## An NAEYC Online Event

**Juanita Copley responded to a selection of questions and comments during an online event from April 25–29, 2011. Read the questions and her responses below!**

My book

*The Young Child and Mathematics**focuses on children ages 3 through 8 and their mathematical learning. The placement of the phrase**young child*before the word*mathematics*in the book’s title is not accidental. It comes first because I believe that the child should be the focus of early education.
Like the first edition, this is a practical book for teachers that communicates the main ideas of the National Council Teachers of Mathematics'

*Standards*within the context of effective, developmentally appropriate practice.
This second edition also reflects recent developments in math education in a wealth of vignettes from classrooms, activity ideas, and strategies for teaching young children about math processes and concepts.

Let’s explore together how to readily and enjoyably make mathematics an integral part of early childhood classrooms each and every day.

I look forward to considering your questions and learning from your own classroom vignettes and examples.

— Juanita Copley

## Comments

## Babies and math

I know a lot about babies, language and early literacy but not much about babies and math. What is there to know about babies and math?

## Thank you!

Many thanks to Juanita Copley her thoughtful and interesting responses and thanks to all who posted questions. I've learned a lot from this Q&A. Thanks to all who participated.

## Teacher-directed vs. child-directed math activities

Hi Juanita,

I am an education coordinator for a Head Start program. We recently purchased a High/Scope math curriculum that includes teacher-guided activities, and I'm already feeling some resistance from staff about how the new curriculum will be implemented. One of the teachers has said that she prefers using a child-directed approach to math (e.g. have baskets with various activities for counting, patterning, etc. that the children select on their own, complete on their own or in pairs, and the teachers would simply check each child's work afterward) and that these "teacher-directed" math activities in the new curriculum are not child-friendly and don't allow the children choices. We're going to go over this at our classroom team meeting this Friday. Do you have any advice how to approach this situation? I feel that, philosophically, staff and I are looking at teacher-guided (not necessarily teacher-directed) math activities from two completely different angles. Thanks!

## Teacher-directed vs child-directed math activities

I understand your dilemma. I grew up in the age of all child-directed activities and I admit that I have adapted my philosophy over the years to accommodate both child-directed AND teacher-guided activities. As a teacher, I intentionally provide experiences that will engage children AND teach to the standards and/or guidelines that have been shown to be foundational to their education. I find that few early childhood teachers ignore literacy standards and that they intentionally teach letter sounds and a variety of literacy outcomes. I have also observed that often these same teachers believe that mathematics is only involved if children choose the activities. From research studies (see the recent NRC volume), we know that children who experience mathematics early are more successful later; in fact, a recent set of longitudinal studies revealed that early math was a significantly better predictor of success in school than any other indicator. I have found that young children enjoy mathematics and that my role as a teacher is to intentionally guide all children (those who choose math activities and those who do not) to learn the mathematics that is foundational to all students.

## STEM - Science Technology Engineering Math

There's a lot in the news about STEM. What are ways teachers can related math to science, engineering, and technology that would make sense for preschool? For early elementary? What's the importance of approaching these disciplines in a coordinated way?

## STEM - Science Technology Engineering Math

I was an elementary science specialist in a school district for 6 years and I found that it was impossible to teach science without mathematics. In my professional life, I have often presented with Karen Worth, the author of many early childhood books. When we have discussed our presentations, we have found that we present the same topics (e.g., problem solving, block building, measurement, data analysis, and patterning) and that the integration of these topics simply make sense and provide authentic learning in real contexts. I was privileged to work on the Curious George series as a mathematics consultant a few years ago. One-third of that series was math related, one-third was science related, and one-third was technology related. When I watch them now, I often have difficulty remembering which ones were math related (the ones I reviewed) and which ones were science or technology related! Do I believe these topics should be integrated in a coordinated way in preschool or early elementary? My answer is simply… yes… sometimes… when it makes sense! Forced integration does not make sense. Let me give a rather silly example of what I mean. Many years ago a school district asked me to consult with them on their integrated program. I visited with the kindergarten team. A local farm had helped them create a rabbit community in an inside courtyard. I was amazed at what children learned from their project. The science and literacy activities as well as the care giving and responsibility attributes toward the rabbits were outstanding! Unfortunately, the mathematics objectives were poorly addressed. Instead, I found that they had made rabbit flashcards with numerals and matching dots to cover the mathematics objectives. Realistically, there certainly are many things that could be counted or measured with rabbits. I only wish the important mathematics had been considered in this approach

## girls and math

I'm curious about the relationship to math engagement when children are young and their being "good" at math later on. As older students are tracked into more difficult math classes in middle school there are more boys in the advanced math classes than there are girls. The same goes for boys and girls in technology related courses - there are fewer girls in these technology classes then boys. What are ways early childhood educators can make sure to engage girls n math that might ave an impact in the later years?

