Author Archives: galenaylor

About galenaylor

Documentation engineer at Facebook. Math tutor. Former aerospace engineer and programmer/analyst.

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.


[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


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.”

Kindergarten Assessments

K_Verbal_Assessment  K_Written_Assessment

Above are the two Kindergarten assessments I created.  Although my initial purpose for the assessments is to use them with older students, I attempted to make them appropriate for Kindergartners.  Full disclosure: I am a secondary math teacher, not an elementary teacher.  However, I have read Knowing and Teaching Elementary Mathematics by Liping Ma (fantastic book, I highly recommend it – thanks Jean M.!)   I have also been researching elementary math tasks so I can better help a 6th grade student with whom I am  working.  I’m sure I will learn more as I use the assessments.

Why two assessments?  Well, some standards need to be assessed verbally (e.g., counting to 100 or counting out loud).  Furthermore, I may want to see and hear the student’s performance.  Specifically:  How fluently is he adding or subtracting?  Where is he hesitating?  Even the written assessments are not intended to be completed in isolation; I plan to watch the student as he completes the problems.  Again, I will see where he is fluent and where he hesitates or skips a problem.  I will gain insight into any misconceptions as well as skills he may be lacking, which will help me decide the concepts I will need to  reteach.

Although the assessments may appear to be cast in stone, my intent is to use them more flexibly.  So, (and I’m thinking out loud here) after the student demonstrates she can count by 1 to 20 or 30, I would stop and ask her to count by 1 from some higher number to 100 (which actually incorporates the second Kindergarten standard, K.CC.A.2).  Ideally, I would then follow up with a question to see if she could explain the counting pattern.

Also, if a student is really struggling with a problem, I would intervene.  First I would start by asking some guiding questions to see if he could then figure it out.  I might change the problem to be simpler or let him skip it all together.  After all, these are not tests; I just want to know where the holes are.  It is paramount that I keep it positive, that I keep the student talking to me.  Teaching is as much art as it is science.  I need to know when to push and when to back off.  I need to know my students.

Still, there may be better ways to assess students on these standards.  If you have suggestions for how to improve the assessments, I hope you will leave a comment!  (I have already seen some assessments that elegantly group multiple standards in a single task, which are intriguing.  Although, as my purpose is to determine which concepts students have yet to master, it might be easier to assess each standard individually.  I’m still thinking about this.)

Finally, I consider these assessments to be “first drafts” and plan to modify or change them after I use them and see how they work.  What do you think?  Am I on the right track?

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…