Category Archives: Uncategorized

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!

Kindergarten Word Problems


Yesterday, a friend commented about the word problems in my Kindergarten assessment and how unbelievable it was that Kindergartners were expected to solve problems of this complexity.  When I looked back at the assessment, I realized that I had only included two word problems and they were not, in my opinion, of the most difficult type.

I was aware of different types of word problems, but it wasn’t until I read Table 1 in the Glossary of the Common Core State Standards for Mathematics (p. 88) that I understood the distinctions. (, p. 88)  Not only are there four categories of word problems, (1) add to, (2) take from, (3) put together/take apart and (4) compare, but each category has different cases as well.  For example, in “add to” and “take from” problems the result can be unknown, the change can be unknown, or the start can be unknown.

Yes, my two measly word problems did not even scratch the surface.

But then I had another thought.  I had been focused on creating problems that were grade-level appropriate.  What was I expecting students to show me for a solution?

I tackled this and I came up with a drawing for each of my two word problems that is straightforward and makes sense to me.  For example here is my take from, start unknown problem:  3 children leave the party.  4 are still there.  How many were at the party before? As a solution to this problem, I drew the picture at the beginning of this post.  Not bad.  Makes sense.

The more difficult problems are the compare problems.  Here is an example of one of the more difficult problems taken from the Common Core State Standards for Mathematics: “Lucy has 3 fewer apples than Julie.  Lucy has two apples.  How many apples does Julie have?” (p. 88).  That was the fewer version of a compare problem with a bigger unknown.  Now, here is the more version:  “Julie has 3 more apples than Lucy.  Lucy has 2 apples.  How many apples does Julie have?”  See how much easier it is to understand the more version?

I tried but couldn’t figure out a simple drawing that would work for the fewer version of the compare problem.  I can solve it numerically, of course, but had trouble coming up with a good visual explanation.  So, I researched models for comparison word problems and came up with several resources that model this kind of problem with a pair of bars. ( has a video along with sample problems.  The Minnesota Stem Teaching Center has a good discussion of several types of word problem models, and a little more than half-way down the page is a discussion of comparison problems using bar models (

Based on the concepts from these resources, I created a drawing for the fewer type of comparison word problem that combines both pictures and bars.  It makes sense to me after-the fact, but doesn’t seem as intuitive to me as the other drawings.


Common Core State Standards for Mathematics, p. 88

I think this approach would have to be taught.  Which brings up another question:  Are Kindergartners or even first graders solving problems like this now?  It’s been several years since my daughter was in Kindergarten, but I don’t remember problems like this.  If Common Core standards in post-primary grades presume these skills, are students prepared?  If not, how are they going to catch up?

I have a feeling some of the Common Core word problems I’m creating may be a bit of a challenge for the students I work with.  As a result, that part of my assessment may be more a jumping off point for teaching how to recognize and solve different types of word problems than an assessment of what students know.  But that’s okay.  After all, my main purpose is to find out where the holes are and fill them – even if the holes are “new” ones that exist because of the emphasis on understanding in the Common Core State Standards.  To quote Martha Stewart:  “It’s a good thing.”

Start at the beginning.

Mathematics.  That word … it makes so many people cringe.  Not me.  I’m someone who always loved math, even though advanced math didn’t necessarily come easily.  So, it was natural that I looked to mathematics when my career seemed isolating and empty and I decided I needed a change.  But not just any change; I wanted to do something I was passionate about, something worthwhile.  I had been volunteering in middle school math classrooms and saw a need for helping students understand math.  Fast-forward a few years, and I have a master’s degree and a secondary math teaching credential.  I can’t decide – do I teach middle school or high school?  Or, do I go back to school so I can be a math specialist in elementary schools because that’s where the trains leave the tracks, if you know what I mean; that’s where kids decide “I can’t do math.”

Then, I’m sent in a new direction by a chance encounter and a phrase that resonated: teach students math all over again, starting with Kindergarten.  This seemed to encompass both my desire to help students comprehend math AND my personal need to know the entire curriculum – from Kindergarten through Calculus.  A broad mathematical understanding is necessary for helping students regardless of grade level, and I knew it would allow me to ascertain what skills and concepts students were lacking as well as how best to position them for the mathematics that lay ahead.

This blog is the account of my journey to understand (and understand how to teach) the entire K-12 math curriculum using the Common Core State Standards (CCSS) as both a framework and a jumping off point.  To begin with, I plan to create a series of CCSS-based assessments that, when given to students who need help, will illuminate missing foundation concepts (beginning  with Kindergarten) and provide a starting point for targeted assistance.  In the end, I hope to have a complete and useful set of assessments, plenty of online references, and a collection of tasks and lessons full of authentic and challenging problems.  Oh, and by-the-way, I’ll also have an intuitive and connected understanding of the whole K-12 mathematics arc.  This should be fun.  All I need now is someone to collaborate with.  Are you with me?  We start at the beginning.  We begin with Kindergarten…