We want children to have a ‘feel’ for numbers—the ability to work flexibly in solving number problems. That is called numbersense. Numbersense is the mastery of number concept, number relationships, and place value and their integration. Mastery means (a) understanding, (b) effective and efficient strategies, (c) fluency, and (d) applicability. Numbersense leads to the mastery of numeracy.
Mastery of numeracy should be an essential outcome of the elementary school (grades K through 4) mathematics curriculum. It is the facility in executing the four whole number operations, including standard algorithms, correctly, consistently, and fluently with understanding.
Poor numbersense in children is due to inefficient strategies such as relying on sequential and rote counting of objects (e.g., blocks, chips, fingers, or marks on a number line). Learning facts and procedures through rote memorization without understanding does not help children in making connections between numbers, arithmetic facts, concepts and procedures. When they encounter new concepts or need to apply mathematics ideas to problems, they find it difficult. And, they give up easily. As a result, many are termed “slow learners.” Often, our pedagogy turns them into slow learners.
Able children are shown and practice efficient, effective, and elegant strategies. Less able or children with special needs simply are not shown the same techniques. With inefficient and less effective strategies, children end up spending enormous amounts of time deriving even the simple facts. This makes the tasks laborious and they either do not succeed or lose interest and lag behind.
Definitions of arithmetic operations such as: addition is counting up, subtraction is counting down, multiplication is only skip counting forward, and division is skip counting backward, do not lead to efficient strategies. For example, less successful children see subtraction as an isolated concept without connecting it with addition. They do not capitalize on learned addition facts. A similar situation happens with division. They end up spending more time on acquiring mastery of subtraction and division facts with limited results. These children have difficulty becoming flexible and fluent in arithmetic facts. Mastery of arithmetic facts is an essential element of numbersense. When addition and subtraction are shown as inverse concepts/operations, the mastery in one reinforces the other. Similarly, after initial introduction of multiplication, children should be taught that multiplication and division are inverse operations.
Mastering arithmetic facts using efficient and effective strategies and models frees children’s working memory. Then, they can engage in learning and mastering higher order thinking skills and applications, easily and effectively. Higher order thinking is dependent on flexible numbersense and the mathematical way of thinking.
The mathematical way of thinking is the ability to: observe patterns in quantity and space, visualize relationships, make conjectures, predict results, and then communicate observed connections and their possible extensions using mathematics language and symbols. Mastering arithmetic facts is a necessary, though not sufficient, condition for higher mathematics. Competence in numbersense translates into effective mental math—the hallmark of mathematical thinking.
Children with numbersense make connections, generalizations, abstractions, and extrapolations of number patterns they observe and processes they have mastered. They link new information to the existing knowledge and develop insights about number and their relationships.
Understanding Number: Spatial and Quantitative Relationships
The fundamental relationships between numbers at elementary level are expressed in two forms:
Spatial: The spatial aspect of number is determining the relationships between numbers by their locations and proximity with each other. The child knows a number when she can locate and place the number on an empty number line in relation to other numbers (to the right of, left of, how far from, or how close to a given number). Being able to point to a number and its place on a number line is not enough to understand number relationships.
Spatial aspect also relates to positional aspect of number: e.g., the second from the start, third person in the row, tenth’s book in the row, etc. The numbers in this form are called ordinal numbers. Children learn ordinal numbers before they learn the quantitative aspect of number.
Quantitative: The quantitative aspect of number is the value of the number. How big? How small? More than? Less than? It is the understanding that a number represents the magnitude of a collection. It is knowing that number is the property of the collection, not just the result of counting. And that the last number used in the count, from any direction, indicates the size of that collection. This value has a unique place on the number line. This is the cardinal aspect of the number (the magnitude, numberness).
Numberness is to know: Is the number bigger than another number? What number is half-way between 10 and 20? Can you place ‘three numbers’ between 45 and 55? What digit is in the tens’ place? What is the value of the digit in the tens’ place? What is 10 more than 67? What digit in the given number has the highest value? What is 8 + 6? What should we add to 9 to get to 17? What is the difference between 17 and 9?
