# Mini Math Bio – Thales, The Father of Geometry

Thales, the Greek mathematician from Miletus, Ionia (today’s Western Turkey), is credited for formulating the very first “Principles of Geometry.”

Imagine, although human beings were alive on earth for hundreds of thousands of years, they had to wait until 6th century B.C. to find out the following interesting rules that guide the behavior of triangles and circles:

Thales’ Rule No. 1: If you draw any triangle inside a circle, if the tip of the triangle is resting on the circumference and the diameter of the circle is forming the long side of the triangle, then the angle formed by the other two shorter sides forms a 90 degree right angle – regardless of the length of the shorter sides. (This rule is also commonly known as “Thales’ Theorem.”)

Thales’ Rule No. 2: The diameter of any circle bisects it into two equal parts and into two arcs of equal lengths.

Thales’ Rule No. 3: If two triangles have one side that is of equal length and two angles that are of equal degrees, then those two triangles are identical.

Thales’ Rule No. 4: Two straight lines crossing each other form four angles around their intersection point. The angles opposing one another are equal angles. That is, two intersecting straight lines form two pairs of equal angles.

Thales’ Rule No. 5: A triangle with two equal sides, i.e., an isosceles triangle, have equal angles where the equal sides intersect with the third side.

We owe the beginnings of our modern-day mathematics and geometry to a man who was so in love with what he is doing that one day, according to Plato, Thales fell into a water well while gazing at the stars. Thus perhaps it won’t be too much off the mark to say that Thales was also the very first “absent minded professor” we had.

# Preparing Students For High School Maths

A Guide For Primary School Teachers

A High School Maths Teacher’s Wish List

What has occurred in recent years as many more students complete high school and seek a tertiary education, is a growth in parents wanting their children to do Mathematics at a higher level. They see Mathematics as a key to tertiary entry and insist that their children be given the opportunity to do the subject at the highest level possible even going against the school’s advice on the matter.

Therefore, high school Maths’ teachers must teach almost all students for all their years at high school irrespective of their innate ability in the subject.

This trend will not go away and high school teachers need the help of primary teachers to prepare their students to enter the rigours of high school Mathematics.

This article is written based on my experience as both a high school Maths teacher and as a Head of Mathematics who often had to advise parents on what was best for their students in the subject. Much of what I write here was presented to primary school teachers in a workshop on the topic.

Most, if not all of the points I make in this article, will be known to experienced primary school teachers so it is aimed more at those new to the profession.

Mathematics is a subject discipline where the student must develop his/her understanding of Mathematics. Learning rules and procedures can take the student only so far. It will not help in the modern world of real life Maths problems in unfamiliar contexts.

To help prepare students for high school Maths, upper primary school teachers need to attempt to develop the following within their students.

1. A work ethic and one which is self-motivating. Often, students in Mathematics will need to work alone and unaided.
2. A homework ethic. The speed of teaching the syllabus requirements in high school is dictated by outside authorities. This means that the teacher must cover a mandated syllabus in a specific time. For the student, this means that homework is an essential part of the learning process if he/she is to keep up with the pace of teaching.
3. A study ethic. It is important that students learn that homework does not equal study.
4. A belief that all students can do some Maths.
5. An understanding that Maths is an essential part of everyday life and we all do Mathematical things successfully every day, often automatically.
6. A belief in students that asking questions in Maths is a ‘cool’ thing to do.
7. A belief in students that Maths is unisexual, not just for the boys.

Below is a list of what I call essential preparation that is not directly Mathematical but will assist students greatly in their study of Mathematics as well as other subjects.

