# Where Football Meets Mathematics

Math is all around us – at our works, homes and definitely in the sports and football predictions.

Frequently we find different connections between math and football, which are used by this game’s greatest specialists. Some of them are basics of the football predictions posted on this website.

The true football fans still remember the Dutchman Dennis Bergkamp and his masterpiece goals. Sports commentators describe him as a genius, who mastered the football game to perfection and possessing extremely accurate shot, but what is he saying about himself.

“When I played in Holland I always tried to hit the ball over the goalkeeper. People always questioned me about this. Why would I want to humiliate the goalkeeper or to demonstrate arrogance? But I always explain: When the goalkeeper leaves his post he makes the angle to it less but opens the space above it. What I do isn’t a show, it is mathematics.”

This advice came from the coach Louis Van Gaal, who teaches strategies for success in the match based on mathematical precision and order.

As his players state about him: Van Gaal isn’t one of those coaches who will repeat to them: “do this, do that”. He will leave the technical part to them, but he is a master of the tactical part and knows what should everyone do in order for the whole system to work properly.

“I hope that I know well the basics of math in football and try to apply them” – Ottmar Hitzfeld as coach of Bayern Munich in 2007.

Ottmar hizfield

Ottmar Hitzfeld graduated as a math teacher and now he is the most successful German football coach at club level. He won twice the FIFA award for best coach in the world and is one of the only three coaches who won the Champions League with two different teams.(Ernst Happel and Jose Mourinho are the other two).

Unlike most Germans, Hitzfeld carried his football career as a player in Switzerland.There he quickly arose as a top striker in the country, helping Basel win the championships in 1972 and 1973. While playing for the Swiss team, Hitzfeld graduated and received his diploma as teacher of mathematics and physical education.

We believe that we have found the next evidence proving that football is mathematics.

Hitzfeld is recognized for his managerial capabilities – the ability to control units in the team to work as one and to develop and apply different tactics. The coach is taught of perfectionism and he sticks to all the details that would lead his team to success. For example, as a coach of the Switzerland national team he led the players to train in specially selected mountain resort where it is estimated that the height above sea level is most favorable to the players to adapt to the specific conditions of South Africa.

Thus, estimating all the factors, Hitzfeld is following his main coaching philosophy – “The next match is the most important match and we must do everything to win.”

Bearing in mind the mathematical education of Hitzfeld, his numerous successes in the football are definitely not accidental, after all this is a game which is often described by specialists as a game of strategies.

# Building a Good Foundation in Mathematics

A solid foundation in mathematics can be crucial for a student’s performance in academics. Mathematics is an essential part of everyday life. Many students in school may have a natural ability to show a good performance pattern in the subject and it is usually an outcome of regular practice during the early stages. As a tutor, one must understand that the subject is purely based on practice and familiarity. Many other students often find the concepts and judgments to be complicated and most of the problem may be pertaining to teaching methods. One can overcome these problems.

Since a strong math foundation may be desired for students at least in the early years, the internet has been filled with courses of such relevance in website texts and videos. Many schools may be reputed to provide instructions which may not be really grasped except by the ones really attentive and sharp. Some of the reasons for a poor performance in mathematics may also result from focusing too narrowly on one aspect.

The different branches of mathematics that are taught as basic knowledge required for professional workmanship before you specialize are Arithmetic, Algebra, Geometry and Probability. Although not a part of the primary level education, probability can be a new area of problem once encountered in the higher grades.

As for the performance in examinations, it is imperative that a student is well prepared with the required knowledge. Starting with simpler examples and gradually increasing your potential to solve tougher problems is the key.

# Assessing Student’s Bookwork In Middle School Mathematics

Some school assessment programs in middle school and junior high school include a mark for bookwork as part of the reporting process in some, if not all, subjects studied.

Below are a set of criteria that I have used to assess student’s bookwork in junior high school Mathematics. (Much of what follows below is my adaptation of ideas shared at a local district meeting of Heads of Mathematics during the late 1990s).

The criteria are:

1. Headings, references and dates are stated for each lesson.

e.g.: Monday Classwork/Homework

5/6/13

Future Maths p.238 Ex52 No.7-9

2. Working and explanations are clearly and logically shown.

3. Work has been checked (ü/x) and corrected if mistakes are found.

4. Cover of the workbook is neat and labelled – Name, Form and Teacher.

5. Hand-outs are glued into book in the correct place.

6. Book is correctly ruled up and work is neat and legible.

At this point, I would like to stress the importance of criterion 1. This is particularly important for students who miss class time through illness. By checking the student’s workbook against another student’s workbook the teacher and the sick student are able to record what was missed during the student’s absence and organise a ‘catch-up’ program.

