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Helping Students to Think Mathematically

What does it mean to think mathematically?  

It means using math vocabulary, language and symbols to describe or interpret mathematical concepts, procedures and to discover relationships among ideas.  Therefore when a student problem solves, they use previous knowledge, skills, and understanding of concepts to solve a problem.  This process might include formulating problems, applying a variety of strategies, or interpreting results.

What can we do to help our students become better mathematical thinkers?  

We can teach and model problem solving strategies.  We can remember and plan our lessons to involve the three stages of conceptual development: concrete, pictorial, and abstract. We can have the students talk or write about how they got an answer either with the class or with a partner.  We can use writing in the mathematics classroom (such as math journals) to allow the students to practice expository writing and show their understanding.  We can exhibit math word walls and have the students use the glossary in their book to write and define terms. 

We can also create a positive and safe classroom atmosphere for problem solving... 

By being enthusiastic and allowing the students to take risks without consequences.  By emphasizing the process as well as the answer, the students may be willing to try unconventional or different ways to solve the problem.  I always tell my students that there isn't just one right way to get an answer which surprises many of them.  In fact, this is one of the posters that hangs in my classroom.

  
As math teachers, let's continue to emphasize problem solving so that all students will acquire confidence in using mathematics meaningfully. But most of all, let's have fun while we are doing it!

$8.00 on TPT
If you are interested in having a math dictionary for your students, check out A Simple Math Dictionary. It is a 30 page student dictionary that uses easy and clear definitions as well as formulas and examples so that students can learn and understand new math words without difficulty or cumbersome language. Most definitions include diagrams and/or illustrations for the visual learner. Over 300 common math terms are organized alphabetically for quick reference.


Defeating Negative Self-Talk

One of the biggest problems with the college students I teach is their math anxiety level. Math anxiety is the felling of tension and anxiety that interferes with the manipulation of numbers and the solving of math problems during tests. In other words - mathphobia! This is a learned condition, not something they are born with and is in no way related to how smart a student is. In my Conquering College class, we have been looking at causes for anxiety which include bad experiences, teacher and/peer embarrassment and humiliation, or being shamed by family members. We've been looking at ways to reduce math anxiety such as short term relaxation as well as long term techniques and managing negative self-talk.

For many of my students, a song is the best way of remembering. I found an old nonsense song by Roger Miller entitled, You Can't Roller Skate in a Buffalo Herd. First we read over the words. Next we watched a video on You Tube and then we actually sang the song. I replaced the words "But Ya can be happy if you've a mind to" with "But ya can be positive if you put your mind to it."

You Can’t Roller Skate in a Buffalo Herd

By Roger Miller

Ya can’t roller skate in a buffalo herd,
Ya can’t roller skate in a buffalo herd,
  Ya can’t roller skate in a buffalo herd,
*But ya can be positive if ya put your mind to it.

 Ya can’t take a shower in a parakeet cage,
 Ya can’t take a shower in a parakeet cage,
 Ya can’t take a shower in a parakeet cage,
 *But ya can be positive if ya put your mind to it.
All ya gotta do is put your mind to it,
Knuckle down, buckle down,
Do it, do it, do it!

Well, ya can’t go swimmin’ in a baseball pool,
Well, ya can’t go swimmin’ in a baseball pool,
Well, ya can’t go swimmin’ in a baseball pool,
 *But ya can be positive if ya put your mind to it.
Ya can’t change film with a kid on your back,
Ya can’t change film with a kid on your back,
Ya can’t change film with a kid on your back,
 *But ya can be positive if ya put your mind to it.


Ya can’t drive around with a tiger in your car,
Ya can’t drive around with a tiger in your car,
Ya can’t drive around with a tiger in your car,
*But ya can be positive if ya put your mind to it.
All ya gotta do is put your mind to it,
Knuckle down, buckle down,
Do it, do it, do it!

Well, ya can’t roller skate in a buffalo herd,
Ya can’t roller skate in a buffalo herd,
Ya can’t roller skate in a buffalo herd,
 *But ya can be positive if ya put your mind to it.
Ya can’t go fishin’ in a watermelon patch,
Ya can’t go fishin’ in a watermelon patch,
Ya can’t go fishin’ in a watermelon patch,
*But ya can be positive if ya put your mind to it.

 Ya can’t roller skate in a buffalo herd,
 Ya can’t roller skate in a buffalo herd,
  Ya can’t roller skate in a buffalo herd,
  *But ya can be positive if ya put your mind to it.

So how did the lesson go? Let's just say that my college students were having so much fun singing the song that the secretary had to come and shut our classroom door. And the response from the students the next day, "I can't get that song out of mind!" Maybe negative self-talk has finally met its match!


