Refining the Numberless Word Problems

Now that summer is beginning to fully set in, and the overwhelming exhaustion that hangs on after the conclusion of a school year has faded, I can begin to truly think about the beginning of the new school year. One of my first focuses will be working with my students to really understand mathematical word problems, and more importantly, begin to see how they might translate to their lives outside of my classroom. One of the most frustrating battles as a math teacher is knowing students can perform the functions necessary to solve the problem, but cannot identify those functions correctly after reading a word problem. It is the Thanos of my classroom, “[It] is inevitable.”

This is where numberless word problems come in. When students are presented with a word problem without numbers they more naturally begin thinking about the events in the problem as though they were taking place in the real world. More simply, students have an easier time imagining the actions in the word problem when the numbers are absent, and when students can imagine the actions actually taking place they have an easier time understanding what mathematical function works best for that problem. Once students have a solid image of the actions in the word problem the numbers can be added back into the problem. My experiences with word problems lead me to believe that the numbers are the primary focus when students begin reading a problem. My goal is to shift their focus from the numbers to the situation being presented.

The basic layout of the numberless word problem is broken down into four different slides:

1. The word problem with no numbers,
2. The word problem with one number added in,
3. The word problem with all of the numbers,
4. And the complete word problem including the question.

Before students see any of the numbers in the problem I ask them to sketch a picture of the problem. This process helps students visualize the problem and talk about their thinking more effectively. After students sketch the problem each slide adds additional information to the problem slowly to give a more complete picture of the scenario. Initially, this protects students from being overwhelmed by too much new information at once; it gives them time to continue their thinking by checking how reasonable their first idea of the problem was. When students reach the final slide they should be prepared to solve the problem and justify their answer.

For this assessment, I have collected five numberless word problems so that students have a new problem to solve each day. As of right now, my class next year has 15 students, so I am able to break students into three groups of five students. The groups will work together to solve the problems and provide feedback to their group mates. On the first day, each student in a group will be assigned a different problem so that all five problems are solved on the first day. Then, the students will rotate through the remaining four new problems each day for the remainder of the week so that each student solves all five problems throughout the week.

To conclude their work each day students will record and post a video to our class Flip Grid explaining the work they did to solve their problem and justify their work. Their video after the first day will be a response to another student’s work where they talk about the similarities or differences in how the problems were solved. And as students are working, I will have the opportunity to meet with students to provide feedback as well as respond to their Flip Grid posts.

This assessment is an effective way to determine how well students understand the application of mathematical processes. Through individual conferences and responses to Flip Grip posts, I will see a student’s thinking as they solve each problem and respond to the process they used as well as the actual content of the problem. And with a new problem, each day students have the opportunity to try again after receiving feedback from me and the other members of their group.

You can view the numberless word problems here and my student instructions here.

 

Please Leave Your Message After the Beep…

This week I watched my wife try to find the time to speak on the phone with one of her best friends. For two days they missed the other’s call by a few minutes and left a voice message (or sent a text which seems to have replaced voices messages largely) in hopes they would find a time both were free to talk. Each time a call was missed the cycle started over until finally, they were both free at the same time. Seeing the subtle frustration after each missed call left me thinking about how students must feel after turning in each assignment. Every day that goes by without feedback is the frustration of a missed call magnified many times over. And that is before the teacher moves on to the next lesson!

This week I added feedback as a critical element of my Assessment Design Checklist. I am continuing to work on these five questions to guide my selection of assessments in my classroom. The aim of this checklist is to ensure that the assessments I select benefit my students in a way that clearly supports their learning.

The question regarding feedback is focused on students correctly completing each task that is assessed. Being in the elementary classroom I find the most important part of my job is guaranteeing that students arrive at the correct answer, and when they do not that they are able to begin the self-assessment process necessary to adjust to those mistakes. The second addition to this checklist looks at student choice in assessment. More than one assessment method allows students to choose how they want to demonstrate their learning, as well as an assessment to return to after receiving feedback if more work is needed.

