Great Activities for Formative Assessment and Problem-Solving in Math

The Mathematics Assessment Project (MAP) is a joint venture between the University of California, Berkeley, and the Shell Center at the University of Nottingham, with support from the Bill and Melinda Gates Foundation. The project aims to “design and develop well-engineered assessment tools to support US schools in implementing the Common Core State Standards for Mathematics.” To meet this goal, the MAP website covers three major content areas: formative assessment, summative assessment, and professional development.

Within each of these areas, there is a wealth of material that can be readily implemented by teachers looking to make formative assessment a bigger part of their classroom. It will, however, take some time to look carefully through all of the lessons and assessment tools to find ones that will fit your class. Furthermore, the material hosted on the MAP site is aligned with the Common Core Standards for grades 6 through 12, so some of the content may not be suitable for students entering at NRS levels 1, 2, and 3.

The best place to start with MAP is with the formative assessment lessons, which are called Classroom Challenges. These classroom challenges are separated by grade level and divided into two different sections—concept development lessons and problem-solving lessons—both of which are aligned with specific CCSS content and mathematical practices. The concept development lessons are intended to build understanding of concepts and content, while the problem-solving lessons draw out student thinking and require students to apply concepts that they have previously learned. All of these Classroom Challenges include problems for students to solve, along with detailed lesson plans and other supplements that help teachers structure the activity effectively. One of the most challenging aspects of teaching from a problem-solving perspective is being able to anticipate areas where our students will struggle, misconceptions they may have, and solution methods that they might attempt. To help us anticipate these struggles and misconceptions, the MAP lessons include photographs of student work accompanied by detailed explanations of students’ thought processes. For each lesson, you will also find a list of common issues for learning and questioning techniques that we can use to address these issues and draw out student thinking. There is a lot of material to look over for each of these Classroom Challenges! But when I first taught the Gold Rush activity, I was thankful for all of the detailed notes and suggestions. They gave me a clear sense of what to expect while my students were solving this problem.

That said, many of the problem-solving activities here might need to be adapted so that they fit your class. Several of them ask students to critique a particular line of reasoning, and many of the problems don’t have what we tend to think of as an “answer.” Rather, they are intended to foster discussion about math and to encourage students to test a theory and draw conclusions. Also, the activities at the high-school level require a great deal of prior knowledge about algebra and geometry. So, while the problems at grades 6 through 8 can be solved with relatively little background knowledge, the problems at grades 9 through 12 would require much more scaffolding. I would recommend trying out  some of the seventh-grade (Maximizing Area: Gold Rush) and eighth-grade (Generalizing Patterns: The Difference Between Two Squares) problems. Both of these problems ask students to analyze patterns, create diagrams/tables, and formulate hypotheses based on the work that they’ve done.

The Summative Assessments are somewhat less useful for our purposes. The assessment tasks—divided into “novice,” “apprentice,” and “expert”—have thus far only been created for high-school grade levels, and many of them, even at the “novice” level, are quite challenging. The summative assessment tests, while they contain several problems that could be extracted and used in an ABE or HSE class, cover such a broad range of material that they are a little impractical for classes that—like many of ours—meet only a few times each week. There are only six “middle-school” level tests, and each one addresses Common Core Standards from grades 6, 7, and 8. The tests at the high-school level contain content similar to what students might see on the TASC, but again, because the tests theoretically cover all of the Standards for grades 9 through 12, these assessments may not be the most accurate in measuring your HSE students’ progress.

While the formative and summative assessment activities provide teachers with exercises and lesson plans, the Professional Development modules on the MAP site offer models for professional development activities to support this style of mathematics instruction. These activities are intended to help teachers get comfortable with formative assessment, problem-solving activities, questioning strategies, and collaborative student work. They are written specifically for trainers, and each module contains several handouts, lists of discussion topics, group activities, etc. While these are interesting to read on your own, they are best used as jumping-off points for professional development activities within your organization. The style of math instruction that MAP advocates is very different from traditional methods of mathematics instruction. These PD modules can be beneficial to organizations that are looking to train their teachers on new ways of teaching math in the era of Common Core.

In all, there’s a lot to look at here, but if you’re able to spend some time looking over the problem-solving activities, you’re bound to walk away with some useable resources and ideas. I would recommend starting out with the Classroom Challenges. Look over some of the questioning techniques, student misconceptions, and questioning strategies, and then pick one or two of them to try out in your classroom. I’d love to hear your thoughts about MAP in the comments!

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About Tyler Holzer

Tyler grew up all over the Midwest and has loved math all his life. When he started college in Nebraska, he declared a math major but then ended up getting a BA in history and an MA in English. Now he lives in South Park Slope with his wife, Rachel, and is the ABE/HSE Coordinator at the Fifth Avenue Committee. He is a founding member of the Community of Adult Mathematics Instructors (CAMI) and a co-creator/moderator of www.mathmemos.org.

