Mathematics Development Chapter 10 Child Development and Education Development in the Academic Domains By: Christina Basso

Mathematics Development -Just a few of the clusters involved in the domains of mathematics -each cluster is used to represent different methods for representing and solving quantitative problems -focuses on the development of knowledge and skills 1Algebra 2Geometry 3Statistics

NUMBER SENSE AND COUNTING -by age 5 or 6 months, infants have some awareness of quantity -by there first birthday, some are able to notice the differences between more and less and in sequences -many infants being counting by or around there third birthday -by there fifth birthday children should be able to count far beyond 10 in the correct order as well -In elementary school, as students begin to write two and three digit numbers they being to master the correct sequence of numbers -when children begin to count they don’t necessarily know that they are or are not doing it accurarely. They tend to say two numbers succesive as they point to objects. 1

-One-One Principle: Each object in the set being counted must be assigned one and only one number word. In other words, you say “one” while pointing to one object, “two” while point to another, and so on until every object has been counted exactly once. -Cardinal Principle: the last number word counted indicates the number of objects in the set. In other words, if you count up to five when counting objects, then there are five objects in the set. -Order-Irrelevance Principle: a set of objects has the same number regardless of the order in which individual objects are counted. -Children apply these principles primarily to small number sets of 10 objects or fewer but within a few years they can apply the principles to larger sets as well. Number Sense and Counting

Mathematical Concepts and Principles -In addition to building on a basic understanding of numbers, mathematical reasoning requires an understanding of many other concepts and principles. -Part-Whole Principle: the idea that any single number can be broken into two or more smaller numbers and that any two or more numbers can be combined to form a larger number. - i.e. 7 can be broken down in 1, 2, and 4 and 4, 2, and 1 can be combined to be 7 -This principle is connected to the child’s ability to understand addition and subtraction properties. -Idea of Proportion: Fractions, Ratios, and Decimals. -As early as 6 months, infants begin to show awareness of proportions by visual representation. -By early elementary school, children can understand simple, specific fractions if they are related to everyday objects. -Middle and High School Mathematics Classes continue to focus on proportional relationships that are more abstract thoughts to students.

2 Basic Arithmetic Operations -Infants have the concepts to understand addition and subtraction before there first birthday -by age 2.5 or 3 children can clearly understand that adding objects to a set increasing the quantity and removing objects to a set decreases the quantity. -by age 3 or 4 children are able to apply this knowledge to counting objects and simple addition and subtraction problems using visual representations -by elementary school, children have developed a variety of strategies to solve simple addition and subtraction problems. -By the end of third grade children are expected to know basic multiplication facts.

-When working with small numbers, they may simply use addition facts to solve the multiplication problems -i.e. 3x3 = or 9 -Sometimes students will count by twos, fives or other numbers -i.e. 5x4 = five, ten, fifteen, twenty or 20 -Others will apply the rules of the times tables -i.e. anything times zero is zero -anything times one is itself -When children start to encounter problems which deal with two and three digit numbers, the concept of place value is essential and must be master for concepts such as “carrying” and “borrowing” -During the preschool and elementary school years, children gradually develop a central conceptual structure that integrates much of what they know about numbers, counting, addition, subtraction and place value. -By age 6 children should be able to understand: -verbal numbers “one”, “two”, “three”, “four”, etc -written numbers 1, 2, 3, 4, etc -the systematic process of counting objects Basic Arithmetic Operations

-by age 8, children have sufficiently mastered how to simultaneously solve mathematical problems -they also have a better understanding of applying the transformation between columns and using concepts such as “carrying” and “borrowing” not only in addition and subtraction but in also in multiplication -by age 10 children become capable in generalizing the relationships between two numbers lines to the entire number system. -Children continue to develop their central conceptual structure as they transition to adolescence but the basic arithmetic operations weren’t studied after age 10 Basic Arithmetic Operation

-As students transition through middle school and high school, they begin to learn a mathematics curriculum based on procedures involving proportions, negative numbers, roots, exponents and unknown variables. -Before students can apply these skills to math problems, they must understand these complex concepts -In other words, children’s understanding of mathematical procedures should be closely tied to their understanding of mathematical concepts and principles -If students can not master these skills, it’s usually an indication they have not mastered the concepts on which the procedures are based on More Advanced Problem-Solving Procedures

