Monthly Archives: May 2013

Dinosaur Math

Because I’m home with a broken foot, my daughter happened to overhear part of my latest tutoring session.  What she heard was my student abruptly asking, “Do you know how long ago dinosaurs became extinct?”  (This type of comment is very common during our sessions.)

Later, my daughter, who is 15, told me she was amazed I didn’t immediately shut down the dinosaur conversation.  I explained to her that listening to this student is an important part of teaching him.  Over several months we have built a relationship; we listen to each other.  As a result, I can challenge him to change how he solves problems, or to try something new.  It works.

It was only later I remembered another reason I listen (at least for a while) to these apparently random, off-topic utterances.  It’s because they frequently spring from a math-related source.  Take the dinosaur comment.  He wasn’t asking because he didn’t know how long ago dinosaurs became extinct.  Rather, he wanted to know if I knew because something about it was bugging him.  He wondered:  If 200 years ago scientists discovered that – 65 million years earlier – dinosaurs had become extinct, then why don’t we now say dinosaurs have been extinct for 65 million two hundred years?  (I chose to ignore the when-did-scientists-know-it question and instead focused on the concept of rounding.)

I used a whiteboard and wrote the numbers 65,000,000 and 65,000,100.  (Okay, I finessed the date a little and said, let’s just use 100 years.)  We talked about rounding and the fact that it would take a really long time before it would make sense to say anything other than dinosaurs became extinct 65 million years ago.  He seemed content and I was happy that I had answered what turned out to be a good question.

It was only a day or so later I realized I had missed an opportunity to make the concept more real.

What if, instead, I had said:  How long is 65 million seconds?  Next tutoring session we start with this one!

First Grade Assessments

1_Verbal_Assessment    1_Written_Assessment

You’ll immediately notice two things about the first grade assessments: They contain fewer pictures and MANY more word problems than the Kindergarten assessments.   First grade does not include the Counting and Cardinality domain or as many other standards that require pictures, such as classifying by color and recognizing shapes.  In the future, I might illustrate the word problems by including a picture of what the problem is about, e.g., use a picture of a toy car if the problem is about toy cars.  But for now, since this is a draft, I will leave it as is.

You may be wondering why there are so many more word problems.  If you read my Kindergarten Word Problems post, you know the Common Core State Standards for Mathematics document contains a table that describes four categories of word problems.  When you consider all cases and versions, there are 15 types of word problems listed.  (This does not cover all types of word problems, just the basic ones of adding, subtracting, and comparing.)  It seems to me a student cannot truly be proficient until she can solve every possible type of word problem; therefore, I have included 12 word problems in the first grade assessment.

Twelve?!?  Shouldn’t that be 15?  Hmm…perhaps I got tired.  I’ll have to fix that later.  In any event, this is a huge increase in the number of word problems (from 2 to 12).  As a result, I will most likely distribute them throughout the assessment before I use it.

For each word problem. I included a description that maps directly to the table in the Common Core standards.  For example, question 12 is described as “word problems within 20, comparing, smaller unknown, ‘more’ .”  If you are so inclined, you can use these descriptions to figure out which word problem types are not represented.

A Non-CCSS Question?

You may have noticed a question in the written Kindergarten assessment that did not map directly to the Common Core State Standards.  It was the last one:  What number goes in the box?  4 + 5 = ▢ + 6.  I added this problem as a result of reading a 1999 article entitled “Children’s Understanding of Equality:- A Foundation for Algebra” [1].  (See link in blogroll.)

What caught my eye in this article is this:  When 145 sixth graders were given the problem 8 + 4 = ▢ + 5, none of them answered it correctly.

Most of the students said “12” should go in the box, indicating they view the equal sign as “a signal to do something” rather than “as a symbol describing a relationship” ([1], p. 236).  So, students are thinking, “eight plus four equals twelve and then we want to add five to it,” instead of “what plus five equals eight plus four.”

I was surprised, until I realized I have seen similar thinking in the work of high school students. When solving a multi-step problem, a student will write and solve one equation, and then – as if forgetting about everything to the left of the equal sign – continue the solution by building onto the right side of the equation.  Of course, this makes the equation no longer true.  Here is an example I created to illustrate what I mean:

004 Equality Misconception

The beginning of the solution – using the formula for the area of a triangle – is just fine.  But multiplying the result by 6, while necessary to determine the area of the regular hexagon, makes the equation untrue.

I think these students know the difference and they are just being careless.  But … maybe not. Maybe they have fundamental misconceptions about equality.  That’s one of the things I hope to find out when I give these assessments.

References:

[1]  Falkner, K. P., Levi, L., & Carpenter, T. P. (December 1999).  “Children’s understanding of equality:- A foundation for Algebra. Teaching Children Mathematics, 232 – 236.  Retrieved from http://ncisla.wceruw.org/publications/articles/AlgebraNCTM.pdf