Assessing Children’s Understanding of Length Measurement:
“I need to teach length to my students but I don’t really know where to start. I don’t really know what’s important for them to understand and what I should focus my teaching on.”
his comment was made by an expe- rienced and highly effective teacher; Tit illustrates an issue faced by many HEIDI BUSH presents teachers when approaching mathematics, and in this case, length measurement. What three different tasks are the important concepts for students to understand? How can I develop this under- that can be used to standing and how can students demonstrate their understanding? assess children’s Three important measurement concepts for students to understand are transitive understanding of the reasoning, use of identical units, and iteration. A focus on Three KeyIn anyConcepts teaching and learning process it is length concepts of important to acknowledge students’ existing knowledge and focus teaching on moving transitivity, use of students on from this point. Assessment tasks that provide insight into students’ identical units and current understanding of or misconceptions about these concepts is, therefore, vitally iteration. important. The tasks presented here were developed to assess students’ understanding of transitive reasoning, identical units and iteration. The Early Numeracy Interview (Department of Education, Employment and Training, 2001) is acknowledged as an excellent tool for assessing students’ mathematical understanding and the tasks presented here are structured and scripted to align with this clinical interview process. The insights gained from these tasks can be used to inform future teaching and learning.
APMC 14 (4) 2009 29 Bush Assessing Children’s Understanding of Length Measurement
Transitive reasoning front of the child.] Please use this straw to check. Transitive reasoning, or transitivity, is a concept Which straw is the longest? based on comparison. For example, if three Tell me how you used this straw to check. lines (A, B and C) are different lengths and A Asking students to check using the loose is longer than B which is longer than C, then straw ensures that even if the answer was we can also say that A is longer than C (Wilson obtained by using a direct comparison of the & Osborne, 1992, p. 102). Understanding this appearance of the two straws or transitive idea can be challenging for young children and reason with reference to the width of the is a skill that develops over time. The following card, it is possible to gain important insight task may be used to determine children’s into their understandings of transitivity. understanding of transitive reasoning. Important observations • Does the student give a reasonable estimate The Straws and the Barrier of which straw is longer? (Modified from Wilson & Osborne, 1992, • Does the student accurately use the loose p. 102) straw to compare the lengths of the two Materials required straws? • 1 piece of A4 card • Does the student place the loose straw • 20 cm straw glued in a vertical position beside each straw in order to make a on the left hand side of the card comparison? • 10 cm straw glued in a vertical position • Does the student’s explanation show on the right hand side of the card evidence of transitive reasoning? (For • 15 cm straw example, if the loose straw is longer than • 1 piece of A4 card folded in half widthways the straw on the right but shorter than the Activity straw on the left, the straw on the left must Take the card with the straws glued down be longer.) each side and stand the folded card between • Does the explanation include language the two straws to create a barrier between such as “longer than,” “shorter than,” etc.? them (see Figure 1). Place the card with straws and barrier in Ideas for teaching and learning front of the child. Teachers need to provide opportunities for Say to the child – [Point to straw on left students to use objects for direct and indirect hand side of the barrier.] comparison in order to make judgements Here is a straw. about length. Here is another straw. [Point to the straw The above task may be replicated with on the right side of the barrier.] larger objects that are not as easily visible or Which straw do you think is the longest? comparable. For example, draw chalk lines [Place the loose 15 cm straw on the table in of different lengths and some distance apart outside. Ask the students to suggest how the lines could be compared if they cannot be moved. If student arrive at a solution involving a third object for comparison, they may be encouraged to decide what sort of object may be used. If the student is having difficulty arriving at a solution, a length of stick or string may be provided as a prompt. How might we use this to find out which line is longer? Figure 1
30 APMC 14 (4) 2009 Bush Assessing Children’s Understanding of Length Measurement
Identical units Ideas for teaching and learning Many tasks involving non-standard units require This concept is an important one for students students to measure an object with a given unit; to understand when measuring with either for example: “How many blocks long is your standard or non-standard units. If an accurate table/pencil case/foot…?” Students, however, measure is to be gained, the units of length should be provided with tasks that require them used to measure an object must be identical. to choose an appropriate unit, or choose a unit The Straw and Mixed Paper Clips task may be from a collection of possibilities. used to determine students’ understanding For example: of the importance of using identical units • We need to measure how long our mat is. when measuring an object. What could we use to measure it? What else could we use? What could we use that The Straw and Mixed Paper Clips might