3 Metacognition in Mathematics -Children should be able to plan, monitor and evaluate their problem solving efforts as they enter middle school. -Children who have a fully developed metacognition will be able to recognize that the sum of could not be 613 -Many students are not able to reflect on this ability and they do not understand how this ability is reflected into their mathematics work

-Virtually all children around the globe have a general awareness that objects and substances can vary in quantity and amount -Children with learning disabilities tend to have the ability to understand number concepts, automatize math facts and solve simple math problems quickly -Others have little exposure to numbers and counting at home and so come to school lacking knowledge in these fundamental areas. -Researchers have also found gender and ethnic differences in mathematics development DIVERSITTY IN MATHEMATICS DEVELOPMENT

Developmental Trends: Mathematics at Different Age Levels

-Average differences between boys and girls in mathematics development tend to be fairly small with some researchers finding a slight advantage for one gender over the other depending on the age-group and task in question. -More boys than girls have very high math ability, especially in high school, and more boys than girls have significant disabilities in math -The slight advantage in boys having higher math abilities on average over girls in addition to being more likely to have significant disabilities in math over girls may have to do with biology and the way the brain develops before and after puberty. -One area Biology might have effected the math development of boys over girls is Visual- Spatial Ability or the ability to imagine and mentally manipulate two-and-three dimensional figures. Gender Differences

-On average boys and men perform better than girls and women on measures of visual-spatial ability. -This appears to be an advantage boys and men have on girls and women when it comes to Mathematics Development and performing certain types of mathematical tasks -Environmentally as well has an effect on the Mathematics Development of both boys and girls. -Mathematics historically has been viewed as a “male” domain and more suitable for men then women. -With this stereotype in mind, many parents and teachers, more actively expect and encourage boys rather then girls to learn math. -Boys tend to have more confidence in their mathematics skills because of this even when the actual achievement between both boys and girls has been similar for their age group. tend to be fairly small with some researchers finding a slight advantage for one gender over the other depending on the age-group and task

-There is consistent research that proves that Asian and Asian Americans score and achieve higher in mathematics then other students from North American cultural groups. -According to research, this may not be because of the ethnicity of the individual but on how teachers of Asian backgrounds were taught and teach mathematics. -Research states and shows that Asian teachers provide more thorough explanations of mathematical concepts, focus classroom discussions more on making sense go the problem-solving procedures, and assign more math homework to apply these essential skills than teachers from ethnic backgrounds do. -The parents of the Asian children as well are more likely to believe that math achievement comes from hard work rather than natural ability in the subject area. -Since they feel this way, they insist their children spend a good deal of their time at home on schoolwork focusing on their mathematical skills. Ethnic and Cultural Difference

-Many theorists speculate that the nature of number words is Asian languages may also facilitate Asian children’s mathematical development. -In the Asian languages, the structure of the numbers from the base-10 number system is clearly reflected in the way you speak them in number words. -i.e. the word 11 is literally “ten-one” the word 12 is “ten-two” and the word for 21 is “two-ten-one” -The same is true for fractions. Words for fractions reflect what the fraction really is. -i.e. the word for ¼ is “four parts of one.” -In English there are too many number words i.e. eleven, twelve, thirteen, fourteen, twenty, thirty, one-half, one-fourth All these words hide the structure of the base- 10 which does not reveal much about the information about the nature of proportions. Ethnic and Cultural Difference

-One thing that Asian and European languages have in common in a system for identifying virtually any possible number. -Numbers and Mathematics are cultural creations that are not universally shared across the cultures. -i.e. The Pirahã Society in Brazil have three words for numbers. One or a very small amount, Two or a slightly larger amount and many. -People from The Pirahã Culture have a difficulty distinguishing among similar quantities greater than three or four, because counting has little usefulness in their day-to-day activities. -Since in their culture and society, counting is not something necessary for their lifestyle, learning mathematics the way we do would be very difficult for a person who would decide to become educated in the American School System. Ethnic and Cultural Difference

-The Okspamin Culture in Papua, New Guinea is another example of how the culture has a different specific nature to Mathematics. -In the 1980’s the Oksapmin people identified the numbers 1 through 29 using different body parts, progressing from the right hand and arm (1 was the right thumb) and the neck and head (14 was the nose) and then down the left arm (29 was inside of the left forearm). -Although this system was sufficiently difficult when it comes to multiplication, division and complex mathematical procedures, the people of this culture did learn to adapt Ethnic and Cultural Difference