Students should be taught:

• Study skills
• How to be powerful listeners
• Checking procedures
• Estimation as a checking device
• Various problem solving techniques
• An effective setting out procedure
• That the answer only is not enough. The students must explain in written Mathematical form how they achieved their answer.
• That there is often more than one way to solve a problem
• An understanding of order convention
• Examination technique

Communicating mathematically is a skill that needs to be taught. It involves students being taught the following:

1. The correct use of Mathematical terms including their spelling;
2. Correct use of all Mathematical symbols;
3. Logical setting out;
4. Justification of each step where necessary;
5. Logical reasoning;
6. The use of neat and clear figures, accurate and appropriate diagrams;
7. To work vertically down the page to allow ease of checking and the elimination of errors in copying;
8. The translation from one form of expression to another, e.g. numerical/verbal data to diagrams/tables/graphs/equations, and
9. Correct and appropriate use of units, e.g. in area, volume and so on.

Lastly, you can give your students a taste of high school classes by doing the following. (You might call these suggestions an Action Plan).

• Set your classroom up with desks in rows and teach a number of “Chalk and Talk” lessons.
• Insist that students work on their own while doing Maths exercises in a quiet environment.
• Use textbook exercises.
• Run some formal, timed examinations in a formal classroom setting.
• Do regular problem solving exercises. Ones in unfamiliar contexts so they get accustomed to the idea that problem solving is an everyday event, not just one that comes up in assessment.

As I alluded to in the title of this article, this is a high school Maths teacher’s wish list. Whatever you can do as a primary teacher to help develop this wish list would be greatly appreciated by Maths teachers but more importantly will help students to step into the rigours of high school Maths more confidently.

# Does Elementary School STEM Career Day Make a Difference?

Stem Career Day at Manchester Elementary in Manchester, Maryland was a day that held excitement and anticipation. The idea was conceptualized in the early part of December. How do we find a variety of STEM Careers to show students the wave of the future? We surveyed parents about their jobs and their willingness to take a day off of work to share their careers’, education, day-to-day requirements, and successes and failures within their lives. We received an eclectic response which included: Hazardous Waste Management, Financial Analyst, Global Production Executive, Software Licensing Manager and IT Program Manager and Nurse to name a few. With these parents willing to come in for the day, the schedule for third, fourth and fifth graders was created and set in place for a February Date.

In December we wanted to get an idea how the students felt about Careers in Science, Technology, Engineering and Mathematics before the day of the event. We sent a pre-survey to all 3rd, 4th and 5th grade teachers to be read aloud to the students, but completed with only the students’ prior knowledge regarding STEM Careers. We also sent a post-survey immediately after the day was completed. In some cases the teacher gave the post-survey the same day as the day of the event.

Our Day was a high-light on the county’s CETV Spotlight on Youth and there were positive comments from students, teachers and parents after the event.

Issues and Trends

The need for STEM careers in 2020 will increase from today’s needs by approximately 50% (Department, 2015). Issues, Trends and Need for community involvement in schools is an issue for today’s school agendas. There are numerous businesses, companies and associations in the areas surrounding schools that have an aspect of STEM (Science, Technology, Engineering and Mathematics) in their day to day processes. But are the elementary schools benefiting from these community connections?

Early exposure to STEM careers does make a difference (Dejarnette, 2012). Many programs are provided at the middle school and high school level, but exposure at the elementary level is necessary to impact students’ perceptions and dispositions. In middle school there is a direct link between perceptions and career interest. By exposing students at an early age their positive perceptions increase (Buldu, 2006). Studies continue to show an increase in positive perception to STEM careers when students are introduced and exposed to 21st century careers. When students in sixth grade are exposed to STEM Professionals a measurable improvement was recorded towards these types of jobs. Pre and post surveys showed a 10% positive increased to the question, “When I grow up I want to be an engineer.” (Bouvier, 2001). Interest must increase in all students including students from groups traditionally underrepresented in STEM-students of color, women, and students from low socioeconomic backgrounds (National, 2011). The President’s Committee of Advisors on Science and Technology assert that improving the interest and attitude toward these careers among young students is as important as increasing the overall level of academic proficiency and attitude in STEM academics. (PCAST, 2010).

