Bloom's Taxonomy: Conceptual Learning and Questioning

Bloom's Taxonomy: Conceptual Learning and Questioning
By Dr. Genola Johnson

Benjamin Bloom's Levels of Taxonomy was created for educators to plan effective instruction. Using the levels during lesson planning and creating assessments assists the teacher in reaching all modalities of learning.

Using Bloom's Taxonomy's helped me understand how thinking was classified. There were certain areas I wanted to reach when teaching a concept and the classifications or taxonomy helped direct my questioning techniques.

To direct the questioning of my lessons, I created questions from the verbs in the taxonomy classifications. If I wanted high, complex questioning I would use words from the analysis, synthesis and evaluation areas.

I always wanted my students to think deeper, use problem solving skills, discuss with peers and seek further information on the concept to be learned.

In my opinion, the foundational idea would be for students to learn a concept using Bloom Taxonomy's and transfer that knowledge to other concepts.

� Understand-Explain ideas/concepts

� Remember-Recall information

� Analysis-Breakdown into parts

� Evaluation-Justify thinking

� Create-New ways, ideas, products of thinking

When creating lesson plans, I would often have the taxonomy close by to ensure I am reaching all levels. Using the assigned curriculum, I would develop my lesson objectives, identify the skills the student needed to learn, and align my objective to the assessment.

All of my lessons contained critical thinking questioning. Sometimes I would build from the knowledge level with questions that were just recall. For example, list the steps in the writing process. If the student can identify the steps, they can begin the process of designing a writing piece.

Today, students need to be able to understand why the need to know a concept. Having the factual knowledge of 2 x 2=4 is essential when you need to import this factual knowledge into an algebraic or geometrical formula when calculating the area of land to build a greenhouse to build a neighborhood garden.

I would often say, "You need to know this information, in order to create or develop, this product." Letting the students know where they are going is essential in getting them to learn the curriculum you are to teach.

Teaching synthesis (creating and evaluating) after teaching the knowledge and comprehension of a concept helps the student put the recall and understanding into a whole part. Students should be able to develop or create something new with the new information they have been taught.

Using the Bloom's Taxonomy to develop your lessons, questioning and assessments helps students and the teacher focus on deeper conceptual learning.

Dr. Genola Johnson has been an educator for 21 years. She uses Bloom's Taxonomy during lesson planning and assessments. For more information about Dr. Genola Johnson, and a list of the Bloom's Taxonomy List of Verbs, visit http://www.gaelcllc.com.

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Solve Math Questions Online and Enhance Your Skill

Solve Math Questions Online and Enhance Your Skill
By Sandy D'Souza

Rigorous practice is the main key to achieve success in math. Research suggests that most students do not spend enough time to practice math on a regular basis. The reasons can be varied, from disinterest to inefficiency. The fact is that when students do not understand the topic properly, they lose their interest and end up disappointed due to poor grades in exams. To solve a mathematical problem accurately, students need to be completely involved. The process of solving a mathematical problem demands several sequential steps. First, students need to find the method involved in the problem. Second, they need to apply the right formula to get the correct solution. Third, they can find the alternate method to solve the same problem.

Practice math questions and answers

To make each learning session more effective, students should practice various problems on the same topic. This gives students more clarity on each topic. Additionally, they can easily find out their learning problems and take required steps to overcome these. However, students have a tendency to stick to a topic which is easy to solve. Experts suggest that they should change this habit and try to solve all kinds of problems to get familiar with the entire curriculum. To become an ace in math, students need to practice math regularly.

Several websites offer math help. When a student feels that he/she does not understand the math concepts thoroughly in a classroom environment and cannot cover the syllabus on time, they can opt for online math assistance. This learning process gives them better understanding of each topic. Most importantly, with this service, students can choose grades, topics and level of difficulties accordance to their preference. They can choose the worksheet which they want to work on. Online math help is fast and easy to use for students. They can find instant solutions related to any topic including algebra, calculus, etc. Students can also use some math quizzes and games available on those websites to make math interesting.

Take online help to solve tricky math problems

Students need to have patience to solve any tricky math problem accurately. However, most students do not practice math regularly and try to memorize some easy methods to solve all problems in exams. This is definitely a wrong technique to prepare for the math exam. Any student can learn math by following step-by-step and detailed explanations. Students can have this facility with online math help. They can choose their preferred tutor along with suitable timings.

