What misconceptions or limited conceptions do students have regarding the equal sign? What causes these misconceptions, and how can instruction clear these up?
Always doing well in algebra and understanding that the key was to write good equations that were in balance so I could solve made this chapter interesting to me. Understanding that so many had/have difficulty with an equation 8+4=__+5 in a 1999 study blew me away. I guess I had teachers very early on that truly helped me understand that I could not solve for an unknown number if I did not have an equal sign that said I had balance between one side and the other. When I have taught in the primary grades or the middle grades I always teach that an equation with an equals sign meant balance between the two sides and therefore what we do to one side we do to the other side to keep this balance. On page 325 is the example of an old scale and illustrating how it is in balance or tilting! I truly believe that hands on seeing a scale and handling objects helps the students understand this balance. From there moving to student written equations (whether they choose candy or fish for the subject) to understand the balance in their life (money in the pocket or an item in that they purchased) for equal value. Having students understand two very different views...the operational view and the relational-computational view is also important. In the equation __ + 7 = 6 + 9, having an understanding that 6 + 9 = 15 so that 7 + the unknown must equal 15 or understanding that seven is one more than 6 so that the unknown must be one less than 9 are two different ideas but EQUALLY important ideas. But after reading this chapter I return to the importance of formative assessment to begin with and not rely on what I believe should be understood! or believe that I am to teach only the standards that are set out for my grade and plow ahead. Its not as overwhelming if I start with the idea of pre-assessment as my format rather than retracing my steps!
I can understand how students misconceive the equal sign as being the "answer". Most early elementary experiences supported my misconception by having two numbers separated by an operation on the left side followed by an equals sign on the right. I don't remember having anything to compare my operation to until I started to learn algebra in 8th grade. The explanation the book makes as to why teachers must help students develop the understanding of = as"equivalent to" instead of "the answer is" makes sense. Teaching students to know that both sides of the equal sign is an "equivalent expression" in each problem will help to solidify the understanding of RELATIONSHIP. Knowing the relationship the is the basis for knowing algebra and finding unknown symbols. When I teach this, I think it important to allow students to play with visual models of equivalence. The seesaw activity (14.12) seems like a fun and engaging way to begin learning this concept. This game can also be used to explain inequalities such as less than and greater than. Further, using money is another way to visualize equivalent expression and greater than or less than. Once we review the visual models/games, I would ask the students to practice some simple mental math strategies such as 2+_=_+2 and get more challenging as they get the concept.
This misunderstanding/misconception of the equal sign stems from the equations students typically encounter in elementary school. A mathematical expression is on the left of an equals sign, and student interpret = to mean "and the answer is". One way to minimize this misconception is to ask students to find equivalent expressions instead of a numeric answer. This approach also supports student application of concepts like part-part-whole and combinations of 10 (for addition), and and reinforces the operations of multiplication (e.g., associative).
Another approach is to have students work with missing-addend and missing-factor equations, as well as equations with the result on the left. I have seen this later example throw students for a loop. They were adamant that the expression had to be on the left, and the answer they were to solve for had to be on the right.
Finally, the text suggests having student write their mental math strategies symbolically. I wonder how proficient a student must be with background understanding to be able to do this effectively.
Yes, Dean, exactly - give students more equations that have missing parts, not just missing answers (and don't forget to mix in some decimals and fractions too)! Students don’t even have to solve the equations; “Rather than always asking students to solve a problem… ask them to instead find an equivalent expression (p. 320).” In other words, instead of solving 9 x 4 = ?, students could write 9 x 4 = 9 x 3 + 9, or = 6 x 6. They will also become more familiar with mathematical properties as they do so. Verbally exploring with students why equations are equivalent, such as 4+6 = 6+4, leads to their understanding of the commutative property, which is key to understanding algebra. It is also important to have students examine equations that are not equivalent, such as 2 x 1 < 2 x 2, which is another big misconception that students have that no one has mentioned yet. Finding equivalent equations will make it easier to solve problems with missing variables, giving even elementary students a head start on algebraic reasoning and skills. Variables are another concept that are often greatly misunderstood by students. If put into context, first graders can begin to explore these concepts. One of the recurring themes I’ve noticed throughout this text is the importance of putting mathematical concepts into context, which is brought up again in chapter 14. “Looking at equivalent expressions that describe a context is an effective way to bring meaning to numbers and symbols (p. 319).” While numeric expressions are a good starting point, the authors say, it is important to explore other symbols that are often misunderstood by students. The equal sign, as you all have said, and the greater than, less than, greater than or equal to, and less than or equal to signs, which I’ve mentioned, as well as variables are all symbols that are surrounded by a lot of misconception. To unlock the meaning of these symbols, students need different ways of looking at number “relationships”, as someone said. They need to talk about it, and use the “correct language” (as someone else said) to build their understanding. They also need lots of practice with different approaches to solving such problems, and need to understand why and how these practices work in order to create generalizations which build an understanding of mathematical properties.
