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.

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Following on the previous two posts, here is the Partner or Folding Pages podcast. With the partner pages, students rediscover, reinforce, and extend the work they did on the graphic pages. They connect manipulation with computation and continue building memory by taking more spins around the city.

Four additional features of the partner pages that support students who are not doing well in math are:
1. They involve cooperative learning with individual accountability
2. Students receive immediate and ongoing feedback
3. They engage teacher, peers, and students in student self-correction
4. They use no-stress timing focused on improving rather than merely measuring performance (See the earlier postings on fluency for a discussion of this.)

Again, because of the lesson design, we can observe and assess how students are doing each step of the way – and make the adjustments our observations indicate.

Vodpod videos no longer available.

more about “Partner or Folding Pages“, posted with vodpod

We were asked if we could post the podcasts referred to in the previous assessment posting, so here is a slightly abbreviated version of the Soccer Teams podcast.

We began that posting noting the importance of designing materials that enable observation while accomplishing learning objectives. The Soccer Teams graphic page is a model of that.

1. It can be used to support students who are not succeeding with the regular text book.
2. It can be used up front to quickly orient students to a concept or procedure, increasing the likelihood that they will succeed with the regular curriculum.
3. It can also be used to efficiently cover prior skill gaps so that students are capable of handling the current lesson.

Students not doing well with multiplication or division may not understand what multiplication and division really mean. Then they try to memorize the things they don’t understand. Students can memorize, “quoth the raven…” Say it enough times and they can fill in the blank sounding like they really know something. But they may very well not know what “quoth” or “nevermore” mean or recognize a raven if they saw one.

With the Soccer Teams graphic, they are able to experience how a single group can be one and many (1 group of 11), and the feeling for magnitude between 1 x 11, 2 x 11,  7 x 11 or 12 x 11. These are some of the conceptual building blocks of multiplication. Times tables are not. By working with the graphic pages, students begin building a memory of these experiences the same way we remember our way around a new city. We don’t sit at home and memorize the streets with flash cards.  We may orient ourselves with a map, but then we go out and drive around. That’s how we learn the city.

With the graphic pages students drive around so we can observe and assess how they’re doing right from the beginning and each step of the way.

Vodpod videos no longer available.

more about “The Soccer Teams Graphic“, posted with vodpod

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.

As we noted in our previous problem of the week, “students’ development of mathematical reasoning involves making generalizations, and deeply felt generalizations are the result of making many specific observations…” So we offer a third approach to Pythagoras. Certain problems are also more compelling to some students than others.

This problem presents an additional feature that an irrational number can be a reasonable and even unavoidable answer to the length of a side or hypotenuse.  Like the previous pythagorean problems, this one allows students to come at the solution by reasoning their way through a consideration of area. It involves students thoroughly noticing the given (the implications of the given information). For instance, what kind of triangle is triangle ADC? Do they stop with one observation or do they notice and consider the implications all of the given information about that triangle? And it enables them to carry forward and reapply previous knowledge like the properties of various triangles, a square, and the angle-side theorems.

So again, imagine that you had never heard of the Pythagorean Theorem (a2 + b2 = c2), and that you did not have a ruler. Using only the drawing below, determine the length of line segment AC. All corners that look square are square. All lines that look parallel are parallel. The answer will be posted next week.

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.

This week’s problem is similar to last week’s. Students’ development of mathematical reasoning involves making generalizations, and deeply felt generalizations are the result of making many specific observations (not just one)—and not merely reiterating the words that we or a book have pronounced. This week’s problem is intended to reinforce and extend the development of reasoning begun with last week’s problem.

Again, imagine that you had never heard of the Pythagorean Theorem (a2 + b2 = c2), and that you did not have a ruler. Using only the drawing below, determine the perimeter of triangle WKX. All corners that look square are square. The answer will be posted next week.