Source:
National Research Council of the National Academies. (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press. (Chapters Introduction, 5-8).
Chapter 6 – Fostering the Development of Whole-Number Sense: Teaching Mathematics in the Primary Grades (Sharon Griffin)
RQs
1. How do the principles of How People Learn apply to the development of Whole-number Sense?
2. How do we teach it in the primary grades?
3. Is it effective?
Notes
1. Teaching Mathematics in the Primary Grades (This chapter focuses on Number Sense)
1.1. You need to know 3 things:
1.1.1. Where you are now (in terms of students’ prior knowledge).
1.1.2. Where you want to go (knowledge you want children to acquire).
1.1.3. The best way to get there.
1.2. Each leads to a question crucial to the design of effective mathematics instruction.
1.3. Each points to a body of knowledge the teacher must have access to in order to answer the question.
1.4. Similar to the 3 principles of how people learn. (p. 257)
1.5. If each teacher sits down and asks these questions on a regular basis, each will be able to construct own set of answers, enrich our knowledge base, and improve mathematics teaching and learning for at least one group of students. The teacher will also embody the essence of what it means to be a resourceful, self-regulating mathematics teacher. (p. 258)
2. Deciding What Knowledge to Teach
2.1. Deciding what to teach is made difficult by all the differing levels of the students and all the different standards in NCTM, state standards, curricula guides, etc.
2.2. Many teachers decide what to teach by choosing one set of understandings that they want all students to acquire.
2.3. Number sense is one of those understandings in the primary grades, especially since it will be used heavily in handling problems in other areas (algebra, geometry, stats, etc.).
2.4. Number sense is easy to see and identify, but hard to define and teach. (p.259)
2.5. Central conceptual structure for whole number has been found to be central to children’s mathematics learning and achievement in at least 2 ways:
2.5.1. Enables children to make sense of a broad range of quantitative problems in a variety of contexts.
2.5.2. It is the building block on which children learn more complex number concepts (e.g. double-digit numbers).
2.6. By choosing number sense, teachers demonstrate an intuitive understanding of essential role of this knowledge network and importance of teaching a core set of ideas that lie at the heart of learning and competency in the discipline (principle 2). (p.261)
2.7. Developmental principles that should be considered in building learning paths and networks of knowledge for the domain of whole numbers:
2.7.1. Build upon children’s current knowledge.
2.7.2. Follow the natural developmental progression (defined from research) when selecting new knowledge to be taught, and thus creating a developmentally appropriate learning path. This will be the base for higher level knowledge networks.
2.7.3. Make sure children consolidate one level of understanding before moving on to the next.
2.7.4. Give children many opportunities to use number concepts in broad range of contexts and to learn the language used in these contexts to describe quantity. (p. 266)
2.8. Central Conceptual Structure Hypothesis
2.8.1. Powerful organizing knowledge network that is extremely broad in its range of application and that plays a central role in enabling individuals to master problems that the domain presents.
2.8.2. Central implies:
2.8.2.1.That the structure is vital to successful performance on a range of tasks, ones that often transcend individual disciplinary boundaries.
2.8.2.2.That future learning in these tasks is dependent on the structure, which often forms the initial core around which the subsequent learning is organized.
2.8.3. Test by Griffin and Chase
2.8.3.1.Two groups (control and treatment), each consisted of kindergarten students of an age that typically have acquired number sense but had yet to do so.
2.8.3.2.Each group given pretest to test understanding of whole number.
2.8.3.3.One group taught via Number Worlds curriculum (treatment), based on teaching the central conceptual structure of number sense, while the other group (control) received a variety of other forms of math instruction for the same 10 weeks. (p. 264)
2.8.3.4.Treatment group improved substantially in all test areas, far surpassing the control group. (p. 265)
3. Building on Children’s Understandings
3.1. Just like anything else, the number knowledge that children have as they enter preschool around age 4 varies.
3.2. To get a picture of their understandings, a teacher can consider the knowledge children typically demonstrate between the ages of 4 and 8 when asked oral questions from the Number Knowledge test.
