S.A.M. Part 2 Stress-Free Math

This article serves to explain the unique approach of SAM (Seriously Addictive Mathematics) in South Africa as a program that supports Singapore Mathematics Pedagogy in a way that makes Math visual and relatable. Children discover how different concepts link to deepen their understanding of Mathematics and encourage them to think creatively. This is done in a way that builds self-confidence and improves self-talk to reduce stress related to solving Mathematical problems.

With a focus in South Africa on inclusivity in the classroom, teachers often teach students of various academic abilities in the same classroom. The inclusion of concrete manipulatives in the classroom benefits students in an inclusive setting, regardless of their individual capacity. Manipulatives are objects used as teaching tools to engage students in ‘hands-on’ learning. Concrete materials in a Math classroom can include various counters, beans, unifix cubes, 10s-frames and base-10 blocks. The idea is not just to make Math visual but also to encourage the active manipulation of ideas and mathematical concepts.

Figure 1. Math-10 Frame

A 10s-Frame can be used to show different combinations of 10 (number bonds of 10). This example shows how 2+8=10. From this concrete manipulative, students can also deduce that 10-2=8, and 10-8 =2. Using a 10s-Frame enables students to link informal approaches with formal ones. This fosters excellent number sense!

Figure 2a. Demonstrate with concrete aids.

Figure 2b. Demonstrate with concrete aids.

The use of physical counters supports conceptual understanding. By using counters grouped together in different ways, students learn different concepts. Figure 2a and 2b demonstrate that multiplication is repeated addition, it also implies that conceptually,  is not the same as, even though it yields the same total.

A study done in Indonesia among Civil Engineering students showed that only 10% of the students could initially describe three-dimensional shapes. After being exposed to concrete teaching aids, they could describe the forms and draw them as well. These students became so interested and excited about solving more complex problems due to their exposure to concrete teaching aids, that they went on to construct their own formulas to work in three dimensions.

According to Educational Psychology research, adding visual representations to text can enhance students’ learning. Pictorial diagrams frequently used in Mathematics tuition range from number lines to pie charts. Number lines foster an understanding of early number concepts, which is critical for a learner’s future success in Mathematics. Learning about the representation of numbers and concepts implies that students acquire representation competencies, knowledge and skills that enable them to visualize and solve problems. Developing these competencies is cognitively demanding and requires instructional support.

At SAM, students are introduced to many more forms of visual presentations, many of them directly stemming from the concrete aids they used in class. The SAM program underscores the strategic use of pictorial diagrams. It supports the construction of diagrams as a learning tool and its application to answer various questions related to the same problem. Visual presentation bridges the gap between concrete presentations and abstract calculations. 

Figure 3. Visual representation of base 10 blocks to teach addition with regrouping.

When students can ‘see’ how 10 ones equal the same as 1 ten, they can progress to the more abstract aspects of problem-solving as they understand their complete operations.

Figure 4. Demonstrate Bar Models used to solve word problems.

This bar model illustrates how abstract words and numbers can be presented visually to foster a clear understanding of the question; it is very beneficial when unpacking information given. Creativity is stimulated by exploring other possible questions that relate to the same visual presentation.

Leonardo da Vinci, an icon of creative thinking, believed that sight played a great role in generating ideas. He also found his main source of creativity in his ability to form new patterns through connections and combinations of different things. Michael Michalko writes in his book “Creative Thinkering” about what makes a genius a genius and explains that they know how to generate connections and associations between dissimilar subjects. Ideas are combinations of something else; geniuses form more combinations than others, especially ideas that are not normally associated. Creativity implies having a rich repertoire of existing ideas and concepts but provides for further connections and variation. Michalko recalls a story of how Leonardo da Vinci threw pebbles in a pond one day, while at the same time he heard a church bell ring. He conceptually blended the sound of the bell with the ripples in the pond to reach the realization that sound travels in waves. Inventions and creativity are combinations of what already existed.

Linking, comparing and contrasting mathematical procedures improve students’ experience of competence during teaching sessions, as it supports solving real-life problems. When different concepts are linked in Math instructions, students develop creative ideas to solve problems. 