7 thoughts on “Great Activities for Formative Assessment and Problem-Solving in Math

  1. I have also found my students really engage with these lessons. I also really appreciate how many great opportunities each lesson has for me to get deep into the reasoning and problem-solving in my own students throughout. I love the way the Classroom Challenge lessons are organized and how focused they are on formative assessment.

    – Most of the Challenges start with a problem that students work on independently for a short time. Then teachers collect that work and formulate questions for students to think about as they review their work – the goal is for questions to be specific and targeting 1-2 aspects of the students thinking. To support this step, the Challenges have charts describing common issues and some suggested questions and prompts to engage students.
    – Then teachers return the work with some brief comments and students reflect on their thinking and use the questions to improve it.
    – Next students work in small groups to create collective answers that are even better than their individual work.
    – Then students have a whole-class discussion looking at the different methods.
    – Then the Challenges have samples of “student” responses to the task your class has been working on. The handouts of “student” responses have prompts to guide your class in an analysis of the thinking presented. These sample responses often show common misunderstandings or novel approaches to the problem at hand.
    – Next there is a whole-class discussion of the “student” samples.
    – The final step is for students to reflect on their work.

    I love how much the ideas of revision, social interactions and sense-making are so present through all of these lessons. I also love how MAP pulls back the curtain on the research looking at how students learn. It takes time to do a formative lesson with students, but “If you don’t have time to do it right, when will you have time to do it over?”

    I agree with Tyler in terms of getting started. There are only about 20 lessons each for 6th, 7th and 8th grade and slightly more than that for high school so it’s easy to start poking around. For HSE level teachers, I would check out the 6th and 7th grade materials first. The first one I ever looked at was a 6th grade lesson called Sharing Costs: Traveling to School. It’s an interesting problem involving proportional reasoning. The problem involves a car pool situation, with people trying to divide up monthly gas expenditures in a fair way, between people who each travel for different distances. what I like about the problem is that there are different ways to think about what’s fair.

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  2. Great article Tyler. I think formative assessments are really underutilized in the classroom. This article has great ideas for assessing the student’s learning without given them the anxiety associated with “tests”

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    1. This is an awesome resource! I’ve used several of the formative lessons with my students and they never fail to engage and challenge them. My favorite one so far is the newly revised Representing the Laws of Arithmetic. My students got so much out of this extended activity. The lesson takes numerous math topics that are typically taught in isolation and gives students the opportunity to explore how they connect. Suddenly, exponents, order of operations, identifying equivalent expressions, distributive property, commutative property, and finding the areas of compound rectangles all come together. Students are charged with finding the relationship between numerical expressions and diagrammatic representations.

      Here’s a picture of how the project came together for my students.

      Thanks, Tyler, for writing about this resource and for pointing me in the direction of the PD resources at Math Shell. I’m definitely going to give that the once over.

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  3. I agree the “classroom challenges” lessons are the best place to start, which helped me to narrow my focus. (For some reason the website’s layout seemed a bit overwhelming to me at first.) There are some really great problems in the middle school band.

    The most useful aspect of this resource for me is the “Common issues” and “Suggested questions and prompts” in the teachers guides. Many other resources have great problems but before implementing in the classroom I need to solve the problems using the different strategies my students might use, anticipate the areas my students will run into problems, and develop strategies for appropriately scaffolding their understanding. For teachers new to these types of math explorations this can be a barrier to actually implementing frequent use of this approach in the classroom. With the teachers guides provided here, a lot of that work is already done which can make implementation more successful (and less stressful for those starting out with these methods).

    I have also found this site useful as a trainer. Invariably at the end of a training teachers will say “Well that’s great but my students are working on XYZ right now. Can you give me problems like this but for XYZ?” On more than one occasion I have used the search function on the MAP website to pull up a lesson on XYZ on the spot.

    I had completely forgotten there were PD modules on the site. Thanks Tyler for the reminder as they may be a useful resource as I do more trainings.

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  4. Great article Tyler, thank you for this! I am excited to share this with my staff at our next meeting. Wonderful resources for all; I look forward to utilizing some of the PD resources myself. There is too much pressure on tests and testing. I always encourage formative assessments to use along the way. So beneficial to our students. As for the teachers, a lot of the work is done for them here – this should make it easier for them to incorporate into their own lesson plan. I look forward to feedback from my staff!

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  5. Check out Tyler’s write-up of the Gold Rush problem (mentioned above) at mathmemos.org. Learn more about Tyler’s experience using the problem in an adult literacy classroom. The write-up includes several samples of student work and Tyler’s analysis of that work.

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