This is part two of a three part series on formative assessment. This post deals with some things you can do between individual lessons based on formative assessment and during a lesson. You can **read part one here**.

**Introduction**

The objective of this post is to describe two possible procedures teachers can use for ongoing, day-to-day formative assessment. The first of these procedures is easier to implement, but gives teachers less information on what students understand. Remember that a primary objective of formative assessment is to create a feedback loop for both teachers and students into the teaching and learning process.

**Example 1**

At the end of your last class you gave an exit slip. One strategy, which is not too time-consuming, is to take the exit slip and first sort it into No/Yes piles, and then sort these piles into 3-4 solution pathway piles, essentially organizing all of the student work by whether or not it is correct and what strategy students used. It may be useful to have an other group, with students whose strategy which are unable to decode.

These groups of student can be used to decide on student groups (recommendation: group by different strategy) for the following day, decide if you need to try a different strategy for tomorrow, and/or find examples of student work to present to students. It can also be used to decide on re-engagement strategies^{1} for the lesson from the previous day, or just decide that you can move onto the next topic in your unit sequence.

**Example 2**

An exit slip is not the only kind of formative assessment you can do^{2}. The most important feature of formative assessment is coming to understand what students are thinking. You can do this by conferring^{3} with individual students during your lesson and asking them questions to elicit their thinking. Of course, this assumes you have given students an assignment which requires them to think!

Imagine students are working on a rich math task^{4} and that you start by initially observing students and see if they are able to get started on the task without your intervention. As the students begin to work, you begin walking around the classroom, and observing them working, and listening to their discussions about the task. Your objective at this time is to gather evidence of what students are thinking about while they do the task.

The three main problems you may have to solve during this time are; students who are unable to get started on their own, students who are going in the completely wrong direction on the task, and students who have completed the task. One of the early tasks during your observation of students working is to figure out which students are in which group. Note that there is a fourth group; students who are not done the task, who may be struggling a little bit, but are making progress. Do not intervene with this group of students!

When you are confused about what students are talking about, or what they are writing, you spend some time clarifying your understanding of what they are thinking, so that you feel completely clear. Now, you choose an intervention^{5} for the student, such that the student is left to do the mathematical thinking of the task, and you do not lower the cognitive demand of the task. During the entire time students are working on the task, you collect information^{6} on what the students do during the task.

In the next post in this series, I will discuss more of the overall objectives of formative assessment, and discuss how the feedback loops created by the process of formative assessment can improve the effectiveness of teaching and learning in classrooms.

1. Re-engagement is an alternative to reviewing material with students. It can be done during any time the unit when you want to consolidate student understand.

2. For other examples of formative assessment, see this presentation that I curated. It has 54 different possible formative assessment strategies in it, some of which are more appropriate for a class focused on literacy skills, and some of which are useful for a mathematics classroom.

3. **This document** describes the process of conferring.

4. A rich math task allows for students to demonstrate mathematical reasoning, is often open-ended, and allows for multiple solution paths. These kinds of tasks generally take students some time to complete.

5. The intervention you choose should not lower the demands of the task you have set the student. You could ask them a question to prompt their thinking, or suggest a way they can interact with one of their peers (do not assume your students know how to collaborate, they may need a prompt to help them orient to each other's work and thinking).

6. It is useful to have anticipated student responses before the task, and solved the task yourself a couple of different ways. Finally, having a template to collect information during the lesson would be critical. Here are two such template designed by my colleague Sara Toguchi: **Descriptive information**, **Specific criteria information**

Newsletter:

David is a Formative Assessment Specialist for Mathematics at **New Visions for Public Schools** in NYC. He has been teaching since 2002, and has worked in Brooklyn, London, Bangkok, and Vancouver before moving back to the United States. He has his **Masters degree in Educational Technology from UBC**, and is the **co-author of a mathematics textbook**. He has been published in **ISTE's Leading and Learning**, **Educational Technology Solutions**, **The Software Developers Journal**, **The Bangkok Post** and **Edutopia**. He blogs with the **Cooperative Catalyst**, and is the **Assessment group facilitator for Edutopia**. He has also helped organize the first **Edcamp in Canada**, and **TEDxKIDS@BC**.

- Creating a WiiMote interactive white board at my school for under $50.
- 20 reasons not to use a one to one laptop program in your school (and some solutions)
- What is Edcamp?
- For whom are Interactive White boards Interactive?
- Forget the future: Here's the textbook I want now
- Mathematics education blogs
- There are no aha moments
- Why educators should blog: A helpful flowchart
- Eight Videos to Help Teachers Get Started Using Twitter
- 15 things kids can do instead of homework
- Paper use in schools
- Online Geogebra training
- What is The Effect of Technology Training for Teachers on Student Achievement?
- Using Google forms for a "Choose your own adventure" style story
- The difference between instrumental and relational understanding
- Ways to use technology in math class
- Why teach math?
- I don't know how to use a fax machine
- A Restitution Guide to Classroom Management
- Reflection of our course discussion about the use of technology in the classroom
- 25 Myths About Homework
- Free tools for math education
- The Role of Immediacy of Feedback in Student Learning
- A Fundamental Flaw in Math Education
- Migrating away from Google Reader

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## Comments

## Anticipating misconceptions with quick sorts

Hi David,

I teach 7th grade and my most diverse learners are in my standard level classes where students' MAP scores range between the 15th-90th percentile. When using exit slips as a quick sort teachers must use that data to regroup students and structure their lesson and classtime to address any misconceptions. To be most effective I need to have tomorrow's lesson prepared with differentiation in mind. Yesterday, a formative assessment helped me identify 14 students who can add and subtract one step equations, but 7 can't do it when it involves combining like terms. Three of the seven can't add negative decimals, and the rest are still combining terms that aren't alike. For Monday I need to have appropriate resources ready for all of my students--from those who don't get it to those that do. I'm the first to admit that some days I'm better prepared for the differentiated classroom than others.

Having taught this topic for several years I know the misconceptions. But as you point out it's what you do with that information that makes the difference. I'm investigating the math workshop model and I think that may help me. To successfully execute the workshop model an entire unit needs to be prepared for all possible "What ifs".

I appreciate your insights on formative assessment and look forward to part 3.

## One advantage you have is

One advantage you have is that you have evidence that at least some of your students are able to do all of the following skills. It might be worth looking into re-engaging students with examples of students work. See http://instructionalactivities.com/activity/re-engagement for example.

The math workshop model you mentioned sounds interesting. Can you describe it in more detail?

## Math workshop model

It runs similar to a reading workshop in that the independent time is quite differentiated. The lesson begins with an open ended task or review problem, followed by a mini-lesson, independent work time where the teacher confers with students and a shareout at the end. After the mini-lesson, for example adding and subtracting one step equations, I would have a quick formative assessment to determine levels of understanding. Students are then immediately regrouped and they practice or are given extensions based on their level of understanding. The key is to have the resources at the ready: combining like terms practice, one step whole number practice, one step decimal/fraction practice, and an extension perhaps on solving equations with variables on both sides. A mini-lesson may not even be the current topic, it could be a number talk. The idea is to create a culture of learning where students become self directed. Here's a link to the book Minds on Mathematics. http://www.amazon.com/Minds-Mathematics-Workshop-Develop-Understanding/d...

Another teacher and I are doing a book study, led by our reading specialist, who also has a keen interest in math. It requires a lot of advance prep and it's not just doing stations.

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