When Drilling Mental Maths Can Be Effective
Drill and practice have long been a part of the mathematics teaching experience, and with the change in the CAPS syllabus to include ten minutes of mental maths in classroom on a daily basis – teachers are finding little time to prepare for these drills. With the move towards e-learning and tablets in the classroom there are many different options available to teachers and parents to improve students marks. However, with a large proportion of the population unable to access any of these resources due to lack of funds, internet access or even regular electricity, a simpler solution should be found that does not use up valuable resources.
With the move towards technology, lots of research has looked at the gains students have made when using computer programs to improve maths (Duhon, House & Stinnett, 2012; Ke &Grabowski, 2007). One of the benefits of computers is that they can repeat exercises endlessly, give immediate feedback and can be used to channel a student’s learning in a specific direction (Torgesen, 1994, in Duhon, House & Stinnett, 2012). However, the results have been mixed and depend on multiple aspects such as age, what is being tested and how it is being tested. Ke and Grabowski (2007) found that game-playing in maths led to an improvement in “cognitive learning achievement”.
An important part of mathematics is the gain of fluency and automaticity in mental mathematics and particularly in the basics of addition, subtraction, multiplication and division. Automaticity refers to the point when “tasks are learned so well that performance is fast, effortless and not easily susceptible to distraction” (Crawford, 2000). Haring and Eaton (1978, in Parkhurst, Skinner, Yaw, Poncy, Adcock & Luna, 2010) suggested three major stages in order to reach automaticity. The first stage is acquisition, where students are learning the answers to facts and the focus is on increased accuracy. The second stage moves towards the development of speed in accurate answering which leads to the final stage of automaticity and fluency. Fluency refers to the point where students can accurately and quickly answer questions.
Why is automatic and fluency important? Research shows the link between the gain of automaticity in basic maths facts and being able to understand more complex tasks. According to Woodward (2006) and information-processing theory, students who cannot retrieve facts automatically “experience high cognitive load” as they perform more complicated tasks, which often leads to errors in procedures and answers. Parkhurst et. al. (2010) suggest that students who have automaticity and fluency in basic maths facts are less likely to have maths anxiety and will have to spend less effort and time on maths tasks. Crawford (2000) notes that students who have to work out basic maths facts become distracted from the more complex problems they are trying to solve and thus experience more difficulty with more complex tasks. Thus, automaticity is important because we want the students’ attention to be focused on the more complex task and not on the more basic one. Woodward (2006) also points out that low-achieving students and those with learning disabilities have “considerable difficulty” in developing automaticity. However. Goldman et al (1988; in Woodward,2006) came to the conclusion that students with learning disabilities take longer to learn the basic maths facts automatically and that this can be addressed through systematic practice.
Furthermore, Woodward (2006) states that frequent timed drills are essential in developing automaticity. Enter the Sharp Scientific calculator range – with a mental maths drill function. The Sharp calculator allows one to practice their mental maths by answering randomly generated sets of questions and getting immediate feedback – the question will ticked and a new question given if correct and marked wrong and repeated if incorrect. The student can also ask the calculator what the correct answer is so that they can learn from their mistakes.
With the above in mind, Seartec set out to see whether the Sharp drill function could in fact help students to improve their mental maths. The hypothesis was that after playing on with the drill function regularly students would see an improvement in their mental maths ability and improve on their automaticity.
Two local schools in Gauteng were selected to participate in the research project. The first school was in a suburban area and the students came from the local community. Two grade seven classes were selected from Kenton Primary and the students came from a range of backgrounds. The second school was a more rural farm school, where students did not pay any fees. The two grade seven classes at Michael Rua took part in the research.
At each school, the students took a pre-test of one hundred questions and were given as much time as necessary to complete the worksheets. One class from each school was selected as a control group and the other as the experimental group. All four classes did the same pre-test and post-test.
Once the students had completed the pre-test the experimental group was shown how to use the drill function on the calculator and had the opportunity to race each other. This group then played with the drill function on the Sharp EL-W535HT scientific calculator on a regular basis.
Following a set period (4 weeks for Kenton Primary – with a week of school holidays in between, and two weeks for Michael Rua due to time constraints), all four classes were given the post-test.
