Introducing a new math concept to
students is very exciting for me, because I don't have to try to unravel
poorly-defined strategies that students may already have. Many students learn
concepts without understanding why those concepts work, and they just take it
as a given that they are going to go through these series of steps to make the
calculation. This lack of deeper understanding of the basis behind the concept
will only be exacerbated in later schooling, resulting in even more lack of
understanding. Algebra is one of the most dominant forms of reasoning that
students will have to do in their later math development.
When I introduce algebra to a student for the first time, the first thing I show them is how much they already know. For example, they know that 3 + x = 5 will give them the answer of two. Any student over second or third grade will be able to reason this one out in their head. 3 + what equals five?
The larger question that they then have to understand is, ‘how did I know to do that? I filled in the blank myself, but what I really did was subtraction.’ This is where the basis of further algebra begins. They need to understand that they are performing the reverse operation. They see the addition sign, and so they now need to subtract in order to find the value of the variable.
I demonstrate this to the student in a number of ways, with addition, subtraction, multiplication, and division. They need to see that what they already know they can do in their head, such as 3X=9, translates to 9÷3. The more they see that they can figure this out in their head and it corresponds directly to performing the reverse operation on paper, the more comfortable they will be with what algebra means.
But it can be a slow process to gain a deeper comfort level. At first anything that looks remotely different will seem unapproachable. The student might completely understand 3x=9, but 1/3 x =9 already looks a lot harder, even though you are dividing by the coefficient each time. I show the student that what this question means is: 1/3 of what is 9? 1/3 of 27 is 9. It's a reverse process in a way, compared to what students are used to doing in math.
The more they practice, though, the more rote it becomes. It’s highly important to explain to students that the reasoning that they can already do in their head will translate to a process on paper, before trying to just teach them the rules of algebra and asking them to memorize them. Providing this deeper understanding of how all math is intertwined, and how addition and subtraction are reverse operations of each other, just as multiplication and division are reverse operations of each other, yields a deeper understanding that will provide stronger math comprehension for the rest of their education.
When I introduce algebra to a student for the first time, the first thing I show them is how much they already know. For example, they know that 3 + x = 5 will give them the answer of two. Any student over second or third grade will be able to reason this one out in their head. 3 + what equals five?
The larger question that they then have to understand is, ‘how did I know to do that? I filled in the blank myself, but what I really did was subtraction.’ This is where the basis of further algebra begins. They need to understand that they are performing the reverse operation. They see the addition sign, and so they now need to subtract in order to find the value of the variable.
I demonstrate this to the student in a number of ways, with addition, subtraction, multiplication, and division. They need to see that what they already know they can do in their head, such as 3X=9, translates to 9÷3. The more they see that they can figure this out in their head and it corresponds directly to performing the reverse operation on paper, the more comfortable they will be with what algebra means.
But it can be a slow process to gain a deeper comfort level. At first anything that looks remotely different will seem unapproachable. The student might completely understand 3x=9, but 1/3 x =9 already looks a lot harder, even though you are dividing by the coefficient each time. I show the student that what this question means is: 1/3 of what is 9? 1/3 of 27 is 9. It's a reverse process in a way, compared to what students are used to doing in math.
The more they practice, though, the more rote it becomes. It’s highly important to explain to students that the reasoning that they can already do in their head will translate to a process on paper, before trying to just teach them the rules of algebra and asking them to memorize them. Providing this deeper understanding of how all math is intertwined, and how addition and subtraction are reverse operations of each other, just as multiplication and division are reverse operations of each other, yields a deeper understanding that will provide stronger math comprehension for the rest of their education.
Alexandra is the Managing Director of Boston Tutoring Services, a tutoring company that offers one-to-one in-home tutoring in Massachusetts. She is also a former Kindergarten teacher who also tutors students in grades K-8, in all subject areas, including test preparation.
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