Differentiated Math Chapter 3: Differentiating for Abstract Math Comprehension February 8, 2008
Posted by mirish in Math Instruction.trackback
This chapter covered 3 teaching strategies that could be used by teachers to help struggling students and to differentiate instruction.
1. The heart of this chapter is the use of a specific teaching strategy labeled CSA. CSA stands for Concrete, Semi-Concrete, and Abstract. This is one way to offer varied instruction and meet the needs of all students. CSA is referred to as the “mental tool” view of math instruction, which suggests that mental images of mathematic equation or problems may best be formed using physical materials or concrete objects. Students need to “visualize” math problems in order to understand them. Clearly, concrete objects or pictorial representations of them can greatly assist in this visualization process.
Learning to use concrete objects and semi-concrete representations is a prerequisite to abstract learning from many students; and for students who struggle in math, use of CSA instructional approaches can be critical. Many students with learning disabilities exhibit visual and auditory perception problems that negatively affect mastery of math. The use of CSA can foster efficient learning of math concepts and provide these learners with strategies to solve problems in every math area.
The concrete level is the lowest level of comprehension. This involves the use manipulative objects during instruction. Some examples would be using tally marks for numbers, manipulatives that can readily be configured to portray the concept , circles divided into fractional parts, or even edibles such a an orange divided into segments. These manipulatives should be accompanies with a worksheet for students to record results of problem solving.
Semi-concrete level uses representation of concrete object is instruction, such a s drawings of objects. This level uses no hands-on support materials. Worksheets for activities at this level should have graphic representations on them, as well as spaces for students to draw their own.
Bender points out that whenever a teacher gives a concrete or semi-concrete demonstration they should stress the relationship to the math concept under discussion.
Abstract thinking should be the teacher’s overall goal. In the abstract phase, mathematics problems without any representations are used, but such worksheets also can be constructed to include space on the sheet for students to develop their own representation of the problem. Bender gives many ideas and examples for teachers on pages 51-59.
2. Another teaching strategy covered in this chapter was the Errorless Learning Procedures. This strategy emphasizes that children require high levels of success in order to be motivated to continue their work in any curriculum area. Psychologists indicate that students need to succeed at least 85 percent of the time in order for learning to take place. An errorless learning procedure is an instructional procedure that precludes students from performing an incorrect response. Thus the students experience much higher success in learning.
These procedures seem most viable in cooperative learning activities and one-on –one teaching situations. Three procedures that were presented were: prompting, time delay, and cover, copy, compare. The last tactic seemed most practical. Copy, cover, compare follows a process where students copy the solution to a math problem such as a math fact or equation, the student then covers all examples and tries to write the fact or equation from memory. Finally the student compares what they wrote with the original fact or equation to make sure it was correct.
3. A third teaching strategy that was presented was called Classwide Peer Tutoring For Differentiated Instruction. This strategy will only be effective if a teacher designates a certain amount of time each day for the tutoring experience. During this time students are put into pairs and one at a time they serve as a “tutor.” The tutor will present math facts, equations, or problems on flash cards and record the responses. Student progress should be charted each day. The teacher is a facilitator and helps students when needed. Obviously, teachers could have each pair working at a different level or on a different concept or skill. Bender points out that this tactic allows teachers to truly individualize their lessons for an entire classroom.
I really enjoyed the errorless learning procedures, described on pages 59-67. Three separate tactics were described and given examples; Time Delay Tactic, Prompting tactic, and cover, copy, and compare tactic. The prompting tactic clearly assures the learner of correct responses, thus giving them the experience of success. The emotional impact makes learning math less threatening! When a teacher prompts a student quickly and then allows more wait time later before the response- indications are that the learning has become more solidified. I am convinced that our students must not fear the subject if we want them to be actively engaged. Giving them confident sense of success will lead them on to more difficult skils.
I also love the idea of classwide peer tutoring. It can’t be used for introducing a subject or skill, but it is extremely valuable during the guided practice and exploration phases of instruction. If modeled and maintianed appropriately- this strategy is essential!
I recently saw first hand what Bender is referring to. My wife and daughter were working on Math homework at our kitchen table. They were working on prolems using long division. After about 15 minutes my daughter was crying and said, “I hate math!”
My wife is not a teacher and probably could never be because of her lack of patience but she is an extremely caring mother. She identified that the difficult problems were frustrating my daughter. She wrote up some simpler problems and helped my daughter. I then took over and fifteen minutes later my daughter understood the steps and was able to complete problems on her own. My wife has had no education classes but realized that her daughter needed to experience success. She differentiated instruction for her daughter and took away the frustration and fear.
The CSA concept appealed to me as a music teacher. Recently, the second grade worked on their instrument identifying unit. We had our field trip to the music store, where they saw the instruments and played some of them (concrete), we identified instruments using web sites and powerpoints (semi-concrete), and on their assessment they had to write down the names of the instruments (abstract). However, they weren’t taught in this order. Next year, I plan on presenting some parts of the unit in a different order, moving from Concrete, to Semi-Concrete, back to Concrete (our field trip), then present the assessment - Abstract. I’m curious if the CSA model works well in that presentation order for all subjects. If we are truly differentiating for different learning styles, I think we need to take into account that some learners need to have material presented in a different pattern. For instance, I would think kinesthetic learners would benefit more from reviewing the Concrete approach before assessing at an Abstract level.
Over the past month and a half, we have been implementing one of the errorless learning procedures described in Chapter 3 in our 2nd grade classroom. It is the Time Delay Score Sheet for memorizing addition and subtraction facts. I revised the score sheet on p. 62 to be more understandable to a 2nd grader, and then paired each student with a math partner for classwide peer tutoring. At designated times during the day they will get with their math partner and go through a set of 20 flashcards that correspond with the level they are on in Rocket Math. Students have been taught the errorless procedure and always enjoy learning when it involves interaction with others. The first tutor displays the flashcard and silently counts to 3. If the tutee has not responded with an answer, then the tutor tells him/her the problem: (8 + 5 = 13). The tutee repeats the answer and marks an “on time correct” response on his sheet. If the tutee answers immediately with the correct answer, he marks an “early correct” response on his sheet; and if he answers incorrectly then he marks “error” on his response sheet. This continues until both students have interacted as the tutor and the tutee. There are very few errors marked on the response sheet because if they honestly don’t know the fact, they are more likely to wait for their classmate to tell them the answer than to have to record an error. This method takes 5 - 7 minutes of classtime and is making a difference in students’ retention of math facts!
The Counting Money: Coins with Stickers is very similar to the Touch Point Money idea. I have used the touch point money concept before and have seen great results with many students. Instead of putting a sticker on the coin labeled with a 1 for nickels, 2 for dimes, and 5 for a quarter, I use the one dot on the coin to stand for 5 cents, two dots for 10 cents, 5 dots for the quarter. The dots are also arranged in a certain pattern, so the student can remember the amounts. Both methods would allow students one more way to learn a concept that may be a challenge for them.
I have been using the peer tutoring for teaching addition and subtraction with regrouping. Both the tutor and student are really enjoying this process. The students seem to be less intimidated telling a peer they are having a hard time and the peer tutor takes pride in what they are doing.