Mathematically speaking, criteria 2 and 3 are the most important. Some teachers may give greater weighting to these two criteria in assessing bookwork.

One final point. The teacher can also train the students to do a self-evaluation of his or her bookwork or have a peer do it for them. Below are the items in a self/peer evaluation to check off with a “yes/no”.

Self/Peer Evaluation

1. All work checked and corrected. yes/no

2. Working and explanations shown. yes/no

3. Ruled up, neat and legible. yes/no

4. Headings, references and dates included. yes/no

5. Hand-outs glued into book. yes/no

6. Book cover is neat and labelled. yes/no

Standard: Almost all “yes” Very Satisfactory

“Yes” sometimes Satisfactory

Mostly “no” Unsatisfactory

With my classes, these self/peer evaluations were done a few weeks before my formal evaluation of the student’s bookwork to encourage the students to improve their bookwork. The evaluation criteria that were unsatisfactory, i.e. had a “no” grading, tell the student where improvement must be made.

# Teachers – Formative Assessment – Informal Assessment of Students’ Content Knowledge in Mathematics

While there can be overlap between some types of formative and summative assessments, and while there are both informal and formal means to assess students, in this article, I will primarily offer suggestions for informal, formative assessment for the mathematics classroom, particularly in the first of the three categories suggested by Clarke & Wilson:

1. The student’s mathematical content knowledge.
2. The student’s mathematical processes, such as reasoning, communicating, problem solving, and making connections.
3. The student’s mathematical disposition, such as attitudes, persistence, confidence, and cooperative skills.

If you agree with the notion that words are labels for concepts, then you will want to use the 1, 2, 3, 4, 5 idea shown below:

Indicate your knowledge of each word by writing a 1, 2, 3, 4, or 5 in front of the word. The numbers signify the following five statements:

1. I’ve never even seen the word/phrase.
2. I’ve seen the word/phrase, but I don’t know what it means.
3. I know the word/phrase has something to do with…
4. I think I know what it means in math
5. I know the word/phrase in one or several of its meanings, including the meaning for mathematics.

———— Unit 2:  Using Measures and Equations ————-

• continuous
• opposites
• line
• length of a segment
• ray
• central angle of a circle
• complementary angles
• vertical angles
• right triangle
• solving an equation
• rational number
• perfect square
• discrete
• scientific notation
• endpoint
• midpoint
• angle
• right angle
• supplementary angles
• acute triangle
• equation
• equivalent equations
• irrational number
• perfect cube
• absolute value
• segment
• congruent segments
• vertex of an angle
• straight angle
• congruent angles
• obtuse triangle
• solution
• square root
• real number
• cube root

I prefer to use this as both an informal pre- and post-assessment. At the beginning of a new unit or chapter (and again at the end), I give students a sheet similar to the one shown above, with vocabulary terms for the unit listed.  [The first time you use this idea, it is necessary to go over the five different levels of word knowledge, but students easily understand the idea that there are words they have never heard of and words that they know in several ways (and everything in between these two).]  It is important to pronounce the words as the students read them and rate their own level of knowledge of the word because there are words that students recognize when they hear them but don’t recognize when they see them. Then, to assess content knowledge, for all words that the students rated as 4’s or 5’s, ask them to write their best understanding of what that word means in mathematics. This is not used for a grade but rather, as formative assessment to give an idea of students’ understandings of the concepts before and after the unit of instruction.

A second way of assessing students’ content knowledge, is giving students a sheet with 5 rows and 4 columns at the beginning of the week. Then, each day, either as students enter class, or as the closing activity for the day, four problems from a previous day’s lesson or homework are given, and students enter each problem (and solution) in the four spaces for the day. The teacher can check these quickly or have a row grader check them. These may be collected each day or at the end of the week, depending on the teacher’s plan for using the assessment information.