## Girls and Math

I wish I could give a good answer to this question. In the early grades, I do not see a difference between girls and boys in math engagement. Both boys and girls seem to be equally engaged in the math I teach. With that said, I am curious about the differences that are seen as students get older. In my research studies, I have investigated what Carol Dweck would call “learning-oriented” students and “performance- oriented” students in problem-solving situations. I have found that more girls (especially those who are gifted) appear to be performance-oriented than boys. Performance-oriented students often over estimate their failures and under estimate their successes and are in danger of becoming “learned helpless.” That idea may provide some of the answer to your question. Perhaps you could investigate your question and send me what you find out…I would love to learn from your work!

## The Young Child & Mathematics 2nd ed. & DVD

Der Ms Copley,

I just wanted to say how much I am enjoying using your newly revised edition. I have used this book for several years with my ECE community college students. Your DVD that accompanies this new edition is fantastic. I think it really models how teachers need to be thinking when they are working with children doing these math games and activities. I feel my students this term are having a higher level of understanding about being math teachers which is quite intimidating for many of them.... Thanks so much ... and my new favorite word ( & for my students as well...) is subitizing!! We are really enjoying your new edition!

## The Young Child and Mathematics 2nd ed. & DVD

Thanks for your kind comments... I am happy that you have found the DVD helpful. As you might assume, I know that what you see on the DVD is far from perfect... In fact, I have never taught the perfect lesson! I have found, however, that real tapes in real classrooms are more helpful than more professional, staged events. I especially like the videoclips because they present prompts for discussion and reflection. My goal (and it sounds likeyours as well) is to get teachers to think about what they are doing and to especially observe how children are learning.

## New DVD

Hi again, I actually thought it was very realistic to show something that you felt could be improved upon and was not presented as perfect or "staged" .... We talked in class about how important it is to be a teacher who is trying to be reflective and improving, no matter how long we have been teaching! Thanks again for some great learniing opportunities for us all.

## Patterning and Common Core for K

Hi Nita! Our Creative Curriculum for Preschool volume on Mathematics that you co-authored continues to help our teachers build a solid foundation in early childhood mathematics. Thank you!

I, too, am puzzled by the omission of patterning in the Common Core Standards for kindergarten children and was wondering if you had any insights. It's such an important concept for building algebraic thinking.

Cate Heroman

## Common Core

I have worked in a PLC with Prek teachers the past 6 years. We created a math map to help pace ourselves and purposefully teach standards at appropriate times as we look at Fall, Winter, and Spring. Can you share best practices for assessments with preschool children?

## Common Core

I have actually waited to address this question because it is such a BIG one! The National Research Council 2008 volume Early Childhood Assessment Why, What, and How provides some excellent review of assessment principles. However, I don’t think your question involves the big principles of appropriate assessment. Instead, I will address the practices that I believe are essential.

First, I assess children’s learning within the daily activities. I prefer to call them “assessment snapshots.” These assessments occur most frequently when I am teaching small groups of students, when I am playing a game or watching them play a game, or when I am observing center work. The objectives I am assessing are intentional and match the instruction that has just been given. I find that these types of assessments are informative to my instruction and immediately at that point, I can adapt my instruction to fit the needs of the students. I intentionally assess some students and some objectives every day.

Second, I assess children’s learning after time has passed. I have always been amazed by children’s development and its effect on how and when children learn a particular concept or skill. Yes, we do know the learning paths for some mathematics objectives; however, their understanding often occurs long after I have introduced a concept or a particular skill. My favorite time of the year is January. I am often surprised at what children have processed by that time and how much they remember.

Third, I try to assess children using different “windows of learning.” In other words, I need to see children in different settings, at different times of the day, in different subject areas, and with different teachers and/or assessors. Previous experiences with the child can prejudice an assessor so I like to see my children and their work periodically with different eyes.