Any question about number relates to both aspects of the number, but questions such as the following mainly relate to the spatial relationships between numbers:
On the whole number line above, what number comes after 17? Place 22 on the number line. What number comes before 45? Is the number 29 closer to 20 or 30? To answer these questions, the child refers to the spatial idea: Where is that number located in relation to other number(s)?
The following questions are related to quantitative relationships between numbers: Without referring to or drawing a number line: Give a number between 23 and 29. What number is 10 more than 54? What is 8 + 6? Give a number between ⅕ and ½? What number is 3 less than 23? What number is ⅔ more than 3? What is the next tens’ number after 53? (Tens number are: 10, 20, 30, 40, 50, 60, etc.)
If a child can answer these questions only by the help of a number line, then that is not indicative of mastery of the number. Applying only spatial, sequential counting to derive arithmetic facts disadvantages children.
Many children use only the spatial aspects of number in deriving and understanding number relationships (facts). When they have to answer questions such as: What is 5 more than 7 or 2 groups of 7 they get the answer by counting on a number line, objects, or on fingers. Truly understanding number relationships and acquiring efficient strategies for mastering arithmetic facts, one needs to integrate spatial and quantitative aspects of number. To learn efficient strategies for mastering numbers facts one needs: (a) sight facts (including making ten), (b) what two numbers make a teen’s number (e.g., 16 = 10 + 6, and (c) what is the next tens after 43, etc.) Instructional models such as Visual Cluster Cards, Cuisenaire rods, and Empty Number Line help in this integration and acquiring effective, efficient strategies.
The concept of place value is an example of this integration. To know the whole number 235 well, one has to focus on the spatial aspects of the digits (1’s, 10’s, and 100’s places; although the numbers increase to the right, the place values of digits in a multi-digit number increase to the left) and the values of these individual digits contribute to the understanding of the value of the whole number itself. For example, both the Standard (5,694) and the Expanded Forms (5000 + 600 + 90 + 4) and later on, Place-Value form (5×1,000 + 6×100 + 9×10 + 4×1), and Exponential form (5×103 + 6×102 + 9×101 + 100) of the number take advantage of understanding and mastery of quantitative and spatial aspects of number. The same concept is then extended to factions and decimal numbers.
Making Numbers and the Number Line Friendly
Daily Counting Using Number Line
To develop number relationships, forming a visual image of a number line is important. This means: (i) mentally locating numbers on the number line, (ii) recognizing the patterns and structure of the number system, (iii) extending those patterns (e.g., 3 comes after 2, so 23 comes after 22, 73 comes after 72, 173, comes after 172), and, (iv) applying these patterns to solve quantitative problems. This competence is the beginning of developing a robust numbersense.
Games and Toys
Children develop number relationships through routine counting while interacting with their environment as part of normal growth and development. Playing with games, toys and remembering number rhymes and stories bring out counting and number relationships. Board games, using dice, dominos, and playing cards are opportunities for learning number relationships. However, informal and infrequent play may be slow or inefficient for the development of number relationships. Formal exposure to appropriate, diverse activities and effective strategies assures efficient development of numbersense in children. For example, Number War Games using Visual Cluster CardsTM (VCC), dominos, and dice are excellent examples of such activities.
Children’s practice of meaningful, strategic counting is an important preparation for developing efficient calculation strategies. Counting is a complex process. It involves several sub-skills and takes considerable time to become fluent and competent. Unitary counting (sequential counting from 1) is children’s first exposure to the structure of number line, but it becomes progressively complex with age and grade. For example, it should progressively include counting by 2s, 5s, 10s, 100s, by a unit fraction, proper fraction, mixed fraction, decimal, etc., starting with any number and moving forwards and backwards. Such progressively complex counting strengthens numbersense.
