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CONSTRUCTING NUMBER LINES: YOUNG STUDENTS EXPLORE CONCEPTS OF UNITS, MEASUREMENT, AND PROPORTION M4YC @ Humberwood Downs Introductions Our Learning Journey Day 1 Focus on Number Sense Difficulties/Challenges Big Concepts Number Knowledge Test Questions for Clinical Interviews Number Knowledge Test A test that is designed to measure the intuitive knowledge of numbers that an average child has available at the age levels of 4, 6, 8, and 10 years. Administered by teachers, graduate students, and researchers from the Dr. Eric Jackman Institute of Child Study, University of Toronto Number Knowledge Test Clinical Interviews Clinical Interviews Research Lessons Human Number Line Three students are asked to line up side by side and the class is asked "Who is in the middle?" The number of students is increased, so one student is in the middle, e.g., 5, 7. The students reflect on strategies such as, having the same number of students on each side determines the middle. Then a challenge is posed with having 5 students. The students are asked "Who is in the middle?" At this point the students reflect on the fact that a person is not the middle, but a space, with the same reasoning that having the same number of students on each side determines the middle. In Kindergarten we used a marker to show the person in the middle, such as a picture of a sun placed above the person in the middle. Research Lessons Tim Hortons’ Lesson The Tim Hortons’ exploratory lesson was one of the early lessons to consolidate the students’ concept of "middle as halfway". The students enjoy "Timbits," and this was used as prior knowledge to extend their understanding of "middle" and to link " middle" and halfway to distance. Butcher paper was used to represent the road from the school to the zoo. A stop had to be made halfway so we could buy some Timbits. Students were to estimate and mark the middle on the road, and then use the folding strategy to prove or disprove their estimate. Research Lessons Connecting to the Hundred Chart In an effort to connect the understanding of a hundred chart to a number line, we used the hundred chart to find the middle, using the strategy of the same number of "things" on each side determines the middle. The challenge was posed when we compared finding the middle with and without the zero. The students had many reflections including the "main middle", indicating a group of numbers (4,5,6) are in the middle of a number line from 0-10, but one was in the "main middle" (5). Research Lessons Connecting to the Hundred Chart Research Lessons The Clothesline The clothesline lesson was one of the early exploratory lessons to give students practice in locating "middle", "proportional spacing", and to consolidate their understanding of "middle as halfway" and to develop spatial reasoning. A clothesline was made with numbered 3x5 cards and clothes pegs. Students were told that a strong wind came and blew all the clothes off the line and only zero and ten remained on the line. The number five card was replaced on the line, and students were asked to help put the clothes back on the line. The emphasis of this lesson was locating middle and equal spacing of the numbered cards on the clothes line. Research Lessons The Freezie Lesson The Freezie lesson was to provide students practice in locating "middle" to determine "half of a whole". A freezie was shared for Ms. Hassen and Ms. James. The students instructed the teacher to cut the freezie in two, so the teacher purposefully cut it in 1/4 and 3/4. The students were furious. They said, "No, Ms. James, you made a mistake, you must give Ms. Hassen a half." The students then suggested ways to find the middle of the freezie. It was then measured with a ruler to locate, the "middle" and to find a "half". The students, Ms Hassen, and myself then had a freezie treat. Research Lessons Solving Addition & Subtraction Problems Children worked with counters and with the number line. The number line displays counting and measurement simultaneously. Using the number line gives students a visual image of the operation being done and a better understanding of the answer. Also, the number line helps students experiment with decomposing numbers. Benefits: Over time, this will enable students to become more capable in performing mental computations. Numbers can be decomposed and the subunits or smaller amounts can be added or subtracted in varying orders, yet still be equivalent. Research Lessons Solving Addition & Subtraction Problems Research Lessons Making Rulers Research Lessons Making Rulers What we learned The linear representation of numbers, which includes presenting numbers on a number line, can access foundational mathematical concepts including; ordinality, spatial awareness and magnitude, concept of middle. Originally focused on counting with precision. Then spatial awareness was greatly influenced by their understanding of middle. Finally students paid more attention to space as well as number counting What we learned Number line offers insights and supports: Counting and counting on, counting down Sequencing Understanding units as equal intervals of distance Use of benchmarks Sense of proportional reasoning Understanding measurement concepts Researchers study the development of estimation on number line Many JK students, even those who can count perfectly from 1 to 10, do not understand the rank order of the numbers’ magnitudes. These children’s number line estimates correlate only minimally with the magnitudes of the numbers they are estimating. Researchers study the development of estimation on number line Even after children learn the rank order of numbers’ magnitudes, they still do not immediately represent the magnitudes as increasing linearly. Their number line estimates often increase logarithmically with the size of the number being estimated. By grade 2 children with experience their magnitude estimates increase linearly Today’s Lesson Now it’s your turn to create your VERY OWN ruler! You will have the opportunity to make three rulers, all will begin with 0. You can choose how long your ruler is going to be AND even more exciting – you’re going to be able to choose what number you want as your end point on ONE of your rulers. The other two will be 0 – 10. The only “rule” is that you have to use a marker to write the zero and the end number. Everything in between you can use a pencil. Let’s Try! Questions for Observers 1. 2. 3. 4. 5. 6. 7. 8. What specific strategies did you notice as students were building their rulers? Do students focus on equal spacing? What strategies do they use to determine the intervals? When making their own rulers, were students able to use an ordinal counting strategy in tandem with their spacing strategy? Did you notice any students using the middle as a benchmark to create the spaces? If so, how did they find middle? How did children show their understanding of half? Did you see evidence of students using gestures and actions to express their mathematical thinking to a peer, the group, etc.? Please describe what you saw. Is there a difference between boys and girls with their accuracy in spacing numbers on the number line/ruler? Do children change the orientation of their rulers? Did you see children imitating their peers? Did the imitating strategy seem to help students?