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Includes different ideas for all ages!

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Early numeracy skills: how to develop them

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Children develop maths and numeracy skills from birth. Build early numeracy skills by counting, sorting, looking at shapes, singing number songs and more.

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Activities, importance, and milestones!

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5.1: Importance of Early Childhood Mathematics and Developing Number Sense

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Previous The Mathematics of Operations

Next Making Sense of Problem Types

Young Children Demonstrate Math Concepts in Everyday Activities

Context and Overview: Young children, even infants, develop basic, non-verbal concepts of quantity: more/less, order, same, and adding/subtracting.  Children develop a rudimentary understanding of these concepts on their own, without much adult help, often using them in everyday life, as in determining who has more or fewer cookies. As young children develop understanding of counting they also learn about addition and subtraction.

Idea of adding as producing more and subtracting less.  Children learn that that:

When you add something to an existing set, the result is that you have more than you had at first.

If you start with two groups of the same number, and by magic (while the child is not looking), one set is now more numerous than the other, you must have added to one or subtracted from the other.

Children don’t have to count to arrive at these judgments concerning more and concerning addition and subtraction: they can solve the problem by reason alone.

Later instruction needs to build on all of these ideas when written numbers are introduced.

Everyday Numerical Addition and Subtraction

Context and Overview: The story now is how concepts of more/less, order, same, adding and subtracting without exact number (for example, adding means making a set larger without knowledge of the exact number), and enumeration get elaborated to create numerical addition and subtraction.  Children learn some of this on their own, but adults can and should help.

Concepts to be learned to understand addition (subtraction is similar):

Addition can be thought of in several ways, including:

Combining two sets, for example, pushing together 5 crayons and 3 crayons to get a new group of 8 crayons

Increasing the size of one set, for example, beginning with 5 crayons and then adding 1 crayon at a time until 3 crayons have been added and the new group has 8 crayons

Moving forward on a number line, for example, moving 5 spaces on a number line and then 3 more to get to space 8

Simple counting is also adding—1 at a time

The order of addition makes no difference (the commutative property)

Adding zero changes nothing

Different combinations of numbers can yield the same sum

Addition is the inverse of subtraction, for example, if taking away 5 from 8 yields 3, then adding 3 to 5 yields 8.  This idea is crucial later on when children learn “fact families.”

Strategies used to add (or subtract):

Children often begin by using concrete objects and fingers to add but gradually learn mental calculation and remember some of the sums

Using concrete objects, children may do the following to solve a simple problem like 3 + 2:

Count all: I have 3 here and 2 there and now I push them together and count all to get 5.

Count on from the smaller: I can start with 2 and then say, 3, 4, 5.

Count on from the larger:  I can start with 3 and then say, 4, 5.

Approaching the problem mentally, children may solve the problem in these ways:

Building on what is known (“derived facts”):  I know that 2 and 2 is 4, so I just add 1 to get 5.

Memory:  I just know it!

More features of numerical addition and subtraction:

It’s always useful to have backup strategies in case one doesn’t work:  If unsure about memory, the child can always count to get the answer

It’s important for the child to be able to check the answer

The child needs to learn different strategies for different set sizes: counting one by one is good for adding small sets but tedious and inefficient for larger sets

The child should also be able to describe how he got the answer: self-awareness is one aspect of “metacognition.”  Of course, remembering what you just did is essential for describing it in words.

Language is vital for describing one’s work and thinking and to convince others; children need to learn mathematical vocabulary

The child should be able to apply the math in real situations or to stories about real situations (like word problems)

Groupings

Fair Shares.  Young children are very much concerned with fairness.  If you give Debbie one cookie, you had better give exactly one to her twin Becky as well.  If you give two to Becky, you must also give two to Debbie.  In this case, you are creating fair shares or groups—1 and 1, 2 and 2.

Suppose that Debbie and Becky have a birthday party and 8 friends attend.  To ensure fairness (and keep the peace) the parent meticulously goes through the fair share routine of giving one grape at a time to each child until all the grapes are gone.  The result is that each of the children (ten in all, including the twins) receives 10 grapes. In this case, the parent has (conveniently for our explanation of the math) created 10 groups of 10 each, so that there are 100 grapes altogether.  The result is eminently fair and mathematically useful, for it provides the basis for several important ideas.

