# 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.

# 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.

# 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!

# 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.

# Strategies For ‘Attacking’ Maths Problems, A Guide For Students And Teachers

In an earlier article, “How to Answer Questions in a Formal Examination-A Student’s Guide”, I discussed how to answer questions to gain the best possible results in an examination. This article continues that theme but this time in relation to answering questions in a formal Mathematics examination. The strategies mentioned in the previous article should be applied to the Mathematics examination as well as the ones discussed below.

It is important to define what I mean by a problem in Mathematics before you start to study the strategies to solve them.

These problems are almost entirely ‘word’ problems. More often than not, the student needs to use a variety of Mathematical skills or ideas to gain a solution. Often, particularly in the senior years of high school, there will be an unfamiliar context in which to use your Mathematical knowledge. Alternatively, there may be a series of sometimes complex steps necessary to achieve a result. Finally, the answer is not one which is obvious.

Below are a list of strategies, if used together, will help you gain greater success in solving real problems in Mathematics not just ones you have practised. However, remember, if you don’t know your basics in Mathematics then no set of strategies will help you solve the problems.

So strategy Number 1 is and will always be:

“Know all your learning work and procedures as well as you can.”

The remaining strategies are as follows:

2. Remember, everything that you need to solve the problem is in the question itself. (So list what data the problem gives you as your starting point).

3. Checking is a compulsory part of every problem you are to solve. Here is a checking procedure to use:

• It is best to check as you do each step in the problem as this saves time often preventing unnecessary extra work.

• Ensure you have done only what you have been asked to do. Check, in fact, that you have actually answered the question fully.

• Check you have copied down all the data for the question correctly.

• Check that your answer (its size, etc.) fits, in a practical sense, into the scenario/context of the question.

4. Make sure you have been neat, tidy, organised, logical, clear, and concise. This will help you with your checking and allow the examiner/teacher to follow your logic easily.

5. This strategy was mentioned in the first article. It is part and parcel of answering any examination problem, especially in Mathematics. It is: List the steps you need to take, in order, to gain a solution.

Below is an example of what I mean by this strategy in Mathematics.

The Swimming Pool Problem

“How long does it take to fill a swimming pool with a bucket?”

Here is how I teach this strategy:

Step 1: I write the above problem on the board.

When I do this, I ask the students for their reaction:

It will be: “We can’t do it”

Their reply: “There are no dimensions”

Your reply: “You don’t need them. If I gave you them to you what would you do?”

Step 2: Now I have the students write down the steps they would use.

Step 3: Then I discuss the steps the class select and list the steps on the board.

e. g. Find the volume of the pool.

Find the volume of the bucket.

How long does it take to fill the bucket and pour into the pool?

How many buckets of water do I need to fill the pool?

Find total time to fill the pool.

Step 4: Now, I make the point that the above steps do not mention the dimensions of the pool. It doesn’t matter what it measures you still follow the same process.

Step 5: Lastly, I emphasise that a correct answer depends upon the correct steps, i.e. method of solution.

As a student, you can’t learn these strategies overnight and expect that they will ‘come to you’ easily in a formal examination situation. You must practice using them. Make a list of the strategies and have them with you as you try each new problem. Evaluate how well you use them and work to improve those you find hardest to use or are easily forgotten. Look to your teacher for help with this process. Remember, in an examination, be disciplined, write out the list of strategies you will use before you start and use them to solve the problems.

# Tutors and Advice to High School Students

As a teacher of Mathematics and, later in my career, as a head of Mathematics department, I was often asked to recommend a tutor by parents or their students.

This often occurred after a student had been absent for some time from class or when a student needed a pass in Mathematics to matriculate into a particular course at university.

These are the points I made to students:

• Maths tutors can’t do the Maths for you, especially homework and assignments. The tutor is there to guide you to build your understanding of Mathematics*.

• They can help improve your confidence; explain areas that you find difficult but they can’t guarantee success. You have to do the work if you are to improve and succeed.

• They may also be able to discover where you began to have problems and work to fix that. Your parents must make that a priority for the tutor.

• You must note down areas in class where you are failing to understand the concept and ask the tutor to go over those areas. The tutor’s explanation will often provide a different approach to the teacher’s approach to the topic that will help you understand the concept/procedure.

• However, success only comes when you work hard in class and works hard with your maths tutor. One doesn’t replace the other.

• You must continue to engage with your teacher, asking questions and seeking advice when it is needed.

• You must continue to work on set homework diligently and do any work set by the tutor.

• It is also important that you accept the idea of tuition and like or respect your tutor. If you don’t like the tutor or can’t follow his/her explanations, tell your parents and seek a replacement.

• Your parents should seek a report from the tutor on progress made and on the efforts of their student, regularly.

Above all, you must be proactive in seeking to improve your understanding of the subject.

*Mathematics here can be replaced by any subject that requires improvement.

# Is The Use Of Calculators Good Or Is The Myth That Says Calculators Make Students Lazy True?

A calculator is a great tool that allows for the mathematical exploration and experimentation and thus enhances the students understanding of concepts. Before I go into the benefits of the use of calculators in education and how to efficiently use them I would like to, first, state the types of calculators available today.

We can classify calculators into two types. The fist type is a calculator that evaluates expressions. This type is used to replace the manual tedious paper and pencil arithmetic. The second type of calculator is the special functionality calculator for example the graphing calculator, the algebra calculator, the matrices calculator… etc. These calculators are used for exploration of concepts. Each type of calculator can fit in mathematics education in its unique way and needs the syllabuses to be specially written to incorporate it in education.

Recent studies show that calculators are evaluable tools for mathematics educations. Instead of the student spending his time in tedious arithmetic calculations he can spend his time in developing and understanding concepts. Many students in the past have been turned off mathematics because of the long tedious calculations and students who were efficient in these calculations were considered good at mathematics. Little attention was made to the dissolve of concepts. They hardly had anytime left to concentrate on concepts. Today with the use of calculators the students spend all their time understanding concepts and the logic behind mathematics. They can relate the concepts to real life application. The overall education experience became richer. This is why calculators are recommended for all education classes from kindergarten to college.

Some may argue that this way the student may become lazy. The reply to this question is consider you are giving a primary school student a problem that he has 100 dollars and went to the market and bought five items of one commodity for a certain price and three items of another commodity for another price and he paid the 100 dollars then what is the remainder that he will receive. Now what is the mathematical quest of this problem? Is the question here how to do arithmetic multiplication, addition, and then subtraction? Or is the question is that the student should know what is going to be multiplied by what and what is going to be added to what and at the end what is going to be subtracted from what? Of coarse the mathematics of this problem is the procedure he is going to do to find the remainder and not the arithmetic process itself. In the past overwhelming the student with the arithmetic operations made many students miss the idea and the concept behind the problem. Some others did not miss the concept but were turned off altogether from mathematics because of the arithmetic operations.

Here I have to emphasize that it is true that calculators are good for education but still one must know how to make them fit nicely in the education process. Students need to know the arithmetic hand calculations. They must study how to do that manually. When the prime concern of the mathematics problem is how to do the arithmetic students should only use the calculator to check for the answer i.e. to see if it matches his hand calculation.

So the rule for using calculators is that the teacher should check the point of the mathematics problem and the concept it is teaching. If the calculator is doing a lower level job than the concept behind the mathematics exercise than it is fine. However, if the calculator is doing the intended job of the exercise then it should be used only to check the correct answer.

In addition, education books should write examples that use calculators to investigate concepts and teachers should lead students in classrooms to show them how to use these examples with calculators to dissolve concepts.