As children become competent in counting, they begin to visualize the number line—number patterns, locations of numbers and their relationships. Crossing the decade/century and realizing the counting patterns is an important achievement for children in understanding the structure of our Base Ten number system. Children’s observed number patterns on real and visualized number line help them develop strategies that give them power to develop and understand efficiency of arithmetic operations. For example, a child observes that 42, 52, 62, 72, and 82 is a sequence of numbers increasing by 10 and they occur 2 after respective decades (tens). She, later, uses it to solve a real problem: what is the change when she has spent 52 cents from a dollar? She discovers that the change could be calculated by counting up by 10 from 52 till she reaches 92 and then 3 more to 95 and 5 more to the dollar (e.g., 4 dimes + 3 cents + 1 nickel = 40 + 3 + 5 = 48 cents). And a little later, she realizes, 52 and 5 tens is 102, that is 2 more than the dollar, so change is 48 cents. Or, 50 + 5 tens = 100, but we should have started at 52, so it 50 – 2 = 48 cents. Similarly, at a later date, to find the product 16×3, a child thinks: 16 is 10 + 6. 10×3=30, 6×3=18, 30+10=40, 30+18=48, so 16×3=48.
What To Do During Counting?
Counting should be a whole class activity, first oral and then in writing. Counting should begin with a number line (with numbers marked and displayed from 0 to a number beyond 100). A portion of the number line, preferably from 0 to 35 should be at children’s eye level and rest on the wall.
During the counting activity, the teacher should emphasize when a decade is complete. She should help children see that something important is happening when they reach a new decade—a new group of tens. Similarly, she should point out what is happening immediately after and before tens numbers (e.g., multiples of 10). She should emphasize what is before and after the new decade (the new ten). Knowing what is before and after that decade is a difficult concept for many children. For example, she should point out that when the count reaches a new tens, e.g., 30, 40, 50, …, the cycle of the count of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 is complete and then repeats again and again.
In counting whole numbers on the number line, children should be able to realize the cyclic pattern of the base-ten system:
…29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, …etc.
This cyclic pattern is the key to understanding number relationships. Initially, counting should be done on a number line in linear form (as seen and discussed above) so children develop the idea that numbers are continuously increasing to the right and decreasing to the left. In later grades they will extend it and understand the idea of positive infinity (+∞) and negative infinity (−∞).
When children see the progression of number in a 10×10 grid form, they see cyclic patterns much more clearly. This understanding of number relationships and structure leads children to arithmetic operations.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, …
Most children develop this structure independently with little help from others. However, for others, it is important to formally develop it.
Mid-line Crossing Problem
It is important to begin counting on the horizontal grid (shown above). Some children, because of their mid-line crossing problem (MLCP) may not discern the pattern easily as they do not see the horizontal numbers on a number line as “equidistant.” For example, many children with MLCP, see the equidistant numbers displayed in the first row (below) as in rows two or three where the numbers are not equidistant. In row two, they are jumbled up in the two ends and in the third row, they are jumbled up in the middle.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (row one)
1, 2, 3, 4, 5, 6, 8,9,10 (row two)
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (row three)
When the numbers are organized vertically, it is easier for them to see the patterns.
Counting Using Number Grid
Number Grids are horizontally (figure one) and vertically organized (below) One). The Horizontal Grid is a 10×10 grid, with entry of 1 in top left most cell. Each row ends with a multiple of 10. The Vertical Grid is a 10×10 grid with top left most cell with entry of 1. Each column ends with a multiple of 10. The procedure for counting using the grids is the same as the number line. Counting on grids can be done horizontally and vertically.
Locating Numbers on an Open/Empty Number Line
Kindergarten and First Grade
On one side of the room hangs a clothes line (low enough so children can reach it and high enough so it does not interfere in their movement). On clothes pins write numbers in dark ink from 1 to through 100. The multiples of ten numbers are written in red. Similarly, the numbers with 5 in the one’s place are written in green.
The numbers: 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 are known as Bench Mark Numbers—important in our Base-Ten system. With practice, children see all numbers in relation to bench-mark numbers. Bench-mark numbers serve as the markers for estimation, location of numbers, and approximating the outcome of arithmetic and algebraic operations.
Place the numbers in two buckets. Numbers 1 through 30 in one bucket and numbers 1 through 100 in the second.
In later grades, children encounter other bench mark numbers, such as: ½ (.5, 50%, etc.), square numbers, certain products of numbers (e.g., multiplication tables), π, standard trig functions (e.g., trigonometric function values of 30°, 45°, 60°, etc.) and important parent functions in algebra.