I appreciate Katherine's explanation of the value of the balance activity on page 325. Using equations to balance a scale give both a visual and conceptual understanding of the equal sign as a balance. It is also important to show students that "equivalent expressions" done always have to be on the left side. They can be on either side. Teachers should make sure to give examples of this when teaching these concepts.
These misconceptions can be caused by the language that teachers use in a classroom. Students understand that when an equation is put in front of them (5+2=?) they are solving for an answer. They can then communicate their answer five plus two equals seven. If a teacher is not using the correct language to establish the relationship of these numbers, students will not understand that 5 plus 2 is equivalent to 7.
Teachers must present problems to students in various ways instead of using equations that are always shown in order as 8+6=____ or 5 x 4=____. To have students think about the equal sign as a sign of equivalence, ask students to find an equivalent expression instead of only asking for the answer. What is an equivalent expression of 8+6? 5 x 4? When students are explaining their thinking highlight that the expressions they found are equal to one another.
I agree with Katherine that the root of most student’s misconceptions regarding the equal sign is that the relationship between the numbers on the two sides is not iterated enough at an early age. I recently worked with 4th graders on balancing equations and it was a whole new way of thinking of an equal sign. I know that it wasn’t until I was in high school balancing equations solving for x, that I realized the true meaning of the equal sign as showing the relationship between two sides of an equation.
Always doing well in algebra and understanding that the key was to write good equations that were in balance so I could solve made this chapter interesting to me. Understanding that so many had/have difficulty with an equation 8+4=__+5 in a 1999 study blew me away. I guess I had teachers very early on that truly helped me understand that I could not solve for an unknown number if I did not have an equal sign that said I had balance between one side and the other. When I have taught in the primary grades or the middle grades I always teach that an equation with an equals sign meant balance between the two sides and therefore what we do to one side we do to the other side to keep this balance. On page 325 is the example of an old scale and illustrating how it is in balance or tilting! I truly believe that hands on seeing a scale and handling objects helps the students understand this balance. From there moving to student written equations (whether they choose candy or fish for the subject) to understand the balance in their life (money in the pocket or an item in that they purchased) for equal value. Having students understand two very different views...the operational view and the relational-computational view is also important. In the equation __ + 7 = 6 + 9, having an understanding that 6 + 9 = 15 so that 7 + the unknown must equal 15 or understanding that seven is one more than 6 so that the unknown must be one less than 9 are two different ideas but EQUALLY important ideas. But after reading this chapter I return to the importance of formative assessment to begin with and not rely on what I believe should be understood! or believe that I am to teach only the standards that are set out for my grade and plow ahead. Its not as overwhelming if I start with the idea of pre-assessment as my format rather than retracing my steps!
ReplyDeleteI can understand how students misconceive the equal sign as being the "answer". Most early elementary experiences supported my misconception by having two numbers separated by an operation on the left side followed by an equals sign on the right. I don't remember having anything to compare my operation to until I started to learn algebra in 8th grade. The explanation the book makes as to why teachers must help students develop the understanding of = as"equivalent to" instead of "the answer is" makes sense. Teaching students to know that both sides of the equal sign is an "equivalent expression" in each problem will help to solidify the understanding of RELATIONSHIP. Knowing the relationship the is the basis for knowing algebra and finding unknown symbols. When I teach this, I think it important to allow students to play with visual models of equivalence. The seesaw activity (14.12) seems like a fun and engaging way to begin learning this concept. This game can also be used to explain inequalities such as less than and greater than. Further, using money is another way to visualize equivalent expression and greater than or less than. Once we review the visual models/games, I would ask the students to practice some simple mental math strategies such as 2+_=_+2 and get more challenging as they get the concept.