3.2.1. Administered individually.
3.2.2. Administered orally.
3.2.3. Prompts to elicit reasoning.
3.2.4. Tool or set of questions a teacher can use to get children’s conceptions about number and quantity and understand the children’s strategies to solve the problems they are posed.
3.2.5. Provides a good picture of entry and exit knowledge.
3.2.6. Provides a model for ongoing, formative assessment (assessment centered classrooms).
3.2.7. Divided into 3 levels. (p. 267) drawn from cognitive developmental tradition that are hierarchical in nature (i.e. level 0 is foundational for knowledge demonstrated at level 1)
3.2.8. Ages associated with each level represent the midpoint in the 2-year age period where the knowledge is typically constructed/demonstrated.
3.2.9. Age norms are age ranges where children in developed societies (typically middle income children) typically pass that level of the test.
3.2.10. Thus, test provides a set of broad developmental milestones for the majority of U.S. children. However, how this holds true for children from vastly different sociocultural groups still must be determined.
3.3. Understandings of 4-Year-Olds (p. 270)
3.3.1. Most have constructed an initial counting schema (well-organized knowledge network) that enables them to count verbally from 1 to 5, use the one-to-one correspondence rule, and use the cardinality rule (last number counted is the number in the set).
3.3.2. Most have also constructed an initial quantity schema that gives them an intuitive understanding of relative amount and of transformations that change that amount. (p.272)
3.3.3. Most can also use words to talk about these quantity relations and transformations.
3.3.4. Most DO NOT use these schemas in a coordinated or integrated fashion, as if they are stored in a separate part of the mind.
3.4. Understandings of 5-Year-Olds
3.4.1. Children begin to solve problems involving single-digit numbers and quantities without having real objects available to count. Usually happens between 5 and 6.
3.4.2. Children behave as if they are using a “mental counting line” inside their heads or with their fingers.
3.4.3. Children are demonstrating their awareness that the counting numbers refers to real world quantities and can be used in the absence of countable objects. (p. 273)
3.4.4. Children are able to tell which of 2 single-digit numbers is bigger or smaller instead of using objects that are physically present. This new competence implies the presence of a sophisticated understanding.
3.4.5. Children successful with these items appear to know.
3.4.5.1.Numbers indicate quantity and therefore…
3.4.5.2.Numbers themselves have magnitude.
3.4.5.3.The word “bigger” or “more” is sensible in this context.
3.4.5.4.That the numbers 7 and 9 have fixed positions in the counting sequence.
3.4.5.5.That seven comes before nine when counting up.
3.4.5.6.Numbers later in the sequence indicate larger quantities.
3.4.5.7.And that 9 is therefore bigger than 7.
3.5. Understanding 6-Year-Olds (p. 274)
3.5.1. Scholars hypothesize that around age 5 – 6, as children’s knowledge of counting and quantity becomes more elaborated and differentiated it also becomes more integrated, eventually merging into a single knowledge network (central conceptual structure for whole-number, or mental counting line structure). This is a blueprint showing the important pieces of knowledge children have acquired and how they are interrelated. (p. 275)
3.5.2. Around age 6 – 7, supported by entry into formal schooling, children typically learn the written numerals.
3.5.3. When written numeral understanding is linked to their conceptual understanding of number, children understand that numerals are symbols for number words, both as ordered “counting tags” and as indicators of set size (numeral cardinality).
3.6. Understanding 7-Year-Olds
3.6.1. Around age 7 – 8, in grade 2, children are able to solve the same sorts of problems using double-digit numbers. (p.277)
3.6.2. Children are also able to mentally use 2 counting lines for single-digit problems like, “How many whole numbers are between 2 and six?”
3.7. Understanding 8-Year-Olds
3.7.1. Around 8 years old, children are able to add double-digit numbers.
3.7.2. Note that the strategies are different. Children with a solid understanding of using benchmarking numbers and the base 10 system are the most successful. (i.e. to answer 12 + 54, a student explains that 10 plus 54 is 64 and 64 plus 2 is 66, so the answer is 66.) (p.278)
3.7.3. Other children will use less sophisticated strategies, such as vertical format mental addition, or counting up by 12 from 54 in fingers.
