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.

The Pros and Cons of Playing Fun Games to Make Math Fun

Sometimes it feels as if students of the 21st century expect to be entertained instead of being taught. Teachers worry that bringing math games into their classroom might just cater to that instead of promoting healthy learning. Playing fun games becomes a problem when they are used as a reward or a game for the sake of a game and not incorporated into the lesson. This article discusses the pros and cons of developing effective ways to make math fun by using fun games as part of their daily lessons.

Here are the Pros of using fun math games as part of every day lessons:

  1. The children think they are getting away with something, when they are really reviewing and reinforcing facts and concepts. You can keep their interest while working them hard.
  2. Students are not fools, they know they still are doing work, but given the choice would rather do a fun activity that boring drills. Motivation is key to learning and games are motivating!
  3. Playing fun games builds cooperative learning when breaking down into small groups to play the games. Games encourage positive interaction between students.
  4. Multisensory reinforcement with seeing, saying and doing is built in to the game playing experience. You “see, hear and say” while playing an educational game, and manipulate pieces and often cards as well.
  5. Playing some full classroom games to make math fun gives students a break and allows them to move around a bit; It activates their sensors!

Here are some Cons to using math games in your every day lessons:

  1. Games create too much emphasis on winning. It just takes one poor loser to ruin the activity for the whole class.
  2. It takes too much time to learn the rules and to get set up into small groups.
  3. Your classroom does not have movable desks to make some mini tables to play these board games.
  4. There will be an active noise level, a buzz in the room and you believe silence is golden.
  5. Teacher preparation time is increased initially to print out the games, and laminate if necessary.You simply do not have the time to make these activities.

Once materials are created or purchased and then organized in the most convenient way, and once the basic rules of the games are learned, children will be able to move desks around and get themselves ready to play fun games activity in no time. They want to participate, so they will move quickly. There is no better reinforcement process than multisensory activities. Using these kind of printable games and activities will not only give some multisensory reinforcement, but will also provide great ways to make math fun and educational at the same time. You cannot go wrong.

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”

You ask: “Why?”

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.

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.

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.

5 Tips to Buy Children’s Apps

With the festival season not very far away, many children are about to receive electronic gifts from their family and relatives. The good news is that apps have emerged as a promising tool to support literacy in general, and science, mathematics and life skills. The challenge for parents is to pick up the right app, more so, if the child has autism spectrum disorder or has special needs.

#1 The education and entertainment combo

Kids learn when they’re engaged. Educational apps like Just Match or Math on the Farm forge a perfect balance between learning and engagement. The Math on the Farm app teaches mathematics skills in a fun way. Here, the child has to answer multiple-choice type questions to score points. The stories in the app are themed on a farm that has flowers, vegetables, domestic animals, and cattle. Bright colors and interactive animation are the highlights of this app. It’s important that the child learns by playing and the Math on the Farm app does just that.

#2 Play with your child

Studies have shown that children learn better if parents join the fun. Take an active role and choose and app that’s likely to hold your kid’s attention. The Just Match app could be perfect for you. This fun educational app teaches matching skills, where you’ll be shown to game figures and an outline which matches only one of them. You’ve to drag and match the figure with the outline. A lively animation will hail your efforts every time you match correctly.

#3 Select appropriate games

Determine whether a fun educational app is correct for your child. Not all four-year old will be equal. So, different apps would appeal to different kids at different times. Ask yourself whether your child will be able to follow the app’s storyline. The touch screen system is a major advancement in the field of communication. Make sure the fun educational app has audio cues and not only words.

#4 Set limits and encourage other playing and learning forms

Well, setting the proper “media diet” is important for your child. It’s almost like balanced food. The more variety, the better it’s for your child. Consider the number of hours the child will spend in front of a screen. A possible rule could be not allowing TV until the homework is complete. The same should apply to a touch screen, unless it’s required in school, which of course is increasingly happening these days.

#5 Download from reliable, trusted sources

Look for established brands that specialize in fun educational apps. Are you comfortable with the app’s characters? Kids imitate popular media characters. Make sure the language and behavior in fun educational apps are appropriate for your kids. Avoid apps that have a lot of violence or are frightening to play. Such apps may have an adverse impact on the child’s mind. The Math on the Farm and Just Match app can fit the bill perfectly. These two apps are sensitive to children’s needs, and are among the best fun educational apps around.

Top Tips on Teaching Times Tables!

Learning times tables can seem daunting to a child, and you can feel daunted just thinking about having to teach your child their times tables! They tend to carry with them negative connotations, the belief being they are generally, boring, difficult and most of all frustrating, but this doesn’t have to be the case!

· Don’t try and bite off more than you can chew! Stick to one at a time and start with the easier ones to encourage progress and to prove to your child that it is achievable and they can do it!

· Rewarding your child’s progress learning might seem like a no brainer! But by making sure encouragement and rewards are constant as your child progresses will encourage them to not only continue, but will also encourage them in other things. They will come to realise if they try and apply themselves to something then it is achievable.

· Use fun games, as a way of learning so it is less boring and more likely your child will get the most out of the time they are learning. There are many games on revision websites, which could benefit your child and will educate them in a fun and effective way.

· Methods of Learning – Everyone learns differently, so its worth while finding out which way your child learns best rather than simply trying to enforce a method on them because if they are struggling this will only discourage them in their efforts and will have more setbacks than actual benefits.

· Learning times tables by route has been a favorite in schools in the past/ Children chanting their times tables over and over again may be affective for some but there are many other methods of learning them which you may not have considered.

· Discuss the patterns you can see so they your child understands how they are coming to the answer they do, rather than just saying what they think you want to hear.

Times tables are the basis on which children learn further math’s skills as they develop and go through school, so it is important they not only learn them off by heart, but also know how they are getting to the answers. This will give them the skills they need to work out other problems they are given in class. The sooner they grasp their times tables the sooner they will be able to work out harder and more complex solutions on their own.

Unfortunately sometimes students can miss out on completely grasping their times tables which can lead to problems in the future when they begin studying for important exams.