Results

The survey was designed to be anonymous. We emphasized to students we wanted their unbiased answers to the questions. The survey began with, “When I grow up I would like to be:” Students wrote down their top 5 choices. Pre-STEM Career Day 24% of students wrote down Careers. (STEM Careers tallied were any job that had correlations to engineering, computer science (technology), or additional science careers.) Post-surveys revealed that percentage was at 33%. As trends and issues would suggest we need to make sure there is particular interest in educating girls at the elementary level in a variety of STEM Careers. The pre-survey showed that 24% of girls and boys listed these Careers. Post-survey results differed from overall results showing that girls listing STEM Careers increased to 33%, boys increased to 39%.

Pre-Survey Results:

“When I want to grow up… ” Overall – 24% Girls – 24% Boys – 24%

Post-Survey Results: Overall – 33% Girls – 33% Boys – 39%

• All percentages have been round to the nearest whole percentage.

Students were also given a rating scale for questions that would determine how they felt about these Careers.

1. I think I could have a STEM Career.

2. I see how STEM careers effect the world today.

3. I think I could be successful in my STEM education.

4. I see how technology is used in STEM careers and I think, “I could do that!”

5. I think I would like to be a Scientist / Engineer when I grow up.

6. I think I could create something important for the world.

The results of two of these question show an interesting result. Although only 24% of girls chose Disagree or Strongly Disagree to having a STEM Career, 49% chose Disagree or Strongly Disagree to becoming a Scientist or Engineer. The boys had a different result. Only 15% chose Disagree or Strongly Disagree to having a STEM Career, but a much larger portion, 52% chose Disagree or Strongly Disagree to becoming a Scientist or an Engineer. This may be due to specific choices for STEM Careers in technology fields exclusive of science or engineering. Part of the education we should be sharing in the classroom is how much technology there is in both science and engineering. Diversifying these careers so that students see the “big picture” in science and engineering is a next step in our educational process.

Conclusion

What can be done at Manchester Elementary School to increase STEM Career awareness? We will continue to provide a STEM Career Day for our school. Next year we will prepare to take on the entire school. The initial planning is to include primary classes with a half-day event with the theme being a “hands-on” day. Intermediate students would have the discussion groups delivered last year, but also include an additional hands-on aspect to the day. When the teachers were surveyed regarding STEM Career Connections they made with their curriculum lessons many teachers limited the number of careers discussed that very closely aligned to the lesson they were teaching. Ex. Teaching Weather – Career Connection, Meteorologist. When in truth teachers could explore Climatologist, Environmentalist, Hydrologist, Information Technology, and Electronic Maintenance. As teachers it is our job not only to teach the lesson, but provide real world connections. Real world connections lead us directly to the world around us and the careers that will be available to the graduates in the 21st century.

References:

Buldu, M. (2006). Young children’s perceptions of scientists: A preliminary study Educational Research, v48 n1, 121-132.

DeJarnette, N. (2012). America’s children: Providing early exposure to STEM (science, technology, engineering and math) initiatives. Education, 133(1), 77-84.

Department of Education. (2015). Science, technology, engineering and math: education for global leadership. U.S. Department of Education. Retrieved from http://www.ed.gov/STEM%20%20.

Hawkins, D. (2015, October 15). Biases and stereotypes at school sideline girls in stem. NEA Today, 60-61.

National Research Council. (2011). Successful K-12 STEM Education: Identifying Effective Approaches in Science, Technology, Engineering, and Mathematics. Board on Science Education and Board on Testing and Assessment, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.

PCAST, President’s Committee of Advisors on Science and Technology. (2010). Prepare and Inspire: K-12 Education in Science, Technology, Engineering, and Math (STEM) for America’s Future. Washington, DC: Executive Office of the President.

# 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.

# A Young Teacher’s Guide To Using Textbooks As Teaching/Learning Tools

At the outset, let me say that teaching from a textbook is a ‘no-no’. It would almost always give the impression to students that you don’t know your ‘stuff’ or you are not confident. Here, you need to remember that you will always know or understand more than the vast majority of your students with the topic you are teaching. There will occasionally be gifted students in your class who will understand it all. Use them as tutors/mentors for fellow students. They will often express difficult ideas in ways better understood by their classmates.