The teacher points to the clothesline and asks each child, in turn, to pick a number from the bucket and place it on the clothes line in its place. Children take their turn placing their numbers. A child can move or adjust the place of the numbers already placed on the line in order to locate his/her number. Teacher should ask children the reason for the placement of their numbers, adjusting the numbers on the number line, and the relationship of their number with other numbers, particularly with the bench-mark numbers.
In the beginning of the year, the teacher should use numbers from 1 to 30. After about 2 months, she should use numbers up to 100 and beyond. After half-year, the teacher should give children an empty number line (ENL) on a sheet of paper where two end numbers are written, and she dictates numbers and children locate the number’s place and write the numbers on the ENL.
Grades Two and Three
The teacher should give children a sheet of paper (2”×11”) each side having an Empty Number Line (ENL) drawn on it with two end numbers. The end numbers change every week.
She dictates 10 random numbers between the two end numbers and children locate the number’s place and write the numbers on the ENL as the numbers are dictated. After children have located the numbers they compare their ENL with their partners and come to agreement on the locations of these numbers. The corrected locations are placed on the ENL on the other side of the paper. This activity should be part of a daily math lesson.
Grades Four through Six
The same activity as in the grades 2 through 3, but the choice of numbers changes. The numbers can be whole numbers, fractions (unit fractions, proper fractions, mixed fractions), decimals, and percents.
Grades Seven through Nine
The same activity as in grades 4 through 6, is repeated but the choice of numbers changes. The numbers can be real numbers (whole numbers; fractions—decimal numbers, percents; integers; rational numbers and irrational numbers).
Every day, before children arrive, the teacher places cards with random numbers written on them (numbers appropriate to grade level). Each child picks a card, and when children line up, each child follows the order by the number on his/her card. Children keep their cards ready; before any classroom activity, the teacher calls on them by specific criteria: The person with the card between 1.5 and 1.6 will answer the next question. The choice of numbers changes every day.
Daily Oral Counting
Daily counting is a warm-up activity for grades K through 8. It could be part of the calendar activity in grades K through 2. The choice of number to count with should be related to the main mathematics concept taught in the classroom that day. For example, when children, in the third grade, have been introduced to fractions, it is a good idea to count by unit fractions. Similarly, when children are adding and subtracting fractions with same denominators, counting should be backward and forward by a proper fraction. It is one of the tools for helping children to have a deeper understanding of number.
Each child should have a Math Notebook where all of his/her mathematics work is recorded. It is the sequential record of classroom mathematics writing: language, concepts, operations, definitions (examples and counter examples), conjectures, proofs, formulas, calculations, constructions, drawings, sketches (geometrical shapes, figures, diagrams, etc.), summary and reflections on class mathematics work. The written work should be done in pencil. Only when the teacher wants a short answer, an example, immediate feedback to a definition, concept, or a procedure, children can use individual white boards. In math notebooks, one can have several examples in succession, so children see emerging patterns. Individual white board work does not leave a history of their work and they may not be able to observe patterns. We, as math teachers, should remind ourselves and our children that: “Mathematics is the study of patterns. It has deep structures.” For this reason, we should help children to observe these patterns in their work and recognize the structures that emerge from these patterns.
Procedure and Language for Counting
Here are the points to consider during the daily counting process (at least five minutes). Each teacher should adapt these to suit her students’ and her instructional needs.
- The teacher should announce the counting number and start number (later children can select the starting number and the counting number). These numbers should change each day.
- During counting, when children give their numbers, the teacher should repeat each number clearly enunciating each word. This is particularly important at the Kindergarten through second grade.
- The teacher should record the numbers from the count on the board creating columns and rows. Children record the numbers on their graph papers in the same way, in columns and rows. The starting number should be placed in the uppermost left corner of the paper (in the first full column of the paper). Leave one column between the columns for comments. As the columns of numbers emerge, the number of entries in each column must be same. For example, begin with 4 numbers in each column. Each day, change the number of rows up to about 10. Having the same number of entries in each column will produce patterns both horizontally (in rows) and vertically (in columns). It makes counting a rich activity. It also provides opportunities for differentiation. “High flyers” can be asked to give numbers horizontally and others vertically.
