CPM is proud to support mathematics teachers by participating in many local, state, regional and national conferences. View **CPM’s 2018 conference schedule** for listed sessions. We look forward to seeing you at one of these inspirational conferences.

Contact Lisa Jasumback at lisajasumbacl@cpm.org or 801-884-2153 for more information

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# Principles of Assessment

Teachers understand that students learn at different rates and through different experiences. The CPM materials have been designed to support mastery over time through a student-centered, problem-based course, and this approach supports students’ different learning styles. But when changing the materials and changing the methodology, the teacher must also change her assessment practices. She cannot tell students she wants them to explain their thinking during class and then assess them with only a multiple choice test. Students will quickly realize that “explaining” is not valued enough to be given the time to be assessed.

Along with the type of assessment questions they ask, we want teachers to consider assessment as important as instruction, texts, and activities. Below are CPM’s Principles of Assessment.

**Teachers need to be involved in the crafting of their assessments.**

The teacher must create an assessment that complements what students did during class. The teacher is the only one who knows the discussions that occurred in his/her classroom, the topics that caused the students to pause and explore more deeply, and the areas that were perfectly clear. Though CPM provides examples of chapter assessments that are samples the teacher should never administer the sample assessment without careful consideration. Rather, these samples should be used to guide the teacher when developing her own assessment. The teacher should never give sample assessments without careful scrutiny. Working with colleagues to develop common assessments is a great learning experience for everyone involved, but even if you are giving common assessments, you may need to adapt the assessment slightly for each class. Adaptations does not necessarily mean “rewriting” but might include letting students to use their notes on part of the assessment, or allowing students to select a subset of the problems so that they may demonstrate their strengths. Students can give justifications for their choices.

**Teachers need to read and work through all assessment items carefully before giving them to students, making sure it is clear what kind of response is expected and that there are no errors.**

When a teacher does not take the time to make sure every question is well phrased and is grammatically correct, we cannot be sure we are assessing the student’s mathematical ability. If the problems are poorly phrased, a student who knows the math may never have a chance to demonstrate his knowledge if he struggles through trying to make sense of the question. Typographical errors can also cause anxiety and distress. Teachers need to take care to be sure they are making the mathematics the key thing being assessed.

**Students should be assessed only on content with which they have been meaningfully engaged, and with which they have had ample time to make sense of.**

Just because the teacher has told the students something does not mean the students understand it. Students need time to process and make sense of the mathematics they are learning, and it is unfair to assess students for mastery before they are ready. The teacher must also be flexible when assessing because students learn at different rates. Using different methods of assessment allows students to demonstrate their strengths, letting you look for what they know rather than focusing on what they don’t know. By balancing skills with problem solving, and revisiting familiar material, assessment can better reflect the student’s understanding. By including old material and newer material together on assessments, students can demonstrate how their learning is progressing. And, by allowing for multiple approaches when solving assessment problems, students can show how they understand the mathematics. Formative and summative assessments serve different purposes, and teachers need to be clear when and why they are giving each. The power of formative assessment should not be dismissed. Teachers understand that formative assessment is more than just pre-tests and exit slips. Teachers use formative assessment daily through questioning of students and providing feedback.

**Formative assessment is a learning experience for both the student and the teacher.**

Teachers adapt their questioning and their lessons based on students’ responses. While formative assessment is on-going, summative assessment happens less frequently. With both formative and summative assessment, teachers insist that students explain and communicate their methods. Teachers take time to understand the student’s approach and to give descriptive, effective feedback to the students on formative assessments to push student learning forward.

**While teachers are required to evaluate and assign grades to their students periodically during the school year, the grading should be flexible enough to allow for variations in when students master the mathematics.**

While some students will have a good knowledge of what is tested at a particular time others may know some things well, but not know others, yet. This is why we recommend cumulative testing with some emphasis on what has recently been learned and more emphasis on what has previously been learned, where there are continuing opportunities to practice and extend, clarify, and consolidate. A grading system needs to be flexible enough to give credit for what students show that they know now, but not permanently punish them for what they don’t know yet. If “benchmark” or “mastery assessments” must be given they should be delayed and given after all students have had multiple opportunities to use and practice the new material and show what they know (several months after the material has been introduced).

# Congratulations!

Two CPM teachers have been awarded the Presidential Award for Excellence in Math and Science Teaching. John Hayes, CPM teacher, Teacher Leader, and math coach, is the math teacher winner from Wisconsin, and Beth Vavzincak, CPM teacher and Teacher Leader, is the math teacher winner from Ohio. This is a well-deserved honor for both John and Beth. On September 8, these teachers visited Washington, D.C. to meet the president and tour the White House. Congratulations John and Beth!

View John's full PAEMST finalist profile **here.**

View Beth's full PAEMST finalist profile **here.**

# What Students Say…

Senior (4th year of CPM)

It's harder to work through the problem than being given the answer. But when I was looking at my required college curriculum, I realized that this work helped me prepare better for college. You understand the methods for solving the problem, instead of a teacher handing you the formula. You understand why it works that way.