-Mathematics is known to the cause of confusion and frustration for children and adolescents than any other academic subject. -Part of this is because of the children not having a full development of the math concepts and procedures at one grade level and the lack of necessary pre-requisities for learning math successfully in later grades. -Children that have difficulty with abstract ideas, proportional thinking, mathematical reasoning and problem solving are apt to struggle in mathematics as they grow other and the concepts of mathematics become more complex for their young minds. -Formal education is proven to be the most effective way to teach mathematics and for children to develop upon the mathematics concepts effectively. Promoting Development in Mathematics

-The following is suggestions for teachers who work with children and adolescents on how to teach mathematical concepts and tasks -Teach numbers and counting in preschool and the primary grades -At all age levels, use manipulatives and visual displays to tie mathematical concepts and procedures to concrete reality -Encourage visual-spatial thinking -As youngsters work on new and challenging mathematical problems, provide the physical and cognitive scaffolding they need to successfully solve the problems. -Encourage children to invent, use, and defend their own strategies Promoting Development in Mathematics

Mathematical Concepts and Principles -A basic understanding of numbers and counting forms the foundations for virtually every aspect of mathematics. -When young children haven’t learned the basics from home, the teachers usually make up the difference so they are on the same level as the other students. -Teachers usually have activities and games involving counting, comparing quantities, adding and subtracting are apt to be beneficial. -i.e. Regular practice in counting objects and comparing leads to improved performance not only in these tasks but in other quantitative tasks as well. -Determining which of three groups has the MOST, LEAST and MIDDLE and middle amount of objects. Teach numbers and counting in preschool and the primary grades

Mathematical Concepts and Principles -Concrete manipulatives can often help children grasp the nature of addition, subtraction, place value, and fractions. -Visual aids such as number lines and pictures of pizzas depicting various fractions can be helpful in the early elementary grades, and graphs and diagrams of geometric figures are useful for secondary students. -Children and Adolescents benefit from concrete manipulatives and illustrations of abstract ideas. -It helps put such a complex idea into simple visual objects they are able to comprehend. At all age levels, use manipulatives and visual displays to tie mathematical concepts and procedures to concrete reality.

Mathematical Concepts and Principles -Although it is said, the boys have a biological advantage in visual-spatial thinking, structured activities and experience is what is known to decrease the gender gap in Mathematics Achievement throughout the age groups of children. -Experiences and Structured Activities that can prevent and decrease the gender gap include: -building blocks -legos -puzzles -simple graphs -basic measurement tools -As children move into the middle elementary grades and beyond, visual-spatial tasks should also include ample work with complex graphs and three-dimensional geometry. Encourage Visual-Spatial Thinking

Mathematical Concepts and Principles -Complex mathematical tasks and problems–those within a child’s zone of proximal development—often put a strain on working memory and in other ways stretch children to their cognitive limits. -For children in the early elementary grades, such tools may simply be pencil and paper for keeping track of quantities, calculations, and other information. -Once children have mastered basic math facts and understand the logic behind arithmetic operations, they might use calculators or computers while working with large numbers or cumbersome data sets. -i.e. A teacher might encourage students to brainstorm possible approaches to problems, model the use of the new problem-solving strategies, and teach students various metacognitive strategies for monitoring and checking their progress. As youngsters work on new and challenging mathematical problems, provide the physical and cognitive scaffolding they need to successfully solve the problems.

Mathematical Concepts and Principles -Students along the way especially young students often invent new ways to add and subtract using objects well before they have formal been taught through instruction. -Instead of ignoring the strategy the child has developed on their own, its important that teachers encourage the fact that the student has come up with the strategy especially if they seem to be effective on the mathematical development of the child. -As children grow, the old strategies they used and became accustomed to will die off and the children will use the new strategies they need for the higher level math they are completing. -An additional way to encourage the students to continue to come up with new strategies that work for them and help develop their mathematical skills is illustrate their strategy using a problem, then in complete sentences describe step by step the strategy and how it worked to efficiently come up with the correct answer. -Strategy Development doesn’t have to be an activity the students participate in. It is just a way to integrate their creative minds into a challenging course. -There is research that has proven the Strategy Development is an effective tool for Mathematical Development in both boys and girls. -It enhances the mathematical understanding of the material for the students in a way they can take the complex material and break it down into a way they are able to process it. -i.e. A second grader might come up with a strategy for adding and subtracting three digit numbers from a two digit number while a high school student might work in groups to come up with a procedure to remember and understand geometric theorems. Encourage children to invent, use, and defend their own strategies.

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