Think about the tricks and techniques that you use to help students transition from concrete arithmetic to abstract algebra. Post some thoughts to the forum, considering:
What has worked for my divers learners in my classroom are hands-on projects, computer technology, and new programs. These methods combined have helped my students transition from concrete arithmetic to abstract algebra.
In the past I have used a lot of discovery activities through the N-spire calculators. Especially when introducing new topics, I let them establish some sort of relationship with the material first. Having them make a connection with a material, rather than me delivering it to them has helped a lot with retention and understanding. I think my students find multiple variables to be most confusing because there is no “answer”. The “answer” is dependent on the input and thus brings me back to why I think CS that focuses on functions will be very helpful in my classes.
In the past I’ve used lots of pictures to help transition students. Students have trouble with using variables and with the variables changing values. Hopefully programming can help students connect tactilely and visually with the algebra.
In the past I have used various methods including, discovery ideas with graphing calculators, short video clips and discussion, activities, stories that then tie into the concept, and real world sports examples or other real world connections that tie into what we are about to study. I want the students to make a connection to the concept in a way that makes sense to them so that I can build on this and add to it. Students really want there to be one way to solve a problem and one answer to that problem and they tend to struggle when this is not the case.
I have been teaching Honors Geometry for the past several years and have not taught algebra in over 10 years. I am switching to a new school and will now be teaching Algebra I and Algebra II.
In the geometry classroom I used the discovery method almost on a daily basis to help students better grasp new ideas and conjectures. I am hoping to bring this method into my algebra classroom as well. I try to use real life examples as much as possible to help students realize how much they do use math (especially algebra) in their every day lives.
I think students find word problems the most challenging and I am really excited to see how I can use this PD in my classroom to help them with those.
I like to let students discover “the rules”. I also like to put concrete arithmetic examples right next to the abstract, especially when working with problems that require formulas. My students generally think of x as a specific numerical value - “the answer” so they have some trouble grasping the concept of x as a value that can be changed. I hope that programming will be a way to drive home the importance of order of operations.
In the past I have used discovery activities to introduce new topics. This helps them dive into new material with me not introducing a lot of information in the beginning. It allows students to actually find information for themselves and ask questions of each other rather than from me, their teacher.
As I looked at other posts, I noticed one person comment on students always needing to find THE ANSWER. This is one of the most difficult things for students because over the years, they are always told that there has to be an answer or to solve and find the answer and not understanding that sometimes learning the process and working through a problem is more important than finding the answer or that may be the ANSWER (knowing what steps to take to solve, what do you do second etc.)
I think, and hope that Programming will be a way for showing students that there are steps and steps can be followed to generate an answer or some finished product. I am excited to learn how programming can literally change the way my students tackle any future problems.
I have utilized video clips, props, short stories, and music to introduce new concepts and relate them to the abstract algebra concepts. I have found that group work helps kids to talk through the new concepts and ask each other questions using mathematical terminology in a safe environment. I love hearing them argue their point and back it up using mathematics.
I know very little about programming, so I am not certain how I will utilize it in my classroom, but I am hopeful it is one more tool that will help my students understand and apply algebra.
I like to help the students make connections to things they already know about mathematics, or some sort of statistics from the real world. If they have that connection, they seem more interested in learning more. This is why I think programming will be so beneficial. I don’t know a lot, but I do believe my students will have an interest.
I like to use application, like showing them that the games they like to play all use mathematics, especially geometry and statistics. Coming from a systems engineering background, the first words I say to the class are, “Math does not exist to torture you, it only exists to model the real world.” They’re skeptical, but when you guide them through examples, it seems to help.
I try to coneect new concept to previously learned skiils, I also use concrete examples as often as possible.
When I transition scholars into Algebraic expressions and equations, I explain to them that they have already been doing it with things that are unknown and review the definitions of what a variable, coefficient, and constants are. through out my teaching I continue to refer back to these definitions. I have also utilized the various properties and explained how they are used to solving linear equations which has helped tremendously. Students never understand the reasoning behind why they have to learn these properties and some educators don’t try to use them to solve equations, I mistake I have made myself , but have corrected .
