Aims, Values and Objectives of Teaching Mathematics (Bloom and Revised Bloom)

What to remember

• Aims are broad and long-term; objectives are narrow, specific and measurable. An aim guides the whole course. An objective guides one lesson and is written in words that can be observed.
• Bloom's 1956 cognitive taxonomy has six levels: Knowledge, Comprehension, Application, Analysis, Synthesis, Evaluation. The 2001 revision uses verbs: Remember, Understand, Apply, Analyse, Evaluate, Create. Evaluate and Create swap places in the revision.
• A good objective is stated in observable behaviour (define, solve, construct, compare), never in vague words such as "know" or "understand".

Aims, goals and objectives

An aim is a general, far-off purpose of teaching a subject. It says why we teach mathematics. A goal is a middle-level target, such as a stage or a class level. An objective is a precise statement of what the learner will be able to do after a lesson. A short route to remember: aims (years) > goals (stage) > objectives (period).

National documents such as NCF 2005 state the main aim of school mathematics as the mathematisation of the child's thinking. This means the child learns to think logically, abstract, make patterns and solve problems, not just to compute. The AP school curriculum names five academic standards: problem solving, reasoning and proof, communication, connections, and representation and visualisation.

Values of teaching mathematics

Values are the benefits that society and the child get from mathematics. They show why the subject is compulsory in school.

The disciplinary value is also called mental training. The utilitarian value is the most easily seen by parents and pupils.

Bloom's taxonomy of educational objectives

Benjamin Bloom led a group of examiners that produced the taxonomy of educational objectives in 1956. It sorts objectives into three domains.

• 1. Cognitive domain (head: knowing and thinking). Bloom's group published this in 1956.
• 2. Affective domain (heart: feelings, attitudes, values). Krathwohl and others published this in 1964.
• 3. Psychomotor domain (hand: skills and actions). Several authors worked on it; Simpson's seven-level version (perception, set, guided response, mechanism, complex overt response, adaptation, origination) is common in teacher education.

Cognitive levels (original, 1956):

Affective levels (Krathwohl): Receiving (attends), Responding (takes part), Valuing (accepts worth), Organisation (builds a value system), Characterisation by a value (the value shapes behaviour).

Revised Bloom taxonomy (2001)

Lorin Anderson and David Krathwohl, with others, revised the taxonomy in 2001. Three main changes:

• Nouns became verbs (Knowledge became Remember, Comprehension became Understand, and so on).
• Synthesis became Create, and it moved to the top. Evaluate now sits below it.
• A second axis, the knowledge dimension, was added.

The knowledge dimension has four types: factual (terms, symbols), conceptual (ideas, classifications, principles), procedural (methods, algorithms, when to use a method) and metacognitive (knowing about one's own thinking). The first three levels (Remember, Understand, Apply) are often called lower-order thinking; Analyse, Evaluate and Create are higher-order thinking.

Action verbs and writing specific objectives

Instructional objectives are best written as behavioural terms. Robert Mager's method has three parts (performance, condition, criterion); the widely taught ABCD format adds the Audience: Audience (who), Behaviour (what they do), Condition (under what situation) and Degree (how well). Example: "Given a table of marks, the Class 8 student will calculate the mean correctly in four out of five cases."

Words like "know", "understand" and "appreciate" are not good for an objective because they cannot be seen or tested.

The RCEM (specification) scheme is also taught in some Indian teacher-education programmes. It groups objectives under Knowledge, Understanding, Application and Creativity (Creativity merging Bloom's analysis, synthesis and evaluation), and each has "specifications" (small behaviours). For example, knowledge covers recall and recognition; understanding covers discriminating, explaining, giving examples and generalising; creativity covers analysing, synthesising and evaluating. (Skills such as computing, drawing and reading graphs belong to the older NCERT/Dave-style scheme.)

The classroom angle

• A teacher begins with the objective, then selects content, method and test items that match it. This is called backward planning.
• One lesson should aim at a few objectives from different levels, not only at recall.
• Test items should be spread over levels. A paper made only of recall questions does not test reasoning.
• Mathematics objectives are not all cognitive. Neat work, patience, cooperation and interest (affective) and use of compass, protractor and scale (psychomotor) must also be planned.
• A teacher should ask higher-order questions during teaching: "Why does this method work?" "Can you make a similar problem?"

Worked illustration: one topic, objectives at several levels

Take the topic "area of a rectangle" in Class 6. A teacher can write objectives at several levels:

• Remember: states the formula for the area of a rectangle.
• Understand: explains why area is measured in square units.
• Apply: finds the area of a room 5 m long and 4 m wide (5 x 4 = 20 square metres).
• Analyse: finds the error when a pupil adds length and breadth instead of multiplying.
• Evaluate: decides which of two plots of the same perimeter has more area, and gives a reason.
• Create: frames a word problem that needs the area formula.

