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Bloom’s Taxonomy in Mathematics Teaching
Bloom’s and Anderson’s Taxonomy
This paper discusses a brief history and background of the original Bloom’s taxonomy established in 1956 and the revised Bloom’s taxonomy by Anderson et al. in 2002. It attempts to explore how the taxonomy of learning in the cognitive domain is applied in a specific teaching field such as Mathematics.
In 1956, Benjamin Bloom headed a committee of educators to establish a systematic hierarchy in classifying educational objectives. They published their findings in a book, Taxonomy of Educational Objectives: The Classification of Educational Goals (Bloom et al, 1956). It is considered a classic educational research emphasizing focus on the higher-order thinking skills among students. The system has been widely known as Bloom’s taxonomy. Bloom’s taxonomy is a guide for educators in developing educational objectives in the cognitive domain. Cognitive domain is the educational aspect focusing on mental/ intellectual skills. The taxonomy helps align goals, the teaching strategy, and assessment methods to be used. It can be best remembered using the acronym KCAASE which stands for knowledge, comprehension, application, analysis, synthesis and evaluation. The competencies are arranged from simple to complex. That is why educators refer to it sometimes as a hierarchy. The Bloom processes enable educators to use appropriate methods of instruction designed for a specific goal.
The first three: knowledge, comprehension and application are referred to as lower order thinking skills (LOTS). Bloom’s taxonomy starts with knowledge. It is simple recall of information and facts such as identification, term definition, enumeration, etc. Key action verbs used in knowledge process are: identify, define, enumerate, select, describe, etc. Sample activities for knowledge area include labeling parts in Biology, and identifying planets of the solar system in Astronomy. Another thinking skill is comprehension which refers to understanding meaning, interpretation, or translation. Comprehension refers to stating an idea or statement into one’s own words and thinking. Connection of ideas is emphasized. Action verbs used are but not limited to infer, interpret, summarize, explain and give an example. Sample comprehension activities include summary writing, phrase translation in a foreign language class, and even translating verb phrases to mathematical expressions in Algebra. Third, application is the use of a learned concept in practical situations in the real world. Sample action verbs used in this category are: manipulate, solve, compute, demonstrate, and sketch. Recommended application activities include demonstration such as in Nursing practice, laboratory experiment in Physics, and using analytical instruments in Chemistry. The lower order thinking skills are essential particularly in targeting learning of content standards.
The last three categories in Bloom’s taxonomy: analysis, synthesis and evaluation are referred to as higher order thinking skills (HOTS). They require a higher degree of involvement Analysis is separating one big idea into component concepts so that relationships between them are known. Focus is given on the overall structure of ideas. It also involves distinguishing facts from inferences. Some action verbs used are outline, compare, structure, and organize. Sample learning activities for analysis are critiquing in Literature, troubleshooting malfunctioning equipment in mechanical technology, dissecting activities in Biology. Another category or skill under this is a synthesis which involves putting component parts together and deriving meaning from it. It emphasizes the creation of new ideas. Action verbs used include in learning objectives include: reconstruct, write, create, compose and design. Activities on synthesis skills include: equipment design in Engineering, songwriting and composing in Music, and writing a short story in Literature. Last is evaluating. Evaluating is making judgment on ideas or material. This requires deeper understanding. Some of the key verbs used are justify, evaluate, defend and support. Examples of activities for this category are choosing a solution from recommended alternatives in Management, justifying an expense budget in Finance, hiring personnel for specific jobs in Human Resources. In sum, higher order thinking skills target performance standards and how person integrates different learning to an appropriate use.
In 2001, Anderson et al. revised the original concept of Bloom’s taxonomy. The shift was from nouns (plain general concepts) to verbs (action verb). The new taxonomy revolves on similar categories that are termed differently to highlight the action part of the learning process. They are remembering (instead of knowledge), understanding (instead of comprehension), applying (instead of application), analyzing (instead of analysis), evaluating (instead of evaluation) and creating (instead of synthesis) (Anderson, et al., 2001). Notice that the last two skills in the revised Bloom’s taxonomy are interchanged from the original. Anderson and his associates believe that creating is higher than evaluating. Both taxonomies are arranged from simple to complex (See Figure 1 for the schematic diagram of the two taxonomies). The HOTS are placed at the upper section of the pyramid (towards the apex) while the LOTS are places at the bottom section (at the pyramid base). Even with the differences, both are considered important in discussing education topics such as principles of teaching, assessment of learning, and curriculum development.
Application in education for Bloom’s taxonomy is varied nowadays. One popular use is in making a table of specification for a summative test. It is also used in lesson planning on creating objectives. Curriculum builders also refer to it in constructing competencies for curriculum guides in different areas. Thus, it has become an indispensable tool for modern teachers in different parts of the world.
Figure 1: Comparison of Bloom’s Taxonomy and Anderson’s Taxonomy
Blooms Taxonomy in Mathematics Teaching
In the development of a lesson plan, good instructional objectives contain the following elements: (a) audience, (b) behavior, (c) conditions and (d) criteria. Bloom’s taxonomy can be used in creating good instructional objectives in Mathematics. To understand the different levels more, the researcher created instructional objectives used in Mathematics teaching (particularly Trigonometry) in Table 1. The objectives are then classified under the different categories in the cognitive domain and discussed in detail.
