A standard in mathematics provides, at the very least, is a baseline or outline to loosely adhere to during the school year. They are at the most though, designed to curricular goals and guidance for the math curriculum (Ferrini-Mundy, 2000). The direction of the future of math standards is equally important. The NCTM is focusing on having every state adhere to the same standards. Traditional teaching and learning is now taking a backseat to an updated common-core driven era because the old ways are dated for the dynamic of today’s classroom. The big difference between a baseline and goal is the minimum requirement and the maximum success rate you are aiming for as a teacher. Just having standards in a classroom and pushing through each lesson to achieve the notion that you made it through each standard produce a sub-par learning experience. There should be goals, not just for getting through standards, but an actual standard of learning each standard. A certain percentage of students should be able to demonstrate a mediocre to high capability of quality work for each standard. Formative and summative assessments could be used to analyze when it is time to move to the next standard.

The separation of standards by state requirements show a difference in in the challenge the standards uphold from state-to-state (GreatSchools). After the NCLB Act of 2002, states were held accountable for the test scores, and even more than scores, the progress of their students. States submit their standards and questions for approval. There was a gap however in the quality of questions from each state. The NCTM is trying to find a happy medium for this. Forty-nine states now have adapted or at least begin implementing the new subject matter standards in mathematics (Ferrini-Mundy, 2000). Classrooms are no longer made of just high and low learners. Classrooms incorporate such a vast and diverse dynamic that not only includes a plethora of students that require differentiated lessons, but also consist of students who learn in all seven styles (Burton, 2010). Being able to transcend information above just delivering it to each student can prove to be challenging. The goal would be to not just deliver, but have students receive, comprehend and apply. Constructivist style teaching and learning offers a gateway to the success of this. Students understand even subconsciously how they learn. Taking an active role in their own learning and mathematical discovery is key to their lifetime learning journey.

Peer problem solving, dynamic small group teaching and think pair share offer an engaging premise for this learner’s accountability (Burton, 2010). This however does not mean every aspect of teaching from previous generations is lost. If it is not broke, don’t fix it applies to anything that was successful from all previous teaching methods throughout time. Traditional teaching methods are ideal for basic levels of learning. This is evident when basic information needs to be construed to the students. How to do addition and subtraction type concepts do not require constructivist style learning. Both styles of teaching provide huge upside but also are handcuffed by cons if used exclusively in the class. Constructivist math programs leave low-achieving students behind. Traditional programs may be tedious to high-achieving students (McDonell, 2008). A combination of both should be used for the greatest success.

Lesson

The objectives of the lesson I observed was to establish two different ways to find the area of triangles. This lesson was used as a base for eventually teaching composite figures and finding not only the area of them, but also the volume. The lessons incorporated problem solving and word problems, heightening the effectiveness of the lesson. The teacher placed the students in group settings. Within each group, students were given two separate problems. After the completion of each problem they discussed how the performed the work and came to find the answer. Once they all agreed on the answer and explanation, they groups were all shifted to a new table which held a new set of questions to solve and discuss. The standards used from the NCTM fall under the measurement and the process categories. It covers a majority of the two standards because of the variety of strategies used in the lessons. Below is all of the strategies used that were pulled from the NCTM website (NCTM, 2014).

Measurements

Grades 6–8 Expectations: In grades 6–8 all students should– understand both metric and customary systems of measurement; understand relationships among units and convert from one unit to another within the same system; understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

Process Standards

Problem Solving

Instructional programs from prekindergarten through grade 12 should enable all students to—

Build new mathematical knowledge through problem solving

Solve problems that arise in mathematics and in other contexts Apply and adapt a variety of appropriate strategies to solve problems Monitor and reflect on the process of mathematical problem solving

Reasoning and Proof

Instructional programs from prekindergarten through grade 12 should enable all students to—

Recognize reasoning and proof as fundamental aspects of mathematics Make and investigate mathematical conjectures

Develop and evaluate mathematical arguments and proofs

Select and use various types of reasoning and methods of proof

Communication

Instructional programs from prekindergarten through grade 12 should enable all students to—

Organize and consolidate their mathematical thinking through communication

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Analyze and evaluate the mathematical thinking and strategies of others; Use the language of mathematics to express mathematical ideas precisely.