3.7.4. Around 7 – 8, children’s central conceptual understandings become more elaborate and differentiated, allowing them to represent 2 distinct quantitative dimensions (tens and ones) in a coordinated fashion.
3.7.5. This is a new structure called bidimensional central conceptual structure for number where children are able to understand place value and are also able to solve problems involving 2 quantitative dimensions across multiple contexts, including time, money, and math class.
4. Acknowledging Teachers’ Conceptions and Partial Understandings
4.1. The Number Knowledge test can also provide an opportunity for teachers to examine their own mathematical knowledge and to consider whether any of the partial understandings children demonstrate are ones they share as well. (p. 279)
4.2. Three Insights:
4.2.1. Insight #1: Math is not about numbers, but about quantity
4.2.1.1.Math is about quantity, and numbers express quantity.
4.2.1.2.Numbers acquire meaning for children when they recognize that each number refers to a particular quantity and that numbers provide a means of describing quantity and quantity transformations more precisely.
4.2.1.3.They realize that numbers are tools that can be used to describe, predict, and explain real-world quantities and quantity transactions, and it gives them a big boost in mastering and using the number system.
4.2.2. Insight #2: Counting words is the crucial link between the world of quantity and the world of formal symbols
4.2.2.1.Numbers are presented orally as a set of counting words and graphically as a symbol system.
4.2.2.2.Provide opportunities for children to connect the symbol system and counting words, which are more familiar to them when entering the formal schooling system from their prior knowledge.
4.2.2.3.Take advantage of the prior knowledge of the counting words. (p. 280)
4.2.2.4.Children need ample opportunities to use the oral language system to make sense of quantitative problems and that they be introduced to the graphic equivalents of the system in this familiar context.
4.2.3. Insight #3: Acquiring an understanding of number is a lengthy, step-by-step process
4.2.3.1.Each new understanding builds systematically and incrementally on previous understandings.
4.2.3.2.Hopefully, each lesson or series of lessons will help a child move up 1 level at a time in his/her understanding and deepen/consolidate each new understanding before moving on to the next, and thus gradually constructing a more sophisticated and higher level understanding.
4.2.3.3.Process takes time, and each student may move at his/her own pace.
5. Revisiting Question 2: Defining the Knowledge that Should be Taught (p. 281)
5.1. Teachers are required to teach a whole classroom of children, so they need a set of general learning objectives for each grade level appropriate for the range of children involved.
5.1.1. All children in the class need to obtain the developmental milestones – the cetral conceptual structures for whole number.
5.1.2. All children become familiar with major ways in which number are represented and talked about so they can recognize and make sense of number problems they encounter across contexts.
5.2. Based on the previous section and the Number Knowledge test, here are learning goals for each grade level Pre-K to grade 2 which are in the reach of the majority of children at each grade level so they can reach developmental milestones.
5.2.1. Pre-K – Children acquire well-developed counting and quantity schema.
5.2.2. K – Children acquire well consolidated conceptual structure for single-digit numbers.
5.2.3. 1st – Children link structure to formal symbol system and to construct more elaborated knowledge network this entails.
5.2.4. 2nd – Children acquire bidimensional central conceptual structure for double-digit numbers that underlies a solid understanding of the base-ten system.
5.3. These goals develop a “number sense” learning pathway – a sequence of learning objectives teachers can use to individualize instruction for children who are learning at a faster or slower rate.
6. How Can This Knowledge Be Taught?: The Case of Number Worlds
6.1. Number Worlds is a program designed to teach whole-number concepts using the How People Learn principles. (p. 282)
6.2. Designed to build specifically on children’s existing understandings (principle 1).
6.3. Designed to help children construct new knowledge, both factual and conceptual, that is organized in a way to facilitate retrieval and application (principle 2).
