Background:
Through my many years of teaching students math, there is a consistent pattern that I always notice. Nearly all of my math students, regardless of their own confidence in their math abilities, claim to hate fractions. Fractions are a beautiful way to represent parts of a whole because of its accuracy, and shouldn't be a stumbling block. Many students that were doing excellently in math usually look for outside help when they reach the chapter of logarithms in Algebra 2 or PreCalculus. It's likely that they find the subject difficult because they learn about logarithms as a completely separate topic that is unrelated to other math topics. Moreover, I've noticed that Algebra itself tends to be a make or break for a lot of students when it comes to pursuing STEM classes later both in high school and college. Students will either make it through Algebra and go on Calculus and Physics, or they will find themselves despising the math and sciences, resolving to never touch the subject in college and go into Statistics to fulfill the math requirements of graduating high school.
The goal of this proposal is to give suggestions on how the math curriculum for grade schoolers can change so that they will not have to go through these hurdles during their math career and see STEM subjects in a different light.
Changes in the curriculum:
- Most of these change have to do with introducing notation early on.
1. Introducing parentheses in Kindergarten.
Kindergartners are learning how to count and add simple numbers with their fingers. Introducing parentheses at this age to addition isn't much of a stretch and it doesn't make the math that they are doing much more complicated. Why wait another 5 years to introduce parentheses only associated with PEMDAS?
Example: 2 + 5 + 3 = ? The answer is ten. What happens when we add parentheses here: (2 + 5) + 3 = ? We add the 2 and 5 first. So it become 7 + 3 = 10. What about 2 + (5 + 3) = ? Now we add the 5 + 3 first. So it becomes 2 + 8 = 10.
An addition worksheet that the students would do would include many problems with the exact same numbers with parentheses moved around. With practice and repetition, students will come to know the purpose of parentheses and how they affect addition. We can extend this to subtraction, multiplication, and division.
2. Introducing parentheses and the dot symbol as other symbols that represent multiplication
We teach students multiplication using the 'x' symbol. Students are so used to using the 'x' to represent multiplication, but abruptly stop using it when starting Algebra. Algebra has so many notation changes that students have a difficult time keeping track of what represents division and multiplication and the 'x' symbol as a variable. Instead, we can use the opportunity to show that there are multiple notations to represent the same thing.
Example: What is 2 x 3? What about 2(3)? (2)3? (2)(3)? 2.3? (Note: I will fix this in Microsoft Word).
A multiplication worksheet that the students would have with problems with similar numbers using the different style of notations so that they can get used to it.
3. Introducing the fraction notation as another form as division
I've noticed that students in the 10th and 11th grade are still confused when it comes to fractions. They have not yet associated fractions with divisions, and that association is not explicitly taught. Fractions tend to be in a completely different category than division. We can resolve this issue by having students make this association early on. Terms such as 'numerator' and 'denominator' are not important at this stage and can be learned later on.
Just like how I propose to teach multiplication using the multiple types of notation, I propose to teach division by including the fraction notation.
4. Introduce Logarithms after introducing roots (square, cube, etc.)
Students learn about exponents and their properties in 5th and 6th grade. They also learn how to go backwards using square and cube roots. It's not a difficult jump for students to learn about logarithms at the same time.
Example: 4^2 = 16. Sqrt(16) = 4 Log4(16) = 2 (Note: Adjust using Word)
By introducing logarithms early, Students won't be as intimidated when they learn about more properties of logarithms later. Just as students are introduced to roots but learn more about their properties, graphs, etc, later, they can learn what logarithms are at the same time but learn more about their properties later on. There's no need to have a three or four year gap between learning about exponents and then learning about logarithms.
5. Incorporating Algebra into Grade School math
At least a year of math (Pre-Algebra and parts of Algebra 1) is dedicated to learning how to manipulate variables and solving algebraic equations that can be easily incorporated all throughout grade school math.
How to change the way students feel about math, particularly algebra? Start by integrating Algebra concepts very early on, with Algebra notation, so that the Algebra notation itself wouldn't be a shocker when they get to 5th or 6th grade. Kindergarten students know 2 + 3 = 5. That means __ + 3 = 5 and __ is 2. So x + 3 = 5 then x = 2. and show that 2 = 5 - 3. (Now I'm getting too much into detail and this should be in the content section) If kids are learning division and multiplication and fractions, have Algebra with fractions and multiplication and division.
Instead of having the answer to a word problem be the main focus of the lessons, possible writing them in algebraic notation be a main focus and solving be more of an afterthought. Because word problems also tend to be a bane of a student's existence. Basically the idea is don't suddenly teach Algebra as a fresh topic in 6th grade, but rather have all of 6th grade Algebra 1 be integrated throughout the 1st through 5th grade material.
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