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Tally Marks

Tally marks are the quickest way of keeping track of a group of five. One vertical line is made for each of the first four numbers; the fifth number is denoted by a diagonal line drawn across the previous four (i.e., from the top of the first line to the bottom of the fourth line). The diagonal fifth line cancels out the other four vertical lines making the entire set represent five.

Tally marks are also known as hash marks and can be defined in the unary numeral system. (A unary operation in a mathematical system is one element used to yield a single result, in this case a vertical line.) These marks are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. They also make it simple to add up the results by simply counting by 5’s. Here is an illustration of what I mean.

  • The value 1 is represented by | tally marks.
  • The value 2 is represented by | | tally marks.
  • The value 3 is represented by | | | tally marks.
  • The value 4 is denoted by |||| tally marks.
  • The value five is not denoted by | | | | | tally marks. For the number 5, draw four vertical lines (||||) with a diagonal (\) line through them.
I have seen many interesting ways to teach tally marks to younger children. Many teachers will use Popsicle sticks so that the students have a concrete hands-on way of making tally marks. Some have even tried pretzel sticks although there is a good chance some will disappear during the lesson. 

But have you ever seen these kind of tally marks?


My husband, who teaches science, received this data collection paper from a student. The students were tossing coins marked TT, Tt, and tt to determine different genetic traits and tallying the results. The ones seen above are Japanese tally marks. (The student lived in Japan.) I was fascinated about how they were made so I asked him to have this student show me the sequence of how to draw the marks.


I'm not sure what they mean or why they are made this way, but if you look at the 2nd mark you will notice that it looks like a "T" for two. The fourth mark sort of looks like an "F" for four, but so does the third one. As you can see, each complete character uses 5 strokes; so, a series of would each represent 5, just like the English ones. However, to be honest, I am at a total lost to what this really means; so, I resorted to the internet. Here is what I learned. 

Instead of lines, a certain Kanji character is used. In Japan, this mark reminds people of a sign for “masu” which was originally a square wooden box used to measure rice in Japan during the feudal period. Here is what the tally marks would look like if we compared the two systems.


The successive strokes of () are used in China, Japan and Korea to designate tallies in votes, scores, points, sushi orders, and the like, much as is used in Europe, Africa, Australia and North America. Tallies beyond five are written like this  with a line drawn underneath each group of five, followed by the remainder. For example, a tally of twelve is written as 正正丅. 

So the next time your visit Japan or go to a Japenese restaurant to order Sushi, look for the tally marks as the waiter takes your order.

Put a LID on It!

There are so many things we consider to be trash, when in reality, they are perfect treasures for the classroom. One that I often use is plastic lids from things like peanut canisters, Pringles, coffee cans, margarine tubs, etc.  These lids can be made into stencils to use when completing a picture graph.

Students must first of all understand what a picture graph is.  A pictorial or picture graph uses pictures to represent numerical facts. Sometimes it is referred to as a representational graph. Each symbol or picture used on the graph represents a unit decided by the student or teacher. Each symbol could represent one, two, or whatever number you want.  This type of graph is used when the data being gathered is small or approximate figures are being used, and you want to make simple comparisons.

Here is what you do to make ready-made picture graph stencils.
  1. Choose the size of lid that you want and turn it over. Then trace a pattern on the plastic lid. Make sure you are using the bottom of the lid so the rim does not interfere when the children use it to trace. 
  2. To make the stencil, cut out the pattern using an Exacto knife. You might choose to do zoo animals: a zebra, a lion, a bear, an elephant or a giraffe. 
  3. Have a large sheet of paper ready with a question on it such as: “What is your favorite zoo animal?” 
  4. The students then select the stencil (picture) that is their favorite animal and trace it in the correct row on the graph. 
Below is a sample of this type of graph. It is entitled, What is Your Favorite Season? A leaf is used for fall; a snowflake represents winter; a flower denotes spring, and the sun is for summer. Notice at the bottom of the graph that each tracing will represent one student.
You could craft stencils for modes of transportation, geometric shapes, pets, weather, etc. The list is infinite. But what if you don't want to or don't have time to make all of those stencils? Then save the strips that are left when you punch out shapes using a die press. They are instant stencils!

If you are interested in additional graphing ideas, check out the resource entitled: Graphing Without Paper or Pencil. You might also like Milk Lid Math. This four page handout contains numerous math activities that utilize a free manipulative. 
 