The groups of 10 form the foundation of our counting system.  There are ten tens and we can count them cumulatively as 10, 20, 30…100.

The grouping creates the basis for division.  In this case, the parent has divided 100 items into 10 groups of 10 each.  Of course, if there were 30 grapes, the division would involve 3 grapes for each of the 10 children.

The situation allows for multiplication as well:  If we know that there are 5 groups, all the same number, with 4 in each, we can multiply to get 20 in all.

Of course we should not attempt to instruct young children in multiplication and division facts and algorithms.  But we can engage them in counting by tens, employing one-one correspondence to create fair shares, determining the fairness of existing collections, and correcting unfairness when they see it.

Number Sense

Context and Overview: Children need to develop number sense, a concept that is notoriously difficult to define in a simple and exclusive way.  I like to think of it as mathematical street smarts, which can be used in just about any area of number, including those discussed above. Number sense, which helps the child to make sense of the world, has several components, each of which undergoes a process of development.

Thinking instead of calculating.  Number sense involves using basic ideas to avoid computational drudgery, as when the child knows that if you add 2 and 3 and get 5, you don’t have to calculate to get the answer to 3 and 2.

Use what is convenient.  Number sense involves breaking numbers into convenient parts that make calculation easier, as when we mentally add 5 + 5 + 1 instead of 5 + 6.

Knowing what’s plausible or impossible.  Number sense may involve a “feel” for numbers in the sense of knowing whether certain numbers are plausible answers to certain problems (if you are adding 2 and 3 you know that the answer must be higher than 3; anything lower is not only implausible but impossible).

Understanding relationships.  Number sense involves intuitions about relationships among numbers—this is “way bigger” than that.

Fluency.  Number sense involves fluency with numbers, as when the child knows immediately that 8 is bigger than 4 or sees that there are 3 animals without having to count.

Estimation.  This involves figuring out the approximate number of a group of objects and is related to the notion of plausible answers.

The Transition to Written, Symbolic Math

Context and Overview: Formal, symbolic mathematics can provide students with more powerful tools and ideas than those provided through their informal everyday math.  Formal math (and its use of symbols) developed in several cultures and is now virtually universal.  Children need to learn it.

Everyday origins and formal math.  Children encounter math symbols in everyday life: elevator numbers, bus numbers, television channels and street signs are among the many.  Often parents, television, and software activities introduce some simple symbolic math—like reading the written numbers on the television or on play cards.

Schools certainly have to teach formal math.  But doing so is not easy.  Even though competent in everyday math, students may have trouble making sense of and connecting their informal knowledge to what is taught in school. Teachers often do not teach symbolism effectively.  If children get off on the wrong symbolic foot, the result may be a nasty fall down the educational stairs.  So the goal for teachers is to help children—even beginning in preschool—to understand why symbols are used, and to use them in a meaningful way—connecting already known informal mathematics to formal symbolic mathematics.  The teacher needs to “mathematize” children’s everyday, personal math—that is, help them connect their informal system with the formal mathematics taught in school.   It’s not ill-advised or developmentally inappropriate to introduce symbols to young children, if and only if the activity is motivating and meaningful.  On the contrary, it is crucial for the teaching of symbols to begin early on, but again, if and only if it is done in a meaningful way.

Here are key issues surrounding the introduction of formal math to young children:

Young children have a hard time connecting numerals and the symbols of arithmetic (+ and -) to their own everyday math. They may add well but be confounded by the expression 3 + 2.  It is as if the child is living in alternate realities: the everyday world and the “academic” (in the pejorative sense) world.  The everyday world makes sense and the world of school does not.  You think for yourself in the former and do what you are told in the latter.

The = sign is a daunting challenge.  The teacher intends to teach = as “equivalent,” and thinks she has, but the child learns it as “makes”  (e.g., 3 + 2 makes 5). This is a tale of how child egocentrism meets teacher egocentrism but neither talks with the other.