However, since textbooks can be expensive, it is important that they be used often and effectively. A good textbook can be a real asset to you as well as for the student. The author was, most likely, a teacher who has written the book on the basis of their own experience.

A good textbook is an essential tool in learning, helping with consolidation and practice of skills. Therefore, take a great deal of time researching various choices to find the one that best suits your needs. A bad choice becomes an expensive mistake.

The writer spent most of his career teaching Mathematics where a textbook is essential. During his years as Head of Mathematics, he introduced several new syllabuses into his school. Consequently, he and his staff needed to research carefully to find the best available textbooks. Sometimes it took up to two years to test out the available text books. His experience in using textbooks and selecting new ones provided the information for this article.

A good textbook is a:

• Reference guide;

• Source of the basics to learn and/or practice;

• Source of rules or procedures to be learnt;

• Source of exercises used to consolidate new learning and ideas on how to practice new skills; (These need to be carefully graded to allow students enough practice to consolidate the basics and then allow them easy movement forward to the more challenging tasks.)

• Source of revision exercises;

• Source of worked examples; and

• As a backup when you teach new and challenging topics.

It should contain the following:

• Chapter summary of ideas to be remembered;

• Chapter review/test;

• Easy to understand diagrams, graphs, pictures and other illustrations; and

• It may also contain a suggested work program as a guide to the time you need to spend on each topic.

Teach your students how to use the textbook effectively as it can be an efficient learning tool at home. Spend a lesson looking at the following list of sections in the textbook, explaining how to use each part.

(a) Contents page

(b) Index

(c) Explanations of new work

(d) Worked examples

(e) Learning work

(f) Exercises to do

(g) Skill practice

(h) Chapter review

(I) Chapter test/diagnosis

(j) Problem solving

(k) Extension work

(l) BOB, back of book – answers

Give students a work outline to go with the textbook. In it, detail the basic exercises the students need to do to gain an understanding of the subject. This is the minimum requirement only. However, specify what needs to be done to extend this understanding to gain the best marks possible.

Insist that students have their textbook with them every day in class. If you know that you will definitely not be using it next lesson, tell the class not to bring it. (Students appreciate not having to carry heavy books unnecessarily.) However, don’t forget to remind the class to bring it to the next lesson. You need to be careful with this process in lower level classes especially in junior high school classes.

When you know that you will use a textbook in a lesson, insist it remains closed until you have finished the teaching part of the lesson unless you plan to refer to it as part of your teaching strategy. It is important to write the number/s of the page/s you intend to use on the board. Then, don’t start referring to what the students must do until you are sure everyone is on the same page. This is also true when you set work from a textbook for the students to do.

Some teachers, particularly, in lower high school classes with less able students, have extra textbooks with them or photocopies of the relevant pages to ensure all students will be able to do the planned work.

In the end, your text must be:

• Teaching tool;

• Teaching aid; and a

• Learning tool

# Can Sudoku Help Students With Math?

Sudoku is a number puzzle game that, on the surface, should be ideal for kids to practice their math skills. Yes and no. Although students manipulate numbers to solve the puzzle, they don’t have to add, subtract, multiply, or divide any of these numbers. So how does playing Sudoku help with math?

First, math is more than arithmetic skills. Many of the skills necessary to succeed in math go beyond arithmetic, and sometimes these skills aren’t practiced enough at school or home. Some of these skills are in use when playing Sudoku. Let’s take a look.

These are some of the skills that Sudoku can help with:

Sorting. The challenge of Sudoku is to arrange the numbers 1 through 9 across every row and column of the grid, along with those same numbers in smaller 3 x 3 grids within the larger grid. No number can appear more than once along any row, column, or 3 x 3 grid. While completing the puzzle, a student needs to sort, arrange, and rearrange these numbers. This requires a level of organization and concentration that has benefits beyond math.