Using Mnemonic Techniques

In my college class entitled Conquering College, we have been working on ways to remember for tests. Of course, mnemonic devices came up. Mnemonics connect new learning to prior knowledge through the use of visual and/or acoustic cues. Such strategies assist students in remembering and recalling larger pieces of information for tests. Included in mnemonics are acronyms, initialism, acrostics, rhyme, rhythm and song and association in addition to visualization using the loci and peg systems. Let's look at four of these categories.


1) Acronyms - A word formed from the first letters of each one of the words in a phrase.
  • HOMES – The names of the 5 Great Lakes – Huron, Ontario, Michigan, Erie, Superior 
  • ROY G. BIV – The colors in a rainbow – Red, Orange, Yellow, Green, Blue, Indigo, Violet 
  • SCUBA - When you’re scuba diving, you’re using a “self-contained underwater breathing apparatus.” 

2) Acrostics – Sentences created from the first letters of key words.
  • Please Excuse My Dear Aunt Sally – for the order of operations 
Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction 

    **I personally prefer the phrase: Pale Elvis Meets Dracula After School. 
  • My Very Earthly Mother Just Sliced Up Neptune.  – the planets in order from largest to smallest: 
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune 

   **I particularly like this one since Earthly gives you a clue that the third planet is earth and Neptune is listed last. This means you only have to know 6.


Free Resource
3) Rhyme, Rhythm, Song – poems, limericks or silly songs – These work well for auditory learners.
  • I before E, except after C and in sounding like A as in neighbor and weigh.
  • In 1492, Columbus sailed the ocean blue.
  • Twinkle, twinkle little star; circumference is 2 π r.       (I actually sing this for my students!)


4) Association – finding a common element. The association is usually coincidental.
  • Litmus Paper: Blue = Base – both begin with “B”. 
  • Arteries: Artery = Away – both begin with “A”. 
  • The principal is my PAL. Helps to distinguish from principle. 
  • Affect = Action (a verb) Helps to separate it from effect which is a noun.
These ideas plus many more are in a free resource called Mnemonic Techniques found on Teachers Pay Teachers. All you have to do is download it!


Using Number Tiles to Problem Solve in Math

Math Activities for Grades 5-8
I prefer using hands-on activities when teaching math. One of the most successful items I have used is number tiles. Because number tiles can be moved around without the need to erase or cross out an answer, I have discovered that students are more at ease and more willing to try challenging activities. There is something about not having a permanent answer on the page that allows the student to explore, investigate, problem solve, and yes, even guess.

I have created several number tile booklets, but the one I will feature today is for grades 5-8. It is a 23 page booklet containing 15 different math problem solving activities that range from addition and multiplication, to primes and composites, to exponent problems, to using the divisibility rules. Since the students do not write in the book, the pages can be copied and laminated so that they can be used from year to year. These activities may be placed at a table for math practice or as a center activity. They are also a perfect resource for those students who finish an assignment or test early. Use these activities to reteach a concept to a small group as well as to introduce a new mathematical concept to the whole class.

Free Resource
Students solve the Number Tile Math Activities by arranging ten number tiles, numbered 0-9. Most of the number tile activities require that the students use each tile only once. The number tiles can be made from construction paper, cardboard, or square colored tiles that are purchased.  (How to make the number tiles as well as storage ideas is included in the handout.) Each problem is given on a single page, and each activity varies in difficulty which is suitable for any diverse classroom. Since the students have the freedom to move the tiles around, they are more engaged and more willing to try multiple methods to find the solution. Some of the problems will have just one solution while others have several solutions. These activities are very suitable for the visual and/or kinesthetic learner.

A free version for each of my number tile resources is listed below. Just click on the link to download the freebie.

"Sum" More Quick Tricks

Sometimes, my students think, I am a magician who pulls answers out of a hat. Over the years, I have learned that mathematicians are ingenious people who are always looking for quick and easy ways to do things. Maybe that's why we now have graphing calculators and computer programs to figure taxes.
I have a friend who teaches math on the college level in North Carolina. In fact, we have been friends since 6th grade, but that's another story. When she read one of my posts, she shared a trick for quickly finding a sum. Her trick has to do with a sequence that begins with any number, with any number of terms as long as they are separated by the same amount. For instance, the series below is a six number sequence with a difference of two between each number.
Here is what you do to quickly to find the sum. Add the first and last terms. 5 + 15 = 20. Now multiply by the number of terms which in this case is 6. 20 x 6 = 120 Finally, divide by 2. So, mentally this is what it would look like.


