Research
Relational Thinking and the Equals Sign
Ambrose, Rebecca C. and Marta Molina. (2006, Sept.). Fostering Relational Thinking While Negotiating the Meaning of the
Equal Sign. Teaching Children Mathematics. Retrieved from
http://sddial.k12.sd.us/esa/grants/sdcounts/sdcounts08-09/fall08/Equal_Sign.pdf
Article Summary
By: Kayla Bieschke
In the article, Fostering Relational Thinking While Negotiating the Meaning of the Equal Sign by Marta Molina and Rebecca C. Ambrose is about a study they completed to look into how students view the equal sign and relational thinking. They focused on three main questions: How do students’ conceptions of the equal sign evolve when considering and discussing varied true-false number sentences? Do students develop relational thinking while the class negotiates the meaning of the equal sign? Do the students retain the new interpretations of the equal sign over time? They did this through five sessions with a group of eighteen diverse third grade students. Their assessment at the beginning showed that all students had misconceptions of the equal sign.Many of their discussion involved the students discussing their ideas and views with their classmates. This helped many of the students with their misconceptions. They also worked on a lot of number sentences and the thing that helped the students the most was writing their own number sentences. Relational thinking is solving number sentences by focusing on the relationships between the numbers in the equation instead of doing the computations. None of their students really did this when they started their sessions but by the end of the five sessions many of the students started doing using this type of thinking. One of the conclusions they found in this study was that understanding the equal sign takes time. This study was over almost a whole school year and most of the students did not really get a full understanding until the end.
This article is important because most elementary teachers will probably need to handle this misconception. Everyone in the class they authors went into had the misconception, so it is a very common issue and it is one that cannot be ignored. Knowing what the equal sign means and how it affects equations is something that needs to be grasped because once students move onto solving equations for variables and other more advanced algebra, they need to know that the expressions on both sides of the equals sign need to equal the same thing. Working on this concept with students could also bring in a few of the practice standards into play. Communication was very important to most of the students in the study because once they got to start talking about their ideas and views on the equal sign they really started to understand what it's purpose is, so they students work on argumentation and possibly reasoning.Regularity may also start to be developed because the students need to understand that this is a property that always holds whether the problem is 1+1=2 or 3(x-8)+12=2x-3. The results of the authors' study may help many teachers realize that this problem exists, we can fix it, it may take time, and allowing students to talk about the equal signs purpose, to solve problems involving equal signs, and opportunities to write their own problems will help them retain the concept.
When should Algebra be taught?
Gojak, Linda M. (2008, Sept). Algebra: What, When and for Whom. Retrieved from:
http://www.nctm.org/uploadedFiles/About_NCTM/Position_Statements/Algebra%20final%2092908.pdf#search=%22algebra%22
Article Summary
by Amanda Orlowski
In the article, Algebra: What, When and for Whom provided by the NCTM (National Council of Teachers of Mathematics) website, it helped make clear when algebra should be taught. When thinking of algebra some may think of hard, detailed equations but algebra is “a way of thinking and a set of concepts and skills that enable students to generalize, model and analyze mathematical situations.” Algebra is important for students to learn from pre-K through grade 12 because by knowing algebra can open doors and make for more opportunities. Algebra is used daily in real-world situations which can be described as a algebraic expression. However, developing algebraic concepts and skills does not happen after one year of being taught material. The article states what teachers should help the students develop during elementary and secondary school years. In the elementary ages, the article states that teachers should help students develop fluency with numbers, identify relationships and use a variety of representations to generalize patterns and solve equations. Then, in secondary school the teachers should help move from verbal descriptions to proficiency in the language of functions and skill in generalizing numerical relationships expressed by symbolic representation. Overall, the article gave an understanding of what should be taught and how algebra is important and it is not just long, difficult expressions.
This article is important because some people may not understand how algebra is important to have starting in pre-K and continuing learning until grade 12. When many people think of algebra they do not think of the background behind learning the difficult material which is taught in higher grades. Students need to grasp the concepts first before continuing into difficult things. Younger students need to learn patterns and solve simple equations such as 10+13=? because once the students learn what “+” means and what “=” means then they can move onto harder material and eventually into long equations.
Teaching Algebra too early? Never!
