What Do Elementary Textbooks Say about Estimation of Length? Division of Science & Mathematics Education and Teacher Education, Michigan State University Kuo-Liang Chang, Lorraine Males, Aaron Mosier, and Funda Gonulates Research Questions We ask the following two research questions in this study: Which of the focusing questions are attended to and to what extent in the instances of length estimation found in elementary school curricula? What do the curricula provide teachers or students to do about the possible foci of attention? Background/Rationale Results Sample Coding of Length Estimation Research suggests that the teaching and learning of length estimation is often associated with vagueness and ambiguity (Forrester & Pike, 1998; Joram, Subrahmanyam, & Gelman, 1998). Some researchers have attributed ambiguity and vagueness of mathematical practice in classrooms to different interpretations of meaning between individuals (Bishop, 1988; Christiansen, 1997; Voigt, 1994). Based on Kasten and Newton’s (2008) literature review on measurement learning, four focusing questions which need to be addressed in order for a prompt involving estimation to be clear are identified as follows: 1. Which dimension should be estimated? 2. Where are the starting and ending points of the estimation? 3. What is the unit of the estimate (e.g. inches, cubes)? 4. How precise should the estimate be? We have chosen to analyze written curricula which have been identified as contributing generally to the poor teaching and learning of spatial measurement (Kamii & Kysh, 2006; Lehrer, 2003). Focus CategoryCodeExample Start and End points indication There may be more than one start or end point when measuring an object. Is the starting and end point specified? Ambiguous Point to Place or Place to Point (library or classroom, or the height of the table where one end of a table has more than one ending point) Place-to-place C Not Ambiguous Point to Point A, B, D, E, F, G, H, I, J, K, L, M Attribute Indication There may be more than one length attribute on an object which can be measured. Is the estimated attribute specified? Ambiguous It is not clear which dimension is indicated. There may be more than 2 dimensions to estimate. A, B, C, D, E, F, L, M Not Ambiguous “width” or “length” defined, or “long” defined by brackets, the following three are regarded as defined: distance, diameter, height. G, H, I, J, K Unit Indication There is definitely more than one standard or non-standard length unit that can be used to measure an object? Is the unit specified? Ambiguous Family of units, a lot of units are given, but no one specific unit specified, paper clips (you can use its width or length) A, B, C, D, F Not Ambiguous Standart units like cm, inch or defined paper clips… E, G, H, I, J, K, L, M Precision Indication There is definitely more than one approximation method to deal with fractional parts when measuring. Is there any approximation method offered? Yes Nearest inch, half inch, quarter inch multiple choice with whole number or fraction units. G, L, M NoA, B, C, D, E, F, H, I, J, K Scott Foresman Addison Wesley Mathematics K 141B A B C Scott Foresman Addison Wesley Mathematics H I J K L M Method Select Elementary Curricula Generate Coding Scheme Identify Estimation Texts Coding Compile Results The curricula appear homogeneous in their attendance to Start/End and Unit Indication. Since they are heterogenous with respect to the other categories, we could not combine the data to make solid, overarching claims about ambiguity. It appears that Start/End and Unit Indication are attended to the most. The curricula are, however, homogenous in the quantity of ambiguous categories per task (Chi-squared=1.38, df = 4). As a result, the curricula can be aggregated for this analysis. Of all instances of Visual Estimation that were identified, less than 2% lacked ambiguity and more than 25% were ambiguous in more than 2 categories. Discussion Given the results, it appears as though concern about the ambiguity of Visual Estimation tasks is justified. With more than 25% of the tasks ambiguous in several categories, there is a lot of room for improvement. A theme that appeared in the data was the lack of reference to precision. Simply attending to precision would have a positive impact on 75% of items. It is interesting to note that Saxon and Scott-Foresman appear to be quite homogeneous in the attention paid to different categories. This is surprising, because they were constructed with different design principles. It should be noted that there was some content that STEM 1 identified as Visual Estimation that did not fit into our conception of the topic. This could have an impact on our results. 1 The analyses reported in this paper were carried out with support from the National Science Foundation (REC , John P. Smith III, PI) to analyze the content of written curricula in the area of spatial measurement. The opinions, findings, and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of NSF. D E F G