Manipulating Math
15:27:00
Hello fellow Mathema-gicians!
Have you ever been faced with a Math problem that was so utterly abstract that you felt as though Williem de Kooning himself created the question?
I changed "turkey" to tofu, sorry. Just trying to make the problem less confusing to my moral compass. Nevermind the fact that a British lady is posing a question about pounds to a group of Canadian students.
Aside from vegetarianism and certain disregard for the Metric system, this wordy problem can certainly cause some initial hesitation and dismay. We know there's a man, a shop, an unspecified diet, some tofu slices and probably a need for fractions and algebra (right?).
Most students will recognize that this problem involves fractions, and they will immediately try to remember "that rule" about algebra and common denominators. Students are going to feel pressured to recall a formula or rule that they once memorized, and if they cannot perfectly recall the steps to finding a common denominator, the panic and frustration usually associated with word problems is going to set in.
As educators, we have to encourage students to trust their intuition and use a manipulative to approach Math that feels too abstract or difficult. It is not so much about memorizing formulas as it is recognizing a general concept and knowing how to manipulate it to suit students' particular interests or comfort levels.
One very simple manipulative can involve Drawing & Representation. When words overwhelm, a visual representation of what we already know can help bring an abstract problem into a much more concrete dimension.
Take a look at how a nine year old student was able to solve the previously mentioned word problem simply by drawing.
She knew that 9 slices of tofu would make one full pound, so she drew 9 slices and then split these slices into 4 quarters to find how many would make up one quarter.
Drawing what you know is not only helpful to visual learners, but it helps all students organize their ideas about solving a problem by moving a problem from abstract words to concrete images that can be manipulated.
Manipulating Mathematics can extend beyond drawing. As educators, it is important that we provide our students with access to various manipulatives that they can choose from. In class we discussed various manipulatives that can be used in the classroom, such as Algebra Tiles, geoboards, and the GeoGebra App.
I even learned how to draw a square using Geogebra as a manipulative!
When students have access to various Math manipulatives, they are given more opportunity to explore what methods best suit their learning styles. If a student has a penchant for the arts, using Drawing & Representation can help them to make sense of wordy math problems. If a student is a tactile learner, they may enjoy using a Geoboard to learn about perimetre and area. Students that are visual learners may prefer to use Algebra Tiles, rather than trying to recall rules and formulas that will lead them to "x". There are also Base 10 Blocks, Tangrams, Linking Cubes ... the list goes on! EduGains has a particularly helpful article about the various manipulatives that can be used for Mathematics, and how to apply them to the classroom.
Manipulatives help students utilize their areas of comfort and also trust their ideas and intuition instead of trying to master Mathematics through memorization.
Have you ever been faced with a Math problem that was so utterly abstract that you felt as though Williem de Kooning himself created the question?
 |
| Kooning, Williem de (circa 1949-50). Abstraction [painting]. Retrieved from https://www.wikiart.org/en/willem-de-kooning/abstraction-1950 |
Really, Williem? What am I supposed to do with that?
Students can feel overwhelmed when presented with a wall of text that demands a solution. For example, let's take a look at one of the problems posed in this week's web modules.
"A man goes on a diet and goes into a shop to buy some [tofu] slices. He is given 3 slices which together weigh 1/3 of a pound but his diet says that he is only allowed to eat 1/4 of a pound. How many of the 3 slices he bought can he eat while staying true to his diet."
I changed "turkey" to tofu, sorry. Just trying to make the problem less confusing to my moral compass. Nevermind the fact that a British lady is posing a question about pounds to a group of Canadian students.
Aside from vegetarianism and certain disregard for the Metric system, this wordy problem can certainly cause some initial hesitation and dismay. We know there's a man, a shop, an unspecified diet, some tofu slices and probably a need for fractions and algebra (right?).
Most students will recognize that this problem involves fractions, and they will immediately try to remember "that rule" about algebra and common denominators. Students are going to feel pressured to recall a formula or rule that they once memorized, and if they cannot perfectly recall the steps to finding a common denominator, the panic and frustration usually associated with word problems is going to set in.
As educators, we have to encourage students to trust their intuition and use a manipulative to approach Math that feels too abstract or difficult. It is not so much about memorizing formulas as it is recognizing a general concept and knowing how to manipulate it to suit students' particular interests or comfort levels.
One very simple manipulative can involve Drawing & Representation. When words overwhelm, a visual representation of what we already know can help bring an abstract problem into a much more concrete dimension.
Take a look at how a nine year old student was able to solve the previously mentioned word problem simply by drawing.
 |
| Retrieved from https://youtu.be/xNT7W9pO6QI?t=17 |
She knew that 9 slices of tofu would make one full pound, so she drew 9 slices and then split these slices into 4 quarters to find how many would make up one quarter.
Drawing what you know is not only helpful to visual learners, but it helps all students organize their ideas about solving a problem by moving a problem from abstract words to concrete images that can be manipulated.
Manipulating Mathematics can extend beyond drawing. As educators, it is important that we provide our students with access to various manipulatives that they can choose from. In class we discussed various manipulatives that can be used in the classroom, such as Algebra Tiles, geoboards, and the GeoGebra App.
 |
| Inrig, Erika (17 October 2017). GeoGebra Square [screenshot]. Retrieved from www.geogebra.com |
I even learned how to draw a square using Geogebra as a manipulative!
When students have access to various Math manipulatives, they are given more opportunity to explore what methods best suit their learning styles. If a student has a penchant for the arts, using Drawing & Representation can help them to make sense of wordy math problems. If a student is a tactile learner, they may enjoy using a Geoboard to learn about perimetre and area. Students that are visual learners may prefer to use Algebra Tiles, rather than trying to recall rules and formulas that will lead them to "x". There are also Base 10 Blocks, Tangrams, Linking Cubes ... the list goes on! EduGains has a particularly helpful article about the various manipulatives that can be used for Mathematics, and how to apply them to the classroom.
Manipulatives help students utilize their areas of comfort and also trust their ideas and intuition instead of trying to master Mathematics through memorization.
2 comments
Hi Erika,
ReplyDeleteYour attention to detail is quite astounding as I didn't make the connection to the British lady and "pounds". Brilliant.
I agree that our students should be encouraged to take a step back and reach for a tool that can help them overcome a difficult math problem. Just like in everyday life, sometimes we need to take a step back and reach for our supports. I like the picture you paint of students seeing a fraction and trying to remember, "that rule." I am seeing a very mathematically confused younger self trying to solve an equation without asking for the help of someone nearby. When I was a younger student I was taught to remember these rules whenever an identifier came up in the wild. We we taught to solve math equations in the same way we were taught to protect ourselves from fire. "What do you do when you see fractions?" Stop, drop, and roll?
I really like the picture you've included with the young student's drawing. I personally would not have looked to illustration to solve that word problem. As a teacher I don't think that I would have taught a method like this one either. It is incredible to see how the minds of others function, and how differently they function from one another. It's important to understand that while peoples' minds function differently, they function nonetheless. Just another means of making us understand how vitally important it is to differentiate instruction for our students!
Great post!
Hi Erika!
ReplyDeleteGreat post this week, and I am not surprised at all that you decided to write your blog on the concept of using drawings to solve math problems. (I did too)
I also like how you made mention of provided access to manipulatives for all our students. I can recall from my evaluator asking my about my choice to use manipulatives in my Micro-lesson. I personally just thought it would help, to which she told me that all students can benefit from using manipulatives, it's not about their ability or academic level, but that visual or physical representations are important to reinforce understanding and to explain the abstract.
Great post!
Also, great visual. Reminds me of Dufy :)