1.  What do you know about circles. (All circles have a center, a radius, a diameter, a circumference, and an area.)

2.  What are some things you know about squares. (All squares have 4 equal sides, 4 right angles, a perimeter, and an area.)

3.  How do you calculate the circumference of a circle? What information is needed? (How about the diameter. Pi x D = C.)

4.  How do you calculate the perimeter of a square? What information is needed? (You need the measurement of one side. 4 x a side = P.)

5. Draw a diameter from left to right through the middle of the circle, parallel with the bottom of the square. Is the measurement of the diameter the same as the width of the square?

6. Imagine that the diameter of the circle is 7 meters. What is the circumference? (Use 22/7 for Pi.)

7. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle?

8. Imagine that the diameter of the circle is 10 meters. What is the circumference? (Use 3.14 for Pi.)

9. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle?

10. Imagine that the diameter of the circle is “D” meters. What is the circumference? (Use 3.14 for Pi.) (D x 3.14)

11. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle? (D x 4)

We have several posts that look at guiding student discovery in a math lesson and thought it would be helpful to organize them here in sequential order. The first two posts provide some background thinking and the next three, accompanied by podcasts, provide a step-by-step description of the process.

This collection of posts is similar to the one in Assessment Posts Organized since ongoing assessment is fundamental to an effective lesson using guided discovery. The initial posts in each collection differ owing to the assessment or guided discovery focus of that collection. Click on a title and you’re off to the post.

The Soccer Teams Graphic – Podcast

Partner or Folding Pages – Podcast

The Progress Chart – Podcast

We looked at ongoing or formative assessment in a math lesson in several posts and thought it would be helpful to use this post to organize them sequentially. So here they are. This collection of posts is similar to the one in A Model for Using Guided Discovery in a Math Lesson since ongoing assessment is fundamental to an effective lesson using guided discovery. The initial posts differ owing to the assessment or guided discovery focus of the collection. Click on the title and you’re off to the post.

The Soccer Teams Graphic – Podcast

Partner or Folding Pages – Podcast

The Progress Chart – Podcast

Since this post is part of a series, it may be helpful to read over and Beginning a Lesson Using Guided Discovery to put it in context. You may also want to look over the podcast posted earlier on the Partner or Folding Pages to see a model of the content of this post.

Earlier this month, we posted an outline approach to integrating assessment into a math lesson. Teacher and student ongoing assessment run throughout that initial step. Equally important, doing something about what is observed, with every student, is also integrated into that step.

This post takes up the second step in the guided approach – working with a folding or partner page of problems. This engages students in quickly reconstructing and re-experiencing the work they did on the graphic manipulative. Both the graphic and partner pages are deceptively simple looking, while being highly flexible and engaging tools for building understanding and fluency together – in a way that can be directly observed.

On the left is an image of a negative number partner page with the problems at one end and the answer key facing the opposite direction at the other end. (Our March 2 post has an image of a Soccer Teams partner page.) The same teaching approach laid out in the post on Guided Discovery should be used here – 1. use short phrases, 2. begin with an action verb, 3. model your directions as needed, and 4. look around to make sure everyone is on track. There is no point in having students construct and build visual memories of a wrong procedure or amount!

These steps for delivering a lesson with the partner pages follow on the approach with the graphic pages. While not prescriptive, they do help keep students actively engaged in the work.

Start by saying:
1a. “Fold the partner page in half and set it down on top of the graphic page.” The partner page takes the place of the blank paper described in the earlier post.

1b. “Read the first problem on the partner page and move it to show that many…” teams, weeks, squares or roots, dollar bills, tickets, shirts, gallons, fractions of a cake, cheese slices, negative temperatures, “quadratic hearts”, or whatever graphic starts the lesson.

Then
2. Ask “How many players (teams, fractions, weeks…) do you see?”

3. Respond to correct answers with a simple “good” or “correct.” Respond to every correct as well as incorrect answer. By doing this, there is no question as to whether the answer is correct or not. For an incorrect action or response, never give the answer. Instead, guide students to correct themselves with: “Try again” or  “Show me on the chart how you got that answer.” As they reconstruct and rethink their solution, many will see for themselves what they did wrong and how to do it right. A few may require further guiding questions.

Do the next problem or two as a class as needed. Then ask the class to work the rest of the problems like that individually. Ask them to read the problem; move their paper; and say the answer in a quiet voice. So that students won’t copy each other, have some work the problems going across the page; some going down (from top to bottom); and so on. Circulate around the room checking on their work responding to correct and incorrect answers.

4. Ask for a volunteer to do the first row of problems for the class. Remind them that it is not cheating to use the graphic page. Ask for others to do the next rows. Have the rest of the class follow along with the answer key to make sure these students are answering each problem correctly.