Junior (3rd year of CPM)

I like the math notes boxes so I can see examples. I also like the closure activities where I can check my own work and find examples if I don't understand something."

Freshman (2nd year of CPM)

I like that the homework is both "review and preview" because I know I will be able to do it on my own at home. I also know that my teammates in class can work together to understand something new that we are learning.

# What Parents Say…

Having the chance to talk about the math with other students in her study team has been valuable for my daughter. This helps her see other students' struggles and how they think about the ideas. It helps reinforce her own skills as she explains her work out loud to others."

My daughter has always struggled with mathematics. But with CPM she began to like math and really understands what she is learning. All of the hands-on learning tools, like the Algebra tiles, have really helped her to finally succeed in math class.

# What Administrators Say…

**Travis, Wisconsin** (Math coordinator)

One of the main things I've heard from the teachers is that the structure of the lessons gives them the ability to quickly identify how individuals, small groups and the entire class are doing during a lesson. As a result the teachers are able to interact with students or facilitate activities to provide immediate responses. Students are less likely to fall through the cracks and students continue to progress at a productive pace."

**Julie, Wisconsin** (Assistant principal)

Students are organized into study teams and work on problem-based applications, team strategies and real-world applications. Many algebra teachers feel reenergized and are having more fun teaching math with the CPM approach. Algebra classes focus on both basic skills and problem solving strategies that are used to help students relate to and understand the concepts behind the problems. Our students are being taught Algebra in a more rigorous and relevant manner. It's really about discovering the math, rather than being told the math.

**Peter, Wisconsin** (Director of curriculum and instruction)

We are already seeing the benefits of the program (after 3 years). Students seem to be better prepared for higher level math courses and stronger results are showing up on various standardized tests.

# Trivial Pursuits

by Ronald A. Wolk

Teacher Magazine, January 2003

It seems obvious that any serious restructuring of a school in need of improvement has to begin with substantial restructuring of the curriculum. The curriculum profoundly influences the way schools are organized and run by determining how time is allocated, how space is used, and how students and teachers are grouped.

In the late 1980s, the fledgling standards movement raised for national discussion a question that gets to the very heart of education: What should students know and be able to do at various points in their academic careers? The founders of the standards movement recommended "parsimony" in establishing the content standards that would undergird curricula. They urged that these new standards focus on the relatively few key concepts that provide a discipline with structure, leaving it to teachers and schools to fill in the blanks.

I saw this as a marvelous opportunity to rethink the purposes and priorities of K-12 education and to restructure curricula accordingly. Unfortunately, the opportunity was missed.

At the national and state levels, in the late 1980s and early '90s, specialists from each discipline were convened to draft standards. Not surprisingly, these writers were inclined to find important virtually everything in their fields of study. Geographers wrote national standards that would challenge Ph.D. candidates, and historians couldn't find a battle they didn't consider crucial knowledge.

This example, taken from a California commission drafting science standards, makes the point: It claims that students should know "that the force on a moving particle (with charge q) in a magnetic field is qvBsin(a) where 'a' is the angle between v and B (v and B are the magnitudes of vectors v and B, respectively)." Furthermore, they should be able to "use the right-hand rule to find the direction of this force." I submit that that is not something every high school student must master.

The sacred cow known as "coverage"–cramming into kids as much knowledge about any one subject as possible–drives these state and national standards. It's no surprise that a federal laboratory in the Midwest estimated that students would need an additional nine years of schooling to meet the national requirements alone.

If we had commissioned historians to write the math standards, scientists to write the history standards, and so on, they probably would have come up with a much better measure of what kids need to know. The slavish commitment to coverage results in facts and information being valued more than reasoning and understanding. It has prompted schools to isolate bits of knowledge rather than connect them in interdisciplinary ways. And knowledge out of context is trivia.

William Schmidt, the U.S. research director for the Third International Mathematics and Science Study, has argued that curricula are crucial to student learning and that the most important characteristic of an effective curriculum is coherence. During a recent conference, he said that "when the internal structure of the discipline is apparent to students and to teachers–whether mathematics or science or history–it becomes clear that the discipline is not an arbitrary collection of topics, but rather a very structured, sequenced outline of the way one proceeds to the deeper structure of knowledge within that discipline." American education, he added, tends to consider curricula arbitrary collections of topics.

It may have been reasonable to expect students to master humankind's accumulated knowledge when all that we knew about science would fit on one shelf in Thomas Jefferson's library. Today, a student can barely scratch the surface of the disciplines, and much of what he or she learns is likely to become outdated quickly.

Any curriculum should favor depth over breadth; it should help students make connections between ideas and concepts; it should help them see the relationships between knowledge within and across disciplines. And students should have the opportunity to apply what they learn in school to their daily lives. If that were to occur, perhaps more of them would know how to go on learning for a lifetime.

© 2002 Editorial Projects in Education; Vol. 14; number 04; page 4