Visualization. While new Sudoku players will write, erase, and rewrite numbers as they solve the puzzle, the more adept player can visualize an arrangement of numbers before writing anything down. Visualization is such an important skill in math. Whether it’s visualizing a geometric transformation or the next step in the solution of an equation, this skill, so underrated, will provide students with a great deal of confidence as they solve problems. Visualization is a higher-order thinking skill.

Problem Solving. A student who completes a Sudoku puzzle has solved a fairly intricate problem. The average Sudoku puzzle is far more complex than a typical word problem in arithmetic, algebra, or geometry. While students may not believe this, most word problems focus on specific skills. Deciphering the text to get at the math concept is the same process as solving a puzzle. In addition, Sudoku encourages students to use these math problem solving strategies: guess and test, look for a pattern, solve a simpler problem, and use logical reasoning. Use Sudoku to enhance these problem solving skills in a low-stakes way.

By far, the most important facet of playing Sudoku is that students will have fun with numbers. This goes a long way toward removing math anxiety, or keep it from ever taking hold. Developing a facility with numbers is a great stepping stone to working with abstract concepts like variables and equations.

# Teachers – Formative Assessment – Informal Assessment of Students’ Mathematical Dispositions

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 third 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.

Disposition is defined as “one’s usual mood; temperament, a habitual inclination, tendency.” With teenagers, it is not always easy to determine their disposition or temperament regarding anything, including mathematics. Too often, if their reference group has decided that “school is not cool,” then it is mandatory that they use all their body language and facial expressions (and sometimes words) to indicate their disdain for our beloved subject. We mustn’t accept this at face value (no pun intended). The ideas in this article will allow you to determine your students’ mathematical dispositions (sometimes without their knowing it!).

The first idea I want to offer came about almost by a fluke. Another teacher and I were creating an assessment instrument for some research we were conducting. We had about half a page left on this eight-page instrument – and didn’t want to waste the paper – so we decided to pose the prompt shown below. It turned out to be the best question of the whole assessment! Consider using this prompt at the beginning of the year – and then several other times throughout the school year to get a sense of the changes taking place.

As a math student in this class, I rate myself on the following scale (put an X on the scale where you rate yourself). 1 = Probably the worst in the class; 5/6 = Not too bad; not too good; 10 = Totally awesome! Maybe the best in the school

The reason I rated myself as a/an ____ on the scale above is because:

I have found that adolescent students are willing and able to be more truthful when asked to write than when asked to share their thoughts publicly. For this reason, I use learning logs as often as possible to learn about students’ dispositions toward mathematics. The term “learning log” is not one that I originated, but it is one that fits my philosophy of how writing looks in the mathematics classroom. The first part of the phrase “learning log” states the purpose of the writing: learning. The second part of the phrase “learning log” connotes a particular format, that is, running commentary. A log is not meant to be a polished piece of writing, taken through draft after draft. Commander and Smith (1996) define the purpose of learning logs as “reflections on specific cognitive aspects of learning…. (emphasizing) the connection or personal engagement with academic skills and techniques” (p. 447).

Using learning logs provides you with a variety of ways to assess students’ attitudes, beliefs, and stereotypes about mathematics. The following are some writing prompts I’ve found useful:

• What does a mathematician look like?
• My ability to do math is…
• When I am in math class, I feel…
• Mathematics has good points and bad points. Here’s what I mean…
• I study, I pay attention, I take notes, I read my math book, but I still don’t get math. True or False? Explain your answer.

Students’ answers to questions such as these provide insight to the teacher as s/he plans instruction. Ignoring students’ dispositions towards mathematics is done at teachers’ – and students’ – peril.