help us measure it more quickly? (Modified from Department of Education, • When asking students to measure an Employment and Training, 2001, p. 42). object with non-standard units, provide a Materials required container with mixed materials, such as • 15 cm plastic drinking straw small and large blocks, paperclips, straws, • 8 large (5 cm) paper clips tiles and so on. This allows students’ • 5 small (3 cm) paper clips understanding of identical units to be Activity assessed. Which material did the student Place the straw and collection of mixed paper choose? Was it suitable for the object clips in front of the child. being measured? Did the student use Say to the child: identical units to measure the object? Here is a straw. • Objects may also be included that are Here are some paper clips. less suitable, such as beans or cotton Please measure how long the straw is with balls, to assess students’ understanding paper clips. [If the child hesitates] Use of appropriate units for length some paper clips to measure the straw. measurement. What did you find? [If correct answer is given (5 or 3) but no units, ask “5 (or 3) what?”] Iteration Important observations • Does the student use identical units to Iteration involves using a unit repeatedly in measure the straw, i.e., all large or all order to find a measurement. Rather than small paperclips? laying multiple units end to end (tiling), • Does the student use a mixture of large a single unit can be repeatedly moved. In and small paperclips to measure the length measurement this involves laying a unit straw? repeatedly end to end along the length of an • Does the student accurately line the object, counting each iteration in order to arrive paperclips up with the beginning of the at a measurement. This concept will not always straw when measuring? come naturally to children and explicit teaching • Does the student lay the paperclips beside of it as a strategy for measuring will be necessary. the straw without gaps or overlap? When the envelope and three paperclips task • Does the student link the paperclips was used with three students from Grades 1, 2 together, therefore creating overlap? and 5, none of the students successfully used • Does the student give the correct unit the paperclips to iterate and find the width of of measure in their response, i.e., “5 the envelope. They chose to use estimation or paperclips” or “3 paperclips”? declared the task “impossible.” It is therefore
APMC 14 (4) 2009 31 Bush
important that we provide students with early of the unit is provided and students merely opportunities to solve such problems, and not count the number of units used. Students assume that it is a concept they will grasp should also be provided with tasks in which too independently. few of a unit are provided, requiring units to The Envelope and Three Paperclips task may be repeated through iteration. For example, be used to assess iteration; that is, are students rather than asking the question, “How many aware that a single unit may be used repeatedly blocks long is your pencil case/desk/foot…?” to measure a length? and providing sufficient blocks to allow a simple count along the length of the object, students The Envelope and 3 Paper Clips should be provided with too few blocks to (developed by the author) make the length of the object. They may then Materials required be encouraged to problem solve a solution, • B4 size envelope (25 cm in width) or a leading to explicit instruction in iteration as a 25cm line drawn on a piece of card measurement strategy. • 3 large (5 cm) paperclips. The concepts of transitive reasoning, Activity iteration and identical units are critical for Place the envelope in front of the child in students to understand when approaching the portrait position. Place 3 large (5 cm) measurement tasks. It is important that we paperclips beside the envelope or line. explicitly teach these concepts to students and Say to the child: not leave their acquisition to chance. With Here is an envelope. this in mind, it is necessary to assess students’ Here are some paper clips. current levels of understanding in order to Please use the paperclips to measure how plan effective learning experiences. The tasks wide the envelope is. [If the child hesitates, described here may be used as a starting point e.g., “There’s not enough,”] Are you able to to determine students’ understanding of three use these paperclips to measure how wide important measurement concepts. Importantly the envelope is? they also represent opportunities for students What did you find? [If correct answer is to construct new understandings. given (5) but no units, ask, “Five what?”]
Important observations Acknowledgement • Does the student use the paperclips to iterate? That is, do they lay the three I would like to thank Ann Gervasoni for paperclips on the envelope end to end encouraging me to write this article. and then move a paperclip repeatedly, laying it at the end of the other paperclips, counting the moves each time until they References
reach the end of the envelope? Department of Education, Employment and Training, Vic. • Does the student place the paperclips (2001). Early numeracy interview booklet. Melbourne: without gaps or overlap? Author. Wilson, P. S. & Osbourne, A. (1992). Foundational ideas in • Does the student correctly measure the teaching about measure. In T. R. Post (Ed.), Teaching width, including stating the unit in their mathematics in grades K–8: Research-based methods (pp. measurement (5 paperclips)? 89–121). Needham Heights, MA: Allyn and Bacon.
Ideas for teaching and learning Generally when working with non-standard Heidi Bush units, students are exposed to a large number Catholic Education Office (Tasmania)
of tiling tasks, in which a sufficient number
32 APMC 14 (4) 2009