In an earlier post, we gave an outline for integrating assessment into a math lesson. And we said that in subsequent posts we would address several aspects in more detail. Here is a closer look at the first step – starting the lesson with a graphic representation of the concept and procedure handed out to each student.

In the post that followed, we put up a video (The Soccer Teams Graphic) that demonstrated exploring the graphic in numerical order and re-exploring it out of numerical order. Through tutoring and classroom application of the graphic pages, we have observed more effective and less effective ways of guiding a lesson using a graphic manipulative. While there are lots of manipulatives and number arrays out there, how they are implemented and integrated makes all the difference. As always, it comes down to effective teaching. Here’s an approach that works when guiding students with a graphic page and a blank piece of paper:

1. Use short phrasessay only what is needed for students to construct, count, and observe on the graphic page. AVOID explaining. “Now you see class, what you just did was…” Guide and allow for their discoveries. Short phrases move the lesson along, keep students actively engaged with the graphic, and help language learners and students who have a hard time focusing.

2. Start with an action verbcount (“Count how many…”), move (“Move your blank paper to show…”), cover (“Cover all but the first team”), uncover, and so on. This aids in fulfilling the first step. Students know right away what to do.

3. Model your directionswhen you ask them to uncover the next row of soccer players or members of a rowing team or weeks with the blank paper, do it with them at first – in a way that they can see your movements. A picture is worth a thousand words. This is helpful for everyone and especially for those learning the language – whether English or math! – and those who have a problem paying attention to aural directions. Your observations will tell you how long to continue modeling. It has proved more effective to model one step at a time rather than to give students an overview to recall and execute.

4. Observe (assess)look around to make sure everyone is engaged and following your direction on the graphic correctly. Since it involves physical activity, it is pretty easy to see who is where they need to be for the direction you gave and who is slow or looking around to see what to do. Assessing determines the speed and direction of the lesson – does a prior skill need to be addressed – who needs a little more support – who is ready for more of a challenge – and so on. Creatively using a graphic page enables you to derive multiple lessons and multiple challenge levels from the same page. Each student matters.

5. Correct student responses nowsince each student matters, something needs to be done when you see inaccurate or continually slow responses. But never give an answer. Guide individual students to correct themselves: “Show me on the chart how you got that answer.”  As they reconstruct and rethink their solution, they see for themselves what they did wrong, and therefore how to do it right. Some may require further guiding questions. It is not efficient to give them the answer and move on. You will have to retrace this same concept or procedure eventually. Do it now before it becomes an unlearned prior skill needed for another procedure the student is trying to learn.

6. Begin the transition to print say “show this many teams” as you write a “5” on the board. Don’t say the word “5.” Or say, “show this on your page” as you write on the board, “7 x 11.” By doing this, students see numbers and symbols as action prompts and begin making the connection between manipulation and computation. It prepares them for the problem pages that follow.

7. Ask questions transition to mini-word problems. “3 teams are coming to the practice tomorrow, how many water bottles will we need?” After the students have some experience with a concept and procedure, then begin to make the process conscious by asking them questions about what they did.

Our experience has been that new vocabulary and notation should be introduced gradually in conjunction with the work on the graphic pages and reiterated during practice with the partner pages that follow. It is better to solidify (talk about) concepts with the correct nomenclature after students have actually experienced using them with the graphic pages. They own something concrete then to hang those words on. We will look more closely at working with the partner pages in a later post.

Summary: Guide work on a graphic manipulative using short phrases. Model how to do it. Make sure they’re doing it correctly; help them if they aren’t. Guide the work using numbers and math notation on the board. Give further written and aural challenges at appropriate levels. Ask them questions about what they did.


In the previous post on this subject, we mentioned the state committee that was uncomfortable with the idea of focusing on formative or ongoing (while teaching) assessment. To answer them, “Yes it does take good teaching!” There is a saying in music that there are no choirs that sing out of tune, only choir directors who allow them to. Several members of the committee seemed to accept as a given a level of out-of-tune teaching.

While recognizing the importance of testing, that post asked if it should be the primary means for seeing how students struggling in math are doing. Tests have an aura about them especially for students already struggling, that ongoing assessment may not.

Formative assessment is popping-the-hood so to speak and seeing what’s going on inside the minds of the students with the current lesson and their facility with related prior concepts and skills. The focus is problem identifying and problem solving.

For instance, students struggling with variables may have an issue with variables or their confusion may come from specific prior knowledge or skill gaps. They may have missed that subtraction and addition, division and multiplication, and squaring and finding the square root are all opposite procedures. Stepping back from that, they may not understand the concept of opposite procedures. Or they may have problems with division or squaring itself.

Formative or ongoing assessment is measurement that is built into the fabric of teaching and learning. With it we notice the problem and can immediately adjust our teaching to uncover and address the cause.

In the next posting on this topic, we’ll look at a few formative assessment approaches.

We recently came across a blog post stating something that seems logical and helpful. Before we start working with students to help improve math performance, it noted that we need to test them to know the areas to focus on. A quick search on the internet reinforced this. A number of sites discussed different types of assessment: diagnostic or prior assessment, formative or ongoing assessment, and summative or post assessment. But when looking for examples of each, ongoing formative assessment received less attention compared with more examples each of diagnostic and summative testing.

This reminded us of when we made a presentation before a state curriculum adoption committee. We explained that since we write intervention materials designed to support students struggling in math, formative assessment is built into the fabric of each lesson in the program. While there are pre, post, and mid-chapter tests, the emphasis is on the problem solving support for these students of ongoing formative assessment. One member of the committee laughed and said, “Well that requires good teaching!”