Both the pre-test and post-test were a set 100 basic mental maths fact questions. These were actually taken from a random drill on the calculator and included sums from zero to twenty, differencing (the opposite of sums), multiplying from zero to twelve and dividing the opposite of multiplying without dividing by zero. (Both the pre-test and post-test are available below to download should you wish to run your own experiment.
At Kenton primary, the control group found out that the experimental group was playing with calculators. They asked to also play with the drill mode and so cannot be counted as a control group. Therefore there was only one control group out of the four classes.
The pre-test and post-test results are given in the table below:
Kenton Primary School results:
Michael Rua Primary School results:
|Experimental Group||Pretest||Post test|
As can be seen from the Kenton Primary results, there was no significant change in results. This may be due to the break of a week’s holiday before doing the test. Secondly, the maths teacher at Kenton Primary already runs a mental maths training program which they spend 10 minutes practicing two or three times a week. It may take longer to see better results.
The feedback from Mr Els (from Kenton Primary) was the students really enjoyed the calculators and their automaticity had improved on the calculators. The students really enjoyed doing the drill as well.
With respect to Michael Rua, you can see the dramatic drop in time with regards to the experimental group. Michael Rua only played on the calculators twice in the two weeks between the pre-test and post-test.
As can be seen from the results, we did not find a significant difference between the pre-test and post-test as well as between the experimental and control groups.
There are several factors that could have caused these results. The first is the time between the pre- and post-tests. With regards to Kenton Primary, the break between the pre-test and post-test and playing on the calculators may have dulled the effects of the drill function. The shortness of the period for the Michael Rua group may have contributed to the lower increase in results. Future research should consider allowing the students to take the pre-test in the first week of the term and the post-test in the last week of the term, giving the students approximately ten weeks to practice with the drill function.
Secondly, lots of research shows the lack of gains across two different mediums (see for example Duhon, House &Stinnett, 2012) such as a paper and pencil pre- and post-test but where the intervention was computer based. This may be because students can answer questions faster and more accurately on a computer than actually spending the time writing it down.
Finally with only one control group, it is difficult to tell whether there was in fact a difference between the pre- and post-tests and whether this affected the test reliability. A further suggestion for future research should consider using the same worksheet of 100 questions a further time apart.
Although the results did not support our hypothesis they do point in the future direction of the where the research should go. We need to give the students more time with the calculators and more time to practice their mental maths. According to Parkhurst et. al (2010) “because practice enhances speed of accurate responding, interventions that occasion higher rates of accurate responding are likely to cause greater increases in automaticity and or fluency”. Crawford (2000-) uses Hasselbring and Goin’s (1988) research paper to suggest that once the student can retrieve a maths fact from memory (i.e. he or she has achieved acquisition) however slowly, the use of drill and practice will lead that student to achieve fluency and automaticity in answering these questions.
Further suggestions for research include changing the pre- and post-tests to a limited time frame of ten minutes each, i.e. 6 seconds per question (although, the expectation is that automaticity occurs in 3 seconds or less), this would also give a better observation of students who were still using methods to find the answers and those that had achieved, acquisition, automaticity and fluency. There is still much to be covered on this topic.
Crawford, D. (2000-) The Third Stage of Learning Math Facts: Developing Automaticity. Retrieved from: http://www.rocketmath.com
Duhon, G.J., House, S.H. & Stinnett, T.A. (2012) Evaluating the Generalization of Math Fact Fluency Gains Across Paper and Computer Performance Modalities. Journal of School Psychology, 50, p 335 – 345
Ke, F. & Grabowski, B (2007) Game Playing for Maths Learning: Cooperative or Not? British Journal of Educational Technology, 38(2), p 249 – 259
Parkhurst, J., Skinner, C.H., Vaw, J., Poncy, B., Adcock, W. & Luna, E. (2010) Efficient Classwide Remediation: Using Technology to Identify Idiosyncratic Math Facts For Additional Automaticity Drills; International Journal of Behavioral Consultation and Therapy, 6 (2), p 111 – 123
Rasila, A., Malinen, J. & Tiitu, H. (2015) On Automatic Assessment and Conceptual Understanding. Teaching Mathematics & Its Applications, 34, p 149 – 159
Woodward, J. (2006) Developing Automaticity in Multiplication Facts: Integrating Strategy Instruction with Times Practice Drills. Learning Disability Quarterly, 29, p 269 – 289