In order to start this unit, I want to have a clear idea of what my students understand about the equals sign. If students have not been taught to think about the equals sign as a symbol that means to balance, I know that trying to master addition and subtraction algorithms, solve equations with variables, understand the concepts of inequality, solve function rules and solve for the unknown as expected in standards 4.NBT.B.4,4.OA.A.1, 4.OA.A.2, and 4.OA.A.1, will be more difficult.
Math Practice Standard 1 expects that students are proficient in making sense of problems. Math Practice 2 expects that my students can reason abstractly and quantitatively, looking at relationships of expressions within an equation. Math Practices 6 & 7 expect students to be accurate and look for structure in their problem solving. These equations and the writing in this quiz will reveal their abilities to do all of the aforementioned. It shows me the level of their reasoning and problem solving using all four operations. It gives me a clear idea of their understanding of solving for the unknown and whether they understand the meaning behind the equals sign.
I opened up this lesson by telling them that I was going to give them a short quiz that would tell me a lot about their thinking about the equals sign. We talked about how the equals sign is something we see all of the time and that we have been using since we first started to add and subtract. The students talked about how way back in Kindergarten and preschool, they were using it to put things together (manipulatives) and that they knew it meant either getting more or less depending on what they were doing.
I asked them to clear their workspace and get ready to show me what they knew!
This short quiz quickly shows me whether or not my fourth graders have mastered understanding of the equals sign.This article from the University of Wisconsin Madison helped me understand the importance of this little test and made me think about how the rigorous expectations of CCSS must have solid foundations and the transition needs to provide this.
This assessment examines four essential things:
1. Placement of the unknown. ( Can the student solve if the box is moved in different places?)
2. Understanding how to use the inverse to solve problems.
3. Using different operations within an equation.
4. Using words to explain what equals means.
I passed out the quiz and hoped it would reveal their understanding.
What am I looking for? I see so many things in just these few samples! The data I gather from this little quiz will help drive my instruction for this unit on place value, addition and subtraction.
In Sample 1 This student simply looked at the equals sign as an infinite horizontal process. Sometimes they will draw another equals sign after the second expression. This student didn't even think about what the second expression meant. It is linear thinking. It is apparent in the subtraction problem, he understood the idea of inverse, which I suspect is because he studied fact families and memorized his facts. That is fine, but it isn't enough as far CCSS is concerned.
The last equation has me stumped. I am guessing that he simply placed the three in the box to have the problem over with! Students are uncomfortable sometimes with leaving things they don't understand, blank. Fill it in with a number and there might a chance you might be right! The written answer is key in understanding that he really does not know what an equals sign means.
For Sample 2, this student can solve the problem. He does not show understanding of why he got the answers he did in his written sentence. This should be an easy thing to fix!
Sample 3 shows no understanding beyond the first problem. Although he addressed the concept of "total" ( which partially relates to the idea of equals), he clearly shows that he does not understand where the total should be in a subtraction equation, an shows no understanding of the inverse. He is not showing thinking about the difference between addition and subtraction as a concept.
Sample 4 shows mastery!
I am going to plan on setting clear learning goals individually or in small groups for students who have not mastered the concepts over the course of the next week. I will record my findings and figure out groups using my Student Checklist. This should make learning how to solve multistep problems much easier for them later on when I need to teach equations and variables.
I am going to advance their thinking by setting up these goals according to my checklist and differentiate. Students will have their own checklist that will look like this. This checklist reflects the standards I mentioned in the the beginning.
*I will be able to explain what equals means.
*I will be able to use the inverse to solve an unknown.
*I will be able to use the four different operations in equations to solve.
*I will learn how to use the numbers I already have to help me solve for the unknown.
Setting up these clear goals will help them master more rigorous work in the future. I will revisit this quiz again about mid year to reflect on their progress and my teaching.
This little clip finalizes my thoughts about my next steps after looking at all of my data.