I was so excited when I discovered the usefulness of Algeblocks, I jumped right into using them. My enthusiasm carried me through, but I soon learned that I had more to learn. Luckily, it was time for me to do the research part of my Master's, and I had just discovered my topic: How to use Manipulatives in the High School Math Classroom. It was also the year of snow storms and I managed to do 80% of my research on snow days! It was meant to be. It is a long paper, but if you really want to read it, I attached the link. Years ago I had an interesting conversation with one of the elementary teachers in my building. She said she uses manipulatives extensively, but she doesn't know how to get the students to make the connections between the blocks and the math they are supposed to do on paper. I realized then that I didn't either. In fact, I still look at that as being the toughest part of teaching with manipulatives, and how you make the connection can be different depending on the student. So, what does research say? First, I will apologize for not citing my sources here. Please reference my paper in the link above for official-ness. Now, here comes the good stuff. There are fundamental steps to follow, steps that of course have variations through the years, but I will stick with the basics. They are the Concrete, Representational, and Abstract stages. I will refer to them as CRA. Concrete is the use of manipulatives ONLY. There is no paper involved. Abstract is the use of paper only, this is where students are using the algorithms we want them to know in the end. Representational is the way you get from concrete to abstract. Representational has many different looks. Concrete Stage This stage will take more time than you want, but it is worth it. Make sure you pick a manipulative that does a good job representing the math you want them to learn. This is important! I use Algeblocks pretty extensively when teaching Algebra, but I have been known to use Cheerios for units and envelopes for x when in a bind. Also, before you get down to business, let the kids play for a few minutes. It helps to keep them from playing when you are trying to do math. To begin, you need to define the manipulatives. It is often the case that the teachers "see" the connection much easier than the students. Therefore, you need to be clear about what you want the students to "see". For example, in Algeblocks, the x is a yellow stick that is a little over 3 green cubes long. Many students want to give the yellow stick the value 3. (I tease them that if they are going to do that, then Pi would be more accurate). But then we have the conversation that it would be physically impossible to create a stick that doesn't line up with some number of green cubes. So we will have to accept that the yellow stick can be worth 5 or -10 or whatever else x is worth. Another thing that might be annoying at first, but turn into some great conversations and "aha" moments, is that you may need to start with earlier concepts to build to what you want to do. For example, before I have students solving equations, we have to understand how to add, subtract, multiply and divide integers. We have to understand why a negative times a negative is a positive, which has nothing to do with bad things happening to bad people! We also need to understand what opposite value means. When we get to actually solving equations, we also have to discuss what the symbol "=" means (Hint: it does not mean "here comes an answer"). The main part of the concrete stage is getting the students to see the physical representation of the math concept you are teaching. Create several examples for them to struggle through. Make problems that will challenge what they already know and think, and make sure you cover several variations so they don't think it all works the same way all the time. Make sure to do them yourself so that you can anticipate the challenges. Above everything else, give them lots of TIME. Do not rush this part of the process. You are trying to get them to internalize the concept. That doesn't happen in five minutes. During this stage, you need to be circulating and challenging students' solutions. Get them talking and explaining. That is one of the cool things about using objects, no matter their level, they can still talk about green cubes and yellow sticks, even if they don't have the vocabulary. I often make them tell me like I am a 5 year old, don't use math words at first, just tell me what you are doing physically. Representational Stage There are several ways to attack this stage. It all depends on your resources, your topics, and your students. 2D version of the 3D manipulative: You can have students use a virtual manipulative via the internet or an app. (Brainingcamp has some great apps for this). One of the cool parts of virtual manipulatives is that they show the abstract along with the manipulation, so it will often "catch" them if they make a mistake. I often just have students draw their work out, see the picture below. This can often be a tool some students use to help them if they are not ready for abstract when everyone else is, which is perfectly fine. Let them be where they are comfortable and understand what they are doing. Although I haven't done much work with it, I would imagine this is the stage for the clothesline math. Simultaneous work with concrete and abstract: When we are nearing the end of the concrete stage, I start introducing the way we write the same math abstractly. For example, I will ask the students to show me 2x + 3 = 5. I write the equation on the board and walk around to make sure they have two yellow sticks and not 2 green guys in front of a yellow stick. I will then ask them to tell me what their first step will be (remember we have already done several problems, so they have already done their exploring and are pretty ready to do it right away). I have them do their step and then I ask them how I could show the same thing on the board. We compare what I have on the board to what they have on their desks. I also have one of the students demonstrate the manipulative work via the document camera while I do the abstract on the board. This allows me to model how I want them to work together. Once I have demonstrated that, I have the students pair up. One will do the manipulative and the other will write. After so many problems, they switch roles. I also insist they talk to each other about what they are doing, they need to do each step together and double check each other's work.