Finally (and there are so many more points), I try to remember that the easiest things to assess are most often the most unimportant and the hardest things to assess are often the most important. I wish you the best as you work on this topic. We all could use ideas on best practices in assessment and it would be great to share them with other early childhood teachers.

## Math in focus

Are you acquainted with Math in Focus which is based on Singapore Math and what do you think od it?

## Math in Focus

As a member of an author team for a major publisher, I will not comment on another program. However, I do want to talk about the bar diagrams that are part of Singapore Math. Many programs use these types of diagrams to help children solve word problems. In fact, the author team I am on has developed a more simplified version that has been very successful. We know from research that training students in using diagrams to solve problems results in more improved problem-solving performance than training in any other strategy. Helping children visualize a problem by looking at the quantitative relationships in a problem should be an important part of any program.

## Teachers who hate math

Hi,

I am a university lecturer in Australia teaching pre-school teachers how to include math in their programs. Any suggestions for those who start the unit with an "I was never any good at math in school and just hate it now" attitude?

This is also part of the topic I am researching for my PhD as it is so frustrating to see young children miss out on many opportunities to develop math skills because their educators aren't interested.

Thanks

Kath

## Teachers Who Hate Math

Oh, I understand this issue! When I first started teaching at the university level, it was my job to teach the math methods to all preservice teachers. Initially, I was so discouraged when the majority of my students told me how much they dreaded teaching mathematics! After a few years, I loved the challenge of introducing them to the way mathematics should be taught and I delighted in the idea that some of the class experiences could open their eyes to the excitement of teaching mathematics. Over the past twenty years, I have been able to talk to hundreds of practicing early childhood teachers. It has been my privilege to introduce them to the importance of mathematics and ideas The recent National Research Council report found that typically early childhood classrooms “are emotionally positive and intellectually passive” and that mathematics was definitely not one of their favorite subjects. In my work with teachers at both the pre and inservice levels, I have found that you can’t make someone “love” math, nor can you make them teach mathematics well. Instead, teachers need some positive experiences with mathematics – both personally and with their students AND they need some information before they truly feel comfortable teaching mathematics.

I use many methods to help them gain experiences. At the preservice level, they experienced teaching small groups of children using easy-to-implement games and activities that were designed by me or another faculty member. At the end of the day, we debriefed all lessons and their observations were particularly exciting. In many cases, they learned some mathematics and in all cases, they gained experiences teaching mathematics. For both inservice and preservice teachers, I have found they need information in the following areas: 1) mathematics content --- in many cases, they have had poor instruction in mathematics and often teachers don’t really understand the mathematics behind the procedures they have learned, 2) child development or learning paths for young children in mathematics, 3) connection of the content to their standards or guidelines, 4) ideas for use in centers, small group instruction, project ideas, circle time, read alouds, routines, and 5) instructional strategies for use in mathematics classes (e.g., questioning strategies, use of manipulatives, in-class assessment opportunities, management of mathematics classes). I teach week – long seminars involving all of these areas and they seem to be effective. In addition, I use real examples of teaching episodes with debriefing opportunities (You can see some of these on the DVD enclosed in the book.) These are useful in my coaching seminars. This past year I have adopted a prekindergarten school with 6 classes. On my days at the school, I teach different lessons in all six classes all around a mathematics topic. Then, we meet for an hour at the end of the day to debrief the lessons. This has been so exciting! The teachers have learned lots, I have had a chance to experiment with new ideas/activities, and the children are enjoying math!

## base 10

It looks like the common core wants kindergarteners to understand that teen numbers are made of ten and extra ones...any extra good ways to teach that? And also...what does it mean when they say for young children to be able to "fluently" add and subtract up to 5? Thanks!

## Base 10

I can tell you have really analyzed the Common Core Standards! Recently, I have spent lots of time analyzing the Standards especially at the Kindergarten level. The National Research Council’s study in 2009, Mathematics Learning in Early Childhood: Paths toward Excellence and Equity, reports research that strongly supports an emphasis on our base 10 system as well as an understanding of addition and subtraction.