Students are told often times " this is the ONLY way to solve this problem" and then when they get to Algebra they find this is not true because now we are showing them multiple ways to solve this one problem. this becomes confusing to some scholars.For example in solving systems you can solve by graphing, elimination, and substitution. Students struggle with the fact that these three ways will get the same result. They think it’s MAGIC…or something. I have to remind them “Welcome to Algebra, there is more than one way to solve…remember that!”
I think programming can help bridge the gaps of the learning process to the true application of the skills taught. Scholars often times do not understand the “Why” but if it is applied and they have an out put they just might be able to make those connections.
I agree with so many of the ideas and methods others have stated. Any time that I can start a process of discovery and have the students find their own way to concept, is when they are the most successful. We are always trying different way, visually, tactile, technological research, literature and You Tube even - any way that works for them to create a relationship that finds meaning to them.
I’ve used visuals or manipulatives for illustrating each step of the way, prior knowledge from their mathematical and nonmathematical courses, and discovery tasks that involve experimentation. For instance, having students physically do a jumping jacks experiment, documenting and graphing their data, creating functions, and analyzing it helps them connect math concepts to their real world, direct task which they have already invested themselves actively in.
I find that I use various videos, as well as hands on materials to teach algebraic concepts. I also found that students understood things like input output machines before we moved to equations and variables. This was much easier for transitioning students to equations with variables and constants. Students made those connections much easier that way. I also love to find videos that help to solidify or aid in making those connections viable to them.
One of the ways to do it, is to teach concepts in a simpler form and have students relate to them by using real life problems that they can understand.
Transitioning from the concrete to the abstract is one of the biggest hurdles to higher mathematics. I have a tough time getting my students to understand that there isn’t a single correct answer in functions. I usually force them to use at least 2 examples of my own deciding (e.g. “This time, I want 276 cookies”)
This doesn’t really get them thinking abstractly, but it does illustrate that the same function can be used for nearly infinite inputs (domains).
What’s worked for you in the past?
I use visual models to illustrate the concrete. I also incorporate writing when students write something algebraically or graphically. For example, we write “number tricks” to create and solve equations. So for an equation like -2x +5 = 21 Students would write the steps show the written meaning next to the algebraic expression: Choose a number. (x), Multiply by -2. (-2x). Add 5. (-2x +5) The result is 21. (-2x +5 = 21).Then students can backtrack starting with the last clue and use inverse operations. My result is 21 (-2x +5 = 21) . Subtract 5 (-2x +5 -5 = 21 -5 ). Divide by -2 (-2x /-2 = 16/-2). My chosen number was 8 (x = -8). When it comes to functions, we usually write what the function notation means. For example f(x) = 3x -10. When we input the value for x, multiply it by 3 , and sibtract 10, we will get the output called f(x).
What do your students find the most difficult?
Recursive function notation and what it actually means. They get intimidated with all of the notation, although they are capable of doing the math. Often times we have to orally dictate what the symbols mean. So I make word cards (inout, out put, previous input, given, base case) to hover above the symbols to help in comprehension.
How do you think programming can help with this transition?
I think programming will be an interesting test of how students apply functions to the real world and what questions they will have when the programming language differs from the mathematical reasoning they were taughts regarding functions (like the example at the end of the video).
When thinking about students transitioning from concrete to abstract mathematics, I have found that many students struggle if there is more than one answer to a problem. They have been taught that for every math problem, there can be only one solution.
Things that have worked for me in the past is showing examples of how using a single function can produce multiple results. Linear functions are a great way to introduce this as they soon find out that based on the input value, the output value will change.
My students find the idea of multiple solutions challenging. This is something that as mentioned above was brought with them from elementary school.
I think programming will help with moving from concrete to abstract thought as students will learn that through the use of functions in programming, there are many ways to solve a problem with one or more solutions.