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Assessment through incredible equations.

Take out your base ten-materials; our number today is eighty-eight. Build that amount, then tell me some of your mathematical thoughts in a sentence to describe what you have in front of you." Or the teacher might have said on another day, "Using your play money, show me one way to make today's number, one hundred thirty-five, and tell me a mathematical sentence to show how you grouped and sorted your play money for that amount."

Young children's understanding of numbers grows out of their real-world experiences, using physical objects to see and find patterns and groupings. They construct internal equilibrium through their own actions on objects. Piaget's research revealed that the cognitive structures of the mind are self-regulating when children are given sufficient time and experience. A good teacher allows them that time and experience. Children's thinking and understanding always begin with experiencing physical objects, but then they need to be able to draw or see a picture of that idea as the concept is assimilated into their mental structures. The teacher's first priority is to nurture and facilitate that process of objects, then pictures, giving students the power to learn for themselves. With the gift of time, they gradually come to understand that the squiggles or symbols we use to represent those same ideas are connected to the concrete objects and pictures they have drawn. This process is how children think and learn.

For example, a child learns to group a pile of loose beans into cups of ten plus loose beans left over. He or she soon learns that it is faster to count by tens and then "count on" the ones left over than to count one by one. Many experiences of sorting, grouping, and counting are needed for the concept of our number system to develop with meaning and understanding. Next, children draw pictures of groups of ten and loose beans left over, and count the beans in the picture. Finally, they understand when they write 10 + 10 + 10 + 4 = 34 or 30 + 4 = 34 that those symbols match the real cups of beans and the pictures that they have drawn. This process is the natural development of children's thinking: to experience objects, then draw pictures for those mathematical ideas, and finally, express the ideas as mathematical thoughts using symbols.

The nature of the mind is always to organize, to order and make sense out of our world. Our activities with children should always create an interest in solutions. They should build confidence, risk taking, acceptance of some frustrations, perseverance, and the realization that not knowing is different from not knowing yet.

Getting Started

I used to have children write and think about "fancy" ways to express the day's date. Then a few years ago, I experienced the many opportunities to teach, reinforce, and extend primary mathematics skills through the calendar. The part of the calendar experience that I want to address in this article is the incredible equations that are generated and recorded each day in many classrooms. Incredible equations grow out of direct concept instruction, concepts reviewed and remembered, and the challenges and new insights of individual children as the year progresses. A teacher can daily assess and evaluate children's learning and understanding by observing what equations the children give, guiding a child to "fix" an equation, and asking probing questions. In this activity the aspects of mathematical knowledge and its connections can be observed and assessed daily as an integral part of instruction and learning. Children are given power when they create fancy incredible equations: they communicate their understanding and reasoning and are led to think, record, and solve problems in new and exciting ways. The discussions, clarifications, and reflections all give evidence of students' progress in learning mathematics. The children's mathematical understanding is the focus, but continuous teaching, learning, and assessing are all going on at the same time.

The NCTM's Assessment Standards for School Mathematics (1995) are being implemented in this simple activity. While students are creating their incredible equations, each of the six assessment standards is involved. For example, students show that they are able and predisposed to apply their mathematical understanding to new situations, and what they construct is an understanding of mathematical principles (the Mathematics Standard). They are able to demonstrate what they know and can do (the Learning Standard). Each student's mathematical power is developed, and those with special needs participate readily (the Equity Standard). Their responses are consistent with the learning goals that students are pursuing in class (the Openness Standard). The equations developed each day, when taken over a period of time, indicate students' mastery of the subject matter (the Inferences Standard). Their work, both group and individual, is collected in journal entries as the curriculum unfolds and is clearly aligned with the curriculum set down by the district (the Coherence Standard). When the assessment standards are clearly understood by the teacher, they can be included naturally on a daily basis through the kinds of activities discussed in this article.

As a Title I Basic Skills teacher, I collaborate with classroom teachers and move in and out of classrooms all day long. One such first-grade room records incredible equations each day during the shared, opening instructional time. Each child has an equal opportunity to demonstrate his or her understanding, and immediate feedback facilitates learning for children. Such a conducive atmosphere empowers students as young as first graders to create incredible equations. In generating incredible equations, some teachers use the number of days that the children have been in school; others, the day of the month; and still others, a particular number chosen for study that day. In this particular room, the number used daily for incredible equations is the number of days that the children have been in school.