The decomposition and composition of number is one of the most important aspects of mathematics for young children. For example, children need to understand that the number 14 can be decomposed into a ten and 4 extra ones; conversely, to make the number 14, you could use a ten and 4 extra ones. This idea is the beginning of an understanding of place value in the base ten system and understanding this idea is foundational to adding and subtracting multi-digit numbers as well as other operations in our system. How do I teach this? In my opinion, the ten frame is the primary graphic model for Kindergarten students. I do lots of activities that involve estimating if a bag of objects contains MORE than 10 or FEWER than 10. After children have made their estimation by circling either MORE than 10 or FEWER than 10 objects, they count out the objects in the bag by trying to fill a ten frame. If the ten frame cannot be filled, there were FEWER than 10 objects in the bag. If the ten frame can be filled and there are extras, there were MORE than 10 objects in the bag. We also play lots of games where we try to fill a ten frame exactly OR try to make a teen number on the ten frame. Children roll a die that has 1, 2, or 3 pips on a face of the die. If the goal is to make 15, the person who places exactly 15 is the first winner of the game. There are many simple games like this and I have no doubts that you will find many of them that have been published or that you can create. The important thing about these games is that children see the teen number as a ten and some extra ones, and not simply as two digits. The research indicates that many young children see a teen number such as 15 as a ONE and a FIVE, not as one TEN and a FIVE. In my experience, this standard is very important and can and should be taught intentionally and appropriately to young children.

The idea of fluency as stated by the National Research Council in their 2009 report is “Fluency means accurate and (fairly) rapid and (relatively) effortlessly with a basis of understanding that can support flexible performance when needed. There are fluency standards at every grade level and you are correct when you say that kindergarteners are to be able to fluently add and subtract up to 5. This standard is best addressed by teaching young children to compose and decompose the numbers 4 and 5. In my prekindergarten classes this past year, I have emphasized subitizing the quantities 1 to 4 as well as stressing conceptual subitizing of the number 5. (Subitizing is explained in more detail in the book) That means that we do many activities with the numbers 4 and 5. We play the hoop game where we toss five counters into a hoop and then record how many are inside and how many are outside of the hoop. We toss 5 counters that are red on one side and yellow on the other side and record how many of each color are face up after the toss. We make unifix trains of 5 using two different colors and record how many of the two colors are in the train. We display five fingers using our two hands and count how many are on each hand. We play, “How many are missing?” games with five tokens and identify how many are hidden. The purpose of all these activities (and there are many more) is to provide many experiences for children with the parts of five. By the end of the year in my six prekindergarten classes, most children can identify the parts of five (0 and 5, 1 and 4, 2 and 3, 3 and 2, 4 and 1, and 5 and 0). With that foundation, children can easily use symbols and identify the operations of addition and subtraction. Fluency for numbers up to five WITH UNDERSTANDING is quite easy when it is introduced in this way. I suggest one caution here. Often, I have had parents tell me, “My child knows all their facts” and they have demonstrated their understanding by giving the answers to flash cards containing addition and subtraction facts. Please note that fluency for numbers up to five is not simply a memorization task. Rather, it is understanding the number quantity and the parts of the number five so that they can demonstrate fluency with the numbers and use them flexibly.

## Grade 3

I am an early childhood educator who is very familiar with the instructional strategies you developed for Teaching Strategies the creative curriculum. I use them everyday in my Pre-k 3 classroom. I am a concerned parent of an eight year old third grader. Although 3rd grade is considered an early childhood grade level, I find the approach his school is taking to meet state standards is not engaging and developmentally inappropriate. In this high stakes world, they are teaching to the test. What steps can I take to ensure that he develop a real world understanding and love of math?

## 3rd Grade

Oh, do I understand your concern! In most states, high-stakes testing is a major concern that typically begins in third grade. I, too, am concerned that third graders are learning things at strictly a procedural fluency level and never think about the conceptual understanding that is so important. If our mathematics instruction is only at the procedural level, we are ignoring the reasoning, higher level thinking that young children can do as well as their disposition toward mathematics. In addition, a focus on the tested items only narrows the curriculum and as children progress through the later grades, they have a very limiting education in mathematics.