6.4. Designed to require and teach metacognitive strategies (principle 3).
6.5. Design Principles
6.5.1. #1 Exposing children to major forms of number representation
6.5.1.1.Numbers are represented in our culture in 5 major ways:
6.5.1.1.1. Objects
6.5.1.1.2. Dot set patterns
6.5.1.1.3. Segments on a line
6.5.1.1.4. Segments on a scale or bar graph (p. 283)
6.5.1.1.5. Segments or points on a dial
6.5.1.2.Children familiar with these forms of representation and language used to talk about number in these contexts have an easier time making sense of number problems they encounter inside and outside of school.
6.5.1.3.Number Worlds provides a way these can be taught and illustrates a knowledge-centered classroom.
6.5.1.4.At each grade level, children explore 5 different lands that correspond to the aforementioned number representations. (p. 284)
6.5.1.4.1. Object Land (real objects – pennies, fingers, etc. – into groups)
6.5.1.4.2. Picture Land (dot patterns representing mathematical sets – dice for example) (p. 285)
6.5.1.4.2.1.This land provides a link between the world of movable objects and the world of abstract symbols.
6.5.1.4.2.2.Children in this stage must make a mental correspondence between 2 sets. (p. 285)
6.5.1.4.2.3.Numerals and tally marks are also part of this land.
6.5.1.4.2.4.Children gradually come to think of these patterns as forming the same sort of ordered series as do the number words themselves.
6.5.1.4.3. Line Land
6.5.1.4.3.1.Representation of numbers along a line like those found on board games like Chutes and Ladders.
6.5.1.4.3.2.Language of distance.
6.5.1.4.3.3.Children learn that a number can refer to a place on a line AND number of moves along a line.
6.5.1.4.3.4.Children learn to treat physical addition and subtraction of objects as the equivalent to movement forward or backward along a line. This is very important in the transition from small countable objects to abstract numbers and numerical operations.
6.5.1.4.3.5.Until children make this transition, they are unable to move from physical to mental operations with any insight.
6.5.1.4.4. Sky Land
6.5.1.4.4.1.Representation of number with bar graphs and scales, such as thermometers.
6.5.1.4.4.2.Always used in a vertical direction, such that bigger numbers are higher up.
6.5.1.4.4.3.Makes convenient context for introducing children to the concept of using numbers as measure, as a way to keep track of continuous quantity in standard units.
6.5.1.4.4.4.Helps develop children’s intuition for the properties of systems.
6.5.1.4.5. Circle Land
6.5.1.4.5.1.Representation of numbers using dials, such as sundials and clocks.
6.5.1.4.5.2.More sophisticated since they incorporate a cyclic quality possessed by certain real-world dimensions like time and natural rhythm of the seasons.
6.5.1.4.5.3.Children develop spatial intuitions that become the foundation for understanding many concepts in mathematics dealing with circular motion (pie charts, time, number bases). (p.287)
6.5.1.5.They are introduced in the order of difficulty, but the goal is to help children appreciate the equivalence of these forms of representation and the language used to talk about numbers in these contexts.
6.5.1.6.Children are encouraged to explore all lands and all number representations early in the school year by beginning with activities in each land that target lower-level knowledge objectives and proceeding throughout the year to activities in each land that target high-level objectives.
6.5.1.7.For adults, different representations are easily seen to be equivalent, but they can seem like completely different worlds to children.
6.5.1.8.Helping children construct an organized knowledge network in which these ideas are interconnected is a major goal of Number Worlds.
6.5.2. #2 Providing Opportunities to link the “world of quantity” with the “world of counting numbers” and the “world of formal symbols”
6.5.2.1.Plus Pup – An Object Land Activity for PreK/K programs
6.5.2.1.1. Provide opportunities for children to count a set of objects and identify how many there are.
6.5.2.1.2. Provide opportunities to recognize that when one object is added, the size of the set increased by 1. (p.288)
6.5.2.1.3. By allowing children to explore, a range of strategies will emerge.
6.5.2.1.4. By prompting children with questions, the teacher can lead students to make sense of the quantity transaction by describing it in their own words.