I also enjoy doing this in groups of four. Student 1 does the manipulative work, Student 2 does the abstract work on paper, Student 3 announces the work of Students 1 and 2 (think sports announcer, most enjoy making a show of it, some just talk normal), and if there is a Student 4, they act as referee and throw the flag if someone isn't doing their job. Abstract Stage The part everyone wants to get to. This is what you normally teach: the algorithms of the concept. No more physical objects, no more drawings; just numbers and symbols. But here is the cool part, you can still talk about those other things. I still reference the "green guys" and zero pairs. Let's say I have a struggling student stuck on solving a two-step equation. Most of the time, I just have to ask if they have any green guys, and they will forget I exist and solve the equation. The other cool part: 2x + 3 is not 5x anymore. In the last 5 years of teaching this, I have seen maybe three kids make that mistake. Why so few? They "see" 2x and 3 as different things, just like we want them to. And those three kids? They are asked to show me 2x + 3 using Algeblocks and explain to me why they shouldn't get 5x.
1 Comment
When I was a student in high school, I did a lot of peer tutoring. It was when I figured out I wanted to be a math teacher. I was struggling to get a peer to understand how to solve a one step equation. I used the method I saw my teachers use, basically just talking about inverse operations, but it wasn't working; no matter how many times I said it. I asked my teacher at the time if there was another way, but was disappointed when I was told there wasn't. I was also skeptical of this answer. How can there not be another way? Fast forward to my early teaching career. I found Algebra Tiles in the cupboard, where they were stashed away from the previous teacher, never used. I tried to make sense of them, but got confused and stashed them back in the cupboard. Fast forward another couple of years. I went to a workshop by Nancy Berkas and Cyntha Pattison in my northwest corner of Minnesota. They introduced Algeblocks (the 3D version of Algebra Tiles). As they explained their use, a lightbulb burst on and lit the path for the rest of my career. I found the "other" way to teach solving equations and more. I found a way that all students can grasp an abstract concept and succeed. I also found a way to get my smarty pants students to think differently and challenge their tendencies to take the easy road of memorizing steps. So, here is the idea, give students something physical that they can manipulate to make sense of the operation/concept you are teaching. They do it frequently in the elementary with base ten blocks. I have yet to meet a math teacher that would condone teaching without base ten blocks in the elementary, but it is difficult to find math teachers that have the same philosophy for the Algebra class. And yet, we complain when students make 2x + 3 into 5x. They do this because they don't understand what 2x is and what 3 is. They need to understand what they are in order to do the operation correctly. Before I continue, let me describe Algeblocks. (I have nothing against Algebra Tiles, they work the same). Constants are represented just like you would with base ten blocks. The kit comes with green centimeter cubes. The number 3 is represented by getting out 3 green cubes, we call them the "Green Guys". X is represented with a yellow stick, it is 1 centimeter by 1 centimeter by a little over 3 centimeters. There are also blocks that stand for y, xy, x^2, y^2, x^3, y^3, xy^x, and yx^2, I think that covers them all. It is always interesting to start a lesson asking students to show me 2x. If this is the first time they have used Algeblocks, I often see two green cubes with a yellow stick to the right of them. If you have Algeblocks or Algebra tiles, try it (after introducing and defining them). It is a huge eye opener that they don't understand what 2x means! It is also interesting to represent x/2, they often put a yellow stick above and 2 green cubes below it. Sometimes they even use another yellow stick in between to represent the fraction bar. I do many things with manipulatives. This post is to just introduce manipulatives in general. I will have future posts to discuss specific uses and lessons. I love how using manipulatives gives me a place to go with students who are struggling, but also gives me a way to challenge students' thought processes when I feel they are just memorizing steps. I use many different manipulatives, not just Algeblocks. I use Base Ten Blocks, Balance Scales, XY Axis Pegboards, Geosolids, Linking Cubes, 1in Square tiles, and Anglegs; all in the high school classroom. I also use digital manipulatives, my favorite being Demos. One of the most difficult parts of using manipulatives is giving up the time to do them. Lots of time is a key factor to teaching with them! But, I promise, teaching with manipulatives saves time in the end. You don't have to take time to reteach misconceptions. Plus, your students actually UNDERSTAND what they are doing, which is always a positive. But be warned, they are not magical. There is a process you need to use (see next post). I often fear when I see an article or blog criticizing the use of manipulatives. But when I read them, I agree with their arguments. You need to know what you are doing. But when done right, they can open doors to understanding! I wouldn't teach without them now that I have experienced what they can do. |
AuthorI teach mathematics for grades 7-12. Teaching mathematics is my passion. Archives
August 2022
Categories
All
|