The National Council Teachers of Mathematics (NCTM) and NAEYC worked together to write a position statement about mathematics at the PK-3 level. In this statement, they defined what would be appropriate mathematics for young children. Perhaps some of the phrases they used would help you take some steps to inform teachers and others about the importance of appropriate and intentional mathematics teaching. The position statement states that quality mathematics for young children should: “…build on children’s experience and knowledge……base mathematics curriculum and teaching practices on knowledge of children’s …development.…provide for children’s deep and sustained interaction…provide ample time, materials, teacher support for…play. I do not think the era of high stakes testing will disappear in the near future; in fact, it may become even more defined in this time of little financial support for schools. As a parent and grandparent, I do everything I can do to help teachers understand that IF you focus on the bigger ideas of mathematics rather than just the tested objectives, children will develop a conceptual understanding of mathematics. If you are in a state that has adopted the Common Core Standards, I suggest that you review the Common Core Practice Standards. These are essential to developing mathematical understanding and must be addressed in this time of high stakes testing. The position statement and the Common Core Practice Standards were both developed from research and can be found as PDF files at NAEYC.org (position statement) and Common Core Standards (practice standards). I would also set up some centers/activities for your child to use. I know you can help him/her develop the foundations that are necessary for success in mathematics. Thanks again for your concern and all of your efforts in this regard!

## Strategies

What are the most effective strategies for the K-3 age group in terms of mathematical understanding? What are the best basis for teaching math?

## Strategies

I especially like your question because it deals with mathematical understanding. I too want my students to UNDERSTAND mathematics, not just know specific procedures or have particular skills. I vividly remember that my seventh grade teacher told me, “To divide fractions, take the second fraction, invert it and multiply, and don’t ask the reason why!” I wanted to know WHY and my teacher ignored my questions. I want the children I teach to understand why they do what they do. I really like the definition of understanding that is described in an NCTM publication, “We understand something if we see how it is related or connected to other things we know.” To help children connect what they know with new learning, I most often use a strategy that is identified as “problem-based interactive learning” (John Van de Walle). Defined simply, it means that I generally start with a problem and then children interact with materials, other children, and/or the teacher and they suggest some possible solutions. Then, I connect their answers directly to the specific objective or standard using as many manipulatives, visual graphics and pictures as possible. We continue to work on similar problems and I differentiate instruction as needed to small groups of children or individuals. In many cases, center activities are used for these differentiated groupings.

## Balancing Discovery and Standards in Integration

One issue that I and my teaching colleagues have observed is how many of us work within a discovery and experimental philosophy toward teaching math, yet get frustrated by the pressure to "cover all of the standards" for our grade levels. What is your take on how to integrate math with other content areas and within academic standards while simultaneously creating experiences where young children can still discover and experiment as they gain mathematical skill and knowledge proficiency?

## Balancing Discovery and Standards in Integration

What a good question… it is one that I continue to ask myself as I teach in early childhood classrooms! First, let me say that I have appreciated (and use) the NCTM focal points and lately the common core standards in my teaching. I am happy that the standards are more focused and give me more clarity about what is important to teach at a skill and proficiency level. I view the Common Core Standards as consistent with research and the learning paths that are based on current research. They help me as a teacher KNOW where I am going, WHAT I should be helping children discover and experience. I also note that the standards document does not define HOW I should teach them and that a discovery and experimental philosophy could (and should) be used to create the experiences children should have to gain mathematical understanding as well as skills and proficiency in those skills. Second, I want to acknowledge that I have the most fun when I am intentionally doing a discovery and experimental activity with children… IF I intentionally create experiences that will allow children to understand mathematics and IF I help them summarize, picture, or verbalize what they have learned, it takes more time, but oh… the children really remember it! Discovery learning fits my way of teaching young children and my philosophy of teaching. In fact, as a beginning teacher, I most frequently taught those type of lessons and expected children to just transfer what they learned from the discovery approach into what they needed to know at the skill and proficiency level (by the way… that didn’t work as well as I wanted it to!) In my mind, there needs to be a balanced approach. Some standards are best taught through a discovery approach; others are best taught with more direction toward the skill to be learned. I do include mathematics throughout the day in my classrooms and I love and often use a project-type approach. For example, In the kindergarten classes I have taught this year, we have introduced measurement and geometry concepts using an advanced curriculum about a frog who is communicating with other frogs in space. Throughout the unit, we have integrated science and writing and the mathematics learning has been outstanding. In another example, the young children in a prekindergarten class grew a vegetable garden, measured their growth, sold them at the local farmer’s market and then had to decide how to share their earnings fairly. I continue to be amazed at what young children can learn when I facilitate their learning in the most appropriate way

## patterning in K and common core

I recently read that understanding patterns is not addressed in the kindergarten common core standards. Isn't patterning an important step in algebraic understanding for children in kindergarten?