Online math help is few steps away from students. Students can access online help anytime and from any place. It enables a good number of students to score well in exams. This innovative learning process also enhances students' confidence. In short, by using this online service, students get adequate learning help in a convenient and comfortable way.

To improve your mathematical skills students can take extra care in some parts like more practice and they can also take help of math tutors or with online math help and also the most important thing is working on the assignments given in regular class sessions. This makes you score good marks and enhances your skills.

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Different Question - Same Answer

Different Question - Same Answer
By Richard D Boyce

One of the skills of excellent teaching is good questioning technique. I was fortunate to be trained initially as a primary school teacher where you needed to be able to ask relevant questions in a range of subject areas. A year after graduation, I was transferred into the secondary school arena where for almost ten years I taught a variety of subjects including English, Science, Mathematics, History and Geography with even a little Physical Education thrown in. This enhanced my understanding of the need to question in different ways. The remainder of my career was spent teaching Mathematics. Therefore, for that reason and because we all did Maths at school, let me use Mathematics to illustrate these points.

• Questioning in different ways can extend students' understanding of the subject.
• It can enhance their critical thinking/problem solving skills.
• It can teach the vocabulary of the subject studied.
• It can develop an understanding of and the use of the language and terminology of the subject.
• It will help develop communication skills as well.

Here are questions from the field of Mathematics.

1. 5 plus 7
2. What is the sum of 5 and 7?
3. Increase 5 by 7
4. What do I get when I add 5 to 7?
5. What is 5 more than 7?
6. Simplify 5 + 7
7. Find the missing number 7+ 5=...
8. Solve 7 + 5 =..?..
9. Two items I want to buy are $5 and $7 each. How much money do I give the sales person?
10. Increase 5 by 7
11. If the answer to a sum of two numbers is 12; and one of the numbers is 7, what is the other number?
12. 7 take 5
13. 7 minus 5
14. Take 5 from 7
15. 7 subtract 5
16. Subtract 5 from 7
17. What is the different between 7 and 5?
18. Decrease 7 by 5
19. What must I increase 5 by to get 7?
20. What do I take/subtract from 7 to get 5?
21. I have $7 and I spend $5 on an ice cream. How much do I have left to spend on lollies?
22. 5 times 7
23. 5 multiplied by 7
24. Add 7 up 5 times
25. What is the product of 5 and 7?
26. The factors of the number I want are 5 and 7. What is the number?
27. Solve 57=?
28. Find the missing number... ?..= 5 7
29. Simplify 5 7
30. 35 divided by 5
31. 35/5
32. If 5 is a factor of 35, what is the other factor?
33. What do I get when I divide 5 into 35?
34. How many times can I take 5 away from 35?
35. How many 5s do I add together to get 35?

Of course, in each group of questions the answer is the same. I'm sure in every learning area at each level teachers could devise a 'similar' quiz. Obviously, it is an opportunity to revise and enlarge the vocabulary of the subject while enhancing learning.

Teachers should keep a record of these questions and enlarge their list as they go. Teachers could ask these questions in the form of a quiz in primary and lower secondary schools where they can also be valuable revision tools.

Our author spent over 40 years in the classroom. Early in his career he taught many subjects as well as Mathematics and mastered the art of asking questions in different ways to test the depth of understanding of his students and to enhance their learning. He has put all his experience in an eBook, "The Question Book". It examines all aspects of questioning providing practical hints for the teacher. The website is http://www.realteachingsolutions.com

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Algebra 1 connections answers

Author: Matthew David

Algebra 1 connections answers

Introduction to algebra:

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics. In this article we shall discuss for need help with algebra 1 math.   (Source: Wikipedia)

Sample problem for need help with algebra 1 math

Need help with algebra 1 math problem 1:

Solve the given linear equation

8x+ 8y = 24,

8y +8 z = –40,

8z + 8x = 16.

Solution:

The given equations can be written as

8x + 8y = 24 ----------- (1)

8y + 8z = –40 -------- (2)

8z + 8x =   16 ----------- (3)

Adding all the equations,

8x + 8y + 8z = 24 + (–40) + 16

Or          8(x + y + z) = 0

In the above equation is divided by 2 we get

8x +8 y +8 z = 0 ----------- (4)

Substituting 8y + 8z = 16 in equation (4) we get

8x + (–40) = 0

8x = 40

X = 5

Substituting 4x + 4y = 12 in equation (4) we get

4y + 20 = 12

4y = –8

Y = -2

Substituting x = 5 in equation (3) we get

8z + 8x = 16

8z + 8*5 = 16

8z = 16–40

8z = -24

z = –3

The solution is x = 5, y = –2, z = –3.