- During the counting, the teacher should ask specific children to come forward and record the number on the board. The child writes the number on the board in the appropriate place. The teacher takes this opportunity to model the writing of multi-digit numbers: Are they of the same size? Are they at the same level? Are the digits equidistance? Are they aligned with each other?
- Counting activity should include counting both forward and backward (not necessarily on the same day).
- Each child should have the opportunity of responding a few times during the counting.
- The choice of a number for counting begins from the easier one in the beginning of the year to bigger and more difficult numbers as the year progresses. For example, one should begin counting by 1 forward and backward in the beginning of the Kindergarten and counting by 10 toward the end of the year.
- Counting by 2 can be assisted by using the Number line, Hundred’s chart, using the Cuisenaire rods’ staircase, the standard number grid, or Vertical Number Grid.
- Counting by 10 can begin concretely, using the Cuisenaire rods and then without them. For example, begin counting by a number, say 7, ask a child to pick up the 7-rod (black). Write the number on the board. The child gives the rod to the child to his right and that child adds the 10-rod and calls out the number (17). The teacher writes the number on the board, starting the first row or column (below is the example of the first row). The process is continued for several more times. And then the teacher encourages children to extend the pattern without the rods. She keeps it on till children can give the next few numbers without the help of rods. Next day the counting by ten can begin with picking another rod. Toward the end of this forward counting begin counting backward from the last number.
- When the teacher begins any counting, she asks who has the next number, and then the next one, till several numbers in the count are generated. This should be done by volunteers first and then by randomly selecting children or the ones who need support. One should take advantage of high flyers’ knowledge of numbers as a starter. Never give the number easily. Try to derive the number with the help of children using decomposition-recomposition process. Someone will come forward. I have never been disappointed in any class, in any school. Some child in every school, in every class comes up with the next number and then others pick up the theme and the pattern and the learning process and counting begins. When a particular child is stuck on getting a number, give him clues: start with the facts he already knows. For example, if the child (Kindergarten level) does not know what comes after 54, go back to the child who gave 51, and continue, most times the child will come up with the number. If he still does not come up with the number, ask him: what comes after 4? If he answers correctly. Ask him: what comes after 14? Etc.
- It is important that the teacher openly acknowledges the child who gives the correct number by children clapping twice in unison. Never leave a child without success. Help each child to taste success, even if it is just what comes after 7 or before 7.
- Once a pattern begins to emerge and children understand the task and the count, ask them to write the next five numbers and place them in the proper places—in the correct columns so that they can observe the emerging pattern in numbers. As children write the numbers, the teacher walks around the room asking each student to give an example. Some children will readily observe the emerging patterns, both vertically and horizontally. Avoid having children give the pattern too soon. Instead, devote enough time discussing the numbers so most of the children see the patterns. It may take several days.
- Do not disclose the pattern, let children arrive at the pattern. Ask teams of two children to discuss the number relationships and the process. Let them arrive at different patterns. Only when most children are able to give the correct entry, then ask a child (not a high flier) to articulate the pattern.
This counting will result in a Dynamic Vertical Grid. In the beginning of the year, the counting will take longer; however, as it becomes a routine, it will take less and less time and more will be accomplished. Soon the counting process becomes an important means for making formative assessment of children’s numbersense.
- During the counting, the teacher should ask a great deal of questions about the numbers—place value, digits, values of digits, location of the number, one before, one after, 10 more, 10 less, what is the next tens, what is the next whole number, etc. These questions instigate mathematics language, concepts, and mathematical ways of thinking.
- The teacher can make up impromptu mathematics problems: What is the difference between 2 consecutive numbers (in the same row, in the same column, both in the same row but few cells apart, both in the same column but a few rows apart, etc.).
Counting Example Grade One/Two:
Following is an example of daily count (for first grade during the middle of the year and in the early part of the year in higher grades):
Teacher: Let us count by 5 forward starting with 49. We will write the numbers in the first column. You do the same on your graph paper.
The first column and the beginning of the second column is derived together.
