Mathematics is another way of exploring and making connections. From the beginning, in all mathematical work, children explain and justify their processes and answers.
In early stages of math development, young mathematicians explore attributes, patterns, and relationships. Young children use concrete objects to develop basic concepts, using symbols (pictures, charts, graphs, and models) to demonstrate their thinking as they advance. They develop number sense by counting, ordering, sorting, classifying, matching, combining, and separating sets. These actions underly concepts of additions, subtraction, multiplication, and division. Children increase their understanding of those relationships as they explore concepts of equality and inequality, counting patterns, place value, and regrouping. In order to predict results and check their answers, they learn to estimate. They match numerals with concepts of number and introduce math symbols as the "verbs." Youngsters find examples of math in our surroundings and explore ways to express mathematical ideas through speaking, writing, and model building with manipulatives. They often rely on using concrete objects or pictures to help conceptualize, organize, and solve a problem, moving from manipulating concrete objects to using symbols. Children develop a sense of "reasonableness" to predict and check solutions to simple problems, estimate quantities through practice, and select and apply appropriate operations in problem-solving. Growing confidence and competence with basic facts allow mathematicians to concentrate on problem-solving, unhampered and efficiently.
As children advance, concrete investigations become representational and ultimately are expressed through abstract symbols. Representations are useful in visualizing operations and relationships, even though they are not generalized or made formal until later. Children at this stage of mathematic prowess identify relevant information and define a problem and the process needed to solve it, use strategies to visualize and explain an approach, and reverse operations or choose another process to check a solution. Self-correction become more automatic if a solution seems unreasonable as they revisit the process, identify the question and steps in reflection, and visually display thinking.
Mathematically-proficient children visualize a posed problem, explain the meaning, and look for entry points for its solution. They analyze givens, constraints, relationships, and goals. They eliminate irrelevant information and estimate a reasonable answer whenever possible. Mathematicians consider and choose from among the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, or a spreadsheet. They make conjectures about the form of the solution, creating a plan rather than simply jumping into a solution attempt. They assess their progress and change course if necessary. They check answers to problems using a different method, and they continually ask themselves, "Does this make sense?" Mathematically proficient students ---strive to communicate precisely to others, using agreed-upon vocabulary and symbols. They work with expanded number lines, arrays (a visual display of multiplication and division), and fraction pieces. Through practice, children gain fluency (speed, accuracy, and confidence) with math facts, algorithms, and computation. Our children explore and use measurement, money, geometry, fractions, and probability.
Strong mathematical thinkers develop abstract thinking capabilities, (acting on information without visual clues). In more complex projects, young mathematicians apply skills to real life math problems they define for themselves and express mathematical thinking through models, illustrations, and stories. Renaissance mathematicians apply the mathematics they know to solve problems arising in everyday contexts, whether describing the movement of materials needs in an engineering project, planning a social event, or analyzing a question arising in the community.
Your child as a mathematician
Our youngest mathematicians
Work at this stage is on-going in respect to beginning algebraic concepts, positive and negative integers, coordinate graphing, statistics, geometry, and the concept of "balance" in equations, the insertion of substitutions, and the relational study of fractions, decimals, and percentages.
Work at this stage includes formulating questions that can be addressed with data; collecting, organizing, and displaying relevant data: selecting and using appropriate statistical methods to analyze data; developing and evaluating inferences and predictions that are based on data.