Connections

Instructional programs from prekindergarten through grade 12 should enable all students to—

Recognize and use connections among mathematical ideas

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole Recognize and apply mathematics in contexts outside of mathematics

Representation

Instructional programs from prekindergarten through grade 12 should enable all students to—

Create and use representations to organize, record, and communicate mathematical ideas Select, apply, and translate among mathematical representations to solve problems

Use representations to model and interpret physical, social, and mathematical phenomena

Standards in mathematics are important because it allows maximum learning. Being able to produce a lesson and then compare the standards allows educators to revamp or add to their lesson plans and implement more then they initially intended. A lesson can be drawn up and leave out simple elements that if added increase learning and meaning. The enhancement of the lesson will lead to a better success rate for the future lessons this one was meant to be a baseline for. A deeper understanding and comprehension of the area of a triangle makes the transition to composite shapes much easier to address. The methods used for this lesson were ideal. Strategies used were group work and a think-pair-share approach to explaining their conclusion of how they came to their answers we very effective. Although the text does not say, whole brain teaching and modeling methods were used for the first half of the lesson. Demonstration effective learning is important in this particular class because the class includes students who fundamentally have problems with simple multiplication even though it is 6th grade. Because of this, she also has to differentiate her instruction. This was done by not only making appropriate group dynamics but also giving low students’ multiplication charts so that they may solve the work on their own. This was not counterintuitive at all because the purpose was to understand solving for area.

The school is low economic status, and technology is scarce. Technology was not used but could have been at basic levels. It could have been used to submit their work, to include their explanations. This would provide a means for accountability. It could have also been used for interactive websites intended for solving area. Technology was not used, but manipulatives were. Each problem consisted of its own cut out to measure. One of the changes I would have made to this lesson would be to allow students to measure something around the classroom. I noticed quite a few triangular shapes in her class to include an awesome Avengers kite. Assessments of the lesson included exit cards for that day and when the section of the lessons was concluded, multiple tests were taken. The teacher used all of these assessments to her advantage. She addressed necessary review time because of them, making the overall lesson an absolute success. Other than allowing students free reign at the end I would not change anything about this lesson. This will be yet another lesson I steal and use for my own classroom.

Resources

Burton, M. (2010). Five Strategies for Creating Meaningful Mathematics Experiences in the Primary Years. YC: Young Children, 65(6), 92-96.

Ferrini-Mundy, J. (2000). Principles and standards for school mathematics: A guide for mathematician. Notices of AMS, 47(8), 868-876. Retrieved from http://www.ams.org/notices/200008/comm-ferrini.pdf GreatSchools Staff (n.d.). State standardized test scores: Issues to consider. Retrieved from http://www.greatschools.org/students/academic-skills/626-state-standardized-test-scores- issues-to-consider.gs

Lee Yuen, L. (2010). The Use of Constructivist Teaching Practices by Four New Secondary School Science Teachers: A Comparison of New Teachers and Experienced Constructivist Teachers. Science Educator, 19(2), 10-21.

McDonell, J. (2008). Constructivist versus traditional math programs: How do we best meet the educational needs of our students?. (Master’s thesis, Carroll University). Retrieved from http://content-dm.carrollu.edu/cdm/singleitem/collection/edthesis/id/2/rec/14 NCTM. (2014). thstandards and expectations. Retrieved from

http://www.nctm.org/standards/content.aspx?id=4294967312

Winstone, N., & Millward, L. (2012). The Value of Peers and Support from Scaffolding: Applying Constructivist Principles to the Teaching of Psychology. Psychology Teaching Review, 18(2), 59-67.

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