Part 1: Exit Tickets — A Tool For Formative Assessment and Immediate Feedback
At a recent professional development event with Julia Payne-Lewis, an assessment specialist at Mind the Gaps, Eureka Math writers looked at research that reveals the powerful impact on student achievement of teachers using formative assessment to provide immediate feedback. Timely, specific feedback as part of daily classroom instructional practices can produce the equivalent of six months’ worth of instruction! (Embedded Formative Assessment (2011) Solution Tree Press)
The Exit Ticket is one aspect of formative assessment contained in each Eureka Math lesson. Given at the end of each day’s lesson, from Kindergarten Module 5 through Pre-Calculus, it is designed to gauge student understanding of the lesson’s desired outcome.
Here’s how we use Exit Tickets in our classrooms to address our students’ needs, both individually and as a whole group.
How do I organize the data?
First, we correct the student work and sort it into piles: Got It; Almost Got It; Not Yet. The Almost Got It group has one or two surface-level misconceptions that can be easily identified and those students need more practice to solidify their understanding. The Not Yet group needs more intensive support because their work shows misunderstandings and often reveals learning gaps.
For the latter two groups, the key is to analyze the work for both weaknesses and strengths. The strengths tell you what the students know and can do well; this will potentially establish the last point of success from which to build on. Using the analogy of the ladder, we want to help students climb up to the highest rung, the lesson’s desired outcome (see fig. 1.4).
What do I do with the data for those who “Almost Got It”?
Using the protocol, find the misconceptions as well as the areas of strength. If the misconception is common among a large number of students, we may put one example under the document camera and do a quick review either during the next day’s morning work time or at the start of the next math lesson. In figure 1.2, notice how the student incorrectly subtracted 40 from 100 to get 40. This can be addressed with a few minutes of fluency, either with the whole class or by pulling aside an individual or two during morning work or during the Application Problem. Students can respond verbally or write the answer on their personal white board. What is 10 – 4, 10 – 3? 10 tens – 4 tens, 10 tens – 3 tens? 100 – 40, 100 – 30? And so on. Another option would be to show the student their work and invite them to do some detective work and identify the error.
If there are students in this category that show more than one error, you may decide to break down the steps needed to solve and look at each one in isolation to see where the weakness lies.
Can the student take out 100 from a three-digit number? If not, start simpler with Take from 10 using teen numbers and gradually build up.
Can the student take a multiple of ten from 100? If not, practice a sequence that emphasizes that the digits stay the same and only the units change. For example, what is 10 ones minus 5 ones? So 10 minus 5 is 5. What is 10 tens minus 5 tens? How much is 10 tens? How much is 5 tens? So 100 minus 50 is? If this is still too abstract, you may need to revert to concrete models, such as the Rekenrek or unifix cubes.
Can the student add multiples of ten and a two-digit number? If not, show it on the Rekenrek. Start simply with 14 plus 10, then 14 plus 20, and so on up to 14 plus 50. Have the student say the sums the Say Ten way (e.g., 6 ten 4) and the regular way (64) to reinforce the fact that the ones stay the same and only the tens are changing. Then ask the student to visualize the movement of tens on the Rekenrek. This can be supported by encouraging the student to make a click sound to mimic the Rekenrek’s sound each time they mentally add a ten.
In the next example (see fig. 1.5), in addition to the error of incorrectly subtracting 40 from 100 in Problem 3, the student is not exercising care when writing her numbers, which highlights the importance of accuracy. This is a place where you could reinforce Mathematical Practice Standard 6: “Attend to Precision”. It’s worthwhile to point out to a student that this practice also extends to how they record their work. If their work is illegible it will be marked incorrect. Often asking the student to read their own handwriting awakens this awareness.
What do I do with the data for the “Not Yet” Group?
Figure 1.3 shows the student was not able to solve any of the problems on the Exit Ticket correctly. What do we observe from his work?
This student has not grasped the abstract “Take from 100” strategy. However, he has persevered by attempting a previously learned strategy (the chip model). Looking at his work with the chip model, he knows that a ten can be decomposed into 10 ones. His base ten drawing is well organized. The trouble seems to be once he starts to subtract. Why did he subtract 8 ones in Problems 2 and 3? Kenneth’s work is an example of how obtuse misconceptions can be and how challenging it can be to uncover the actual weakness.
The “Take from 100” strategy both strengthens place value understanding and is an application of the associative property, important mathematics with implications well beyond Grade 2. However, because Kenneth is struggling at this point in the module with the vertical method, we decide not to focus on the “Take from 100” at all. We want him up and running with at least one subtraction method. Since the vertical method will work for him in every situation, we decide to teach it to mastery and then see if he has the capacity, and we have the time, to extend his toolkit. Let’s start the process by forming questions to determine the last point of strength.
