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Preventable errors: From little things, big things can grow.

In introducing his book on educational psychology, David Ausubel penned what is for me one of the most pertinent quotes for teaching.
   The most important single factor influencing
   learning is what the learner already knows.
   Ascertain this and teach him accordingly. (1968, p. vi)


This principle is at the heart of teaching and learning cycles, and is crucial in planning to teach. Ideally, in our classrooms we seek to build a bridge between what our students currently know and what we want them to learn. To establish what the learner already knows we need to be clear about what it means to say that a student knows something in mathematics.

What does it mean to know something?

Rather than starting with the general case of what it means to know something, I will consider a very specific example--what it means to know 1/4 = 0.25.

Let me nail my colours to the mast. When I use the expression "knows 1/4 = 0.25", I mean the student uses 1/4 and 0.25 interchangeably. In particular, the student doesn't associate 0.25 with a fraction not equal to 1/4 and doesn't tell me 1/4 =0.14 or some other decimal not equal to 0.25. In short, when I say a student knows something, I mean both what it is and what it is not.

When should a student know 1/4 =0.25? The Australian Curriculum: Mathematics (ACARA, 2015) states students are taught to make connections between fraction and decimal notations up to two decimal places in Year 4 (ACMNA079). Consequently, it is a reasonable expectation for students to "know 1/4 = 0.25" by Year 5 and certainly a very reasonable expectation before Year 7.

However, in 2014 approximately 20% of Year 7 students in NSW didn't appear to know 1/4 = 0.25. That was the percentage of students who wrote 1/16 = 0.25 in Q32 of the Year 7 Non-calculator NAPLAN assessment. As the same constructed response question occurred on the Year 9 paper, it is clear that the proportion of students who knew 1/4 = 0.25 did not improve, with 20% of Year 9 also providing this answer. I do not know what the figures were for all of Australia, as the responses to constructed response items are not kept by all states.

Does it matter?

In day-to-day operations involving quantity, 1/2 (or 0.5 as a decimal) is our most familiar fraction. It is so frequently used that Kath Hart (1988, p. 216) jokingly described it as an honorary whole number. Its frequent use has led to it being described as a benchmark value for fractions. We use 2 as a reference point: "about halfway, half a kilogram, glass half full, ..." If we apply this benchmark fraction to itself, we establish a new benchmark of 1/4 as 1/2 of 1/2. In telling time, an essential life skill, the half hour and quarter hour are basic units. It is not surprising then that within Level 2 numeracy of the Australian Core Skills Framework (Mclean, Perkins, Tout, Brewer, & Wyse, 2012) learners are expected to identify and use "everyday fractions, decimals and percentages, e.g., 1/4, 1/10, 50% or 0.25" (ACSF, p. 132). So, does it matter if you don't know 1/4 = 0.25 or 1/2 = 0.5 = 50%? My answer is clearly "yes".

The National Assessment Program in Literacy and Numeracy (NAPLAN) provides a useful large-scale picture of numeracy acquisition. When almost one-third of all Year 5 students selected the answer 1/5 = 0.5 in the national numeracy assessment in 2016, I wondered why. Sadly, when I reviewed the question and the options students selected, I realised I knew why. I knew why approximately 28% of Year 5 selected 0.15 as being equivalent to 1/5. I also realised that over 89 000 Year 5 students, approximately one-third of the entire cohort (ACARA, 2016), didn't really know that 1/2 = 0.5 and almost 27 000 students chose 1/5 = 0.25, a clear indication that we hadn't been successful in enabling them to establish robust benchmarks for 1/4 or 1/2.

Improve fraction benchmarks: Yes we can!

When you have the means to prevent something bad from happening, I believe you have an obligation to do it. I view the errors associated with fraction benchmark values (1/2 = 0.5 and 1/4 = 0.25) as preventable errors. The errors that students make with decimals have been well researched in Australia (e.g. Stacey, 2005; Steinle, 2004). Moreover, students respond well to teaching designed to address these errors (Durkin & Rittle-Johnson, 2015; Helme & Stacey, 2000). We have the means to identify and remedy this problem!

Begin by finding out if any of your students associate 0.5 and 0.25 with values other 1/2 and 1/4 respectively. You might simply ask which decimal is equivalent to 1/5 and provide options of 0.15, 0.2, 0.25 and 0.5, as was asked of Year 5 students. If every student has a robust understanding of the benchmark values for 0.5 and 0.25 they have a foundation for estimating the results of calculating with everyday fractions, decimals and percentages.

One lesson I have used to develop robust fraction benchmarks is having students estimate where three-eighths of the length of the board would be and justify the estimate (Gould, 2007). The lesson is based on estimation with fractions, from the Mathematics Curriculum and Teaching Program Activity Bank by Charles Lovitt and Doug Clarke, which is out of print. To strengthen the link to decimals and percentages, have your students record the fraction values used in estimating as decimals and percentages.

Let us set a simple additional national goal for 2018. The goal is to be able to confidently say every student in Year 6 to Year 10 knows 1/2 = 0.5 = 50% and similarly 1/4 = 0.25 = 25%. For those who believe this to be "too easy", why not add in that they know 1/10 = 0.1 = 10%?

References

Ausubel, D. P. (1968). Educational Psychology: A Cognitive View. Holt, Rinehart and Winston Inc. New York.

Australian Curriculum Assessment and Reporting Authority (ACARA). (2015). Australian Curriculum: Mathematics (version 7.5). Retrieved 3 November 2017 from http://www.australiancurriculum.edu.au/ Mathematics/Curriculum/F-10

Australian Curriculum Assessment and Reporting Authority (ACARA). (2016). NAPLAN achievement in reading, writing, language conventions and numeracy: National report for 2016. Retrieved 3 November 2017 from http://nap.edu.au/docs/default-source/default-document-library/2016-naplan-nationalreport.pdf?sfvrsn=2

Durkin, K., & Rittle-Johnson, B. (2015). Diagnosing misconceptions: Revealing changing decimal fraction knowledge. Learning and Instruction, 37, 21-29.

Gould, P. (2007). Reasoning about three-eighths: From partitioned fractions towards quantity fractions. Retrieved 3 November 2017 from http://www.crme.kku.ac.th/APEC/PDF%202007/Peter%20Gould.pdf

Hart, K. (1988). Ratio and proportion. In J. Hiebert & M. J. Behr (Eds.), Number concepts and operations in the middle grades (pp. 198-219). Hillsdale, NJ: Erlbaum.

Helme, S., & Stacey, K. (2000). Can minimal support for teachers make a difference to students' understanding of decimals? Mathematics Teacher Education and Developmentt, 2, 105-120.

Mclean, P., Perkins, K., Tout, D., Brewer, K., & Wyse, L. (2012). Australian Core Skills Framework. Commonwealth of Australia. Retrieved 3 November 2017 from http://research.acer.edu.au/ transitions_misc/12

Stacey, K. (2005). Travelling the road to expertise: A longitudinal study of learning. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. 19-36. Melbourne: PME.

Steinle, V. (2004). Detection and remediation of decimal misconceptions. In B. Tadich, S. Tobias, C. Brew, B. Beatty, & P. Sullivan (Eds.), Towards Excellence in Mathematics (pp. 460-478). Brunswick: The Mathematical Association of Victoria.

Peter Gould

<pjgould109@gmail.com>
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Title Annotation:opinion
Author:Gould, Peter
Publication:Australian Mathematics Teacher
Article Type:Essay
Geographic Code:8AUST
Date:Dec 22, 2017
Words:1314
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