The solution.  We should not avoid teaching symbols but need to introduce them in a meaningful way.  This means taking account of what children already know and relating the introduction of symbols to that prior knowledge.  It also means motivating the use of symbols.  Thus if you want to tell a friend how many dolls you have at home, you need to have counted them with number words (symbols) and then use spoken words (“I have five dolls”), written words (“I have five dolls” written on a piece of paper or a computer screen), or written symbols (5) to communicate the result.

Manipulatives can help.  Use of manipulatives can be effective in teaching symbolism and formal math.  But they are often used badly.  The goal is not to have the child play with concrete objects but to use these objects to help the child learn abstract ideas.  The goal of manipulatives is to get to the point where children do not need them by putting them in the child’s head to use as needed in thought. For example, suppose the child learns to represent tens and ones with base ten blocks.  Given the mental addition problem 13 plus 25, the child may understand that each number is composed of tens (the 10 by 10 squares) and some units (the individual blocks), and that solving the problem involves adding one ten and two more, which is easy, and then figuring out the number of units.  The mental images of the tens and ones provide the basis for her calculation, part of which may be done by memory (one plus two is three) and part of which may be done by counting on her fingers (five fingers and three more give eight).

Conclusion

The basics of number are interesting and deep.  Although young children develop a surprisingly competent everyday mathematics, they have lot to learn and teachers can help.

Download an expanded version of What Young Children Know and Need to Learn about Counting and Operations below

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What Children Know and Need to Learn about Operations

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Erikson Institute Early Math Collaborative

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NUMBER OPERATIONS | Erikson Institute Early Math Collaborative

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Math Fact Fluency: Everything You Need To Know

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Math Fact Fluency: Everything You Need to Know

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Memorizing basic math facts is an important aspect of early math education, but it can be a hard process for some students. Some common strategies, such as flashcards and timed tests, can cause kids to become frustrated. What about choosing fun activities to make the experience more enjoyable for your child? Below, we're sharing 11 effective and engaging strategies you can use to help your students learn math facts. But first, let's take a deeper look at why math fact fluency is so crucial to success.

Why is Math Fact Fluency Important in Mathematics Education?

While it may seem old-school and unnecessary to have students memorize math facts in the age of calculators, fact fluency is still very important for creating a strong foundation for mathematical success with topics like algebra.

Instead of relying on counting strategies to figure out a math fact, your child should be able to recall the fact instantly so they can focus on larger processes. If they expel all of their mental energy figuring out the answer to the math fact, they'll likely either get lost in the calculations or burned out. This can result in them struggling further as they progress in their math education.

Students who achieve math fact fluency have stored basic math facts in their long-term memory and can recall them with automaticity. This "lessens the mental load" and allows them to use their working memory to gain a deep understanding of higher-order concepts and solve multistep problems.

Oftentimes, issues such as students falling behind in math and experiencing math anxiety stem from them not mastering their math facts.

4 Strategies to Encourage Fluency of Basic Facts

Rather than focusing on rote memorization to have your child learn their math facts, use these strategies to help them understand number relationships and improve their recall long term.

1) Start with Adding Doubles

Learning foundational addition, like doubles, such as 3 + 3 or 8 + 8, can be an effective strategy to ease into addition facts. Since there are just ten doubles to memorize, it's a manageable goal for a child who struggles with math. Practicing doubles is also a natural lead-in to help your child learn neighboring facts, such as 3 + 4 and 8 + 7 (as the answer is just one more or less than for doubles).

2) Explain Part-Whole Relationships

2) Explain Part-Whole Relationships

Part-part-whole relationships involve seeing numbers as being made up of two or more parts. This strategy works best when used for addition or subtraction facts, as students can see the number relationships when you add two parts together or remove a part from the whole. This is a great way to illustrate how numbers form fact families, which is explained further down below.

Note: when working with subtraction, the whole and one part must be present. Here's an example of how it works.

3) Teach the Commutative Property

The commutative property states that numbers can be added or multiplied in any order and still get the same answer. Based on this property, if your child knows 4 + 2 = 6, then he can solve 2 + 4 = 6. This approach reduces the number of addition and multiplication facts that they need to master by half, since if they've mastered it one way, they have math fact fluency it the other way as well!