Affective objectives (shows interest in measuring the classroom) and psychomotor objectives (measures with a tape correctly) are planned in the same lesson.

Why objectives are needed

Objectives give direction to teaching, help in choosing content and method, make tests valid because items match outcomes, tell pupils what is expected, and allow the teacher to judge whether teaching worked. Without objectives, teaching drifts toward whatever the textbook happens to contain.

Exam traps

• Aim vs objective: aims are general; objectives are specific and measurable.
• Disciplinary vs utilitarian value: mental training versus daily-life use.
• Original vs revised top level: the original top level is Evaluation; the revised top level is Create.
• Synthesis vs Create: the same idea; the revised name replaced it.
• Bloom (cognitive 1956) vs Krathwohl (affective 1964): do not swap the authors.
• Affective vs psychomotor: feelings and attitudes versus physical skills and handling tools.
• Comprehension vs Application: explaining an idea versus using it in a new problem.
• "Understand" in an objective: it is a vague verb; the corrected form would be "explain" or "classify".

One-liners

• 1. Mathematisation of the child's thinking is the central aim in NCF 2005.
• 2. Bloom's cognitive taxonomy appeared in 1956.
• 3. The revised taxonomy was published in 2001 by Anderson and Krathwohl and others.
• 4. The revised cognitive levels are Remember, Understand, Apply, Analyse, Evaluate, Create.
• 5. The knowledge dimension has factual, conceptual, procedural and metacognitive types.
• 6. The affective domain was described by Krathwohl and others in 1964.
• 7. The five affective levels run from Receiving to Characterisation.
• 8. Mager's ABCD stands for Audience, Behaviour, Condition, Degree.
• 9. Higher-order thinking skills are Analyse, Evaluate and Create.
• 10. Mental training is the disciplinary value of mathematics.
• 11. Daily-life use of mathematics is its utilitarian (practical) value.
• 12. The RCEM scheme uses Knowledge, Understanding, Application and Creativity.

Practice questions

1. Which statement best describes an aim of teaching mathematics?

   1. A broad, long-term purpose of the subject
   2. A test item for one topic
   3. A precise outcome of one lesson
   4. A list of teaching aids for a unit
   Answer

   A. A broad, long-term purpose of the subject

   Aims are general and long-term; objectives are specific.

2. Mathematisation of the child's thinking is the central aim stated in

   1. The Hunter Commission report
   2. The Kothari Commission report (1964-66)
   3. NCF 2005
   4. The Wood's Despatch of 1854
   Answer

   C. NCF 2005

   NCF 2005 names mathematisation of thinking as the main aim of school mathematics.

3. Mental training through reasoning and accuracy is the

   1. Utilitarian value
   2. Disciplinary value
   3. Recreational value
   4. Cultural value
   Answer

   B. Disciplinary value

   The disciplinary value trains the mind.

4. Using mathematics to handle shopping bills and measurement is its

   1. Moral value
   2. Utilitarian value
   3. Aesthetic value
   4. Cultural value
   Answer

   B. Utilitarian value

   Daily-life use is the utilitarian (practical) value.

5. The beauty of symmetry and tessellation illustrates which value of mathematics?

   1. Aesthetic
   2. Vocational
   3. Disciplinary
   4. Social
   Answer

   A. Aesthetic

   Pattern and symmetry give the aesthetic value.

6. The cognitive taxonomy of educational objectives was first published in

   1. 2001
   2. 1964
   3. 1938
   4. 1956
   Answer

   D. 1956

   Bloom and colleagues published it in 1956.

7. Which is the top level in the original Bloom cognitive taxonomy?

   1. Application
   2. Analysis
   3. Evaluation
   4. Synthesis
   Answer

   C. Evaluation

   In 1956 the order ended with Evaluation; Synthesis was fifth.

8. In the revised taxonomy of 2001, the top level is

   1. Analyse
   2. Create
   3. Evaluate
   4. Apply
   Answer

   B. Create

   Synthesis became Create and moved to the top.

9. The affective domain taxonomy was developed by

   1. Simpson
   2. Mager
   3. Krathwohl and colleagues
   4. Anderson
   Answer

   C. Krathwohl and colleagues

   Krathwohl and others described it in 1964.

10. Which is NOT a level of the revised cognitive taxonomy?

    1. Comprehension
    2. Remember
    3. Analyse
    4. Understand
    Answer

    A. Comprehension

    Comprehension was renamed Understand in the revision.

