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. (http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf, 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. Thinkingblocks.com (http://www.thinkingblocks.com/TB_AS/tb_as3.html) 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 (http://scimathmn.org/stemtc/resources/mathematics-best-practices/modeling-word-problems).
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.
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.”