Now ask children to write the next five numbers in the count. They should begin from the cell marked by “*”. As they are writing their numbers, the teacher should go to the child who is struggling and help generate one or two numbers. For example, she asks:
Teacher: What are we adding to 99?
Teacher: What is 1 more than 99?
Teacher: Good! We have added 1 to 99. Where did we get 1 from?
Child: Did it come from 5?
Teacher: Yes! That is good. Now, what is left from 5 to be added after adding 1?
Teacher: 4 is added to what number?
Teacher: Very good! What is 100 + 4?
Teacher: Good! So, 99 + 5 is what number?
Teacher: Now continue. What is the next number?
Child: That is easy. 104 + 5. I know 4 plus 5 is 9. So, 104 + 5. That is 109.
Then the teacher moves to another child.
If a child has finished writing 5 numbers, she checks his work. If it is correct, she asks him/her to check other children’s work as they finish the task. If two children have finished writing the numbers, you ask them to compare their entries with each other and make corrections. If they have any disagreements, they should come up with consensus by supporting their arguments. When many more children have written all the five numbers, ask them to compare the answers in pairs. Keep on making pairs to correct each other’s work and checkers to check other children’s work. The teacher, with the children’s help, writes the five numbers on the board. The teacher should begin with the child who was struggling and then move on to others.
Children who are able to complete the task earlier, are asked to write 7, 8 or more entries. The numbers in red are the entries provided by children.
Then the teacher asks children to work in pairs to identify the numbers in the place indicated by “___”.
When children (in teams) have found the number in the indicated place, she asks them to supply the numbers. She records them on the board in a separate place than the grid. She discusses their answers and asks for their methods of finding these numbers. Children defend their answers. Exact answers are identified. Then, the most efficient methods for finding the exact answers are identified. Entry in the place is made. If time permits, she creates more places with the red line (___).
The Modified Vertical Grids are effective in helping children improve the numbersense and to assess if children have acquired the structure of the number system. In vertical grids some of the entries are left blank to perform formative assessment. Children are asked to give the missing numbers by counting horizontally and vertically.
Counting Example at Grade Three/Four Level:
Numbersense is a constantly evolving skill for a child. One of the processes is to relate the new numbers being introduced to the numbers the child already knows. In the third grade a new set of numbers (fractions) are introduced in earnest. Therefore, it is important to begin to relate fractions with whole numbers and with each other.
In third grade, counting using whole numbers (1, 2, 5, 10, 100, and 1000), starting from any number should continue. However, towards the end of the year, when the concept of fractions is being introduced to children, the teacher should introduce counting by a unit fraction. A unit fraction is a fraction with numerator as 1. Following is an example of counting by a unit fraction (e.g., ⅕) starting from 4. All other elements of counting procedure remain the same as before.
Counting Example at Grade Four/Five Level:
During grades four through six, our focus is on understanding and operating on fractions (and all the other related concepts). In the fourth through sixth grades, we need to relate fractions to each other and whole numbers, decimals, and percents. Although in fourth grade counting using whole numbers (1, 2, 5, 10, 100, 1000, and unit fractions) starting from any number continues, children should begin counting by proper fractions. In the fifth grade, they should add counting by mixed fractions. Following is an example of counting by a mixed fraction (e.g., 1⅖) starting from 4. All other elements of procedure remain the same. In these grades, they can also count by decimals.
The objective of the daily counting tools (Number line, Open/Empty Number Line, Horizontal and Vertical Number Grids, Modified Number Grids, Hundreds Chart, Modified Hundreds Chart, Dynamic Number Charts, etc.) is to develop and improve numbersense. This objective can be achieved if children are given enough practice in counting using these tools and if they achieve the bench-marks in this activity at their grade level.
Each classroom should have clear display and use of these tools. However, they should not be overused for deriving addition, subtraction, multiplication, and division facts and operations. Their overuse makes children dependent on counting as the only strategy for developing arithmetic facts.
 See Games and Their Uses in Mathematics Learning by Sharma (2008).
 See the goals of counting at each grade level in Part One of this series on Numbersense.
 For the numbers to be used at each grade level see previous posts on Numbersense on this blog.
 See the Numbersense Part 1 in this series of posts.
 See first post in this series on Numbersense.