11. 'Define a parallelogram' is an objective at the level of

    1. Analyse
    2. Create
    3. Understand
    4. Remember
    Answer

    D. Remember

    Defining is recall of a fact.

12. A student frames a new word problem for the equation 3x + 5 = 20. This is at the level of

    1. Understand
    2. Remember
    3. Apply
    4. Create
    Answer

    D. Create

    Producing something new is the Create level.

13. Which of these is a properly written specific objective?

    1. Knows linear equations
    2. Solves a pair of linear equations by substitution
    3. Understands the idea of variables
    4. Appreciates algebra
    Answer

    B. Solves a pair of linear equations by substitution

    Only the first uses an observable verb.

14. Mager's ABCD in objective writing stands for

    1. Aim, Behaviour, Content, Domain
    2. Analysis, Basis, Concept, Drill
    3. Audience, Behaviour, Condition, Degree
    4. Attitude, Belief, Concept, Domain
    Answer

    C. Audience, Behaviour, Condition, Degree

    Mager's four parts are Audience, Behaviour, Condition and Degree.

15. The four types in the knowledge dimension of the revised taxonomy are factual, conceptual, procedural and

    1. Metacognitive
    2. Affective
    3. Experimental
    4. Psychomotor
    Answer

    A. Metacognitive

    Metacognitive knowledge is the fourth type.

16. Which pair of verbs belongs to the Analyse level?

    1. List, name
    2. Judge, criticise
    3. Design, devise
    4. Differentiate, find the error
    Answer

    D. Differentiate, find the error

    Differentiate and find the error are analysis tasks.

17. Seven levels from perception to origination describe the

    1. Cognitive domain
    2. Affective domain
    3. Psychomotor domain (Simpson)
    4. Knowledge dimension
    Answer

    C. Psychomotor domain (Simpson)

    Simpson's version of the psychomotor domain has seven levels.

18. Which level of the affective domain is highest?

    1. Responding
    2. Characterisation by a value
    3. Valuing
    4. Receiving
    Answer

    B. Characterisation by a value

    Characterisation is where a value shapes behaviour.

19. Using a protractor to draw an angle of 70 degrees mainly serves which domain?

    1. Affective
    2. Knowledge dimension
    3. Cognitive recall only
    4. Psychomotor
    Answer

    D. Psychomotor

    Handling instruments is a motor skill.

20. A Class 8 student shows more interest in solving puzzles after a term. This is a change in the

    1. Affective domain
    2. Psychomotor domain
    3. Cognitive recall
    4. Knowledge dimension
    Answer

    A. Affective domain

    Interest and attitude belong to the affective domain.

21. Which statement shows the difference between an aim and an objective?

    1. Aims are for one period; objectives for the whole course
    2. Aims are set by pupils; objectives by parents
    3. An aim is general; an objective is observable and specific
    4. An aim is measurable; an objective is not
    Answer

    C. An aim is general; an objective is observable and specific

    Aims are broad; objectives are specific and measurable.

22. Judging which of two methods of solving a problem is shorter and explaining why is at the level of

    1. Evaluate
    2. Remember
    3. Understand
    4. Apply
    Answer

    A. Evaluate

    Judging against a criterion is evaluation.

23. Explaining in own words why a square is a rhombus is at the level of

    1. Apply
    2. Understand
    3. Create
    4. Remember
    Answer

    B. Understand

    Explaining a relationship is understanding (comprehension).

24. The RCEM scheme of objectives lists Knowledge, Understanding, Application and

    1. Skill
    2. Synthesis
    3. Interest
    4. Attitude
    Answer

    A. Skill

    The specification scheme uses Knowledge, Understanding, Application and Skill.

25. Which value of mathematics is meant by the statement 'a wrong answer cannot be hidden'?

    1. Utilitarian value
    2. Moral value
    3. Recreational value
    4. Vocational value
    Answer

    B. Moral value

    Truthfulness and honesty make up its moral value.

26. Choosing content, method and tests only after fixing the objective is called

    1. Rote teaching
    2. Incidental teaching
    3. Drill method
    4. Backward planning
    Answer

    D. Backward planning

    Planning starts from the desired outcome.

27. Study the statements. 1. Aims are specific and limited to one lesson. 2. Objectives should be written in observable behaviour. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Statement 1 is false (aims are broad); statement 2 is true.

28. Study the statements. 1. Synthesis in the 1956 taxonomy became Create in 2001. 2. Create is placed below Evaluate in the revised taxonomy. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Create is the top level in the revised version, so statement 2 is false.

29. Study the statements. 1. The revised taxonomy added a knowledge dimension. 2. The revised taxonomy changed levels from nouns to verbs. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are real changes made in 2001.