Oishi, L. (2011, September 28). A new age for algebra. Retrieved November 19, 2012, from District Administration solutions for school district management: http://www.districtadministration.com/article/new-age-algebra-1
Article Summary
by: JD
The key idea of this article is if we push students too soon to Algebra we may lose them all together in math. A lot of students in the past have moved into Algebra in middle school without being properly prepared. Mike Shaughnessy, president of NCTM states that “NCTM encourages schools to introduce algebraic thinking, concepts and subskills from the earliest elementary grades.” This is good advice. A prepared child makes for a more confident child. Common Core is helping to lead educators to notice that it is not the quantity that we teach, but the quality. One publisher, Big Ideas Math feels that a narrower and deeper curriculum is what will work. Another thing that will work is for districts to invest in their workforce. “Enhancing teachers’ skills will take time and requires them to develop both content and pedagogical knowledge. “ This will be so much better than taking a one-size fits all approach.
This research is important because teachers need to understand that we need to teach students the right information at the right time. If we push children too early, they may lose interest in math. If we don’t make math problematic, we could also lose the children that are at a higher level with a concept. The new Common Core Standards tell us that the quality of the task is more important than the quantity. Giving students more to do doesn’t mean they will come away with a deeper understanding. They need to be given tasks that meet their needs and are rich in mathematics.
Developing Algebraic Reasoning in the Elementary School by: Adam D
Carpenter, T. (2003). Developing algebraic reasoning in the elementary school . Retrieved from
http://www.wcer.wisc.edu/news/coverStories/developing_algebraic.php
The article I read discussed the implementation methods of algebra in the early grades of elementary education. Students need to be thinking “algebraically” when/while they are thinking of arithmetic. By using generalizations and discussions (in large group, small group, or partners) students are comprehending the relationships/representations rather than just “doing the problems”. The only “yes but” in this article is that the teacher must provide proper problems to students to engage them successfully in the thinking process.
The article I chose is important because students need to be subjected to algebra at an early age but the subject needs to be taught jointly with other aspects of mathematics (i.e. arithmetic). We cannot continue to allow students to think of mathematics as a collection of separate entities. It’s vital that aspects of algebraic thinking are used throughout all of the different mathematics lessons, so that students in the elementary grades can begin to engage in meaningful discussions about proof and make progress in understanding the importance. We need to continue developing their algebraic reasoning to reflect their ability to generate, represent, and justify generalizations about fundamental properties of arithmetic. Students need to be taught how to adapt their thinking. By becoming successful with this, students can integrate these aspects into other subject areas.
http://www.wcer.wisc.edu/news/coverStories/developing_algebraic.php
The article I read discussed the implementation methods of algebra in the early grades of elementary education. Students need to be thinking “algebraically” when/while they are thinking of arithmetic. By using generalizations and discussions (in large group, small group, or partners) students are comprehending the relationships/representations rather than just “doing the problems”. The only “yes but” in this article is that the teacher must provide proper problems to students to engage them successfully in the thinking process.
The article I chose is important because students need to be subjected to algebra at an early age but the subject needs to be taught jointly with other aspects of mathematics (i.e. arithmetic). We cannot continue to allow students to think of mathematics as a collection of separate entities. It’s vital that aspects of algebraic thinking are used throughout all of the different mathematics lessons, so that students in the elementary grades can begin to engage in meaningful discussions about proof and make progress in understanding the importance. We need to continue developing their algebraic reasoning to reflect their ability to generate, represent, and justify generalizations about fundamental properties of arithmetic. Students need to be taught how to adapt their thinking. By becoming successful with this, students can integrate these aspects into other subject areas.
Choreographing Patterns and Functions
Hawes, Z., Moss, J., Finch, H., and Katz, J. (2012). Choreographing patterns and functions. Teaching Children Mathematics,
Vol. 19 (No. 5), pp. 302-309.
Article Stable URL: http://www.jstor.org/stable/10.5951/teacchilmath.19.5.0302
Article Summary
By: NB
Yes, you can dance algebra! This article details how a teacher in a 1/2 classroom did that and also describes how they got to that point through geometric patterns and function rules. The article is written by researchers who followed the classroom as they went from students just barely secure in addition and subtraction to students who understood basic algebra. The
unit began with students growing geometric arrays using tiles and visually seeing the patterns and discerning rules from those patterns. The students also had the opportunity to design their own patterns. The instruction moved on to function machine activities. The teacher had created an actual “machine” with cards that students would input and discern the rule. Once that was secure, the dancing began. The teacher began the unit with his own dance and had the students try to figure out what he
was doing. The unit culminated in a recital where all the students performed their function for the class and the class had to solve for the function. This is a great unit that includes visual and kinesthetic learning.
This article is important because it provides an outstanding example of how to teach algebra in the younger grades. Evident are the mathematical processes of looking for and making use of structure and attending to precision. All of the activities are
ones that all students can access. Students chose their own numbers when creating their arrays, functions
and dances so they could select numbers they were comfortable with. The unit also included a good mix of conceptual development along with the actual practice of the process.