The Pros and Cons of Testing

Tests are here to stay whether we like it or not. As I read various blogs, I am finding more and more teachers who are frustrated over tests and their implications. I am seeing many of my former student teachers leave the teaching profession after only two or three years because of days structured around testing.

High stakes tests have become the “Big Brother” of education, always there watching, waiting, and demanding our time. As preparing for tests, taking pre-tests, reliably filling in bubbles, and then taking the actual assessments skulk into our classroom, something else of value is replaced since there are only so many hours in a day. In my opinion, tests are replacing high quality teaching and much needed programs such as music and art. I have mulled this over for the last few months, and the result is a list of pros and cons regarding tests.

Testing Pros

  1. They help teachers understand what students have learned and what they need to learn.
  2. They give teachers information to use in planning instruction. 
  3. Tests help schools evaluate the effectiveness of their programs. 
  4. They help districts see how their students perform in relation to other students who take the same test. 
  5. The results help administrators and teachers make decisions regarding the curriculum. 
  6. Tests help parents/guardians monitor and understand their child's progress. 
  7. They can help in diagnosing a student's strengths and weaknesses. 
  8. They keep the testing companies in business and the test writers extremely busy. 
  9. Tests give armchair educators and politicians fodder for making laws on something they know little about.  
                                           **The last two are on the sarcastic side.**

Testing Cons
  1. They sort and label very young students, and those labels are nearly impossible to change.
  2. Some tests are biased which, of course, skew the data. 
  3. They are used to assess teachers in inappropriate ways. (high scores = pay incentives?) 
  4. They are used to rank schools and communities. (Those rankings help real estate agents, but it is unclear how they assist teachers or students.) 
  5. They may be regarded as high stakes for teachers and schools, but many parents and students are indifferent or apathetic. 
  6. They dictate or drive the curriculum without regard to the individual children we teach. 
  7. Often, raising the test scores becomes the single most important indicator of overall school improvement. 
  8. Due to the changing landscape of the testing environment, money needed for teachers and the classroom often goes to purchasing updated testing materials. 
  9. Under Federal direction, national testing standards usurp the authority of the state and local school boards. 
  10. Often they are not aligned with the curriculum a district is using; so, curriculum is often changed or narrowed to match the tests. 
Questions That Need to Be Asked
  1. What is the purpose of the test?
  2. How will the results be communicated and used by the district? 
  3. Is the test a reflection of the curriculum that is taught? 
  4. Will the results help teachers be better teachers and give students ways to be better learners?
  5. Does it measure both a student's understanding of concepts as well as the process of getting the answer? 
  6. Is it principally made up of multiple choice questions or does it does it contain any performance based assessment? 
  7. What other means of evaluation does the school use to measure a child's progress? 
  8. Is it worth the time and money?

How Many Sides Does A Circle Have?

Believe it or not, this was a question asked by a primary teacher.  I guess I shouldn't be surprised, but in retrospect, I was stunned. Therefore, I decided this topic would make a great blog post.

The answer is not as easy as it may seem. A circle could have one curved side depending on the definition of "side!"  It could have two sides - inside and outside; however this is mathematically irrelevant. Could a circle have infinite sides? Yes, if each side were very tiny. Finally, a circle could have no sides if a side is defined as a straight line. So which one should a teacher use?

By definition a circle is a perfectly round 2-dimensional shape that has all of its points the same distance from the center. If asked then how many sides does it have, the question itself simply does not apply if "sides" has the same meaning as in a rectangle or square.

I believe the word "side" should be restricted to polygons (two dimensional shapes). A good but straight forward definition of a polygon is a many sided shape.  A side is formed when two lines meet at a polygon vertex. Using this definition then allows us to say:
  1.  A circle is not a polygon.
  2.  A circle has no sides.
One way a primary teacher can help students learn some of the correct terminology of a circle is to use concrete ways.  For instance,  the perimeter of a circle is called the circumference.  It is the line that forms the outside edge of a circle or any closed curve. If you have a circle rug in your classroom, ask the students is to come and sit on the circumference of the circle. If you use this often, they will know, but better yet understand circumference.

For older students, you might want to try drawing a circle by putting a pin in a board. Then put a loop of string around the pin, and insert a pencil into the loop. Keeping the string stretched, the students can draw a circle!

And just because I knew you wanted to know, when we divide the circumference by the diameter we get 3.141592654... which is the number π (Pi)!  How cool is that?

Only $1.90


If you are studying circles in your classroom, you might like this crossword puzzle. It is a great way for students to review vocabulary.