4) Show Them How to Skip Count

Skip counting is a simple strategy that your child can use to memorize foundational multiplication facts. Counting by 2s, 5s, 10s, and so on can be fun to do together as your child begins to hear number patterns. You can also use a hundreds chart while skip counting so your student can begin to visualize those patterns.

For instance, when skip counting by 10s, see if your student notices that all the numbers end in zero. This strategy is particularly effective when using a chart that begins with zero and ends in 99. You can print our free chart here.

Please note that this strategy should only be used temporarily until your child achieves true math fact fluency. If your student relies on skip counting to determine the solution for a math fact, then they haven't yet mastered that fact.

7 Activities to Achieve Math Fact Fluency

Sprinkle some of these activities into your day to support learning, reduce math anxiety, and make working toward math fact fluency a far more enjoyable experience.

1) Fact Family Cards

This activity is most effective when your child understands the relationship between addition and subtraction or multiplication and division. Fact families are simply three numbers that are related and make a set of math facts.

For example: The numbers 9, 3, and 12 make up a fact family.

The addition and subtraction facts they make include:

Another fact family is 3, 4, and 12. They make the following multiplication and division facts:

You can cut out math fact family cards in the shape of a triangle or hexagon to practice and play games.

2) Math Toss

Toss a balloon or ball around the room with your child. As you toss, call out a math fact. The goal is for them to say the answer before catching it. If this is too challenging at first, modify the rule slightly to call out the solution before they pass the balloon after catching it.

3) Jump and Hop Basic Math Facts

This is a great activity for elementary students to develop their math fact fluency. Draw a hopscotch board with sidewalk chalk and write a math fact inside each square. When a child hops on a square, have them read the fact aloud and shout out the answer. You can also recite math facts while doing something rhythmic such as jumping on a trampoline, jumping rope, hopping on one foot, or bouncing a ball.

4) Trading Cards

Invite your students to create math fact trading cards, specifically with themes and colors that they like or involving an interest or hobby. Display the cards or develop your own trading card game that involves practicing the math facts.

5) Apps and Computer Games for Conceptual Understanding

There are plenty of captivating computer games and apps for mobile devices that focus on math fact practice and building math fact fluency. Many of them are free or very low cost. Work them into your weekly routine for short periods of time so your child gets some extra practice.

6) Fact Flash Tag

For this activity, you will need a chalkboard, whiteboard, or a blank wall with sticky notes and a flashlight. List the answers to several basic facts on the board or on the sticky notes (either arranged in rows or randomly). Darken the room and ask your child a math problem. We suggest choosing one operation to focus on at a time, such as multiplication, to limit frustration. The goal is to find the correct answer and shine the flashlight on it as quickly as possible.

7) Fact of the Day

If your student is having difficulty obtaining math fact fluency with just a few facts, choose one of them at a time and make it the "Fact of the Day." Post the fact in a place where it will be seen often. Then, spontaneously ask your child for the answer throughout the day. You can even award a sticker or other small token each time they give the answer quickly and correctly.

Another way to have fun with the fact of the day is to make it the "password" to do certain things or a substitute for someone's name.

Need Some Extra Help? Try AIM!

Helping your student develop math fact fluency will build their confidence, reduce math anxiety, and improve their ability to learn more advanced math skills in the future. If your child struggles to recall basic facts accurately and automatically, check out our math intervention program Accelerated Individualized Mastery (AIM). It was specifically designed to help students learn to visualize number relationships so they can commit facts to long-term memory.
This blog was updated on November 4, 2022.

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11 Effective Ways to Promote Math Fact Fluency

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Does your child need some extra help memorizing their math facts? Try out these 11 simple strategies to boost math fact fluency!

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How Do Teachers Support Problem-solving?

Although students do much of the talking and questioning in a TTP lesson, teachers play a crucial role. The widely-known 5 Practices for Orchestrating Mathematical Discussions were based in part on TTP. Teachers study the curriculum, anticipate student thinking, and select and sequence the student presentations that allow the class to build the new mathematics. Classroom routines for presentation and discussion of student work, board organization, and reflective mathematics journals work together to allow students to do the mathematical heavy lifting. To learn more about journals, board work, and discussion in TTP, as well as see other TTP resources and examples of TTP in action, click on the respective tabs near the top of this page.