The checklist is still a work in progress, but my hope is for these four questions to focus on different aspects of each assessment so that when they are combined they maximize the effectiveness of my students’ work.

 

It’s Friday, so we Test

I just wrapped up the last day of my school year as a fifth-grade math teacher. Now, my focus shifts from the daily routine of plan, teach, adjust, and repeat as a classroom teacher to more of a reflection on the year as a whole. And having just started a course focused on assessment I am taking a closer look at the assessment I was instructed to give most often throughout the year, the Focus Quiz. This is the same assessment I looked at to identify assumptions two weeks ago, but now I am interested in the genre of this assessment and considering adjustments that could be made to the assessment itself, my instruction in light of this assessment, and how this assessment works with technology.

“Focus Quiz” is the title used by my school district, but as a genre, this assessment is best described as an end of unit quiz. Every two weeks, almost like clockwork, I received a class set of quizzes from my instructional coach and passed them out to my students. For one class period, my students work on ten to fifteen-word problems; some of those problems were multiple choice, some of them were free responses, all of them were modeled after the state assessment. The quiz focused on the material that I taught in the preceding weeks only, so if I taught multiplication and division when the numbers students worked with included a decimal every problem assessed their ability to multiply or divide with numbers that had a decimal. The data from students’ scores and the most missed questions were then used to plan future lessons and group students based on their needs.

I do not think I have met or worked with anyone that has not experienced the end of unit math assessment. And for good reason. Each unit tends to build on the previous unit in math, and it is critical that students master the skills as they go; it becomes increasingly difficult to catch up in math once you fall behind. This is reason enough to keep the end of unit assessment as a foundational element of a mathematics course. Frequent assessments targeting the newest skill produce the necessary data to adjust plans for students that need more time with a skill they did not master. However, when these assessments are used only as a summative assessment the data indicating student performance is lost. These assessments should be used formatively as well where the data gathered guides future planning.

At the conclusion of each lesson, a teacher could use one or two questions similar to the questions from the end of the unit assessment to check for student understanding. The data gathered from student performance at the conclusion of each lesson can be used to adjust plans for the next day if more time is needed for the student to master the content before moving on. This gives both the teacher a clear picture of each student’s progress and skill mastery with time to reteach or practice skills before the summative assessment. The end of unit assessment functions as a final “gotcha” rather than a checkpoint. When used as a checkpoint this assessment can give students confidence when they show mastery of the skills taught or can be used to scaffold student learning as they continue to learn.

To help myself plan for more effective assessments I started this checklist of questions. These questions (soon there will be five) check the validity of the assessments I consider adding to my plans. The first two questions check if the assessment aligns to the content I taught and if my students understood what exactly I was teaching. Wiggins and McTighe, in their book Understanding by Design, explain that learning is best supported when the students can explain what they are learning, why they are learning it, and how the assessments are supporting their learning. To support students’ understanding of the “why” in learning I should explain “what” exactly they are learning so that students can also explain it. These two questions struck me as obvious and basic when I added them to my checklist, but they rest at the center of good instruction and assessment. If the answer to either of these questions is “no” when considering an assessment it is clear that assessment should not be used.

As I consider the typical end of unit assessment, and specifically the focus quizzes I used last year, the answer to each question is yes. Although I feel some of the “real world” scenarios presented in the questions are not something most people would actually encounter, they are at least a reduction of real-world scenarios to a level most fifth-grade students can understand, so it is possible to explain the “why” of these questions. And by having students reflect on these questions so that they are responsible for explaining what they learned these assessments can be used to check that students know the “what” of their learning.

Taking the end of unit assessment and spreading it out to formatively to check learning throughout the unit is likely the best use of this assessment, and it is actually my students favorite part. Two of the tools I used regularly in my classroom were Kahoot! and Plickers. These two sites turn the questions from an end of unit assessment into a quick game that my students loved. Their answers are recorded and feedback is given immediately. The quality of each question is entirely up to me and the rigor matches that of the formal assessment. By adding both of these technologies this assessment is more engaging and the data gathered from it is easier for me to use in planning.