Chapter 1: Teaching Mathematics in the Era of the NCTM Standards
NCTM StandardsDiagram 1 below summarized the various principles and standards that govern the teaching of mathematics today.
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| Diagram 1 |
Excellence in mathematics education involves more than content objectives. The National Council of Teachers of Mathematics' (NCTM) release and updates of Principles and Standards for School Mathematics is being used as the guide to reform mathematics education throughout the world. The six principles are: Equity, Curriculum, Teaching, Learning, Assessment and Technology. These standards should be intertwined with the mathematics programme.
5 Content Standards
There are five content standards. These are Number and Operations, Algebra, Geometry, Measurement as well as Data Analysis and Proficiency. Four grade bands have also been created and each has its set goals and expectations. The four grade-bands are: pre-K-2, 3-5, 6-8 and 9-12. E.g. For the pre-K-2 grade band, under the content of number and operations, they must count with understanding and recognise "how many" as compared to grade-band 3-5 where they must know place values.
5 Process Standards
Following the five content standards, there are five process standards which are integral components of mathematical learning and teaching. These are Problem Solving, Reasoning and Proof, Communication, Connection and Representation. To teach in accordance with these process standards will enable us to meet the learning objectives set in the grade-bands of the content standards. Problem Solving: solve the problem rather than using a method to get the answer. Reasoning and Proof: use logical thinking to decide if the answer makes sense. Communication: be able to describe and explain the idea. Connection: know that part and part makes a whole, and that fractions are also part of a whole, like in percentages. Representation: the use of charts, symbols, graphs, manipulatives and diagrams to express mathematical ideas.
7 Teaching Standards
These teaching standards enables the teachers to resource and evolve the classroom environment, teaching tools and strategies to support the principles and standards.
The teaching standards are Knowledge of Mathematics and General Pedagogy, Knowledge of Student Mathematical Learning, Worthwhile Mathematical Tasks, Learning Environment, Discourse, Reflection on Student Learning and Reflection on Teaching Practice.
Becoming a Teacher of Mathematics
As summarized in the diagram, a teacher of mathematics must enjoy challenges. One must have profound, felxible and adaptive knowledge of mathematics content. The teacher must model persistence to work out the problem with investigations. The positive attitude of the teacher towards mathematics will influence the student to enjoy mathematics. Always be ready to change and embrace new invented strategies anytime. Take on a reflective disposition. Contine to learn content and methodology of teaching mathematics.
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| Diagram 2 |
The elaborations presented in the book brings me to view mathematics in a different light. To me, mathematics has always been finding anwers to a mathematical problem and that many of the disciplines can be used in real live, especially in counting. According to J.A.Van De Walle, K.S. Karp & J.M.B.Willaims (2010), "Doing mathematics means generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense..... mathematics is a science of concepts and processes that have a pattern of regularity and logical order....Finding and exploring this regularity or order, and then making sense of it, is what doing mathematics is all about."
Asking simple questions like "What is the pattern ? to 6+7 is the same as 5+8 and 4+9" easily throws up concept that -1 +1 to each set of numbers.
I agree to the statement that "Studying relationships .... helps students understand what they are doing, therefore increasing their accuracy and retention."
The Language of Doing Mathematics
I tried using some of the verbs listed and could immediately see the children engaging themselves thinking rather than working out the answers. I will definitely work towards using these verbs in my mathematics programme.
explore justify construct develop
investigate represent verify describe
conjecture formulate explain use
solve discover predict
The Setting for Doing Mathematics
Create the spirit of inquiry, trust and expectations. The productive classroom culture for mathematics should include:
- Ideas expressed have the potential to contribute to learning and warrant respect and response.
- Respect the need to understand their own method and recognize that there are other methods to the solutions.
- Mistakes give opportunity to examine errors in reasoning, thereby raising level of analysis.
Introducing calculators must be done together with how mathematics operations work, so that students do not use it simply for quick answers.
Experiences to maximize learning opportunities
- Constructivist Theory: Knowledge cannot be poured into a learner. The student has to construct knowledge by reflective thought and active thinking in order to make connections and contruct knowledge.
- Sociocultural Theory: Learning is dependant on the student's social interactions with other learners within and beyond the classroom.
# Provide Opportunities to Talk about Mathematics: Encourage students to talk about their strategies. Hear them reflect and interact with peers.
# Build Opportunities for Reflective Thought: Use problem-solving approach and have the students explain and justify their solutions.
# Encourage Multiple Approaches: Take simple counting off a table of dots arrange in 5s. Once the solution is revealed, have children explain their strategy. Some will be counting in 1s, others in 2s while yet others by 5s. Student's reflective thinking on these strategies will already construct new knowledge.
# Treat Errors as Opportunites for Learning: Asking the student to regroup and use another strategy will help the student realize the misapplication of prior knowledge and thus accommodate a new learning.
# Scaffold New Content: Challenging the student to the next level of learning by providing structure, content and assistance.
# Honor Diversity: Value the uniqueness of each individual. Establish a classroom culture where ideas are valued and respected.
Understanding Mathematics Defined
Understanding is a measure of the quality and quantity of connections that a new idea has with the existing ideas. The greater the number of connections to a network of ideas, the better the understanding. Thus the two ends of the continuum of understanding: Relational Understanding (high end) and Instrumental Understanding (low end).
Mathematics Proficiency
- Conceptual understanding and procedural understanding is the basic knowledge about relationships or foundational ideas of a topic.
- Five Strands of Mathematical Proficiency:
operations and relations
2) procedural fluency - flexible, accurate, efficient and
appropriate execution of procedures
3) strategic competence - ability to formulate, represent, and
solve mathematics problems
4) adaptive reasoning - able to think logically, reflect, explain and justify
5) productive disposition - have the habit to see mathematics
as sensible, useful, and worthwhile, coupled with a belief
in diligence and one's own efficacy.
Add all these up and you get the mathematical intertwined strands of
proficiency as illustrated in Diagram 3 below.
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| Five Strands of Mathematical Proficiency Diagram 3 |
Implications for Teaching Mathematics
Instead of asking "Does she know it?" I will replace with "How does she understand it? What ideas does she connect it?" This way I will be able to see the different ideas and understanding of each student. Example, for additions, instead of counting counters, or using traditional algorithm of lining up and adding the ones and then the tens, with the understanding of place-value concept, I will also show them the invented approach. (add 10s to the root number and count foward the ones).
Effective learning of new concepts and procedures, less to remember, increased retention and recall, enhanced problem-solving abilities and improved attitudes and beliefs are the reasons why relational understanding is so important and beneficial.
Models and Manipulatives
Educators are promoting the use of concrete and hands-on methods to teach mathematics. This reading impacts upon me the effective use of manipulatives to teach concept by letting the student think, reflect, show and explain, to solve the problem, and not asking them to mimic the teacher's procedure to reach an answer. Manipulatives could include countable objects, rods, strips and squares, sticks, dot grid, integers as well as calculators.
Conclusion - My CallingPlan my programme to focus on creating opportunities for students to develop their own network of blue dots. Include a comprehensive treatment of mathematics. Design my instructions to elicit prior knowledge, reflect the social and cultural background, challenge them to think critically and creatively. Continue to use manipulative models and introduce use of technology.
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