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Although students do much of the talking and questioning in a TTP lesson, teachers play a crucial role. The widely-known 5 Practices for Orchestrating Mathematical Discussions were based in part on TTP. Teachers study the curriculum, anticipate student thinking, and select and sequence the student presentations that allow the class to build the new mathematics.

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Walk into any classroom today and you'll find great variation in student growth and kids with a variety of strengths, academic experiences, and needs. Superintendents now report an average of half of all their students are behind grade level, and other research suggests as many as 10% of students may be two or more grade levels behind.

The sheer variety of what students bring to school can be a major challenge for educators. Because these students' needs are different, they'll likely need different supports to be successful. At the same time, teachers are also tasked with ensuring every student has equal access to grade-level standards.

With only so many hours in the day, how can educators both differentiate to meet student needs and give all students access to the grade-level content they need to keep up and continue to succeed?

What the research tells us about helping kids succeed

It's been my privilege to conduct research focused on meeting this intense challenge. And what my colleagues and I have found is heartening: there are many strategies teachers already use that have proven to be especially effective in helping children make the academic gains we know they can make. We're calling these the Transformative Ten.

1. Provide supplemental learning time for targeted retrieval practice

2. Mix whole-group, small-group, and individual activities

3. Adjust student groups in real time

4. Share students and strategies within a grade level

5. Differentiate tasks within a unit

6. Provide targeted practice for foundational skills

7. Teach from multiple standards at once

8. Create opportunities for self-directed learning

9. Use student discourse as formative assessment

10. Explicitly teach academic vocabulary

Each of these strategies optimizes learning time, maximizes the amount of content students receive, and promotes student autonomy and empowerment. In our guide titled "The Transformative Ten," you can read a detailed description of each approach along with relevant research and specific examples. Think of the paper as your toolkit for getting started addressing the tricky balancing act of differentiation vs. providing access to grade-level content.

About our studies

My research began with a commitment to understanding how schools generate high levels of student growth and to continuing the work of my late colleague, Andrew Hegedus. In 2019, Andy released provocative work demonstrating that schools that produce above-average student growth come from all corners of the country; in fact, 60% of the lowest-income schools showed growth above average.

My colleagues and I picked up where Andy left off and identified 789 schools that produced above-average growth for students in all achievement levels for multiple years. We then partnered with two of those schools, located in Schiller Park, Illinois. The Illinois schools are similar to others across the country: 62% of their students receive free or reduced-price lunch, 55% of students are non-white, and per-pupil spending in the district is below the state average. Yet these schools produced higher-than-typical growth across all student achievement levels six years running.

Getting to the heart of what effective instruction looks like requires close partnership. I visited the schools-Kennedy Elementary School and Lincoln Middle School-four times over the last 15 months. Each time, we conducted observations of classroom instruction and interviews with our four focus teachers, adding up to around 87 total hours of data. None of that could have happened without the willingness of teachers, principals, and a district willing to open up their practice for the benefit of other educators, and Schiller Park's teachers and administrators deserve the bulk of the credit for making this work possible.

The purpose and intent of an effort like this departs in some important ways from other kinds of educational research. Unlike qualitative research, our intent was not to show conclusively that the strategies documented here can or will lead to high growth; other scholarship, particularly the pioneering work of John Hattie, meets that need. Instead, our intent was to pair existing research with concrete, in-the-moment details of professional practice that allow teachers and administrators to extract the best practices from high-growth schools and apply them in their own contexts and to their own students. Taking this approach allowed us to get much more specific about what high-quality instruction looks like while still drawing on past effectiveness research and the previously certified successes experienced in Schiller Park.

For the kids

Schiller Park teachers describe their commitment to student growth in five words: "All kids are our kids."

While the challenges associated with meeting the needs of diverse learners grew during the pandemic, they were present long before the pandemic and aren't likely to disappear anytime soon. By focusing on student growth and learning from the schools that achieve that growth, we stand to gain the specific research-based strategies teachers need to help all students succeed, not just those on the margins of grade-level proficiency.

We believe learning can be both individualized and equitable, and the Transformative Ten are first steps for getting there.

