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The Long and Short of It

My remedial college math class is currently working on fractions. When we discussed how to change an improper fraction to a mixed numeral, long division came up. I showed the class a shortcut I was taught many years ago (approximately when the earth was cooling) and none, no not even one student, had seen it before.  I wonder how many of you are unfamiliar with it as well. First let's look at long division and how most students are taught today. We will use 534 divided by 3.

Now if that doesn't make your head swim, I don't know what will. Everything written in the third column is what the student must mentally do to solve this problem. Then we wonder why students have trouble with this process. There is another way, and it is called short division for a reason. This is the way I learned it.......
I don't know about you, but I would rather have my students doing mental math to solve division problems than writing everything out in the long form. And the paper and frustration you will save will be astounding! So what will it be.....long division or short division?

Divisibility Rules Resource
As a Side Note: Since many students do not know their multiplication tables, reducing fractions is almost an impossible task. The divisibility rules, if learned and understood, can be an excellent math tool. This resource contains four easy to understand divisibility rules and includes the rules for 1, 5, and 10 as well as the digital root rules for 3, 6, and 9. A clarification of what digital root is and how to find it is explained. Also contained in the resource is a dividing check off list for use by the student. If you are interested, just click under the resource title page.

Accentuate the Positive - Eliminate the Negative

Have you ever wondered why a negative number times a negative number equals a positive number?  As my mathphobic daughter would say, "No, Mom.  Math is something I never think about!"  Well, for all of us who tend to be left brained people, the question can be answered by using a pattern.  (Have you noticed a reoccurring theme in my articles?  All Math is Based on Patterns!

Let's examine 4 x -2 which means four sets of -2.  Using the number line above, start at zero and move left by twos - four times. Voila!  The answer is -8.  Locate -8 on the number line above.  Now try 3 x -2.  Again, begin at zero on the number line, but this time move left by twos - three times. Ta-dah!  We arrive at -6.  Therefore, 3 x -2 = -6.

Here is what the mathematical sequence looks like.  Moving down the sequence, observe that the farthest left hand column decreases by one each time, while the -2 remains constant. Simultaneously, the right hand answer column increases by 2 each time.  Therefore, based on this mathematical pattern, we can conclude that a negative times a negative equals a positive!!!!

Isn't mathematics amazing?

Examining Multiple Choice Questions

According to Ron Berk (a keynote speaker and Professor Emeritus of Johns Hopkins University)the multiple choice question "holds world records in the categories of most popular, most unpopular, most misused, most loved and most hated" of all test questions.  Because of the many students teachers see each day and the little time teachers have to make tests and then grade them, multiple choice questions have become one of the favorite type of testing questions in education.  We see them on state assessments, national assessments, ACT tests, college tests, driving license tests, etc.  However, those who consistently use them aren't all that crazy about them and with good cause.

First of all, answering multiple choice questions doesn't teach students how to formulate answers; it teaches them how to select answers.  Many times choosing the right answer is more a literary skill rather than of content knowledge.  Multiple choice questions promote guessing, and if a guess is right, students get credit for something they didn't know.  Moreover, the instructor is deceived into thinking the student understands the concepts being tested.

Many multiple choice questions do not challenge students to think.  Instead they encourage the students to memorize.  In my opinion, test bank questions are the worst.  A simple analysis of this type of question in a variety of disciplines suggests that about 85% of the multiple choice questions test lower level knowledge, levels I (remembering) or II (understanding) of Bloom's Taxonomy.

When I first started at the community college where I teach math, the math assessments which the department used were mostly multiple choice.  I asked about testing our students using levels V or VI of Bloom's and the reaction I received was disheartening.  One instructor implied that our math students would be unable to answer such questions.  I guess his expectations were a great deal lower than mine. I am positive he hadn't read an article by two professors at Kansas State University.

According to Victoria Clegg and William Cashin of K.U., "Many college teachers believe the myth that the multiple choice question is only a superficial exercise - a multiple guess - requiring little thought and less understanding from the student.  It is true that many multiple choice items are superficial, but that is the result of poor test craftsmanship and not an inherent limitation of the item type.  A well designed multiple choice item can test high levels of student learning, including all six levels of Bloom's Taxonomy of cognitive objectives."  (Idea Paper No. 16, Sept. 1986)

So what are some things that make challenging multiple choice questions? Let's take a multiple choice test to help us answer that question.

Choose the best answer.  Which multiple choice question is the hardest to answer?

a) The one where it’s absolutely obvious that all choices are wrong answers.
b) The one where the question and/or answers are so badly written that two or more answers could be correct depending on how the student interprets the question.
c) The one where the list of possible answers are true or false; it depends on how the the student reads the question.
d) The one question where it is really two questions in one, but the options only answer one part of the question.
e) All of the above – except that there is no "all of the above" option given.

I trust you see the humor in this question.  Unfortunately, I have seen one or all of the above on math tests given in our department.

Now let's look at two different multiple choice questions from a mealworm test available on Teachers Pay Teachers.  The first one is pretty straight forward and requires little thinking on the part of the student. On Bloom's, it would represent a level I question - remembering.

Mealworm Test
1) Which tool will help you best see the mealworm up close?

a) Ruler
b) Mirror
c) Hands Lens
d) Eyedropper

The next question from the same test requires the student to understand what a good scientific investigative question is. This would be a level IV question which is analyzing.

4) Which question can be answered by investigating?

a) Will the mealworm eat the fruit?
b) How far can a mealworm travel?
c) Will more mealworms go to the paper with an apple slice on it or to the one with no fruit on it?
d) Why do mealworms move?

How about this one from a butterfly test?  (also available on TPT)  What level of Bloom's does it represent?

3) How does the life cycle of a butterfly differ from the life cycle of a frog?

Butterfly Test
a)     Only the butterfly has an egg.
b)     Only the butterfly has an adult stage.
c)      Only the frog has a tadpole.
d)     Only the frog has a pupa.

Again it is level IV because the student is asked to compare; yet, this test is for the grades 3-5 while the meal worm test is for grades 5-8.  I give you examples from both so you can see that challenging multiple choice questions can be written for most grade levels.

If you would like help writing good, challenging questions of all kinds, you might check out Bloom's Taxonomy Made Simple.  It is a five page handout that breaks the six levels of Bloom's down into workable, friendly parts, using the familiar story of The Three Little Pigs. Examples of good ideas of how to write assessment questions using all six levels of Bloom's are given.  For your practice, a follow-up activity of 16 questions is included.