We decide to start entirely at the pictorial level, where he was successful to some degree, and then cycle back through the work with the written vertical method. We also decide NOT to show him his work on the Exit Ticket, but rather make a fresh start. Why? His own work might confuse him and the “Take from 100” method may have confused his fragile understanding of place value units. We want a clean slate from which to work.
How to begin? We start by just seeing if Kenneth can answer the original problem at the pictorial level without the complexity of the written vertical method or the Take from 100.
T: Kenneth, represent 176 on this place value chart.T: Show me how to subtract 9 tens.Watch what he does.
Unsuccessful: Suppose he decomposes a ten and subtracts 9 tens from the ones place? We suspect he may do that because he only decomposed the ten on the Exit Ticket. After all, this is being formally taught the next day in Lesson 24. Perhaps proceed with the following questioning to see where his place value understanding is weak.
T: (Start with a clean place value chart.) Please draw 176 again on this place value chart.T: (Assuming the student represents 176 correctly.) Perfect. (We already know he can do that.) Show me how to subtract 3 ones from this number.T: (Assuming the student crosses out three ones.) What number is left?T: (Assuming the student says 173.) Show me how to subtract 3 tens from this number.T: (Assuming the student crosses off 3 tens.) What number is left?T: (Assuming the student says 143.) Show me how to subtract 5 ones from this number.
Watch carefully. Does the student decompose a ten into ten ones and subtract? Does he instead abandon the picture and say there are 2 ones left or go to the hundreds place place rather than the tens for a unit to decompose? Adjust questioning accordingly.
T: (Assuming the students decomposes a ten and subtracts 5 ones effectively.) What number is left?
T: (Assuming the student says 138.) Great.
At this point, Kenneth has shown what he can and cannot do! We may want to move on to the numerical level and cycle through the exact same set of problems numerically as far as possible, rewriting the problem each time and referring to the pictorial level only as necessary (i.e., 173–3, 173–30, 143–5).
These are specific suggestions! What generalizations can we make?
· With “Not Yet” students, eliminate complexities as needed. In this case, we eliminated the complexity of the Take from 100 and, at first, the numerical work.
· Start by representing the original problem in a simpler way and move back down “the ladder.” This allows remediation to be efficient. If we start too low and move up, we may waste valuable instructional time.
What resources are available to help address other errors?
The answer depends on what your students’ work reveals. If there is underlying foundational work that can be addressed through fluency activities, then we adjust the next day’s fluency to incorporate an activity targeted to that foundational standard. We use every precious moment to engage students in these small but mighty fluency activities that can be done as students enter the class, leave the class, during morning work, or for 2 minutes during the Problem Set time. You can send some of the activities home (for example, TargetPractice, Make Ten Using the Double Ten-Frame). Students especially love taking the white board home and playing teacher!
The Eureka Math Digital Suite website provides an excellent resource for subscribers. Here you will find links to fluency activities by standard. To use this resource, login to greatminds.org and select “my dashboard” (fig. 2.1). Next, click on “launch product” under “Eureka Navigator”(fig. 2.2). You can select the grade level and module you are working in (fig 2.3) and you will find the “module standards” on the right (fig. 2.4). Underneath is a drop-down menu for Focus Grade-Level Standards and Foundational Standards (fig. 2.5.1 and 2.5.2). Clicking on either will take you to links for the individual standards (fig. 2.6). By clicking on an individual standard you will land on a page that lists all the fluency activities with direct links to those activities.
Within a given topic of lessons, you can address misconceptions that indicate a need for further concept development during the next lesson’s 10 minute Problem Set time. While proficient students work on the Problem Set independently, you can pull a small group and start with a simpler sequence of problems to address misconceptions as you guide students up the ladder to the desired outcome.
By analyzing student work, you identify common misconceptions and learning gaps. Tailor your feedback to address student errors in a clear, specific, and timely manner. As Doug Lemov says in Teach Like a Champion, “The shorter the delay between recognizing a gap in mastery and taking action to fix it, the more likely the intervention is to be effective.” In the long run, the more you study your students’ work, the more you will be able to predict errors that may occur in future lessons. The knowledge of student thinking will enable you to address those errors in the Concept Development before the students actually make them.
This is the first part of a continuing series on identifying how to use formative assessment when implementing Eureka Math. Upcoming blogs in this series will focus on Fluency Activities, Application Problems, Responses to Questions during Interactive Lessons, Listening in on Student Collaboration, and Creating Quizzes.
(Access more information about the Atlas Protocol, here. The steps are laid out on pages 3 and 4.)