Recently, I had the privilege of working one-on-one with 115 urban 1st and 2nd graders during a sabbatical that I used to assess primary students' mathematical skills and reasoning abilities. As a professor of mathematics education at the elementary level, I wanted to observe first-hand how primary students attempt to solve mathematical problems. I especially wanted to see how English as a Second Language students engaged in the process, so I interacted with students, teachers, and parents at my college's partnership school in Chicago, Illinois.
Although I could have measured students' abilities in number recognition, classification, patterns, counting, addition, subtraction, and problem solving through paper-and-pencil tests, one-on-one evaluations provided me with a window into their verbal abilities, organizational patterns, and levels of engagement. For the teachers who participated in my study, I wrote reports summarizing which students had a solid foundation in such concepts as time and money, but I could not possibly document in detail the processes through which individual students deduced their correct and, more important, partially correct answers.
How a student arrives at a solution reveals far more about the effectiveness of instructional methods than the total number of items that he or she answers correctly on tests. Whereas general classroom interaction allows teachers to observe some of their students' mathematical perceptions in a given lesson, it is extremely difficult to assess all students' comprehension this way. And knowing students' perceptions and misperceptions of a concept is key to developing their mathematical reasoning.
What Individual Assessment Can Reveal
Informal individual assessment gives teachers a broader and more robust means of determining whether a primary grade student has a firm grasp of foundational concepts. I often wanted to change places with classroom teachers so that they could appreciate how such factors as students' experience, personality, interests, and learning styles shape their insights and development. Below are some examples.
Confidence levels. Maria, a 1st grade student, watched me intently and waited for some kind of affirmation between each math question I gave her. She answered nearly every written and oral item correctly, even when I only made eye contact with her. But when I purposely did not respond at all, she became extremely unsure of her work and stopped frequently to erase her answers.
Maria was far more dependent on an instructor than was her classmate, Juan, who answered fewer of the items correctly but plowed ahead irrespective of my feedback. Even though Maria had greater accuracy, Juan scored slightly higher because he completed more items. Their test scores alone revealed limited information for instructional purposes, but their testing behaviors revealed significant insights.
Metacognitive awareness. In another example, a teacher dictated a series of two-digit numerals for Miguel and Luis to write down. Miguel easily wrote them from memory, but Luis couldn't mentally picture the configurations of 50 and 18, so he located them on the number line hanging on the classroom wall. Even though he didn't have a solid understanding of two-digit numerals, he had learned to use resources to solve problems.
Likewise, when Luis could not remember every numeral in the series of 0–15, he succeeded in writing all of them by asking himself, “Where is an 11?” and then finding one on another section of the test. He did the same for 12. By the time he wrote 13, he remembered the pattern and continued without error. This rather sophisticated metacognitive awareness helped him fill in the gaps of his knowledge.
Divergent and convergent thinking. One-on-one evaluation gives students the opportunity to reveal their strengths in divergent reasoning and convergent thinking. When I asked them to sort a collection of various Barbie and Ken doll shoes into different groups, nearly all of the 1st graders successfully created two separate piles of boots and shoes. When next asked how the two piles were different, most students figured that boots are worn in cold weather and shoes are worn year-round. Apparently remembering a recent growth spurt, one girl responded that boots (which are seasonal) stayed the same size but that shoes “shrink on your feet.” One boy looked at the two piles and surmised, “Shoes are worn by people, but boots are worn by princes.”
When I asked the students how boots and shoes were similar, most had difficulty. After I prompted them with the question, “Would we put them on our ears?” most replied, “No, we'd put them on our feet.” But one girl eyed me oddly and answered, “I'd put them on a doll.” Another replied, “I'd put them in the closet.” None of these delightful responses, grounded in reality and fantasy, is revealed by merely placing a checkmark next to the tasks “analyzes,” “synthesizes,” or “sorts by attribute.”
Assessing mastery. As a supervisor of student teachers, I observe many 1st and 2nd grade math lessons in which teachers instruct students to complete a workbook page. I strongly caution my student teachers against using this work to determine who comprehends the concept and who does not. The guided portion of the lesson primarily measures how well a young child can follow directions; the “independent work” portion too often measures how well a child can get the answer from looking at a neighbor's page. Informal individualized assessment is a much better tool for determining a child's level of mastery.
Interests. In the middle grades, teachers can tap into academic and social areas of a student's life through written interest inventories and assessments. The gap between oral and written expression is so significant at the K–2 level, however, that oral interviews are far more productive. Even in 2nd grade, many students tire in the writing process before they can list all their experiences, interests, and goals.
One outstanding kindergarten educator requests a student teacher each fall semester, in part so that she can have more time to assess students' interests, needs, and abilities early in the school year while the novice teacher learns how to manage the classroom. After interviewing students and studying reading and math ability indicators, the teacher selects literature and develops math centers to address students' specific talents and limitations.
Learning styles. Different learning styles surfaced quickly in my assessments. Strong auditory learners commented frequently on their progress, asked me to repeat directions, and made up elaborate stories when posing problems. Visual learners created models to solve problems and looked for other representations of numerals or shapes before answering questions. Tactile learners used manipulatives more often than other 1st grade students. They often picked up pattern block pieces to identify their shape, placed objects in their hands when counting, and built structures with many of the materials before reluctantly moving on to the next item.
How Do You Find One-on-One Time?
- Establish the practice of one-on-one time with students as a priority for the school. I work with one district that hires substitute teachers each fall so that kindergarten teachers can administer individual assessments for every child.
- Set up small-group learning centers. Even if a teacher can only schedule 20 minutes per week for students to work collaboratively in small groups or with partners, he or she can assess two students' progress each week during this time period.
- Make effective use of student teachers. During a student teacher's block of independent, full-time teaching, arrange to work one-on-one with four different students each day.
- Take advantage of time periods in which a portion of the class is working with another teacher, such as during computer instruction. But do not use times when the whole class is engaged in such weekly subjects as music and physical education classes that are generally taught by specialist teachers. To do so would devalue these important subjects, keep students from participating in activities they look forward to, and reduce already limited preparation time.
- Invite a different student to lunch one day a week to briefly discuss his or her interests and experiences. Even informal conversations about math and reading preferences will give insight into a student's comprehension and motivation.
- Have volunteers or paraprofessionals serve as “guest readers” to the whole class rather than having them assess skills or read individually with students. Do not use “Drop Everything and Read (D.E.A.R.)” time for anything but engaging in and modeling the importance of reading.
Although these options aren't perfect, I urge primary teachers to make the effort to work individually with each of their students. It is a costly investment in terms of scheduling and time, but one-on-one interactions and assessment yield great dividends for educators and students alike.