6.5.2.1.5. By playing, children come to realize they can use the counting numbers themselves, with or without their fingers. The teacher can then move onto predictions and proofs. (p.289)
6.5.2.1.6. By encouraging problem-solving and communication, children’s thinking is visible and ongoing assessment opportunities are provided.
6.5.2.1.7. This game serves as a conceptual bridge between an increase in quantity in the real world, and the +1 symbol that describes this increase in formal mathematics.
6.5.2.2.Minus Mouse
6.5.2.2.1. Introduced after Plus Pup (p. 290)
6.5.2.2.2. Identical to Plus Pup, except the mouse takes one cookie away.
6.5.2.2.3. Most young children prefer the comfort of the familiar and thrive on the opportunities the similarities provide for them to anticipate what happens, and confidently make predictions.
6.5.2.3.Plus Pup Meets Minus Mouse
6.5.2.3.1. Make the problem more complex by adding Plus Pup and Minus Mouse to the same activity. Either Plus Pup or Minus Mouse will surface based on a card drawn from a pile.
6.5.2.3.2. The challenge is to interpret the icon with its associated symbol to determine the action that should be performed and to figure out how to solve the problem of how many cookies are in the bag.
6.5.2.3.3. Scaffolds on the development of whole-number sense.
6.5.2.3.4. In all of the activities, teachers can assess each child’s level of understanding based on the solutions constructed (or not) for each problem posed, the explanations provided, and the strategies employed.
6.5.2.3.5. These assessments can provide the teaching with an appropriate starting place for the next activity (how many cookies in the bag) and the sorts of questions that should be posed to individual children to help them advance their knowledge. (p. 291)
6.5.3. #3 Providing visual and spatial analogs of number representations that children can actively explore in a hands-on fashion
6.5.3.1.Spatial contexts created for the Number Worlds program often take the form of game boards where numbers are depicted as positions on a line, scale, or dial and on which quantity is depicted as segments on these representations.
6.5.3.2.By using pawns to represent “self” and moving through these contexts to solve problems, children gain understanding of the relationship between moving along a line, scale, or dial and increases and decreases in quantity.
6.5.3.3.The Skating Party Game
6.5.3.3.1. Played in Circle Land at the K level.
6.5.3.3.2. Designed to help children realize that a dial is another device for representing quantity, and that the same relationships that apply between numbers and movement along a number line also apply to numbers and movement in this context.
6.5.3.3.3. Game is a circular path that includes 10 segments (0-9) to help children understand the cyclical nature of the base-ten number system.
6.5.3.3.4. Objectives:
6.5.3.3.4.1.Identify or compute set size, and associate it with position on a dial.
6.5.3.3.4.2.Associate increasing quantity with moving around a dial.
6.5.3.3.4.3.Compare positions on dial to identify which have more, less, or the same amount.
6.5.3.3.4.4.Use this knowledge to solve a problem. (p. 292)
6.5.3.3.5. Children take turns and roll a die and count the dots to move their pawns around the dial. Award cards are collected each time they go around the dial and the one (or group) with the most cards wins.
6.5.3.3.6. Questions posed during game play should be tuned to the child’s current level of understanding.
6.5.3.3.6.1.Who is furthest?
6.5.3.3.6.2.How much further do you have to go?
6.5.3.3.6.3.How do you know?
6.5.3.3.6.4.How come everyone had the same number of turns, but [name] has the most cards? (p. 293)
6.5.3.3.7. There are some variations, such as using +1/0/-1 cards.
6.5.3.3.8. Encourage children to construct their own answers by paying close attention to the activity. (p. 294)
6.5.3.3.9. Children gradually build up their intuitive understanding of links among worlds of quantity (in spatial context), counting numbers, and formal symbols.
6.5.3.4.Rosemary’s Magic Shoes
6.5.3.4.1. Developed for Line Land for 2nd graders to help children build understanding of base-ten number system.
6.5.3.4.1.1.The prop is the Neighborhood Number Line, which is comprised of 10 blocks of houses, each containing 10 houses.
