# When is a Line Not a Line?

After a long hiatus, I am finally getting back to my assessment project.  The end of the third grade assessment is in sight!  As I work to understand each standard and turn it into an assessment question, I do a fair amount of checking to make sure (1) I’m using appropriate levels of problems and (2) my assumptions are good.  (I don’t always succeed, but that’s another story!)  This check is particularly important with elementary topics because my credential and most of my experience is with secondary math.

Two of my favorite sites for checking my assumptions are IXL Math and Illustrative Mathematics.  They are both very organized and easy to use.  IXL Math has lots of practice problems for each standard and Illustrative Mathematics has amazing tasks and a great interactive graphic showing the domains across grade levels.  (I’ll be using both of these sites over the summer with my 3rd and 4th grade tutoring students.)  There are others, of course, and I also use Google to research topics.  (If  you have a favorite site, please share it in the comments!)

Thank goodness I checked my assumptions about line plots (3.MD.B.4).  I don’t know about you, but when I think of line plots my brain constructs an image of coordinate pairs connected by line segments.  You know, something that has lines in it.  So, I was a bit surprised when I Googled “line plots for 3rd grade” and images of stacked X’s came up on my screen.  (See example above.)

Okaaaay.  Hmm.  Well, the making-a-line-plot standard doesn’t have an image of a line plot.  So, I did a little digging and discovered that, apparently, in the 3rd grade world a line plot looks like stacked X’s.

Don’t get me wrong.  I think the stacked-X’s-as-line-plot is an outstanding stepping stone between bar graphs or pictographs and traditional line plots.  I understand it’s an important bridge, which will greatly help students make the transition.  In fact, I hope teachers facilitate this transition by eventually asking students to draw a dot at the top of each stack of X’s and connect those dots with line segments.

It’s just the name that gets me: line plot.

I realize I’m a very literal person, and third grade was a LONG time ago, but I think even third grade me would be confused.  “Where are the lines?”

When is a line not a line?  When it’s a third grade line plot, of course.  Anyone can see that.

# Assessments Update

Okay, this is becoming a little daunting.  I knew the fun pictures of soccer balls and ice cream cones I used for the Kindergarten assessment were going to be replaced by more number-dense problems, but I’ve just finished the third grade Operations and Algebraic Thinking domain and I already have 29 problems.  Yikes!  My philosophy of including each word problem variation may not survive third grade.  But how do I choose?  Will it be the Goldilocks solution: one easy, one hard, and one “just right?”  Or is there a better way?

I just looked back at my 3rd grade assessment to see how many word problems I have and noticed  many of my other problems have multiple parts.  For example:  Write a related multiplication fact for each division expression has 4 division expressions to solve.  So, my assessment is really much longer than 29 problems.  (My count: 11 word problems for 3.OA.3 and 7 for 3.OA.8.)

Oh, yes, and looking back at my notes from a month ago before I broke my foot (but that’s another story), I had already eliminated some of the word problem variations.  Here’s what I wrote: “Naïve to think I could do them all… hard and easy array problems, array versus area problems, measurement versus non-measurement, [and] compare problems.”  And that’s not even changing the location of the unknown in the problems, which makes a huge difference in complexity.

Well.

I think I will continue to create the assessments as a complete mapping of the CCSS-M (including all word problem variations) because it is a way for me to more deeply understand the standards.  However, it’s clear my original (and, apparently, naive) idea of creating a series of assessments that would illuminate concepts not mastered in previous math classes, and that would be practical to give students, is in need of some rethinking.

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.)

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:

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

# Kindergarten Assessments

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?