On the first day of school the idea of writing equations or mathematical sentences was introduced. Such simple sentences as 1 + 0 = 1 or 2 - 1 = 1 were discussed, experienced through groupings of children or built with learning tools, and then recorded on chart paper in picture and symbol form. The discussions, experiential learning, and recording of equations have continued throughout the year, the numeral each day being one more than that of the previous day. The numeral is expressed as mathematical thoughts in as many sentences as the class can generate or until the chart is full. The activity is very exciting for the children and has held their interest day after day. After about day 50, they begin to ask, "Do we still need to draw pictures, or can we just write the number sentences on the chart?" They are assured, "If someone wants a picture, it will be drawn, but you as students will lead." This whole discussion shows the growth and understanding of the class. The children do apply what they have learned day by day in direct mathematics instruction as they create and expand equations each day. Some children thrill in writing long mathematical sentences and are out of breath at the end of reading them. These sentences provide an opportunity to talk about writing them in a simpler way. The most powerful part of incredible equations is that a child is never wrong. The child can be guided to "fix" any equation.

What It Looks Like

In the group-instruction area is a container to which one more cube is added each day. These cubes are used to make groups of hundreds, tens, and ones, often by the child in greatest need of understanding that concept. The area also has bundles of straws, a place to draw a picture of the numeral of the day, cards containing pictures and numerals, expanded-notation flip cards, a chart with 200 pockets, odd-even cards, money, a place to record dollars-and-cents notation, and so forth. In the calendar area of the room, many visuals and concrete materials are always available and ready for use if a child needs help. Each day, a card with the numeral for the days spent in school is added on the pocket chart and on the number line.

Figure 1 shows the pocket chart after one more numeral was added on day 157. Using the chart, the children quickly become aware of a row of ten and readily use the chart as a visual reference in writing their equations. For example, on day 157, the sentence 200 - 43 = 157 was derived when the child started at 200 and thought backward, "Minus ten, minus ten, minus ten, minus ten, minus three more gets me back to one hundred fifty-seven." When the child was asked how he arrived at such an incredible equation, he came to the chart and showed with a sweep of his hand how he went backward from 200 in rows of 10 and then 3 more to get back to 157. He obviously knows what "take away" and "subtract" mean.

Figure 2 shows the recorded sentences for day 157. The children's work reveals that they understand and use addition and subtraction operations; have acquired the concept of names for ten; use counting by tens; are comfortable with the use of hundreds numbers; have some familiarity with parentheses for groupings; and can employ expanded notation that shows evidence of an understanding of hundreds, tens, and ones. All of this learning has been fun for the children while allowing them to show their understanding of number concepts in written form, how this understanding is communicated orally, and how talk is written down so that it can be shared with others. The children have communicated their mathematical knowledge and understanding; they are authors, and at the same time, their teacher is assessing what they know and have learned.

I was in the same first-grade classroom on day 176, when the incredible equations were being generated once again by the class. They had had some direct instruction and experience with money the week before, and I observed quarters and half-dollars being used in students' incredible equations on that day. The sentence 200 - 10 - 10- 4 - 0 = 176 was given by one of my Title I students who felt empowered to give such a long sentence. Sure, she repeated the previous sentence, but she added - 0 to make it her own, and in doing so, she showed me that she understood the concept of zero. Another child gave the sentence (200 - 32) + 8 = 176. He had really been thinking hard and really struggled to add back the 8 to get to 176. Watching him think and work out the equation on the chart was amazing and renewed my understanding of wait time, both for teachers and for other children. The child was not wrong; he just had to be given time and be guided to get back to 176. It is always revealing when a child says, "Will you write it up and down too?" What a wonderful opportunity to increase vocabulary with words like horizontal and vertical and to show first graders how to line up the ones, tens, and hundreds carefully. As I left her classroom that day, I asked the teacher if I could lead the incredible equations the next day and try to stretch the children's mathematical understanding and vocabulary. As always, she was gracious and glad to have some peer coaching.