Figure 5a. Demonstrate by relating number size to length.

Students engaging in addition and subtraction problems understand concepts like long and short. When a connection is made where ‘Longer numbers’ present a bigger numbers and ‘shorter numbers’ present smaller numbers (Figure 5a), students understand subtraction order, but link the answer to words like ‘difference’ while creatively making the connection that adding the ‘shorter numbers’ the total should be equal to the ‘longer number’.

Figure 5b. Demonstrate that by connecting fraction size to the area.

This pictorial diagram links equivalent fractions to equal areas, where a whole is presented visually with the same sized rectangle. The SAM Worksheets are written to foster connections between different ideas to deepen understanding, and ultimately enable students to make new connections of their own in the world of problem-solving.

When students have the opportunity to make sense of what they are learning, they grow in conceptual understanding. Working with manipulatives encourages talking and explaining; it sets the stage for wilful engagement in verbal explanations between students and facilitators. Facilitators can ask students questions like “Can you represent this in another way?” to stimulate active reasoning.

Facilitators can also encourage active reasoning when prompting students to self-explain their methods or pictorial models used to solve problems. Using the correct language for different approaches and operations encourages students to use the correct terminology. Students recognize what they’re asked to solve in questions when exposed to other words relating to the same procedure. Facilitators can use similar terms like “add, addition, more than, together and plus” to support students’ more profound understanding. 

A coaching approach is followed at S.A.M.; the facilitator prioritizes asking questions to stimulate conversation and thinking. Students are also prompted to ask questions and explain their steps when solving problems and communicate different ways of solving the same problem. 

When students move to more abstract ways of solving problems, they still argue and think, but silently, from the bank of acquired Math language they’ve picked up on the way.  

In an article published in the International Journal of Science and Mathematics Education, on “Multiple Solutions for Real-World Problems, Experience of Competence and Students’ Procedural and Conceptual Knowledge”, the researchers found a direct link between students’ feelings of competence and their self-belief and intrinsic motivation. They also found that it influenced their academic outcomes positively.

At S.A.M. each learner undertakes a Placement Assessment used as a starting point in crafting an individualized program. The idea is that when a student engages with Mathematics at a point they are comfortable but not bored, mentally stretched, but not frustrated due to the difficulty level of the work, they would thrive. When learners can successfully complete a task, they grow in confidence and can move to the next level of complexity. They build a sound foundation of prior knowledge and feel more competent at solving problems. Positive self-talk drives intrinsic motivation, and feelings of competence give a student the courage to progress to the next level.

Post Script

The Singapore Mathematics Framework allows for instruction in Mathematics by focussing on skills, concepts, attitudes, metacognition and processes through a focus on problem-solving. Seriously Addictive Mathematics as a business venture, and a means to introduce Singapore Mathematics to young children, aims to improve the Human Capital of South Africa, one student at a time. For more information, please get in touch with the author, Karma Palmer, at karma@seriouslyaddictivemath.co.za/ +27 (0) 82 071 5669, or follow the enquiry link on www.seriouslyaddictivemath.co.za if you are interested in enrolling your child or starting a SAM Franchise in your area. 

Reference and Further Reading

·     Achmetli, K., (2019). Multiple solutions for real-world problems, experience of competence and students’ procedural and conceptual knowledge. International journal of science and mathematics education, 17(8)

·     Dewi, M. L., Hakim, A. R., Setiawan, A., Adhisuwignjo, S., & Rohadi, E. (2018). Mathematics teaching aids to improve the student’s abstraction of geometry in civil engineering of state polytechnic malang. IOP Conference Series. Materials Science and Engineering, 434(1)

·     Gelb, M. J. (2000). How to think like Leonardo da Vinci: Seven steps to genius every day. Dell.

·     Michalko, Michael. (2011). Creative thinkering: putting your imagination to work. Novato.

·     Witzel, B. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings. Learning Disabilities: A Contemporary Journal, 3(2).