It appeared that everyone in the room recognized that good assessment is systematic and ongoing.  But several members of the committee saw repeated testing as the means of insuring ongoing assessment.

We won’t go into possible reasons for the apparent preference for testing. Rather, in this post, we want to lay out some problems with relying on testing as a primary means of assessment, especially for students challenged by math. In the next post, we will think through the value of formative assessment.

Going back to the opening paragraph, diagnostic or prior assessment is essential. We’re flying blind without it. But favoring tests over integrated ongoing assessment may miss part of the potential power and value of assessment and even create some of the very problems our better teaching are trying to address.

When taking tests, students challenged by math will continue to write some incorrect answers. They will leave some answers blank. And a test is usually graded by someone else, not the students, and is usually handed back at a later time disconnected from the experience of that assessment. So what’s the problem with this?

Students who are not doing well may not give us the best indication of what they know and can do by taking another test. Continuing to write incorrect answers can reinforce memories of those wrong answers, reinforce faulty procedures leading to the incorrect answers, and reinforce partially understood or misunderstood concepts. Having to leave some answers blank can reinforce a negative self-image, and frustration with and resistance to the study of math. It also means that needed practice on those problems is missed. When the test is graded by the teacher and handed back at a later time, the students are no longer engaged in the problems and many will not retrace their steps in arriving at the wrong answers.

Assessment has multiple uses for teacher and student. It gives the teacher some evidence of student preparation and performance, which informs what is taught and how it is taught. It also provides students with feedback. While testing is an essential component, should it be the primary form of feedback for the intervention teacher? The teacher may not know from a test, which problems the students labored over or solved quickly whether they got them right or not. The test may not reveal which prior conceptual, procedural, or simple nomenclature gaps led to an incorrect answer. Depending on when a test is given, it may be too late to go back and address certain problems. And because of all of this, it may not be the best form of feedback for students struggling in math either.

In a later post, we will look at a different approach growing out of one meaning of the word assessment, to sit beside and observe.

Standard timing practice has more to do with measurement than improvement. Measurement is useful, so is improvement. They can be accomplished together.

The standard approach to timing along with the fact that it is typically done without any graphic or manipulative support to refresh knowledge of the facts creates more problems then it solves.

Students write some incorrect answers, which can reinforce memories of wrong answers. Students leave answers blank, which does little for their interest in mathematics, reinforces a negative self-concept, and causes them to miss needed practice on certain problems. Getting the paper back sometime later when the students are no longer engaged in the exercise means they probably will not go over missed problems to see where they went wrong. When it is corrected by someone else and not self-corrected, they are another step removed from the whole process.

Again, measurement and improvement can be accomplished together. Consider the posting “Measuring AND Improving Fluency.”

How to learn the way around a new city? In a more traditional approach, the teacher, in the classroom, explains some basic turns and landmarks and even models how to get to a specific location, then hands the student the keys to find someplace specific for homework. In this approach, the teacher is pulling on past experience with the city, and the explanations and modeling on the board are a short-hand for that experience. The student needing extra help (intervention) has no or at best a mixed experience to make accurate sense of the short-hand.

In a second approach, the teacher and student get in the car together but the teacher drives, pointing out the turns and landmarks he or she knows and again asks the student to find a specific location for homework. Better, but it is a second-hand experience for the student.

A third approach is to have the student drive with the teacher in the passenger seat not telling where or when to turn. The teacher first orients the student to a map of the city to be experienced. With good questions, the student notices landmarks on the map and is able then to begin making connections between the map and what is experienced while driving, “Oh this turn coming up is right here on the map.” “Good,” says the teacher, and that’s it, no lecture on the history and use of gazetteers. The teacher keeps the student in the driver’s seat.

If the student makes a wrong turn, the teacher simply asks the student to show on the map what they did and where they wanted to go. The student retraces the steps and notices that a left turn was needed not a right turn. “Good,” says the teacher and the student turns left. The city is slowly internalized as it is experienced, guided by the teacher and the map.

A good graphic representation of the math, not explained but oriented to and guided through, can be the map of the conceptual and procedural processes students have to navigate through.

What exactly is fluency? With languages, fluency is the ability to clearly express thoughts without having to grope for words. With mathematics, fluency refers to a student simply knowing a fact or procedure, without having to stop and think about it. Many first-graders can instantly tell you that 2 + 2 = 4. But too many older students hesitate with say, 9 + 5.

If the students’ minds are burdened with having to go back and reestablish 9 + 5, when they should be free to think about more complicated concepts (x + 9)(x + 5), their progress will be hampered. Their feelings about math in general will take on an aura of stress and clutter, and the feelings of curiosity, the joy of discovery and confidence will fade away. Tasks such as homework and tests will take longer than they should.

The first graders and older students may understand the concept. While understanding provides a basis for fluency, it does not automatically produce it. What enables learners to acquire fluency? Practice. But practice of the right sort is required. See our post, A Model for Using Guided Discovery in a Math Lesson.

There are two levels of accomplishment with understanding and fluency. The first level: students slowly solve problems with understanding and accuracy. The second level: students quickly solve problems with understanding and accuracy.

Many think that the job is done when students have reached the first level. After all, they got the problems right, and understood what they were doing—does it really make a difference if they were fast or slow? It does. Students who instantly know that 9 + 5 = 14 have a clear advantage in more complicated contexts: 49 + 65 = ?   91 + 52 = ?   98 + 57 = ?  976+458 = ?  9x + 5x = 280   (x+9)(x+5)= ?

As students grow older, 9 + 5 and all the other basic facts do not go away. They are used repeatedly throughout the upper grades.