Need help with algebra 1 math problem 2:

Solve the given linear equation

10x+ 10y = 32,

10 y +10 z = –50,

10z + 10x = 20.

Solution:

The given equations can be written as

10x + 10y = 32 ----------- (1)

10y + 10z = –50 -------- (3)

10z + 10x = 20 ----------- (3)

Adding all the equations,

20x + 20y + 20z = 32 + (–50) + 20

Or          20(x + y + z) = 0

In the above equation is divided by 2 we get

10x + 10y + 10z = 0 ----------- (4)

Substituting 10x + 10y = -50 in equation (4) we get

10x - 50= 0

10x = 50

X = 5

Substituting 10z + 10x = 20 in equation (4) we get

10y + 20= 0

10y = –20

Y = -2

Substituting x = 5 in equation (3) we get

10z + 10*5 = 20

10z = 20–50

10z = -30

z = –3

The solution is x = 5, y = –3, z = –3.

Algebra 1 connections example problem 1:

Solve the linear function y = 2 - x and x - y = 10

Solution:

Given equations are,

y = 2 - x --------- (equation 1)

x - y = 10 -------- (equation 2)

Substitute the equation 1 into equation 2, we get

x - (2 - x) = 10

Rearrange the above equation, we get

x - 2 + x = 10

2x - 2 = 10

Add 2 on both the side of the equation, we get

2x = 12

Subtract the above equation by 2, we get

x = 6

Substitute x = 6 in equation 1, we get

y = 2 - 6

y = - 4

Answer:

The final answer is x = 6, y = - 4.

Algebra 1 connections example problem 2:

Simplify the given [removed]4x + 43) + 12x = 52 - 2x

Solution:

Given expression is (4x + 43) + 12x = 52 - 2x

Expand the above expression, we get

4x + 43 + 12x = 52 - 2x

16x + 43 = 52 - 2x

Subtract (52 - 2x) on both the side of the equation, we get

18x - 9 = 0

Add 9 on both the sides, we get

18x = 9

Divide the above equation by 18, we get

x = `(9 / 18)`

Answer:

The final answer is x = `(1 / 2)`

Algebra 1 connections example problem 3:

Find the x intercept of the given polynomial equation f (x) = x2 - 225

Solution:

The given polynomial equation is f (x) = x2 - 225

Plug f (x) = 0, for finding x intercept

0 = x2 - 225

Rearrange the above equation, we get

x2 - 225 = 0

Add 225 on both the sides of the equation, we get

x2 = 225

Take square root on both the sides, we get

x = ± 15

Answer:

The final answer is x = ± 15

Practice problems for algebra 1 connections answers

Algebra 1 connections practice problem 1:

Solve the linear function y - 2 = x and 4x + y = 22

Answer:

The final answer is x = 4, y = 6

Algebra 1 connections practice problem 2:

Factorize the equation x2 + 22x + 40 = 0

Answer:

The final answer is x = - 20 and x = - 2

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Between, if you have problem on these topics Acute Angles, please browse expert math related websites for more help on Geometry Tutor and different math topic.

Identify the Correct Statement

Author: Omkar

Introduction to identify the correct statement

In this lesson we will see how to handle multiple choice questions effectively. These questions differ from other detailed problem solving as the student is provided with choices of various answers and the student is required to identify the correct answer. Normally four to five alternatives are provided, out of which usually one is correct but occasionally some multiple choice questions will have more than one correct answer. Normally, examinations with only multiple choice questions come with time constraints and in some cases a penalty is imposed for wrong answers to avoid wild guessing. It is therefore important that this section is attempted quickly and accurately.

As said above, the success in attempting these questions will depend on the ability to identify the correct answer quickly. It might not be required to solve the problem from beginning to end. The student might have enough hints to identify the wrong choice. Some of the problems will require solving up to a stage and then eliminating the wrong answers. In some cases it will be good to try working from the alternatives given into the questions and eliminate the wrong ones.