Now, how many of you went back to add up 5 + 7 + 9 + 11 + 13 + 15? Did you get the answer of 60? Isn't it amazing!?! Maybe math teachers are magicians after all!

   

Quick Times - Multiplication Tricks

I am always looking for different strategies when working with my remedial college students since many of the ways they were taught to do math aren't working for them.  I came across this "Quick Times" method and thought it would be another approach I could share with my mathphobics for multiplying.  They love anything that is different, quick and makes them look astute when doing mathematics.

Let's assume we have the multiplication problem of 41 x 12.  In the Quick Times method, first start by multiplying the first digit of 41 by the first digit of 12 to get the first digit of our answer.  We then multiply the second digit of 41 by the second digit of 12 as seen below to get the last digit of our answer (the ones place).

Now we need to find the middle digit of the product.  This is done by multiplying the outside digits, then the inside digits, and adding those two products together as shown below.

This quick method will only work when multiplying two digit numbers by two digit numbers, but it does cause the students to do mental math.  My students like the challenge of doing all of the computation in their heads.  Let's try another one that is a little different.  Let's do 63 x 41.  Again we multiply the first digit of each number and then the second digit of each number to get the first digits of the answer and the last digit of the answer.


As before, multiply the outside digits, then the inside digits, and add the two products together.
Now we must put the 18 into the middle spot, but there is only room for one digit in the tens place.  YIKES!!  What do we do now?  Very easy....because we can only have one digit where the question mark is, we must regroup (carry) the one in the tens place of the 18 and then add it to the 24.

Have you figured out the final answer?  It is.....

You are probably thinking the old method works so much better, but that is only because that is the method you are use to using.  Why not try the ones below using the Quick Times method and see if you get the correct answer.  Use the old method or a calculator to check your answers or go the the answer page above.

a)  36 x 21       b)  24 x  12      c)  48 x 29       d)  59 x 18       e)  63 x 13     


Making Gifts - Not Something I Normally Do

At this time of year, we have many friends retiring or celebrating those "up-in-years" birthdays. Many of the invitations read, "No Gifts, Please." I understand at this point in our lives, we have more than we need, but it is always nice to bring something to show your friend that you care. We just attended a 70th birthday party for someone we have known for years. Not only is he our friend, but he is someone both my husband and I have taught with. I looked on Pinterest (where else?) and found several ideas that I combined. Here is what I came up with - a large birthday card that was editable!!


Here is what I purchased to complete the giant card.

  1. A large folding poster (You need heavy poster board to hold all of the candy!)
  2. A Nestle's Symphony Bar
  3. A Snickers Bar
  4. Nestle Crunch
  5. One package of EXTRA chewing gum
  6. 100 Grand Candy Bar
  7. Butterfinger
  8. Skor Candy Bar
  9. Mr. Goodbar
  10. Package of Milk Duds
  11. Package of Whoppers
I hot glued each of the candy bars or packages of candy onto the poster. I then used rubber cement to attach the phrases. I created my own phrases that sort of matched the candy, but if you are making a card, get creative and make up your own. You might even find some better candy bars or items to put on the card.

I have to say this birthday card was a real "hit" and even became a center piece of the party. Also, the party goers thought it was extremely yummy!


Plant Mathematics and Fibonacii

Oak Leaves
We continue to look at Fibonacci numbers and how nature continually exhibits this pattern. As stated in my last post, this number pattern can be linked to ordinary things we see every day such as the branching in trees, the arrangement of leaves on a stem, the flowering of an artichoke, or the fruitlets of a pineapple. BUT were you aware that scores of plants, including the elm or linden trees, grow their leaves, twigs and branches placed exactly half way (1/2) around the stem from each other?

Similarly, plants, like beech trees, have leaves located 1/3 of a revolution around the stem from the previous leaves. In the same way, plants like the oak tree have leaves positioned at 2/5 of a rotation. Plants like the holly continue this pattern at 3/8, while larches (conifers) are next at 5/13. The sequence extends on and on. Looking at these fractions side by side (see below) do you see a number pattern in the numerators?


Likewise pay attention to the precision of a similar pattern in the denominators. Interestingly, in the numerators and denominators, if you add the two sequential numbers together, you create a Fibonacci series where all numbers in the series are the sum of the two preceding numbers.



Mathematicians recognize this unique pattern as the Fibonacci sequence. Since patterns such as this one are commonplace in botany as well as other areas of science, they are regularly studied so we can better understand the relationship between mathematics and our world. In my opinion, such mathematical precision and accuracy can only be the product of an intelligent Designer. "Through Him all things were made; without Him nothing was made that has been made." (John 1:3 - NIV)