30. Study the statements. 1. Remember, Understand and Apply are lower-order levels. 2. Analyse, Evaluate and Create are higher-order levels. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    The usual grouping is lower order (first three) and higher order (last three).

31. Study the statements. 1. The utilitarian value of mathematics concerns daily-life use. 2. The disciplinary value concerns training of the mind. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both statements are standard.

32. Study the statements. 1. Krathwohl's taxonomy covers the cognitive domain. 2. Bloom's 1956 work covers the affective domain. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    Bloom's 1956 work is cognitive; Krathwohl's 1964 work is affective. Both are swapped.

33. Study the statements. 1. 'Understand' is a precise verb for writing an objective. 2. 'Explain' is an observable verb for writing an objective. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    'Understand' is vague; 'explain' can be observed.

34. Study the statements. 1. Metacognitive knowledge means knowing about one's own thinking. 2. Procedural knowledge means knowing how to do something. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both describe the revised knowledge dimension correctly.

35. Study the statements. 1. Mager's objectives include a condition and a degree. 2. An objective such as 'The teacher will teach fractions' is learner-centred. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    It describes teacher action, not learner behaviour, so statement 2 is false.

36. Match: (a) Utilitarian (b) Disciplinary (c) Cultural (d) Aesthetic with (p) Mental training (q) Daily-life use (r) Symmetry and pattern (s) Heritage of mathematics.

    1. a-s, b-p, c-q, d-r
    2. a-p, b-q, c-s, d-r
    3. a-q, b-p, c-r, d-s
    4. a-q, b-p, c-s, d-r
    Answer

    D. a-q, b-p, c-s, d-r

    Utilitarian is daily use, disciplinary is mental training, cultural is heritage and aesthetic is pattern.

37. Match: (a) Remember (b) Understand (c) Apply (d) Analyse with (p) Find the error (q) State a formula (r) Explain a concept (s) Use a rule in a new case.

    1. a-q, b-r, c-s, d-p
    2. a-r, b-q, c-s, d-p
    3. a-s, b-r, c-q, d-p
    4. a-q, b-r, c-p, d-s
    Answer

    A. a-q, b-r, c-s, d-p

    Remember is stating, understand is explaining, apply is using, analyse is finding an error.

38. A teacher says: 'After the lesson the pupil will be able to construct a bisector of a given angle with compass.' Which feature does it MOST clearly show?

    1. Only an aim
    2. Condition and behaviour in observable form
    3. Only an attitude
    4. A vague hope
    Answer

    B. Condition and behaviour in observable form

    It names the condition (compass) and an observable behaviour.

39. A question that asks pupils to design an experiment to test a pattern belongs to

    1. Remember
    2. Understand
    3. Apply
    4. Create
    Answer

    D. Create

    Designing something new is Create.

40. Which of the following is a metacognitive item?

    1. A pupil copies a table
    2. A pupil reviews her own mistakes and decides how to study
    3. A pupil recalls a formula
    4. A pupil lists terms
    Answer

    B. A pupil reviews her own mistakes and decides how to study

    Metacognition is awareness and control of one's own learning.

41. A lesson test has 20 questions: 15 recall, 3 application, 2 analysis. The main weakness is

    1. No recall items
    2. Too few questions
    3. Over-emphasis on the lowest level
    4. Too many analysis items
    Answer

    C. Over-emphasis on the lowest level

    Most items test only recall; levels should be spread.

42. In a plan, mathematics objectives for a Class 6 geometry lesson are best selected from

    1. Cognitive domain only
    2. Cognitive, affective and psychomotor domains
    3. Psychomotor domain only
    4. Affective domain only
    Answer

    B. Cognitive, affective and psychomotor domains

    Mathematics teaching needs all three domains: reasoning, attitude and instrument skills.

43. The five academic standards of the AP school mathematics curriculum include problem solving, reasoning and proof, communication, connections and

    1. Representation and visualisation
    2. Memorisation
    3. Speed calculation
    4. Rote drill
    Answer

    A. Representation and visualisation

    The fifth standard is representation and visualisation.

44. Which of the following best describes the vocational value of mathematics?

    1. It gives joy through puzzles
    2. It develops a love of patterns
    3. It gives a record of ancient scholars
    4. It provides a base for careers such as engineering and accounts
    Answer

    D. It provides a base for careers such as engineering and accounts

    Career preparation is vocational value.

45. The value of mathematics linked to puzzles, magic squares and games is

    1. Recreational
    2. Moral
    3. Vocational
    4. Disciplinary
    Answer

    A. Recreational

    Puzzles and games give joy, so recreational value.