Approaches to identify the correct statement

Main approaches

Identify and eliminate wrong alternatives
Find the range of values for the possible answer or the sign of the number etc and eliminate the alternatives that are outside the range
Try plugging the alternatives in the conditions mentioned in the problem statement and see if all the conditions are met. This will help in eliminating the wrong alternatives quickly
Let us analyze the various approaches to identify the correct statement without actually spending time to solve the problem and arrive at the final answer

Ex 1: What is the value of `sqrt(52.4176)`

A) 6.94

B) 3,88

C) 7.86

D) 7.92

Sol: It will be extremely time consuming to actually find the square root of the number 52.4176 without a calculating device. Moreover, the chances of making a mistake in calculations are also high.

Step 1: Let us first take the integer part and then identify the perfect squares near by.

The integer part of 52.4176 is 52 and the perfect squares near by are 49 and 64.

`sqrt(49)` = 7 and `sqrt(64)` = 8.

So `sqrt(52.4176)` lies between 7 and 8.

Step 2: This will eliminate the first two choices. We are now left with choices 7.86 and 7.92. One of these numbers if multiplied by itself should get 52.4176.

Note, that 52.4176 ends with 6.

Step 3: So the if we try multiplying 7.92 by 7.92, the end digit will have 4 ( as 2 x 2 = 4) and not 6. So 7.92 is not the right answer. The only alternative left is 7.86 and when multiplied by itself will get a number ending with 6. This is the correct choice

Ans: (C) 7.92

The above approach will considerably save time and effort to identify the answer. Note, we identified the answer, we did not work out the answer. In multiple choice questions this approach is very important

Another approach is to work from the alternatives that satisfy the conditions in the question. This approach will be faster in many cases

Let us now try another example

Ex 2: Given b = 2a, Find the values of a,b and c if, `(21a)/(c) = (b+c+1)/(a)= (2c+5a)/(b)`

A) a= 3,b= 6,c= 7

B) a= 2,b= 4,c= 7

C) a=4,b=8, c=2

D) a=1,b=7,c=6

Sol: It will be too time consuming to solve the equations and to arrive at the values for a, b and c. It will be easier if we plug in each alternative into the conditions of the question and eliminate the ones that does not satisfy.

Step 1: First condition is b = 2a, we can easily see that alternative (D) does not satisfy this condition and can be eliminated. We are now left with (A), (B) and (C) only.

Step 2: Let us try the alternative A: `(21a)/(c) = 21*3/7` = 9 and `(b+c+1)/(a) = (6 +7+1)/(3)` = 4.67. These are not equal and hence alternate (A) is not correct

Step 3: Let us try the alternative B: `(21a)/(c) = 21 * 2/7` = 6 and `(b+c+1)/(a) = (4+7+1)/(2)` = 6 and

Step 4: `(2c+5a)/(b)= (2*7 + 5*2)/(4) = 24/4` = 6.These are all equal to 6 and hence alternate (B) is correct

Step 5: To complete let us try alternate C as well

`(21a)/(c) = 21 * 4/2` = 42 and `(b+c+1)/(a) = (8+2+1)/(4)` = 2.75. These are not equal and hence alternate (C) is not correct

Ans: (B) a= 2,b= 4,c= 7

We will look at one more approach to identify the correct statement

Let us consider another example

Ex 3 : What are the roots of the quadratic equation, 3.1x2 –2.1x – 6.9 = 0

A) 1.47, 3.30

B) 2.1, -3.6

C) –3.2, -1.8

D) 1.87, -1.19

Sol: If we solve the problem using the quadratic formula, it will take a long time as it will involve find the square root of fractional numbers etc. To identify the correct statement among the above four, this is not required either. If we use the formula connecting the roots of the quadratic equation, we can eliminate the alternatives easily

Step 1: We know Sum of roots is `-b/a`

Product of roots is `c/a`

Step 2: If we apply this for the above equation we get

Sum of root of the equation 3.1 x2–2.1 x – 6.9 = 0 is `2.1/3.1` an dpreoduct of the root is –6.9/3.1

Step 3: The product of roots is negative. This means that we will have one root with positive sign and another with negative sign. This will eliminate alternatives (A) and (C). We now need to pick from alternatives (B) and (D)

Step 4: Sum of the roots is positive, this means that the absolute value of the positive root is higher than the negative root. This will eliminate alternative (B)

The only alternative left is (D)

Ans: (D) 1.87, -1.19

Thus we could identify the correct statement without doing any calculation

Exercise on correcting statements

Pro 1: What is the value of Sin 470?

A) 0.31

B) 0.94

C) 0.731

D) 0.26

Hint: Value of Sin 0 increases from 0 to 1 as theta moves from 0 to 90

Ans: C

Pro 2: Which of the following triplets that best forms the sides of a right angled triangle?