It’s not in anyone’s best interest to think that formative assessment is something that is ‘added on’ to our already full curriculum. Formative assessment is part of good teaching. There should be a seamlessness between instruction and assessment. Keep in mind that the word “formative” comes from the Latin word meaning “shape or form.” Formative assessment has as its purpose to shape upcoming instruction. Use and/or modify the ideas offered in this article. You will find that your instruction is more targeted and more effective. Then, design more ideas of your own – and share them with as many other teachers as possible.

# Three Tips On Ordering and Comparing Fractions

Comparing and ordering fractions

All fractions are not same in value. One fraction may be smaller than the other fractions and it may be larger than some other fractions. Hence, kids need to know comparing fractions. Comparing can be subdivided into three sections. So kids need to know three tricks to learn this skill.

Trick Number 1:

First trick to compare fractions is to see if they have got the same numerators. If the numerators are same then the fraction with the largest denominator is smallest. For example; consider the following fractions:

3/5, 3/4, 3/8 and 3/7

As all of the above fractions have the same numerator (3), so to compare them we need to compare their denominators. The largest denominator makes the fraction smallest, therefore 3/8 is smallest of all and 3/4 is the largest. Let’s rewrite all of the fractions in an order from smallest to largest as shown below:

3/8, 3/7, 3/5 and 3/4

The above order (smallest to largest) is also known as ascending order.

Trick Number 2:

The second trick is same easy as the first one. This trick is about comparing fractions, when they have same denominators. When the denominators are same, then the fraction with the smallest numerator is smallest and one with largest numerator is the largest. For example;

Consider we want to compare 3/9, 1/9, 7/9 and 2/9; write them in ascending order.

Look at the given fractions, all of them have the same denominator (9). So, 1/9 is the smallest because it has the smallest numerator and 7/9 is the largest with largest numerator. Below they are written in ascending order.

1/9, 2/9, 3/9 and 7/9

Trick Number 3:

Above two tips explain the comparing fractions with either same numerators or same denominators. But most often the kids are asked to compare and order fractions with different numerators and denominators.

In such a case they need to make denominator of all the fractions same. To do this they need to know the least common factor (lcm) of all the denominators also known as least common denominator (lcd).

Consider the following example on comparing fractions:

Write the following fractions in descending order (largest to smallest)

2/3, 1/4, 5/6, 3/4 and 1/2

Solution: Look, most of fractions got different denominators. Write all the denominators as shown below and write first six multiples of all of them.

2 = 2, 4, 6, 8, 10, 12 3 = 3, 6, 9, 12, 15, 18 4 = 4, 8, 12, 16, 20, 24 6 = 6, 12, 18, 24, 30, 36

Now, look at the factors of all the numbers and find the smallest and common in all, which is 12 in this case. Hence the lcm or lcd is 12. The next step is to rewrite all of the fractions into equivalent fractions with denominator as 12. This step is shown below:

2/3, we need to multiply its denominator (3) with 4 to change it to 12. But to keep the value of the fraction same, don’t forget to multiply the numerator (2) with the same number 4. Let’s do it;

(2 x 4)/(3 x 4) = 8/12

Similarly write all the fractions with denominator equal to 12 as shown below:

1/4 = (1 x 3)/(4 x 3) = 3/12 5/6 = (5 x 2)/(6 x 2) = 10/12 3/4 = (3 x 3)/(4 x 3) = 9/12 1/2 = (1 x 6)/(2 x 6) = 6/12

Now all the fractions have been written into equivalent fractions with same denominator 12 and it’s easy to compare these. Write all the equivalent fractions in descending order (largest to smallest)

10/12, 9/12, 8/12, 6/12 and 3/12

But these are not the fractions asked to be compared. So, this is not our answer, but now it’s very easy to write the original fractions in the required order by looking at above order. We know 10/12 is equal to 5/6 and 3/12 is equal to 1/4 hence write the original fractions in order

5/6, 3/4, 2/3, 1/2 and 1/4

Finally, it can be said that to compare and order fractions, kids need to keep above three tips in mind. Of course, the knowledge of least common multiple (lcm) is the key to compare two or more fractions with different denominators.

# 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.