6.5.3.4.1.2.The character for this game is a monster-tracker name Rosemary who has a pair of magic shoes that allow her to leap over 10 houses in a single bound.
6.5.3.4.1.3.For the magic shoes to work, she must tell how many times to jump 10 houses and how many times to walk past 1 house.
6.5.3.4.1.4.Children take turns picking a number tile that indicates a house where the presence of a monster has been suspected.
6.5.3.4.1.5.Using the shoes, they move to the house as quickly and efficiently as possible, check for monsters, (by drawing a card), and place a sticker on it, showing it’s monster-free. (p. 295)
6.5.3.4.1.6.In later versions, children are required to keep a record of RoseMary’s movements using the formal symbol system.
6.5.3.4.1.7.Children need to watch each other’s moves closely to see if other methods are more efficient and to share thinking with the class.
6.5.3.4.2. Children realize that they can leap over 10 houses from any number sequence, not just decade markers.
6.5.3.4.3. Children realize they don’t always need to move forward, as it might be more efficient to move to the closest 10’s marker and back a few steps.
6.5.3.4.4. Children gain fluency in computing the distance between 2 numbers in the 1 – 100 sequence and in moving fluently from one location to the next, using benchmark values to do so.
6.5.3.4.5. Children gain an appreciation of the relative values of numbers (92 is a long way from 9) and that 9 + 2 cannot be 92.
6.5.3.4.6. This is an acquisition of fluency, factual knowledge, and conceptual understanding that are greatly facilitated by spatial analogs that can be explored in a hands-on fashion.
6.5.4. #4 Engaging children’s emotions and capturing their imagination so knowledge constructed is embedded not only in their minds, but also in their hopes, fears, and passions
6.5.4.1.Dragon Quest
6.5.4.1.1. Number Worlds program uses imagination in its activities, but this activity uses it to a greater extent.
6.5.4.1.2. Addresses major learning goal for 1st grade of helping children link their central conceptual structure for whole number to the formal number system. (p. 296)
6.5.4.1.3. Developed for Picture Land in the 1st grade program.
6.5.4.1.4. Children must work most with numerals and operation signs.
6.5.4.1.4.1.The game is based around a fire-breathing dragon that is terrorizing their town.
6.5.4.1.4.2.The children are heroes chosen to put out the dragon’s fire.
6.5.4.1.4.3.The children need 10 (or 20 or more in later phases) pails of water, and if they enter the dragon’s area with less, they are taken prisoner and must be rescued by another player.
6.5.4.1.4.4.Children take turns rolling a die and moving along the colored game board where they can land on a well pile in which they draw a card from that indicates with images and symbols a certain number of pails of water.
6.5.4.1.4.5.The first child to reach the dragon’s layer with 10 pails of water can put out the dragon’s fire and free any prisoners.
6.5.4.1.5. Children become capable of performing a series of successive addition and subtraction operations. (p. 297)
6.5.4.1.6. This activity gives children the opportunity to use formal symbol system in more efficient ways and to make sense of quantitative problems they encounter in the course of their own activity.
6.5.5. #5 Providing opportunities for children to acquire computational fluency as well as conceptual understanding
6.5.5.1.Computational fluency is given special attention in the warm-up period of each Number Worlds lesson.
6.5.5.2.In PreK and K, these activities typically take the form of count-up and count-down games played in each land with a prop appropriate for that land.
6.5.5.3.It makes it possible for children to acquire fluency in counting, and at the same time, to acquire conceptual understanding of the changes in quantity associated with each successive number (up or down).
6.5.5.4.Sky Land Blastoff
6.5.5.4.1. Used after children have become reasonably fluent in count-up activity that uses the same prop.
6.5.5.4.2. Children use a large, specially designed thermometer with a moveable red ribbon.
6.5.5.4.3. Children pretend to be a rocket ship and count down while the teacher (or student) moves the ribbon.
6.5.5.4.4. Seeing the “liquid” level drop gives children a good foundation for subtraction (p. 298) by allowing them to see that quantity decreases in scale height.