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10 ways to meet the learning needs of all students - Teach. Learn. Grow.

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Recent research from NWEA highlights 10 classroom strategies educators can use to support student growth for all kids.

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The Graduate School of Education and Human Development

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The Graduate School of Education and Human Development

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Meeting the Needs of Every Learner: The Role of Special Education Professionals | The Graduate School of Education and Human Development | The George Washington University

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Special educators and administrators must focus on recognizance, collaboration, and inclusion to ensure equitable learning environments for all students.

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Meeting the Needs of All Learners: A New Learning Domain – BetterLesson

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You have a wide variety of students, all of whom you care about and want to succeed. Learn how to support all of their diverse needs in our new Learning Domain: Meeting the Needs of All Learners.

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Facilitating Discourse | Carnegie Learning Support

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All students should be prepared to share their thinking and reasoning, whether correct or incorrect, at any point in a lesson. Holding groups and individuals accountable for their explanations and justifications will add further value to the learning activity. As you have surely discovered, when you truly understand a concept or idea yourself, then you are able to explain that concept or idea to someone else. In the book, 5 Practices for Orchestrating Productive Mathematics Discussions, by Margaret S. Smith & Mary Kay Stein (2011), a framework is presented for facilitating mathematical discourse. In a collaborative classroom, discussions should stem from the work produced by students (and not from the teacher). A productive mathematics discussion will surface mathematical ideas and misconceptions and lead to deeper student understanding. The teacher can utilize the following instructional practices to design a string of classroom dialogue. Anticipate likely student responses to challenging mathematical tasks How might your students approach this problem? What types of misconceptions do you predict? Monitor students’ actual responses to tasks How will you track what students are doing and saying? What questions will you ask students as they work? Select purposefully particular students to present their work during the share phase. What methods and reasoning do you want to emphasize in the discussion? How will you choose specific student work to drive the discussion? Sequence purposefully student responses that will be displayed in a specific order. How will you order the presentation of student work to best meet the mathematical goals? How will students share their work? (e.g., chart paper, document camera, white boards, verbally, video, online discussion boards, etc.) Connect different students’ responses and connecting the responses to key mathematical ideas What questions will you ask to make the key ideas visible to students? How will you help students make connections across multiple approaches? In a collaborative classroom, the 5 Practices still allow for the teacher to maintain control over the content discussed by students. Teachers should be mindful of holding all students accountable for being actively involved in the discussion. Students should be expected to present their ideas, ask and answer questions, and comment on others’ ideas. These practices should be embedded within your lesson planning to generate thoughtful discourse on a daily basis. A Note About Students’ Written Responses When working in the Student Text, students should always try to answer questions with complete sentences. Full sentences should also be encouraged in written answers to Assignment or Assessment questions. Complete sentences help students reflect on how they arrived at a solution, make connections between topics, and consider what a solution means both mathematically as well as in context. In the Teacher’s Implementation Guide, answers are provided in the form of complete sentences whenever appropriate. Note that many answers in the Teacher’s Implementation Guide are samples of correct answers. Students may present correct responses that do not exactly match the sample answers provided in the teacher’s materials. Actual students’ responses may include alternate solution paths or interpretations of a given question. This variety provides teachers with a rich opportunity for classroom exploration and discussion. Because correct student responses may vary, we recommend showing only student responses to the class and not the sample answers from the Teacher’s Implementation Guide.

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5 Surprisingly Flexible Strategies for Subtraction Everyone Needs to Learn - The Naturally Creative Classroom

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5 3 digit subtraction regrouping strategies that you NEED to be teaching. Students can learn strategies that are flexible and fun!

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Multiplication Strategies Progression | Department of Education

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Welcome to the asynchronous module, Multiplication Strategies Progression. At your own pace, read through the materials, watch the short video clips, and make sense of the pictures. This module is approximately 1 hour in length and can be completed in one sitting or in smaller parts. When you have competed the module, click the link to the questionnaire in the box on the right. After successfully submitting the questionnaire, your contact hour certificate will automatically be emailed to the address provided in the questionnaire. If you should have any questions about this process or the content in this module, please contact Jen Robitaille at jennifer.r.robitaille@maine.gov.