First is knowledge (or remembering). Through a mnemonic strategy, students can define the trigonometric functions of a specific angle. For instance, by remembering SOH, a student can define the sine function as opposite side over hypotenuse. The objective describes simple recall of terminology. The next level, comprehension relates prior knowledge to understand what is asked. For instance, given a graph of a unit circle and with prior knowledge of angle measurement (zero to 180 degrees), the student can locate the special angles that are just multiples of 30°, 45° and 60°. The student interprets the graph based on what he understood in the past lessons. Another level is application which emphasizes using information in a new way. For instance, a student is given a right triangle with two known parts (sides or angles). Using either Pythagorean Theorem or trigonometric functions, the student can determine the other three unknown parts. The student applies the given conditions by substituting them to the formulas.
The next level is analysis. This requires more critical thinking than the previous three levels. Analysis is basically dividing the whole into parts. In the example in Table 1, a word problem is given concerning a right triangle. The student needs to divide the whole problem into comprehensible sentences. By dividing the problem into manageable parts, he can convert the sentences one by one into mathematical expressions. Once he has the mathematical expressions, the student gets a clearer picture of what the problem really is. Then there is synthesis. This level puts parts together into a new form. This just means that a new concept or idea is formed or created. Given a sine function, a student solves for the amplitude and period. With these two values, he can create the graph easier. The last level is evaluation which focuses on judging on presented data or information. Given a picture of an oblique triangle given three parts, the student evaluates whether she will use Sine or Cosine Law. By understanding if the given data is side-angle-side (SAS), side-side-side (SSS), angle-side-angle (ASA), etc., the student can use the appropriate formula to use. This results in lesser trial and error. The last three levels indicate that the student has to think first before attacking the math problems.
The discussion is quite definitive in their respective classes; however mathematics teachers have quite some trouble in this aspect. Thomson (2008) studied 32 highs school mathematics teachers in southeast United States. The teachers were tasked to create an examination for Algebra I and classified them under the different thinking skills in Bloom’s taxonomy. The study suggests that most math teachers have difficulty interpreting the thinking skills in Bloom’s taxonomy. They also have the tendency to make test items that cater to the higher order thinking skills. This is quite advantageous for students so that their critical thinking skills will be developed. However, an appropriate mix of LOTS and HOTS are needed to make a good assessment of learning.
There have been developments in mathematics teaching credited to the approach by Bloom and Anderson. One study is that of Darlington (2013) on the application of Bloom’s taxonomy in advanced mathematics questions. She patterned a hierarchy in mathematics learning assessment as composed of three levels: (a) routine procedures, (b) using existing mathematical knowledge in new ways, and (c) application of conceptual knowledge to construct mathematical arguments. In the first level, knowledge, comprehension and application skills are required. Routine procedures involve factual knowledge, recognizing examples, and routinary use of procedures. In the second level, focus is given on new approaches of solving problems. Existing knowledge is required since this level involves information transfer and new situations. This level is somehow related to Bloom’s analysis. Lastly, application of conceptual knowledge to construct mathematical arguments relate to synthesis and evaluation. The last level usually involves justifying, interpreting, implications, conjectures, and comparison. Examples of this level include proving a theorem, recognizing limitation, assessing mathematical models, etc. Furthermore, Darlington explained that the last level is the area where undergraduate mathematicians lack. It is in the last level that one can develop to become a practicing mathematician or a real-world problem solver (Darlington, 2013).
Conclusion
Bloom’s taxonomy underwent developments particularly in 2001 when Anderson introduced his revised taxonomy. However, the two models are particularly related and have been in use by the teachers of the modern use. The taxonomies have applications in teaching principles, learning assessment and curriculum development. It can be integrated in the teaching methodologies for the sciences, mathematics, and business concepts.
The paper focused on Mathematics teaching and provided brief examples of how a math teacher can use Bloom’s taxonomy. Modern teachers are focused on higher order thinking skills and it is on this cognitive level that students become real-world mathematicians. Real world mathematicians are important to society since they are good analysts and problem solvers.
References
Anderson, L., Krathwohl, R., Airasian, P., Cruikshank, K., Mayer, R., Pintrich, P., Raths, J., & Wittrock, M. (Eds.) (2001). Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom’s Taxonomy. New York: Longman.
Darlington, E. (2013). The Use of Bloom’s Taxonomy in Advanced Mathematics Questions. Proceedings of the British Society for Research into Learning Mathematics 33 (1): 7-12. Retrieved from http://www.bsrlm.org.uk/IPs/ip33-1/BSRLM-IP-33-1-02.pdf (December 2, 2014).
Bloom, B., Engelhart, M., Furst, E., Hill, W., & Krathwohl, D. (Eds.) (1956). Taxonomy of Educational Objectives: The Classification of Educational Goals, Handbook 1: Cogitive Domain. New York: David McKay.
Thompson, T. (2008). Mathematics Teachers’ Interpretation of Higher-Order Thinking in Bloom’s Taxonomy. International Electronic Journal of Mathematics Education Vol. 3 No. 2. Retrieved from http://www.iejme.com/022008/d2.pdf (December 2, 2014)
