“Mathematical knowledge develops in the ordinary environment, usually without direct instruction.” – Ginsburg, et al, Handbook of Early Childhood Development

Picking up from last week’s Note on early literacy development, this week we are looking at early math development. As with last week, this Note builds on the three prior notes (archived here), which detailed cornerstones of our school’s educational theory (curiosity, competency, and emergent curriculum), and likewise includes these three areas:

Overview of how our educational theory leads to academic development

List of specific areas of learning for each age in the school

Tips for following through on this at home

Math learning is a “privileged domain” in education, which means that it is something that we are hard-wired for – our brains have evolved in a fashion that equips very young children with an early capacity (and desire!) to begin learning math on their own. Our pre-historic evolutionary ancestors needed such math skills as patterns, bigger/smaller, and higher/lower in order to survive. This evolutionary trait is why math develops innately in the ordinary environment – because our brains were designed to create math knowledge through basic interaction with the world. This works within our Reggio-inspired, emergent curriculum context because it allows us to use our students’ evolutionary mathematical capacity within play-based projects which arise from children’s interests.

I watched three year olds recently create their own currency in a dramatic play “grocery store”, using only bottle caps. They organically developed a system in which big bottle caps were “one” and small bottle caps were “half”. The fractions were off of course, but the deep understanding of big/small and more/less was there. Likewise, even the toddler’s penchant for lining up toys in long rows itself has a hidden math agenda. Sequencing, linearity, and congruity are necessary components for later mathematical understanding, and children are drawn to exploring these ideas already at 24 months. Toys are “out of place” that are not in line, and a doll does not belong in a line of trucks, the same way an even number sticks out when placed amidst a list of odd numbers.

Building on this notion – that children are mathematical before we instruct or expect them to become so – I will use an extended quote from Jo Boaler, Professor of Mathematics Education at the Stanford Graduate School of Education (and a good Twitter follow!):

“All children start life being excited by mathematics, and parents can become a wonderful resources for the encouragement of their thinking”. Notice that Boaler does not see parents as resources for “knowing the right answer” but very intentionally focuses on parents encouraging thinking. Boaler writes that when children are playing with blocks or any shapes, “parents need to be around to marvel with their children, to encourage their thinking, and to give them other challenges. One of the very best things parents can do to develop their children’s mathematical settings and to explore mathematical patterns and ideas with them. The best sort of encouragement does not involve sitting children down and giving them extra math work, or even buying them mathematical books to work on. It is about providing settings in which children’s own mathematical ideas and questions can emerge and in which children’s mathematical thinking is validated and encouraged.”

Before children learn discrete math information, they need to be encouraged to explore math-rich environments, such as blocks, shapes, and collections of materials (beads, buttons, shells, etc). This is because in math learning, as with all learning, there are three discrete yet overlapping elements involved (National Research Council, Eager to Learn):

Learning processes, such as memory, attention, and observation

Cognitive skills, such as reasoning, comparing and contrasting, classification

Specific information, such as number recognition and shape identification

In order to learn specific information, children need time spent in early childhood that allows them to develop their learning processes and cognitive skills. This time includes parents and teachers encouraging children to explore and talk about their environment. At our school, this type of learning is embedded in play and authentic activities that children see as purposeful and organic. When done right, this sounds conversational and fluid: “Let’s think about how we can get the bridge to stand on its own”; “I wonder why the Lego piece won’t fit in there”; “I need a few more pieces, do you have any?”.

Notice this is in direct contrast to the “known-answer-quizzing” that nearly all adults lapse into: “What shape is this?”; “How many pieces do you have?”; “Can you tell me what number this is?”; “I have two and you have two, how many do we have together?” Here’s the deal – children see right through it. They know that you know the answer to the question already, and that they are expected to answer for your satisfaction. They know the question, and the answer, are serving a didactic, pedagogical, acontextualized function. Keeping in mind intrinsic motivation, we embed our math-talk and math-thinking in areas that matter to children, in contexts which are necessary for the child to continue their play and their passions.

Several years ago our teaching staff wrote an in-house document, “The Developmental Framework for Young Children”, that outlines opportunities for learning at each age. Please note that the word “opportunities” is used intentionally here instead of “milestones.” Each student reacts to these opportunities in a unique fashion and each student learns these skills at his or her own pace. The list below is excerpted from the “Mathematical Understanding” section for each age group; this is only a small sampling of a longer list of math skills at each age group found in the Framework. Following the list, I will outline some tips for using this information at home.

