles:

  1. Exposing Children to Major Form of Number Representation - There are 5 major ways numbers are represented.  Children are exposed to all of them, called various "lands" (object, picture, line, sky, and circle-land).  Children engage in activities, such as games, that demonstrate each of these representations. 
  2. Providing Opportunities to link quantity to counting numbers to formal symbols - For example, children play games where physical objects are exchanged and they have to describe the events abstractly. 
  3. Visual and Spatial Analogs of Number Representations - Games such as circle land, where figures race around a circle, then students are asked who is winning (the number of times around the circle must be added to the position on the circle, forcing students to represent a time around the circle with another symbol), are played. 
  4. Engaging Children's Emotions and Capturing their Imagination - In a dragon game, children are very engaged and have to perform increasingly difficult conceptual task, ending in an informal (yet conceptually developed) proof. 
  5. Opportunities to Acquire Computational Fluency as Well As Conceptual Understanding - A specially designed thermometer game allows children to count up and down in different intervals, all the time seeing change in number.  It has opportunities for increased conceptual difficulty. 
  6. Encouraging the Use of Metacognitive Processes - Question cards prompt students to reflect upon their own understanding, and this is facilitated by roles (such as question poser) that allow students to play important roles and open safety in a community-centered environment.  This slowly helps students become better at asking questions. 

        The Number Worlds program has been evaluated on 3 criteria: 1.  Conceptual and procedural knowledge, 2. number sense, 3.  interest and attitude towards mathematics.  Children in the program, when compared with control groups, showed significant differences in all three areas in nearly all measures.  This success also opened the opportunity for a longitudinal study that again showed clear results. 

        Creating such a program that meets all three learning principles takes continuous effort over a long period of time, and even teaching it effectively still requires a great deal of work.