ReplyDeleteThis misunderstanding/misconception of the equal sign stems from the equations students typically encounter in elementary school. A mathematical expression is on the left of an equals sign, and student interpret = to mean "and the answer is". One way to minimize this misconception is to ask students to find equivalent expressions instead of a numeric answer. This approach also supports student application of concepts like part-part-whole and combinations of 10 (for addition), and and reinforces the operations of multiplication (e.g., associative).
ReplyDeleteAnother approach is to have students work with missing-addend and missing-factor equations, as well as equations with the result on the left. I have seen this later example throw students for a loop. They were adamant that the expression had to be on the left, and the answer they were to solve for had to be on the right.
Finally, the text suggests having student write their mental math strategies symbolically. I wonder how proficient a student must be with background understanding to be able to do this effectively.
Yes, Dean, exactly - give students more equations that have missing parts, not just missing answers (and don't forget to mix in some decimals and fractions too)! Students don’t even have to solve the equations; “Rather than always asking students to solve a problem… ask them to instead find an equivalent expression (p. 320).” In other words, instead of solving 9 x 4 = ?, students could write 9 x 4 = 9 x 3 + 9, or = 6 x 6. They will also become more familiar with mathematical properties as they do so. Verbally exploring with students why equations are equivalent, such as 4+6 = 6+4, leads to their understanding of the commutative property, which is key to understanding algebra. It is also important to have students examine equations that are not equivalent, such as 2 x 1 < 2 x 2, which is another big misconception that students have that no one has mentioned yet.
DeleteFinding equivalent equations will make it easier to solve problems with missing variables, giving even elementary students a head start on algebraic reasoning and skills. Variables are another concept that are often greatly misunderstood by students. If put into context, first graders can begin to explore these concepts. One of the recurring themes I’ve noticed throughout this text is the importance of putting mathematical concepts into context, which is brought up again in chapter 14. “Looking at equivalent expressions that describe a context is an effective way to bring meaning to numbers and symbols (p. 319).” While numeric expressions are a good starting point, the authors say, it is important to explore other symbols that are often misunderstood by students.
The equal sign, as you all have said, and the greater than, less than, greater than or equal to, and less than or equal to signs, which I’ve mentioned, as well as variables are all symbols that are surrounded by a lot of misconception. To unlock the meaning of these symbols, students need different ways of looking at number “relationships”, as someone said. They need to talk about it, and use the “correct language” (as someone else said) to build their understanding. They also need lots of practice with different approaches to solving such problems, and need to understand why and how these practices work in order to create generalizations which build an understanding of mathematical properties.
I appreciate Katherine's explanation of the value of the balance activity on page 325. Using equations to balance a scale give both a visual and conceptual understanding of the equal sign as a balance. It is also important to show students that "equivalent expressions" done always have to be on the left side. They can be on either side. Teachers should make sure to give examples of this when teaching these concepts.
ReplyDeleteThese misconceptions can be caused by the language that teachers use in a classroom. Students understand that when an equation is put in front of them (5+2=?) they are solving for an answer. They can then communicate their answer five plus two equals seven. If a teacher is not using the correct language to establish the relationship of these numbers, students will not understand that 5 plus 2 is equivalent to 7.
ReplyDeleteTeachers must present problems to students in various ways instead of using equations that are always shown in order as 8+6=____ or 5 x 4=____. To have students think about the equal sign as a sign of equivalence, ask students to find an equivalent expression instead of only asking for the answer. What is an equivalent expression of 8+6? 5 x 4? When students are explaining their thinking highlight that the expressions they found are equal to one another.
I agree with Katherine that the root of most student’s misconceptions regarding the equal sign is that the relationship between the numbers on the two sides is not iterated enough at an early age. I recently worked with 4th graders on balancing equations and it was a whole new way of thinking of an equal sign. I know that it wasn’t until I was in high school balancing equations solving for x, that I realized the true meaning of the equal sign as showing the relationship between two sides of an equation.
ReplyDelete