The third suggestion for formative assessment of content knowledge is performance assessment.  Entire articles (and books) have been written on the next suggestion for formative assessment of mathematical content knowledge, but even though I cannot fully explain it in the context of this article, I would be remiss not to mention the idea of performance assessment. Performance assessments are assessments “in which students demonstrate in a variety of ways their understanding of a topic or topics. These assessments are judged on predetermined criteria” (ASCD, 1996, p. 59).  Baron (1990a, 1990b, and 1991) in Marzano & Kendall (1996) identifies a number of characteristics of performance tasks, including the following:

• are grounded in real-world contexts
• involve sustained work and often take several days of combined in-class and out-of-class time
• deal with big ideas and major concepts within a discipline
• present non-routine, open-ended, and loosely structured problems that require students both to define the problem and to construct a strategy for solving it
• require students to determine what data are needed, collect the data, report and portray them, and analyze them to discuss sources of error
• necessitate that students use a variety of skills for acquiring information and for communicating their strategies, data, and conclusions (p. 93)

Begin exploring various formative assessment tools with your students to determine their content knowledge in mathematics.  You will learn a great deal – and then be able to help your students learn even more!

# What Children Need to Succeed in Mathematics

All children can succeed in mathematics. How do I know this? My empirical experience approaching thirty years tells me this is so. My intuition tells me this is so. You just need to know what to give your children, and more importantly, what not to give them.

Children are natural problem solvers. If you leave them alone, they will figure out the most amazing things. If you nurture them properly and give them the space to grow, they will become natural mathematicians. The trick is to let them believe—at least initially—that learning and education are fun and will lead them to have more fun in life. How simple that sounds! Yet this premise is that simple.

Unfortunately most parents get bogged down in their own problems and do not subscribe to this philosophy. Such parents become frustrated when their child shows a lack of interest in school work and school related tasks. These problems spiral and eventually become overwhelming. Rather than address the problem and correct the negative programming that has been instilled on the child, parents resort to criticism and reprimand.

No child can ever succeed in mathematics with criticism. The one thing that this world has way too much of is that one word—criticism. If you do not believe this statement, try this experiment for one week: refrain from all criticism. Do not criticize your coworkers, your friends, your relatives, yourself. Do not criticize the government, the world, the planet. Watch if your life does not somehow take on a whole new dimension of vibrancy, peace, and enthusiasm.

Pass this enthusiasm onto your child. Tell your son or daughter how creative he or she is. Instill in your children that they are leaders, capable of solving any problem that presents itself. Mathematics is a subject which is self-propelled by high self-esteem. Children who have high self-worth, high self-esteem tend to be better problem solvers. Why this is so is self-evident: a child who believes in himself will approach tasks and problems with a gusto that says he can lick the task. Consequently, this child approaches the problem with the attitude that he will win and the problem will lose. End result: more success in whatever task at hand.

In conclusion, every child can succeed in mathematics. Give your children a never-say-quit attitude. Give them encouragement. Give them love. Do not give them criticism. These three former give, this latter withhold. This is a secret for success in mathematics, in school, and indeed life. Use it.

# Teachers – Summative and Formative Assessment in Mathematics – What Are the Differences?

I’m a big fan of using definitions as a starting point for thinking about a topic…so let’s look at a definition of assessment from the National Council of Teachers of Mathematics (1995):

Assessment is…the process of gathering evidence about a student’s knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes (p. 3).

Depending on your age, this definition may describe the experience you had with assessment in mathematics during your school career, but for most readers, “testing” was really the only kind of “assessment” we knew. Like clockwork, at the end of every few sections of the math book, there would be a quiz (for a GRADE) and at the end of every chapter, there would be a TEST (for a MAJOR GRADE). Then, no matter what grades any of us received, we would go off to the next chapter, where the cycle began again.This type of testing (of which there are many varieties) is known in today’s parlance as “summative assessment,” defined as

“a culminating assessment, which gives information on student’s mastery of content” (Association for Supervision and Curriculum Development, 1996, p. 60).

The principal characteristics of summative assessment are that it:

1) occurs at the conclusion of a learning activity,

2) is to make a final judgment,

3) may compare students to other students, and

4) often results in a grade or some other ‘mark.’

In contrast, the principal characteristics of formative assessment include that it

• occurs during learning activities/experiences,
• is for the purpose of improving the learning, and
• will inform the teacher so that s/he can make adjustments if needed.

A useful definition of formative assessment is

“assessment which provides feedback to the teacher for the purpose of improving instruction” (ASCD, 1996, p. 59).

