Curious About Problem Solving + Supportive Environment – Teacher Directed = ?
Chapter 1: Teaching Mathematics in the Era of the NCTM standards
According to Van De Walle, Karp and Bay-Williams (2010) mathematics education has undergone steady changes for the last two decades in both the content of school mathematics and teaching methodology due to various sources including knowledge of research. There are two important factors behind these changes; the professional leadership of the National Council of Teacher of Mathematics (NCTM) and the political pressure due to U.S. students not performing as well compared to international studies (Van De Walle et al., 2010). I agree with Van De Walle et al. (2010) that although high expectations for students are important, testing alone is not appropriate to improve students’ learning. Teachers need to support students to reach their fullest potential in math by focusing on mathematical thinking and reasoning (Van De Walle et al., 2010). Also, I believe teachers need to look at some approaches to math assessment that would give a more comprehensive understanding of student accomplishment and growth in math concepts and processes. In addition, teachers need to know the research-based best practices in teaching of mathematics for students and how to use technology in order to support and enhance students' learning in mathematics.
The six principles are as follows (Van De Walle et. al., 2010):
- Equity
- Curriculum
- Teaching
- Learning
- Assessment
- Technology
According to Van De Walle et. al. (2010) the principles clarify that excellence in mathematics education is much more than listing content objectives.
The Five Content Standards are as follows (Van De Walle et. al., 2010):
- Number and Operations
- Algebra
- Geometry
- Measurement
- Data Analysis and Probability.
As stated by Van De Walle et. al. (2010) instead of using different sets of mathematical topics for each grade band, the authors agreed on a common set of five content standards throughout the grades and each content standard has a small set of goals applicable to all grade bands.
The Five Process Standards are as follows (Van De Walle et. al., 2010):
- Problem Solving
- Reasoning and Proof
- Communication
- Connection
- Representation
These five processes direct the methods of doing all mathematics and it is an integral component of all mathematics and teaching (Van De Walle et. al., 2010).
Among the ideas in Mathematics Teaching Today are six shifts in the classroom environment to allow students to develop mathematical understanding (Van De Walle et. al., 2010). The six shifts are as follows (Van De Walle et. al., 2010):
- Communities that offer an equal opportunity to all students.
- A balanced focus on conceptual understanding as well as on procedural fluency.
- Active student engagement in problem solving, reasoning, communication, making connections, and using multiple representations.
- Well-equipped learning centers in which technology is used to enhance understanding.
- Mathematics authority that lies within the power of sound reasoning and mathematical integrity (NCTM as cited in Van De Walle et. al., 2010).
Since the late 1960s the United States has collected data on how students are doing in mathematics through the National Assessment of Educational Progress (NAEP) (Van De Walle et. al., 2010). The data provide important information for policy makers and educators to measure the overall improvement of U.S. students over time and it examines both national and state level trends (Van De Walle et. al., 2010). I think that it is important to have some form of assessment over time in order to know what needs to be changed in a system and what works best for children. However, my concern is that a standardized test alone may not show an accurate picture of the child’s mathematical knowledge and understanding.
The results of an International Mathematics and Science study where 41 nations participated in 1995 and 1996 showed that 11 countries have significantly higher scores than the United States (Singapore, Hong Kong, Japan, Chinese Taipei, Flemish Belgium, Netherlands, Latvia, Lithuania, Russian Federation, England, and Hungary) (Van De Walle et. al., 2010). One may wonder what these students do differently in order to have higher scores or is it that they are just better test takers. I agree with Van De Walle et. al. (2010) that it is important to focus on conceptual understanding and making connections to other mathematics strands rather than traditional emphasis on mathematical procedures. “We can predict that there will be work that requires interpreting complex data, designing algorithms to make predictions, and using the ability to approach new problems in a variety of ways.” (Van De Walle et. al., 2010). Thus, it is my belief that teachers need to ensure that they support, guide and inspire students and prepare them for the demands of a changing technological society where new careers and knowledge are constantly being created.
What does it means to do mathematics? Doing mathematics means not just solving a problem but discovering the many ways there are to arrive at the solutions. I strongly believe that teachers need to challenge students forward along a path to solve a problem, and then go backward to find alternate methods to the solution. Let math become an adventure! As stated by Van De Walle et. al. (2010) mathematics is a science of concepts and processes that have a pattern of regularity and logical order.
Also, it is important that students have a conducive and productive classroom environment where students are respecting each other’s ideas. Students should feel safe and supported while learning through trial and error to understand mathematical concepts.
According to Van De Walle et. al. (2010) in the real world of problem solving there are no teachers with answers and no answer books – doing mathematics is about using justification as a means of determining if an answer is correct. Hence, a supportive environment is crucial where students are allowed to try a variety of ways without fear of making mistakes.
What does it mean to learn mathematics? As stated by Van De Walle et. al. (2010) we find the answers in current theory and research on how people learn, for example, constructivists theory – learners are not blank slates but creators of their own learning. People construct their own knowledge based on their prior knowledge (Van De Walle et. al., 2010). This made me reflect on the importance of differentiated learning in the classroom and to ensure each student is appropriately challenged. What would be the next level for this student and how can I as a teacher make it interesting. Equally important for learning is the social interactions in the classroom, students can reach the next level understanding with support (zone of proximal development, ZPD) and the culture within and beyond the classroom (Van De Walle et. al., 2010). Students should be given opportunities to talk, reflect and be encouraged to think of multiple approaches when doing mathematics (Van De Walle et. al., 2010). I agree that we should honor diversity and each student’s ideas should be valued and included in classrooms discussion of the mathematics (Van De Walle et. al., 2010).
What does it mean to understand mathematics? Each child is unique and brings to the classroom different prior knowledge and understanding of math. According to Van De Walle et. al., (2010) we can think about understanding such as it exists along a continuum from a relational understanding – knowing what to do and why – to an instrumental understanding – doing without understanding. Also, it is important to know the difference between conceptual and procedural understanding. Conceptual understanding is knowledge about the relationships of a topic and procedural understanding is knowledge of the rules and procedures used in carrying out mathematical processes (Van De Walle et. al., 2010). Students need to have the conceptual understanding if math is going to be fun (Van De Walle et. al., 2010). I agree! However, while conceptual and procedural understanding is essential to being mathematically proficient, they are not sufficient, we have to take into consideration the five strands which are interrelated and interwoven with each other (NRC as cited in Van De Walle et. al., 2010):
- Conceptual understanding
- Procedural fluency
- Strategic competence
- Adaptive reasoning
- Productive disposition
Teachers have to be aware of how the student understands it and what ideas he or she connects it with (Van De Walle et. al., 2010). It is important to observe and assess the student to figure out what strategies he or she is using, which will then lead us to see how much the student really comprehend.
The teacher needs to focus on supporting the students in their understanding so that they build more connections and are more likely to connect new ideas to the existing conceptual webs they have (Van De Walle et. al., 2010). Thus, students have less to remember, increased retention and recall, enhanced problem-solving abilities and improved disposition towards mathematics (Van De Walle et. al., 2010).
If there are more ways for children to think about an idea the better chance they will develop a relational understanding (Van De Walle et. al., 2010). To help students understand, teachers may use models and/or math manipulatives. I agree that teachers need to constantly think about a variety of creative strategies to reinforce student's understanding. It is when the student has grasped the concept that defines success!
Curious About Problem Solving + Supportive Environment – Teacher Directed = Students Who Love Math!
Reference
Van De Walle, J., Karp, K. & Bay-Williams, J. (2010). Elementary & middle school mathematics. Teaching developmentally (7th ed.). Boston, MA: Allyn and Bacon
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