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.

Our youngest mathematicians

• understand the concept of whole numbers
• identify patterns and use them to assist in simple calculations
• identify shapes and describe spatial relationships
• match numerals to numbers

Developing mathematicians

• identify measurable attributes and compare objects by using these attributes
• identify and use number symbols to express numbers
• create simple math equations using symbols
• use concrete objects to represent and check calculations
• select strategies to check their work
• use language skills to express mathematical thinking
• rely on fluency in one-to-one correspondence, place value, and the correspondence between concept of number and concrete representation
• estimate with growing accuracy
• develop fluency in the following areas: measurement (including time), money, geometry, basic fractions, and graphing to tell a numerical story

Proficient mathematicians

• calculate with accuracy
• increase fluency (speed, accuracy, and confidence) with math facts
• identify problems that can actually be resolved using mathematics
• apply math to engineering challenges and other scientific contexts
• develop an understanding of decimals, including the connections between money, fractions, and percentages
• develop an understanding of area, perimeter, and volume
• estimate reasonable solutions with growing accuracy
• use precise mathematical language skills to explain thinking

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.

Accomplished mathematicians

• visualize a problem and use language cues in order to set up the solving structure, focusing on essential information
• select a solving strategy and monitors its use, amending the selected process when necessary
• use a checking strategy to verify solution
• use the concept of regrouping to perform addition, subtraction, multiplication, and division problems with ease
• develop and use the interchange among fractions, decimals, and percentages
• use language in conjunction with illustrations to express mathematical concepts
• read, understand, and create statistical information: mean, median, mode, range, graphs (line, bar, scatter, pie)
• extend the number line and its meaning to negative numbers confidently
• express geometric and story problems and solutions in algebraic terms
• solve algebraic problems
• understand and identify proportions and ratios
• create and extract from simple examples the necessary heuristics to plan for and check more complex calculations

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.