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Mathematical Thinking

Math is a language for talking about numbers, measurement, patterns, shapes, and comparing and sorting objects, and data. Children use this language as they experience math daily in formal and informal experiences. The math component of SEE Every Child is based upon the Building Blocks Pre-K (Clements & Sarama, 2013) curriculum, with a particular emphasis on embracing mathematics as a language through which children can better understand and discuss their daily lives. Children will learn basic math skills aligning with the standards set by the EEC:

  • Numbers and operations
  • Geometry
  • Measurement
  • Patterns and algebra
  • Data analysis and classification.

These skills are practiced and expanded upon through a variety of math experiences including exploration of manipulative materials, group activities emphasizing mathematical thinking, and incorporation of math concepts into Big Ideas and Projects. Math skills are also promoted through the following practices (the Building Blocks pacing guide contains more information about each):

  • Whole Group math instruction can include explicit presentation of math materials and concepts, routines, calendar, songs, movement and games. Whole Group lessons build as the week progresses, repeating particular activities based on children’s experiences, and building complexity.
  • Small Group times targeted learning for each child according to individual development. Small Groups occur while other children are engaged in Hands On self-directed math activities (see below) or during Choice Time.
  • Hands On are activities that children can do independently or with minimal support from an adult, in order to deepen their mathematical understandings.
  • Math Throughout the Year are recommended routines that build on the mathematical skills highlighted each week. These routines appear throughout the year, are repeated to reinforce concepts, and can be done during transitions, as warm ups during Whole Group time, Small Group, and adapted for Hands On.
  • Key Reflection Questions appear throughout each week and support children’s reflections on mathematical content. Strategic Key Questions have been chosen to align with the concepts and content for each week. When children talk in their own words about what they did during an activity they develop mathematical reasoning and skills such as turn taking, listening, and speaking.

In addition to Building Blocks we draw upon the work of the Erikson Institute’s Early Math Collaborative as well as our own “Math Vitamins” that we send out a few times a month.

In addition, here are some important nationally recognized early math resources.

  • About Teaching Mathematics: A K-8 Resource by Marilyn Burns
  • A Collection of Math Lessons: From Grades 1-3 by Marilyn Burns (both these books have nice modifications for preschool)
  • Developing Math Concepts in Pre-Kindergarten by Kathy Richardson
  • The Young Child and Mathematics by Kathy Richardson
  • Math Their Way by Mary Barrata-Lorton
  • The Math Their Way Summary Newsletter

Sorting and Classifying

Rocks. Jewels. Shells. Keys. Collections. All children have them – just ask. Remember dumping out a box of buttons or shells and putting them into categories? Then perhaps you named each set. In order to do this you had to identify some kind of attribute of each group – an attribute describes a property that some things have in common. Exploring these collections helps children to observe similarities and differences, compare objects by attributes such as color, size, shape, develop language skills, and connect their observations of attributes to other real world experiences. Understanding that objects can belong to more than one category helps children to develop flexible thinking. Asking children to name the categories and describe attributes supports vocabulary development. Eventually, as children sort and classify, they naturally move to counting or patterns, making sorting a foundational part of children’s math learning.

Pattern

Patterns cut across all the other areas of mathematics. When we look for and identify patterns we can begin to predict results and actually train our minds to make sense of seemingly unrelated information in our world. Our number system is based on patterns (think counting by twos or fives, or our base ten system). But patterns also represent a predictable, repetitive, organizational structure. It is important to surround children with a range of patterns so that they do not simply see a pattern as an ABABAB configuration. Searching for patterns in the environment, artwork, and creating and reading the pattern in children’s designs and structures helps children to generalize the idea of patterns into many realms.

Geometry

Children actually have many experiences with shapes before they enter school. Geometry is one of the most visible strands of mathematics. Playing with objects, understanding how they feel when grasped, learning their properties and how they fit together is part of a child’s early experience. Teachers must build on these experiences, exploring and naming shapes intentionally, and designing problem solving experiences that help children to develop spatial reasoning. Geometry also does not exist in a vacuum. Teachers must connect the study of shapes to pattern, measurement, and numbers. Materials such as geoboards, pattern blocks, unit blocks (the block area), and other types of parquetry blocks are manipulatives children should use every week. Connecting these materials to real world experiences via shape walks, art projects, and construction projects helps children to develop a deep knowledge of geometry.

Number Sense

Numbers are a way to express mathematical relationships in our environments. Children first experience number through sorting and classifying objects, which is why having collections of objects in baskets or boxes is so important. Eventually children will want to count what they have and begin to notice relationships between quantities – which has more, which has less, the idea that an amount gets larger when I add more to it, that the last number I count is the number in that set, that I have to touch each object and only count it once to get an accurate count. All of these kinds of experiences happen without numerals or symbols being introduced. Eventually children can be shown the names of the numbers and play games associating quantities and symbols so that the idea that a numeral stands for or represents a specific quantity is understood.

Graphs and Data

Children’s job is to make sense of the world and to sort out the things they see and do. They are like little data analysts, categorizing and making judgements about their interactions with objects and people. Graphing is a way for children to intentionally collect data, see it, reflect on it, and then ultimately interpret and make sense of it. For young children, making graphs as concrete as possible is key. “Real graphs” where children can use their own shoes to make a graph of the kinds of closures (laces, velcro, slip ons) help children to see the similarities and differences in something they can relate to. Eventually graphing can be done on paper – large, easy to read charts are best. Yes or no questions such as “I like dogs.” or and sorting questions such as “What color are your eyes?” help children to consider a range of answers. Reflect on data with a debrief – “What do you notice about this graph?” is a good starting question. Guide children to notice categories that have more, less, the same amount and then interpret what that might mean for the class as a group helps children to apply their observations to the real world. Have clipboards and small graphs available for children to make their own surveys and walk around the room asking, “What color do you like best?” This is an opportunity to teach tally marks or other ways of collecting data in a more abstract way.

Measurements

Measurement for young children is really about the idea of comparison. Is the red pipe cleaner longer than the green one? Which carrot is the longest? The shortest? This is really an exercise in evaluation and estimation – important real worlds skills. Using rulers may be an important skill, but being able to judge length, weight, volume, size, etc. is developed through experiences with matching and comparing objects. Children also need to practice using the tools of measurement such as scales, cups, thermometers, and rulers. But don’t rush for those rulers too soon. Young children benefit from “non-standard units” of measure. An “inch” is a very abstract concept – but measuring with unifix cubes, blocks, body parts, or whatever is easily available and familiar is more meaningful and teaches the big idea that objects can be represented by units of measurement.

Operations

Addition is when we join things together. Subtraction is when we take one number away from another number. Multiplication is when we add the name number over and over. Division is when we break a number apart and share it equally. These are simplistic definitions but for young children who are beginning to play with numbers and understand the ways in which numbers can be manipulated, they characterize the key concepts in operations. For young children, good literature or song are initial entry points to operations. Having children draw a number story and begin to quantify their drawings and add numerals helps children to symbolically represent what may have been in their minds for some time. Revisiting collections and telling stories about them is a way to begin to help children to represent the idea that numbers can be joined and separated in particular ways.