One of the most common misconceptions for students new to algebra occurs in simplifying and evaluating algebraic expressions, that is the idea that 5x+4 equals 9x. It is understandable how students might simplify this expression based on their prior knowledge of adding integers, but once again they fail to recognise x as a variable term. Since the second term only has a numeric part, 5x and 4 are not like terms, and so they cannot be combined into a single term. Even higher-level algebra students make this mistake, it is not uncommon to see the expression 3x2+4x3 simplified to 7x5 rather than recognising that these terms cannot be combined since they have different exponents. I came across a simulation on PhET that would help to addressing this issue. Following is a suggested pedagogical approach using a graphic that depict the cyclical nature of GEM and its phases,
Generate relationship – the students will generate pictorial representation of algebraic expressions using the simulation on PhET. They will use different representation such as circles, triangles, and coins to generate graphical expressions. The aim is to have them categorise items and generate expressions such as 
. Then generate the relationship between the variables in occurrence bananas and apples in this expression, the students should generate an idea on how to simplify it to 
.
Evaluate relationship – the students will replace variables in pictorial representations with variables such as x, y, or z. They will then evaluate the expression, comparing their results with the pictorial representation. They are expected to produce x+y+y+y+x+x+x equals 4x+3y and establish an analogy with what they have done in the previous step.
Modify relationships – the students are also encouraged to substitute a random value in for any variables. In this case, x will have a random value, and y will have a different random value. They will evaluate and see whether the algebraic expressions are equivalent or not.
Reference:
Khan, S. (2011). New pedagogies on teaching science with computer simulations. Journal of Science Education and Technology, 20(3), 215-232.
Hi Vivien,
I cannot stress enough how “combining like-terms” has been the biggest struggle of my life teaching grade 8 math students. I have used pictures and words and other representations to give examples of adding apples with apples and bananas with bananas but it does not seem to have ever worked.
I love the simulation that you have used in your post and think it is going to be a great source for my future garde 8 math students.
Although, I am curious whether you will do all three 3 steps of GEM in one lesson or divide it into multiple lessons? As I think the lest step maybe a bit too overwhelmeing for students learning to combine like terms for the first time. What do you think?
GK
Hi GK,
I think 50 minutes will be sufficient to run the activity. For this activity, the students will benefit from first practicing step one and two cyclically with basic algebraic expressions. Therefore, they can generate an idea on how to group the variables when they have simple algebraic expressions. I am thinking of step three as the step through which they would verify and reinforce the idea they constructed in step one and two. This step would take more time for challenging algebraic expressions. But, hopefully after running the whole activity once, only step three will be regularly used to verify their results and potentially modify their understanding.
Vivien
Hello Vivien,
I enjoyed reading your clear and concise posting. I saw that were several examples, where assessment FOR learning occurred. Assessment For Learning is the process of seeking and interpreting evidence for use by learners and their teachers to decide where the learners are in their learning, where they need to go and how best to get there. This is where I think GEM is a beneficial framework because it encompasses formative assessment throughout the process.
Especially with the use of pictures, it engenders thinking without repetition and it leads to rich mathematical conversation.
I am consistently asking students to explain how they got their answer, your T-Gem lends to students critically thinking about how they got their answer. I wonder once students become more familiar with the T-Gem framework, would be less quick to give a quick fact/answer and provide their thought process.
Mary
Hi Mary,
You are right that GEM framework encompasses formative assessment, and I think this informs both the teacher and the student on potential learning achievement. As collaboration teacher-students, students-students goes throughout the GEM process. Feedback would inform students’ knowledge construction. I am not sure that this would lead to students giving less quick facts, but I believe that relatively to their level of understanding over a topic, they would provide stronger and consistent arguments to their answers.
Vivien
Hi Vivien,
This is the topic that I am working with a group of grade 7s on right now. They are really struggling with combining like terms and simplifying the equations. Would you chunk these into separate lessons? Also, would you give the students some exploratory time before you begin?
I’m looking for any and all ideas:)
Nicole
Hi Nicole,
I introduce this activity with a game where the students need to use symbols to represent numbers of physical quantities. We have a disposal of multiple items such as geometric shapes. The instructions are written on a piece of paper. An example is “arrange in this order: four squares, three triangles, one pentagon, two squares and two rectangles.” The students are set in groups of four. Two students are in charge of using numbers, symbols, and letters (no words and verbal communication) to help the other two students to display the shapes as prescribed. The aim of this game is to have the students categorizing the shapes, and generate some sorts of representation to communicate. The game can take about 20 minutes. I would not chunk the three steps of the activity into separate lessons. I would work the students through the whole process step1, step2, and step3 a couple of times. I believe about 50 minutes practicing these steps is sufficient to have the students understanding the cycling process. Also, I think it is important to run the three steps with the students a couple of times in one lesson. So they can experience knowledge over the whole process. The cyclical manner they will go through with various algebraic expressions will be informed by their understanding and the challenge they face. My guess is that many will stop using step 1 once they understand how to categorize variables.
Vivien
Hi Vivien,
I like the fact that many of these examples are very nostalgic for me. I remember doing algebra in elementary school and enjoying it.
I wonder if enjoying the algebra had to do with the teacher or was it something else?
A good next step might be to explain to students why 5x+4 is important to their day to day life.
Christopher
Hi Christopher,
If I was fortunate enough to have a prepared answer all the times that I heard students saying, “When am I going to use this in my life?”. I would have contributed to change some of my students’ negative attitudes towards maths. I also think that explaining to students why maths is important in their day to day life would engage them.
Vivien