A) 13.1,16.7, 28.51

B) 15.2,16.7, 30.4

C) 17.8,19.6,35.7

D) 24.3,15.2,28.66

Hint: Use the principle that sum of any two sides of a triangle is greater than the third side. This will help in eliminating the alternatives

Ans: D

Pro 3 : what is the value of 6.812-3.922?

A) 28.635

B) 31.097

C) 15.637

D) 38.927

Ans: B

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Quizzes To Develop Communication Skills

Quizzes To Develop Communication Skills
By Richard D Boyce

All subjects we teach have their own language and terminology. These help communicate meaning to what children learn and help them communicate what they have learnt more effectively. Consequently, we, as teachers, need to teach and test the language/terminology of our subject areas. The quiz in its various forms is a way to enhance that learning in a time efficient and fun way.

Mathematics has always been an area where the quiz has been used to great effect. Below are examples of how it can be used in Mathematics as an example for other teaching areas.

1. Subject Language/Terminology Quiz

Here the teacher simply gives students a series of terms. The students write out in their own words an explanation of each term.

e.g.: concurrent parallel congruent collinear skew (Maths) constellation galaxy planet satellite orbit (Science) revolution democracy nationalism parliament cabinet (Hist.)

Students should be encouraged to use other subject terms/words to help explain what the terms mean. Diagrams should be encouraged to enhance explanations.

2. Terms Quiz

This really is the reverse of the Language Quiz above. Here the teacher gives a "definition" and asks the students to give the appropriate term to fit the definition.

e.g.: (i) Lines which meet at the same point are said to be (Concurrent)

- What name do we give to numbers which are non-terminating, non-recurring decimals? (Irrational Numbers)

- What name do we use to describe figures in Geometry which have congruent angles in one to one correspondence but no equal sides? (Similar Figures)

- What names do we give to the ratio of vertical change to horizontal change? (Gradient, Slope, Tangent of an Angle)

- Name the part of Mathematics which studies the relationship of the angles of a right-triangle with the sides of that triangle. (Trigonometry)

3. Spelling Test

Students must be encouraged to use terms correctly in the justification process which has become an important part of modern teaching. This means that the spelling of terms must be seen as important as well.

During my own school years and in my early teaching years, our teachers often broke up words into their parts. They explained what each part meant and how the word got its meaning. We should retain this idea:

I run adjective ending

e.g.: con / curr / ent

Prefix - "with or together"

Therefore "concurrent" literally means, "I run together".

Obviously, these spelling tests should be included when the topic is revisited or after it has been taught.

These tests should not be more than five questions at a time.

Other ways to improve spelling of the terms of the subject include:

1. A list of incorrectly spelt terms and ask student to write out the list correctly.

e.g.: Rewrite the Maths terms, spelling them correctly:

i) equotion (ii) paralel (iii) multiplication

(iv) numarel (v) integer

2. A list of say '5' spelt a number of ways and ask student to select the correct one.

e.g.: Read each list of Maths terms and select the one correctly spelt:

(i) intecept; intercept; intecerpt; intencept

(ii) equation; equtation; equotion; equitation

(iii) linner; linear; linnear; lionear

(iv) function; funition; function; founction

(v) quadatic; quadratic; quodratic; quodrotic

4. Symbols Quiz

This is an excellent way to revise work in Geometry and to ensure symbols in proofs/explanations/working answers are well understood and used correctly.

e. g. Explain what each of the following symbols mean in Geometry:

or In the following Mathematical equations a symbol has been used incorrectly or has been omitted. Cross out the incorrect symbol and replace it with the correct one or add the missing one, e.g.:

(a) 6... 4 + 3 = 100... 5... 7

(b) 6... (4 - 3 ^ 42

or In Chemistry, you could test the chemical symbols.

or In physics, you could test the accuracy of formulae.

or In English, you could test abbreviations.

This article explains one of a series of different types of quizzes that our author has used, to great effect, during his career in high school classrooms. In his early career, he taught several subjects to junior high school classes where he learnt the art of using the quiz as a revision tool and as an introduction to a new topic where he reviewed past knowledge. You will find two eBooks on his website http://www.realteachingsolutions.com explaining his use of different types of quizzes. The titles are, "The Quiz in Middle School Mathematics" and "The Quiz as a Teaching Strategy".

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