This concept of assessment meshes nicely with the NCTM definition shown above (i.e., “the process of gathering evidence about a student’s knowledge of, ability to use, and disposition toward, mathematics and of making inferences from that evidence for a variety of purposes”). Formative assessment – with or without that name – has always been around – depending on individual teacher’s attitudes towards this.  For the teacher who believes, as Grant Wiggins does, that “Good teaching is inseparable from good assessing,” there has always been an ongoing cycle of teaching, assessment, of the teaching, reteaching (as necessary), assessment, teaching, and so on. “Assessment should serve as the essential link among curriculum, teaching, and learning” (Wilcox & Zielinkski, 1997, p. 223).

So, the next time you hear others talking about assessment, ask if they are referring to formative or summative assessment. That will help you know what questions to ask next.

# A Young Teacher’s Guide To Homework In Mathematics In High School

Most of what appears below was the advice that I wrote for teachers who taught Mathematics in my department when I was its head. It appeared in my department’s handbook.

Homework was an accepted part of what we did as Mathematics teachers for all classes except those with special needs students.

How And When To Set Homework

• It should be set daily or after each lesson.

• Write the assigned homework on the board.

• Ensure the students write it in their school diaries at the end of the lesson. In junior classes, you may stand at the door checking the homework is written in their diary as they leave.

• Discuss how long the work should take and any necessary advice.

• Lastly, early in the school year, teach your students how to use their textbook to help them do their homework.

What Homework Should You Set?

For students to achieve their full potential in Mathematics at high school, homework must be done on a regular basis. Homework, based on current class work, is meant to be an extension of the lesson and is needed for the re-enforcement of concepts.

In high schools, homework in Mathematics may consist of:

• Written exercises set for practice of skills and concepts. These are based on classwork.

• Learning work, e.g. rules, vocabulary and theorems.

• Assessment tasks – these usually count towards Semester reports.

What About Students Who Don’t Do Their Homework?

Teachers should record in their diaries the names of defaulters. Parents must be advised when a pattern of missing homework becomes evident.

Teachers should develop a process for dealing with homework defaulters.

What If Students Can’t Do Their Homework?

As most homework is based on the work done in class that day, this is not usually a problem for most students. However, if a student has difficulty in beginning homework, teach these strategies:

• The student should look for a similar problem in the work done in class. This is usually all that is needed to jog the memory.

• The student should look for an example in the textbook prior to the exercise. Each different type is usually done in full with an explanation.

• If students still have difficulty, they should see their teacher the next day BEFORE CLASS and arrange a time for individual help. Most teachers are available for a “homework help” time at lunch time or before and after school. Your teacher will tell you when he/she is available.

What If A Student Tells His/Her Parents That They Never Have Homework?

Often, there are complaints from parents who tell us that their students never have homework. This is clearly not the case! If a student has no written homework, (which is unlikely) then we would suggest that the parents set one of the following to be done:

• Ask the student to write a summary of the rules for the current unit and to work an example of each type of problem. The textbook will be useful here. Look for chapter summaries.

• Look at the student’s exercise book and find an exercise that caused difficulty. Set this exercise to be done.

• In each textbook, there are chapters on basic skills. Students can do any of the exercises from this chapter.

• Often there are chapter reviews and practice tests. These can be done.

The Review Process

Homework should, wherever possible, be reviewed during the next lesson for the greatest impact on learning to occur. This learning may, in fact, be the basis of the next lesson. A full description of a review practice can be found in the Article “Reviewing Homework in High School Classes” to be found on this website.

Even though there is a continuing debate as to the merits of homework, the advice here will help the young Mathematics teacher deal with homework successfully.

# How to Solve Problems in the Mathematics Education Using Calculators

Mathematics is a very important element in the development of the value and standard of the life of the individual and society. Mathematics enters in everything from science, technology and engineering to arts and social sciences to economics and decision making.

It is very important that society constantly produces new generations of well trained mathematicians. This means that the school should increase the student's interest and develop his skills in the subject. To achieve this goal the school should see the reasons why students fear mathematics and why many students do very poorly in the subject.

The fist reason for the difficulty of the subject for students is that mathematics is abstract. It is not tangible they can not hold it. To make them understand a pure concept at this early stage is very difficult. They need to relate it to real life problems that they face in their daily lives. This way they will be stimulated by its importance for them and will see that it is useful and fun. Moreover, the use laboratory experiments using calculators would make concepts more understandable. There are many new innovative software calculators built just for this purpose.