2s turning 3:

Imitate counting using number names

Rote count to three (by memory)

Discriminate by size (big and little)

Begin to identify and sort items by single attributes, such as color or size

Classify objects by type, during play and at clean up time (i.e. animals with animals, Legos with Legos)

Identify basic geometric shapes (circle and square)

Stack blocks in group of 2 or more

Organize toys in a line

3s turning 4:

Rote count to ten

Count out six items using one-to-one correspondence

Identify, sort and classify objects by additional attributes (shape, size and color) when working with specific materials, or when involved in daily activities, such as room clean up

Identify what does and doesn’t belong when looking at a group of objects

Replicate a simple design using blocks

Replicate, extend and create an ABAB pattern a variety of materials (shapes, blocks and colors)

Understand and use language for putting objects in order (first, next, last)

Describe three events in sequential order

Understand and use comparative language: big/bigger/biggest, full/empty, tall/short

Understand and use language relating to concepts of directionality (behind, beside and under, up/down, over, above and through)

4s turning 5:

Rote count to twenty (by memory)

Count ten items using one to one correspondence

Recognize numerals in print from 0 – 20

Order written numerals from 0 – 12

Match a written numeral with a specific quantity (from 0 – 10)

Demonstrate understanding of ordinal nature of numbers when counting and putting written numerals in order (use correct language, i.e. first, second, third, fourth etc.)

Identify practical uses for counting

Read and interpret information from a variety of graphs

Identify and group objects by more than one attribute at one time (i.e. such as by putting large, blue Legos in one pile and small, blue Legos in another)

Use blocks to experiment with balance, equality, weight, estimation, spanning and bridging, height, slopes, perimeter, area and types of roofs and enclosures

Identify geometric shapes in the environment

Begin to develop understanding of non-standard forms of measurement (i.e. using unifix cubes to measure self or using string to measure the environment)

Begin to understand that when groups of objects are joined together the result will be a larger quantity; when we put things together we are “adding them to make more” and if we take some away there will be less

Use language to describe mathematical observations and solutions, including finding ways of documenting observations (i.e. recording on a graph, writing numerals on a sign)

Create and replicate both simple and more complex patterns using three or more colors or shapes

Replicate both simple and more complex block designs from both two dimensional and three dimensional models

Begin to understand conservation of matter (i.e. if a clay ball is turned into a “snake” it still contains the same amount of clay or if a pitcher of water is divided into different containers it still is the same quantity)

How can you support this learning at home? You probably already are, just by playing and talking with your child! Here are some ways you can heighten your child’s math learning at home:

Blocks! This is an obvious one but is always our starting place. To quote Boaler again, “Children’s play with building blocks in the early years has been identified as one of the key reasons for success in mathematics all through school. Any sorts of play with building blocks, interlocking cubes, or kits for making objects is fantastically helpful in the development of spatial reasoning, which is fundamental to mathematical understanding.” If you buy one block set, I recommend this basic set of classroom “unit blocks”. For babies, you can start with this set. For all children, you can add this cylinder-block puzzle and these Cuisenaire rods as well. As you read above, play with your child – don’t just quiz them on shapes. In your play, make interesting patterns and stimulating designs. Share your curiosities and wonderment as you play.

Talk about multiple characteristics of objects. This is the idea that a block can be red and a rectangle – and knowing that those are distinct qualities of the block. Qwirkle, Set, and Set Jr are all great games to explore objects with multiple characteristics (and a helpful reminder that even as an adult this is hard!). And again, embed this within everyday, casual language – “Your red hoodie has a zipper; your blue hoodie does not have a zipper.”

Use math language intentionally (but casually!) around the house. Sequencing words such as first/second/third and first/next/last, as well as comparison words such as larger/smaller, up/down, heavy/light, wet/dry, and long/short. Use numbers in productive, pragmatic ways – “The store is four blocks that way”; “We need three plates on the dinner table”; etc.

Keep “collections” in your child’s play area. Caches of buttons, beads, shells – really any trinket – encourages children to think mathematically as they sort, organize, and pattern the items. Collections should have at least a couple dozen of the item, and can be stored (relatively) easily in something like a jewelry or bead organizer.

I would love to hear your feedback on the topic, and as always, any questions you have as you consider these ideas.

Shabbat shalom,

Noah