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Math manipulatives are used in many early years math programs as a way to help young children visualize the mathematical concepts they are learning. Beads or blocks are often used to help children learn to count and often there is sometimes a whole math playground with toys and games for children to discover simple math ideas in a physical space.
While these math manipulatives are common in math classrooms for younger children, there is a clear trend of less of a focus on visualizations as students get older. Perhaps books will have a diagram or teachers will attempt to explain a way to visualize a higher level math concept, but rarely do math textbooks and other math exercises for middle and high school explicitly focus on visualizations.
Instead, most math materials focus instead on rote memorization of steps in a process which can allow students a foolproof method of getting to the answer even if they have no idea what is actually happening as they robotically follow the prescribed steps in the math formula.
Math manipulatives are often relegated to only being used for students with Dyscalculia or other learning difficulties. However, new research suggests that math manipulatives are vital for efficient math teaching at any age or ability level.
New Research on the Importance of Math Manipulatives
A new study has recently been released showing that students who had a better understanding of how to represent mathematical problems in their head were better able to choose efficient strategies when solving math problems. (Gros et al., 2024) This was found to be true regardless of age as both adults and children had similar results.
The participants were given several math problems and asked to solve them using the fewest steps possible. Afterwards, they were asked to draw a diagram of how they had solved the problem so that the researchers could analyze their method of thinking.
They were interested in seeing whether they used a cardinal or ordinal approach to solving the problems. Cardinal approaches group items into sets, such as 5 blue beads and 8 white beads, while ordinal approaches focus on the number and where a number sits in an order, such as traveling for 3 hours at 5:30 PM.
Each problem had a fast one step way and a slow way to solve the problem. The researchers found that the style of drawing and mental model used to solve the problem was highly predictive of whether the student was able to find the faster way to solve the problem or not.
Interestingly, the style of thinking did not need to necessarily match the style of the math problem itself as ordinal mental models were more strongly associated with finding the efficient solution regardless of problem type. According to the researchers themselves, “ drawings showing ordinal representations were more frequently associated with a one-step solution, even if the problem was cardinal.” (Gros et al., 2024)
Connections to Classrooms and A Math Playground
These results suggest that it may be best for math teachers to try and teach their students to visualize in order to be able to efficiently solve problems and have a better understanding of what they are actually doing when solving math problems. Additionally, it suggests that math teachers should perhaps focus on ensuring students can visualize ordinally as this technique was most closely associated with solving problems efficiently.
The researchers also noted that their results suggested that the analysis of various diagrams or visuals could be a very effective way of teaching math concepts and how to intuitively choose efficient methods to solving math problems rather than simply relying on mindless memorization of steps in a process.
All too often, a large cohort of any math classroom will graduate not understanding why it was so important to be able to solve all of these random puzzles with seemingly arbitrary rules and go on to not be able to utilize math in their lives to solve problems efficiently. Because of this, teachers should put a greater focus on the inclusion of math manipulatives and models for older students just as they do for younger students.
The results of the study showed that the importance of visualization is not reliant on age and so the need for high school students to be able to visualize is equally important as it is for a preschool student. The problem used to be that while it is easy to visualize simple math concepts with most everyday objects, it is significantly harder to visualize more complex math concepts.
Nowadays, this is no longer an issue as there are a plethora of online math manipulatives and simulations that allow for students to see what is going on in even the highest level math concepts. These online tools can allow teachers to create a math playground for their older students to discover intuitively how math concepts work just as a simple physical math playground works for young children.
There are already many websites that have collections of fantastic online math manipulatives that teachers can assign for students to do in class or as homework to deepen their understanding of the ideas being taught. As long as students have a smartphone, tablet, or access to a school computer they can access these materials and begin learning immediately with little to no preparation.
One example of this is the website “Math Playground” which has online math manipulatives for various math concepts ranging from kindergarten all the way up through algebra. Especially useful for many young learners are the “Thinking Blocks” activities which improve upon simple physical blocks by adding in audiovisuals to enhance learning in preorganized activities that can save teachers tons of time in planning.
For even higher level mathematics, here is a website for calculus manipulatives, here is a website for trigonometry manipulatives, and here is a website for online manipulatives covering ideas in statistics, probability, and personal finance. A quick google will reveal many other websites for even more specific concepts. Teachers need only to do a quick search on the internet before jumping into any new concept and try to make sure they have an online math manipulative for each concept they teach over the course of the year.
This small addition to each lesson plan will make a huge difference in the ability of students to not only understand what is happening when they do these complex math problems, but research shows that they will be more efficient at problem solving, making them get correct answers more often and in a faster time.
This approach not only makes them more prepared for traditional testing situations, but also will help them connect their math knowledge more readily to the real world as well. This, along with the very easily accessible nature of online manipulatives means there really is no excuse not to be using these crucial learning tools.
A final, but not to be underestimated benefit may be that math class becomes more enjoyable for students as well. Rather than spending class drilling through workbooks, they’re able to play in an online math playground during class that helps them gain a better understanding of how the world works.
Just like a young child loves playing in a playground with spinning letter blocks, interactive light switches, and sliding counting beads, older children will find a math classroom full of interesting interactive visualizations and challenging math puzzles will not only learn more and be more efficient at problem solving, but also look forward to coming to math class rather than dreading it.
Conclusion
Online math manipulatives are a crucial teaching tool for helping students to understand basic and highly complex mathematical concepts. Research shows that the ability to visualize math concepts is linked to more efficient problem solving as well as a greater understanding of how these ideas work in the real world.
The researchers suggested that the analysis of diagrams and other visualizations should be considered as a useful teaching approach. This can be accomplished as the researchers did by having students draw and discuss their problem solving strategies or through the use of online math manipulatives that can help the teacher clearly visualize ideas in preorganized ways.
These can save teachers a lot of time planning and explaining high level ideas and allow students to explore the ideas at their own pace and through a learning method that matches the way their brain was built to learn. This research, along with their easy to access nature, suggests that teachers should try to include at least one online math manipulative for each new concept introduced. This will ensure students can understand the significance of their work and be more efficient in choosing problem solving strategies.
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References
Gros, H., Thibaut, J.-P. and Sander, E. (2024) ‘Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults’, Memory & Cognition [Preprint]. doi:10.3758/s13421-024-01523-w.