5. Ask for two volunteers to model working in partners, one doing the problems, the other checking the accuracy with the answer key. The partner checking the answers follows the model in step #3, never giving answers and saying “good” for every correct answer. And the partner simply says “try again” if the answer is incorrect. They have the graphic page to refresh their understanding of the procedure and the basic facts. The rest of the class follows to make sure these two are doing it correctly.

6. Now have everyone work in partners following the model in step #5.

Summary: The partner pages build understanding and fluency together – in a way that can be directly observed. They involve cooperative learning with individual accountability, students receive immediate and ongoing feedback (but no answers), so they engage teacher, peer, and student in student self-correction. Going in different directions on the partner pages enables non-rote repetition. Like music students, math students benefit from going over a familiar exercise more than once. Getting better at what they are starting to get good at builds confidence along with competence.

At this point, students have achieved the first two levels of success: 1. they understand what they are doing, and can accurately solve problems slowly, with the aid of the graphic page; 2. using a partner page, they connect manipulation with computation and gradually wean themselves from the graphic page, accurately solving problems with greater fluency. The next level, developing independent speed, accuracy, and understanding, using the progress chart, will be addressed in an upcoming post.

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.

As we noted in the two earlier posts on this subject, ongoing or formative assessment is as valuable or more valuable than testing to the teacher or parent of students struggling in their study of mathematics. Here is an outline of an approach to integrating assessment into a math lesson. It involves designing materials that allow for observation, while accomplishing learning objectives, and then setting up many opportunities for the teacher and students to observe (assess) throughout the lesson. In subsequent postings, we will address several aspects of this lesson in more detail.

1. Start the lesson with a graphic representation of the concept and procedure handed out to each student. Asking questions, the teacher guides the students in a discovery of the core concepts. (Click on Soccer Teams or take a look at the next post for a model of this in teaching multiplication by 11s.) (See our later post Beginning a Lesson Using Guided Discovery for how to deliver a guided discovery lesson.) Since students have their own graphic page, the teacher is able to see who is moving their paper to construct problems. Conversely, it is pretty easy to observe who is looking around to see what to do, who is not moving their paper, or who has moved it too far for the given directive. This is easier to observe than if students are asked to jointly say the answer – it’s difficult to tell who knows the answer and who is copying.

Here the teacher assesses right at the beginning of the lesson, and can adjust in the moment. If students are struggling with say missing addends, the teacher can use the graphic for teaching missing addends to refresh basic addition. Then transition right back to the intended lesson. Or if the students are struggling with a lesson on squares and roots, the teacher can use the squares and roots graphic to refresh their understanding of and facility with multiplication, then transition right back to squaring. (See our podcast on The SquareMaker for an example of graphically representing squares and roots.)

2. Once the students are oriented to the graphic, help them make the connection between manipulation and computation with problem pages linked to the graphic page. The problem pages have the problems at one end and the answers at the other. The pages are folded in half so students see only the problems at first. (To see an example, click on Partner or Folding Pages or take a look at a subsequent post for our podcast covering partner pages.) The students do the first few problems together as a class so the teacher can assess if the class has the approach down. Then they finish the page on their own saying the answers quietly. If a student makes a mistake, they have the graphic to reconstruct the problem. The teacher walks around checking in on them. Noticing a mistake, the teacher simply asks the student to show how they got that on the graphic to see if they understand what the problem means and if they can construct the right answer.

Here the teacher continues assessing and the students begin to self-assess. Wrong answers do not linger for very long. And students are able to immediately retrace their steps on the graphic and see what a correct approach to that problem looks like.

3. Then one student models the top line of problems, constructing them on the graphic and saying the answers out loud, as the rest of the class follows along to make sure the answers are right. Another student completes the next line and so on. Here teacher and students are assessing.

4. Two students then model partnering with each other while the rest of the class makes sure both are doing it right. One student works the problems while the other looks at the answers and says “good” for correct answers or “try again” for incorrect answers. The student working the problems has the graphic to refresh the experience of the problems. They self-correct in real-time so no one leaves a page wondering if they got the answers right or not. And all of the problems are done so no needed practice is missed. Here students are assessing and self-assessing with observation and support from the teacher.

5. Then everyone works in partners with the teacher walking around, listening in and asking guiding questions where needed. Again, here students are assessing and self-assessing with observation and support from the teacher.

This approach involves more teacher assessment in the beginning moving to more student assessment and self-assessment through the lesson. It includes ongoing observation, immediate response when the students are still engaged in the learning, and varied, repeated practice that is not rote (students can work the problems on the page in a different order each time).  Like music students, math students can derive benefit from going over the same exercises more than once, as long as it does not become rote. At any point the teacher can adjust the lesson for the class or a student or group of students to address a prior skill needed for success with the given lesson as described above.

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.