6.5.5.4.5. Also lays the foundation for measurement.
6.5.5.4.6. Complexity can be added by changing the position of the ribbon, or asking children or make redictions about where the ribbon will be if it is dropped X degrees. In this way, the teacher exposes children to a learning path attuned to their understanding and allows them to develop a network of conceptual and procedural knowledge. (p. 299)
6.5.5.5.In programs for 1st/2nd grade, higher level computation skills are used in warm-ups, such as Guess My Number.
6.5.6. #6 Encouraging the use of metacognitive processes (e.g. problem solving, communication, reasoning) that will facilitate knowledge construction
6.5.6.1.Additional supports are built into Number Worlds that support problem solving, communication, and reasoning.
6.5.6.1.1. Question cards
6.5.6.1.1.1.Developed for specific stages of each small group game were designed to draw children’s attention to quantity displays they create during the game.
6.5.6.1.1.2.Prompts them to think about these quantities and describe them.
6.5.6.1.1.3.Prompts them to reflect on reasoning and put it into words.
6.5.6.1.1.4.Initially for use by the teacher, children can gradually take over the function, taking greater control over their own learning. (p. 300)
6.5.6.1.2. Dialog prompts included in teacher’s guide.
6.5.6.1.2.1.Provides more general set of questions than those provided in the game.
6.5.6.1.2.2.Useful in prompting children to use metacognitive processes.
6.5.6.1.2.3.Do NOT give guidance on how a teacher should respond to answers children give at different age levels.
6.5.6.1.2.4.So in order to provide more/better follow-up questions and answers to children’s thinking, it is important that teachers have a better understanding of the answers that children give at these levels, and create increased opportunities for children to express their thinking to build experience and expertise.
6.5.6.1.3. Wrap-Up periods provided at the end of each lesson.
6.5.6.1.3.1.Children assigned as “Reporters” for their problem-solving group describes the mathematical activity that the group did that day and what they learned.
6.5.6.1.3.2.The “Reporter” takes questions from the class, and any member of the team can help answer them.
6.5.6.1.3.3.Most significant portion of learning occurs here because children have the opportunity to reflect, explain concepts to their peers, and acquire more explicit understanding of the concepts.
6.5.6.2.With practice, teachers become increasingly skilled at asking good questions to get things going and taking a back seat. It is a skill that takes practice and patience. (p. 301)
7. What Sorts of Learning Does This Approach Make Possible?
7.1. Number Worlds was created to address 3 major learning goals:
7.1.1. Conceptual and procedural knowledge of number.
7.1.2. Number sense.
7.1.3. Interest in and positive attitude towards mathematics.
7.2. Program evaluation has been focused on the extent to which children exposed to it have been able to demonstrate gains on any of these fronts.
7.2.1. Groups of students followed have always been from low-income, predominantly inner-city communities.
7.2.2. Assumption is that if it works well for them, it will work just as well, if not better for affluent communities.
7.2.3. Several different forms of evaluation have been conducted.
7.2.3.1.Number Worlds program students vs. matched control group that had taken part in different math readiness programs. Number Worlds students consistently outperformed control group on tests of mathematical knowledge, developmental measures, and experimental measures of learning potential.
7.2.3.2.Number World program student follow-up 1 year later (after completing K program) using a double-blind test. Number Worlds students were again found to be superior in virtually all measures including teacher evaluations of “number sense”. (p. 302)
7.2.3.3.Number Worlds children tracked over a 3-year period vs. 2 other groups having superior/high level achievements in math.
7.2.3.3.1. Number Worlds program students caught up with and gradually outstripped higher achieving students of the magnet school (acclaimed school with special math coordinator).
7.2.3.3.2. Number Worlds program students outperformed the other group from end of K onward.
7.2.3.3.3. Number Worlds program students compared favorably with students from China and Japan that were tested on the same measures.
7.2.4. Findings provide clear evidence that programs based on the principles of How People Learn works for the population of children most in need of effective school-based instruction.
7.2.5. Teachers and students using the Number Worlds program consistently report a positive attitude toward the teaching and learning of math. (p. 3003)