New Ways of Thinking and Writing Equations

My first probing question the next day was "Could anyone express one hundred seventy-seven in terms of money and tell me what to write?" Immediately six hands went up, and the first child used quarters and pennies and the word cents as shown in figure 3. I then showed the students how to write the dollar notation with the decimal point. I asked if someone could give me another money sentence, and I used the dollar sign and decimal point to record their thoughts. A little girl had her dollar bill, a fifty-cent piece, a dime, and a dime with two pennies. After giving her sentence, she suddenly yelled out, "Here's a nickel too!" We then "fixed" her equation very easily. We also had a chance to talk about how to read numbers, saying "one hundred seventy-seven" without using the word "and." I helped the students understand that when a decimal point is used, saying the word and is reserved for its occurrence: "One dollar and seventy-seven cents." How quickly these excited first graders were learning from one another and their teacher. I then returned to the first sentence given and asked how many times I had written down 25 [cents]. Again a number of hands shot up, and one boy said, "You're going to write a times sentence like you showed us before!" We continued to talk about how many times the 1 [cent] had been written down, and I showed the students how to use multiplication in their equations in several different ways. I talked about groupings and how I would use the parentheses to keep the quarters together and the pennies together - language that students could understand.

The last part of my "leading and stretching" students for that day was with a review of the clock. They knew that an hour has sixty minutes. Again using the 200-pocket chart as a reference, they quickly figured out that 2 hours would bring them to 120 minutes, and then by adding more 10's and some 1's, they arrived at 177 minutes. I continued to stretch their thinking by asking, "Could we make the first clock sentence into a shorter sentence?" A bright and perceptive child used times to group the 10's (5 x 10). My last comment was, "I'm going to use words to express this clock time."

The chart paper was full, so I told the students that they had thought so hard for me that it was their turn to write a few more sentences for 177 on a clean chart, without my help. Note the incredible equations in figure 4 from these end-of-the-year first graders. I especially liked the fourth equation. This child understood how parts of a sentence cancel each other out, but had to go back and add in the - 9 when he was rechecking and reading his sentence to the class.

Assessment Continues

This first-grade classroom truly is implementing and modeling the assessment standards through this daily group activity. As discussed earlier, all six standards have been celebrated through this activity. On the 100th day, the classroom teacher had students write individually as many sentences for 100 as they could. This task was an individual written assessment for all children. Each child's paper went into his or her portfolio to be shared with parents at the spring conference. The written assessments contained many patterns like 100 + 0 = 100, 99 + 1 = 100, 98 + 2 = 100, 97 + 3 = 100, and so forth. ([ILLUSTRATION FOR FIGURE 5 OMITTED] for an example of children's individual work.) I have encouraged more journal writing and individual recording as further ways to assess the children's thinking, understanding, and problem-solving abilities in this classroom. The continuous monitoring of students' progress must become natural in the teaching-learning process. Also, helping students to be more responsible for their own learning, evaluate their own work, and set their own goals must be constant goals for the teacher. In collaborating with classroom teachers. I gradually get to "lead and stretch" my colleagues in mathematical instruction, learning, and assessment.


This article attempts to articulate that writing equations as a daily group activity is a golden opportunity for assessing children's thinking and progress on a regular basis. By keeping assessment cards handy to record individual thinking of children on a daily basis, the teacher will note that children learn from one another and that their equations, comments, and questions increase as they grow mathematically. This approach is one way to foster a close relationship between instructional activities and ongoing assessment. A skilled teacher encourages and supports alternative ideas, probes the questioning mind, allows time to think, opens and frees the mind, appreciates the thought process, and thrills in the joy of small steps of discovery. All this can happen during the generation of daily equations by children in the classroom. May we as teachers ever travel with a new and broader vision of assessment through our observations as we instruct.


Burk, Donna, Allyn Snider, and Paula Symonds. Box It or Bag It Mathematics. Salem, Ore.: Math Learning Center, 1988.

D'Aboy, Diana A. The Place Value Connection. Palo Alto, Calif.: Dale Seymour Publications, 1985.

National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Restart Va.: NCTM, 1989.

-----. Professional Standards for Teaching Mathematics. Reston, Va.: NCTM 1991.

-----. Assessment Standards for School Mathematics. Reston, Va.: NCTM, 1995.

Washington State Commission on Student Learning. Essential Academic Learning Requirements. Olympia, Wash.: Washington State Commission on Student Learning, 1996.

Diana D'Aboy is a Title I Basic Skills teacher at Lidgerwood Elementary School, Spokane WA 99207. She was Washington state's 1991 Presidential Awardee for Excellence in Elementary Mathematics Teaching, The author wishes to thank Esther Rimbey, a first-grade teacher at Lidgerwood Elementary School in Spokane, WA 99207, for the exciting atmosphere of learning in her classroom, open collaboration, and access to her students. She demonstrates daily the practical application of the concepts discussed in this article.
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Title Annotation:mathematical equations
Author:D'Aboy, Diana A.
Publication:Teaching Children Mathematics
Date:Oct 1, 1997
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