## Patterning in K and common core

You are right! Patterning is an important step in algebraic understanding. I have often heard people say that patterning is not in the standards in kindergarten and that is correct if you are only looking at the Common Core Content Standards. However, if you analyze the Common Core Practice Standards, patterning is listed there as a practice with all mathematics. As you know, there are patterns in nature, there are literacy patterns, there are patterns in language (i.e. days of the week. Most of us (please hear the word, “us”) have spent a great deal of time on color patterns (e.g., blue, red, red, blue, red, red) or action patterns (e.g., snap, clap, snap, clap). What we need to be emphasizing are the patterns specific to number and geometry, the two important areas in content for young children. The seventh practice standard states, “ 7. Look for and make use of structure. Mathematically proficient students look closely to DISCERN A PATTERN OR STRUCTURE. Young students, for example, might notice that three and seven more is the same amount as seven and three more…” In my opinion, the Common Core Standards do address the importance of patterns. They just need to be focused on patterns in number and geometry.

As an example, the counting patterns are important. While our verbal number patterns are not good patterns especially in the teens (i.e., eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty should be ten-one, ten-two, ten-three, ten-four, etc.), our written counting numbers follow a definite number pattern (i.e., the ones digits repeat… 1,2,3,4,5,6… and the decades increase in a similar pattern the teens, (1__, 2___, 3___, etc.). Similarly, the +1 pattern is a critical to understanding addition. (1 more than 3 is 4, 1 more than 17 is 18, etc.) I believe that the 100s chart that displays this pattern using the written symbols should be in early childhood classrooms and the numerals pointed out as children count. Yes, patterns are important to algebraic reasoning and that the Common Core Practice Standards address and emphasize their importance.

## Math Sequences

I understand that children learn math concepts in sequence. What are some commonly overlooked steps that teachers should be aware of. Thank you

## math sequences

I am glad that you used the word “concepts!” One of the steps that I believe that we often overlook is “conceptual development.” Instead, we tend to teach the procedures without developing the meaning or connecting the information to what the child already knows. Let me illustrate this with two different topics that I see missing in the early grades.

Measurement is often a topic that is relegated to the end of the year, ignored entirely, or taught by showing children how to measure the length of something using paper clips, plastic links, or some other unit. Then, in third grade children in most states are tested on measurement and children have not had the experiences that are necessary to really understand what they are doing when they measure. Children need a chance to discover the concepts of conservation, iteration, the use of same size units, and the inverse relationship between unit size and number of units before they are taught the specific measuring procedures. I have been teaching classes in prekindergarten and kindergarten this year. Recently I used licorice sticks to measure how far cars traveled when they were released on a ramp. We selected the car that went the farthest distance using licorice sticks; it traveled 8 licorice sticks. Then, I brought in MY car bragging how far MY car would go! At the end of the day, MY car traveled down the ramp and only when a short distance. When children cheered, I reminded them that we needed to measure it first. This time when I measured it, I used very short licorice sticks and it went 10 licorice sticks. While a few children didn’t see a problem, most of the students told me I did it wrong. They were able to tell me what I had done incorrectly (i.e., used a different size unit) and they also told me what I needed to do right! They were beginning to understand the importance of using the same size units to compare and measure. I didn’t need to teach it… they remembered it every time we measured lengths.

Counting is a familiar procedure that we all address. However, we often overlook some very important concepts and procedures. For example, we have children count items in a row, a particular pattern like a ten frame or a dice/domino or in a random order. We often forget that counting OUT a specific number of items is important as well. When I assess four year olds as I have done much of this year, I find that many children can count items that are displayed but have difficulty counting OUT that same number of items and putting them into my hand. Counting has many other important ideas that should be addressed as well: 1) You can count in any order and you will have the same number of items. 2) The last number said answers the “how many” question, 3) It is important when you are counting that you keep track of what has been counted and what has not been counted, 4) The + 1 pattern is really a counting sequence. In other words, one more than a quantity is the next counting number.