Learning to measure length The problem with Michael Drake Victoria University of Wellington the school ruler <[email protected]> Ever wondered why children have difficulty using a ruler? In this article Michael Drake investigates some of the difficulties students encounter and provides some ideas for teaching about and learning to use rulers. The desire to quantify our world and experience, understanding of why students might find linear that is to measure, is a uniquely human trait. measurement difficult to understand. Within this desire, the measurement of length One task that frequently has been used to holds a special place, not only as one of the test student understanding of linear measure- simplest attributes to measure but also as one of ment is called the broken ruler problem. Figure the most interesting. How tall am I? How high 1 (National Center for Education Statistics, have I climbed? How far have we walked? Do I 2012) shows one version of the problem. Other have enough material to make this dress? How- versions may show an object that is offset so the ever, international assessments and research over left end is not aligned with zero, or may involve several decades have indicated that, worldwide, physical measurement with a broken ruler. So some students have difficulty understanding the why do some students find the broken ruler measurement of length — regardless of which problem difficult? What can we do differently system of measurement is being used. Yet, that might help? This article seeks to answer understanding linear measurement is important, the first question by closely examining some not just because it is a common measurement rulers to see exactly what it is that students are activity but because it is used in many branches struggling to use. To answer the second question, of mathematics and the sciences, where scales the insights gained from examining rulers are based on linear measurement are commonly combined with insights from research to create found on measurement instruments and in suggestions for teaching and learning. graphs. Therefore, it is important to develop our What is the length of the toothpick in the figure above? Figure 1. The Grade 4 toothpick problem from NAEP (National Center for Education Statistics, 2012). APMC 19 (3) 2014 27 Drake Figure 1 shows an American version of the the end of the ruler (e.g., Figure 2). However, my broken ruler problem that was used in the experience is that some students still have trouble main National Assessment of Educational Progress understanding linear measurement and all that (NAEP) from 1996 to 2003. Results for this the extra space has succeeded in doing is intro- item are shown in Table 1. They indicate that ducing a new error. Bragg and Outhred (2000) even by Grade 8, roughly a third of students report similar results. When measuring with such were not able to answer the question correctly. a ruler, a common error among younger children Other versions of the broken ruler problem is to align the end of the ruler (rather than zero) have been used in the NAEP long-term trend with the edge of the object, as they would line up assessments roughly every four years since the their hands if using them to measure the width of 1980s. They show similar results (Kloosterman, their desk. Older students more commonly make Rutledge & Kenney, 2009). Thus we have useful the mistake of aligning the ruler at 1, probably data against which the effectiveness of regularly- because counting starts at one. voiced teaching suggestions for improving student understanding of rulers can be measured. Table 1. Percentage of Grade 4 and 8 students who correctly answered the NAEP toothpick problem. Year 1996 2000 2003 Grade 4 24 25 20 Grade 8 64 64 58 Source: Males, Sweeny, Gilbertson, & Gonulates (2011). Common errors Figure 2. A selection of New Zealand school rulers. One common error for this problem is that The second common error with the broken some students simply read the end-point that ruler problem is to produce an answer which aligns with the toothpick, producing the answer is one more than the correct answer. For the 1 1 103 2 inches. In NAEP 2003, 14% of Grade 4 toothpick in Figure 1, this answer would be 3 2 and 7% of Grade 8 students gave this answer inches; in NAEP 2003, 23% of Grade 4 and 20% (US Department of Education, 2003). Hiebert of Grade 8 students gave this answer (US Depart- (1984) suggests that this is because some ment of Education, 2003). One common way to students learn a procedure for reading a ruler obtain this answer is to count the mark that aligns without really understanding that the answer with the start of the toothpick as 1, then continue represents the distance between the beginning to count the marks. Traditional wisdom to rem- point and the end point of the object. One of edy this error is to get students to focus on count- his suggestions to overcome this was to show ing the units (the spaces) rather than the marks, a how individual unit-long pieces could be lined seemingly simple solution. Unfortunately, NAEP up to make a measure, then introduce students results suggest that students’ success rates with the to a ruler as a way to abbreviate this tedious broken ruler problem have changed little since the process. The other was to place zero a short 1980s (Kloosterman et al., 2009). distance from the end of the ruler because on such a ruler students must directly deal with the Problematic issues idea that the length of the object is the number of units between the endpoints. Early teaching What are we missing? Let us look more closely at approaches built around the construction of our rulers. Go to Figure 1 and observe the length rulers have been echoed in research for some of the marks on the ruler. You should notice that years now, and New Zealand and Australian the unit marks, half marks, and quarters are all rulers have their start point a short distance from different sizes, which suggests that the size of the 28 APMC 19 (3) 2014 Learning to measure length: The problem with the school ruler marks provide important scaffolding information with the second tape? Where is the start point for users of this ruler. Also note that to use this of this scale? For this tape you also might note ruler to correctly obtain a half unit, a student that each centimetre is only partitioned into five must identify that the end of the object aligns pieces rather than the usual ten. with the mark that indicates halfway. It seems that with this ruler, students must be able to work effectively with the marks, not just attend to the units. Now focus on the top scale (gauge) of the ruler in Figure 3, in which the unit is millimetres. It too has marks of different sizes, but it makes little sense to focus on the units (or spaces) instead of the marks as they are so small. Here the numbering indicates that each interval is 10 mm, so the mark that scaffolds halfway (e.g., between 120 and 130) is 125 mm. For this gauge, it is important again to focus on the Figure 4. Another collection of linear measures from marks, but here it is the interval, the way it is around the home. numbered, and how it is partitioned into pieces Looking at how such instruments handle of equal size that is important. the transition at one metre was also interesting (Figure 5). For example, the top gauge restarts with 1, 2, 3, 4, …, so to measure with this tape it is important to check how many whole metres have been measured before switching to centi- metres. The second gauge simply continues in centimetres (100, 101, 102, …) while the third gauge works in multiples of 100 millimetres (1000, 10, 20, 30, …). Figure 3. Part of a metric school ruler. Now consider Figure 2. Notice that there are four totally different scales (gauges) on the rulers. On the top ruler, there is a simple gauge in centi- metres. On the bottom ruler is a gauge in inches, partitioned into halves and tenths. There are two gauges in millimetres but with different lengths of mark, and two in centimetres with millimetres Figure 5. Transitions at one metre. nested between them. One of these latter two indicates that the large lines are centimetres, This recognition of the variety between rulers and the small lines millimetres. The other (out is important, as it highlights that students who of view) indicates the gauge is in centimetres. learn about only one form of ruler may be Now look at the gauges at the top of each ruler. assessed on something quite unfamiliar. It also Notice that only one of the three starts with zero. begs the question: when students are learning Considering these are three rulers I happened to to use a ruler to measure length, what do they have at home (and not ones deliberately sought need to learn? Plainly not to use the school out), this is considerable variability, so I cast my ruler. Rather, they are starting to learn about a eye wider and found other tools for measuring class of instruments, each of which may contain length in different parts of my house (Figure 4). significant differences. We should also remember Even more variability is evident. For example, that each example from this class was created while the top rule is fairly standard, how easy by a professional who understood the concepts would it be to measure a line that is 13 cm long involved with length measurement and who was APMC 19 (3) 2014 29 Drake creating a tool to meet the needs of a particular cedural approach lasted many centuries before the circumstance.