The conventional system of teaching mathematics contributes significantly to the poor performance of students. This is because many teachers teach students how to implement algorithms without teaching them the idea behind the algorithms and there is no stress on the understanding of the concepts. This causes the interest level in the subject to decline and soon students develop a disassociation with the subject. To solve this problem the schools can create a mathematics society and a forum where students can communicate with each other discussing material and asking each other questions. This kind of social behavior may increase the student's interest in the subject.

Students pass early mathematics classes and they are not really fit in the subject. This is because the exams and grading did not reflect the student's true ability. This causes the students to have very big problems in senior mathematics classes which are dependent on the early classes. For example if we look at the early pupil stage we can imagine a pupil who does not understand the concept of divide and multiply. That is to say when should he divide and when should he multiply. Now can this pupil at a later stage know to solve a word problem? The answer is surely no. He will not know what we are talking about. We can also imagine a student who does not understand the concept of the algebraic equations. This student could not be asked to plot a graph because simply he will not have an idea about what we are speaking about. The teachers can use calculators to teach pupils the concepts of multiply and divide. They can let them experiment on their own with many numbers using the calculators. From the observations the pupils will understand the concepts of multiply and divide. Moreover, teachers could use calculators to teach students the concepts of functions and graphing. It is easy and fun using calculators.

Finally, those who are teaching mathematics may have sufficient knowledge in the subject. They usually have a mathematics degree but this is not enough. To be a teacher you certainly got to be strong in the subject but this is only one major factor in being a successful teacher. The teacher should know about education and student psychology. he should also have a strong personality and be a good leader.

# 20 Tips And Tricks To Teach Mathematics At The Primary Level

The primary Math education is a key determinant and I must say the very foundation of the computational and analytical abilities a student requires for a strong secondary education. It is the very base on which secondary education is built on. This is why it is mandatory that the teaching techniques and methods we employ as teachers and educators be of such rich quality that the development of a child with respect to his mathematical abilities be wholesome, practical and balanced.

Being a Math teacher is not easy. It is usually the favourite of a few and the nemesis of many. It has been observed that children mostly try to escape doing Math work. While there is a section of students who absolutely love mathematics enough to pursue a career in it, many students live in fear of it. Today we are going to give our teachers some helpful tips and tricks to make teaching math an enjoyable and interesting experience not only for the kids.

20 Tips and Tricks to Teach Mathematics at the Primary Level

1. Ambience plays a very significant role. It is your responsibility to see that a classroom is properly ventilated with ambient light.
2. Ensure that Mathematics class is neither before lunch break (when children concentrate more on the Tiffin than studies) nor the last period where students wait more for the bell to ring (not to mention start feeling sleepy!) Keep Math class when the children are active and fresh.
3. Cultivate the students’ interest in Mathematics by letting them know about the power, structure and scope of the subject.
4. Hold the students’ attentions from the get go! Introduce the topics with some fun facts, figures or interesting trivia
5. Chalk out the lesson plan effectively keeping time and content allotment in mind
6. Use audio and visual aids wherever possible
7. Draw on the board if required (especially, lessons like geometry, shapes and symmetry)
8. Call students to work on the blackboard (engagement of every child is necessary and not just a select few!)
9. Ask for a student’s opinions and thoughts on concepts and mathematical ideas.
10. Give them time to discuss important concepts and study the text of the chapter too before taking on the problems themselves.
11. Teach more than one way or approach to solve a problem.
12. Give regular homework exercises making sure that the questions are a mixed batch of easy, medium and difficult) Children should not feel hopeless. Easy problem questions evoke interest.
13. Reward them! Whenever students perform well, be generous and offer them an incentive to continue working harder.
14. Let children enjoy Mathematics and not fear it.
15. Instill in them the practice to do mental math.
16. Also, never give a lot of homework. Children are already burdened with assignments to work at home in almost all school subjects, it is thus your duty to make sure that the homework you delegate to them is fair sized or little. (This trick will inculcate in them the motivation to complete math homework first)
17. Present challenging questions to students so as to develop their analytical and deduction abilities
18. Keep taking regular tests to cement knowledge.
19. Teach at a consistent pace. Do not rush with any topic. Before proceeding, be confident that the students are clear with the prior topics.
20. Play games to create a fun filled classroom teach and learning experience.