Today I wanted to discuss the importance of thinking of math in non-mathematical ways – this can be a very valuable strategy to students of all ages. Math tends to hold an image that only linear thinkers can understand and accel at it; however, this is not true at all. It is important for educators not to create an atmosphere that believes this, as it will result in a fixed mindset. Anyone can accel at math, and there are various ways to understanding math without only focussing on numbers and formulas. Having students first look at problems in a way that is more visual may be a necessary step for many students before they can arrive at understanding a complex math formula.
The Dot Problem
The Dot Problem is a good example of this – instead of having students come up with the formula that represents the growth pattern, have them take the time to see the pattern in their own unique way and describe it through images, words, numbers and personal connections. This way, students can construct a formula that makes sense to them. While this way of arriving at an answer may take longer, this level of learning is much more meaningful.
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| Some ideas generated from students in class |
When discussing this in class, we were told to ignore our innate desire to create a formula right away. Instead, we were to represent the pattern using images, words, manipulative and making connections wot what it represented to us. Once we each had our way of seeing the pattern, it made it easier to visualize what changes were happening at each stage. If looking at the image growing by 4 corners each stage, we could represent that as +1 +1 +1 +1 on each corner. Breaking the problem down into manageable pieces made it easier to construct, and actually make sense of the expression.
The representing Process Expectation for grade 4 students is to "create a variety of representations of mathematical ideas (e.g.,by using physical models, pictures, numbers, variables, diagrams, graphs, onscreen dynamic representations), make connections among them, and apply them to solve problems" (pg. 65) which applies directly to this method of solving the Dot Problem. While constructing the formula and getting to the final destination is obviously important, the ways we plan, organize, connect and represent are just as valuable.
The Power of Math Manipulatives
Additionally, I wanted to discuss the importance of math manipulatives. While any opportunity for hands on learning is beneficial for students, some manipulatives are better than others, depending on the problem. It is the teacher’s responsibility to assess the various forms of manipulatives, and offer ones that benefit student learning, not inhibit or restrict them.
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| Christine used marshmallows as her manipulative! |
For example, in a lesson about geometric shapes, Christine used toothpicks and marshmallows as her manipulatives. Not only were they very engaging for students of all ages, but they also offered flexibility, allowing students to explore geometric shapes creatively, with no limitations. In contrast, while Maxwell’s lesson was clear and helpful, his choice of linking cubes were not the appropriate choice, because they only allowed students to explore rectangular prisms, and that is it. I found it very frustrating with these limitations placed on what shapes I could explore and what ones I was unable to; thus, considering how beneficial the manipulatives are to student learning is essential before assuming that they will automatically help because they are a manipulative.
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| The math textbook suggests many math manipulatives |
As shown here, the math textbook offers numerous manipulatives that are beneficial for student learning and engagement. However, the teacher should be cognizant of choosing an appropriate manipulative, so they better fit the activity and improve student learning.
Thanks everyone for checking out my blog!
– Madeleine
