These questions are huge! They create a lot of emotion and controversy among educators. So let's take each question and I'll address my reasoning behind my thoughts.
Question: Why would we want to support and even encourage students to come up with different strategies to solve problems?
Answer: Mathematics needs to move from procedure and memorization to truly understanding the concepts behind the operation. When we learn a skill, such as division of fractions (Don't ask why, just invert and multiply) we may be able to do this skill and get the correct answer while we are working on this unit. A large majority of our students are confused by this process even during the division of fraction unit and invert the wrong number. But let's imagine all students are capable of mimicking the procedure during that lesson of study. This short-term memory does not insure long-term memory. Go away from the unit of study and come back to it a few months or years later. Research shows that if we have memorized a process that didn't make sense to us, we will often mix up several different procedures over time. We get confused when we invert, when do we change the sign, when do we cross-multiply.
Question: Won't straying from the traditional algorithm be confusing for students?
Answer: Well it depends on how a variety of strategies are learned by the students. I used the word learned intentionally vs. being taught. Teachers often feel like it is their job to teach a variety of strategies, but these strategies should never be taught by the teacher. This is why students become confused. It is just more ways and processes to remember. If students are solving problems and using math tools to solve problems it is natural for a variety of strategies to emerge naturally from the students. For example:
If students solve the following problem: Jaden has 56 cents and finds 37 more cents, how much money does Jaden have now? If Jaden uses base ten blocks and naturally adds the 5 tens to the 3 tens to make 8 tens or 80 and then puts the six units and the seven units together to make 13 and then add 80 + 10 to get to 90 + 3 more to get 93, Jaden has created and shared an algorithm or strategy that makes sense for her. Camden may take four cubes from the 37 (to make it 33) and now knows the 4 cubes added to 56 gave them 60 and then can easily add 33 to get 93. This strategy is now one Camden understands. When students share out their different strategies, it allows them to have a deeper number sense but they also have to be able to explain their strategy. If we understand the processes and nor merely memorize them there is a much better chance of it being put into long-term memory.
Question: I learned how to calculate problems this one-way and I'm doing fine, why shouldn't my students?
Answer: Congratulations! You are one of few. We have an epidemic of math illiterate people in our country. The U.S. Department of Education reports that 58% cannot calculate a 10% tip. For people who did well in mathematics and had a great memory, may have gotten by fine. We know everyone learns differently and what works for one, may or may not work for another. If we do not change the way we do business and teach mathematics we will continue to receive the same dismal results in the future. In other words, to answer Dr. Phil's question, "how's it working for us", my answer is not very well. We need to change our beliefs and values in teaching mathematics.
Estimate Before You Solve
This strategy is an underused strategy in our classrooms. Why? I think it's because we aren't very good at it and we really don't know how to help our students become better estimators. We often turn it into a procedure, like rounding, or front-end adjustment methods. For more information on teaching estimation and ideas to use check out Chapter 8 - Estimation and Mental Math. The purpose of this strategy is not to "teach" estimation, but rather to use estimation to determine the understanding of our students. Let's go back to the division of fraction problem discussed earlier this chapter. Try this for yourself: Estimate what the answer to the following problem should be:
3 1/3 divided by 5/8 = ?
If you are like many Americans, you might be thinking, "I don't know, but I can calculate it with a procedure I memorized." That's the point. If you don't know what the answer should be about, how do you know your calculated answer is correct or even makes sense? I remember, stating to students, "Is your answer even in the ballpark?" If they don't know they don't have the understanding behind the strategy.
Make estimating before you solve a "Way we do Business" in the classroom. Students will not be good at it at the beginning. This information will help you to know the level of understand your students have or don't have.
Draw a picture or model the problem
If students are unable to estimate the problem above or a problem you are working on. Ask students to draw a picture and/or model the calculation. This can be done with base ten blocks, number line, a drawing, etc.... If students still struggle, you may want to put into a real-world context. Such as: If I have 3 1/3 yards of fabric, and I want to make snack bags that require 5/8 of a yard each, how many snack bags could I make. By asking students to model the variety of strategies will begin to emerge. Teacher must serve as a great questioner to get at the thinking. Just don't miss it! A teacher must always be on high alert to find, recognize, encourage, and support the thinking of their students. It is also critical that we encourage the sharing of these different ideas created by students.
Show Me a Different Way
If students always solves similar problems the same way, to push their thinking a little deeper, ask them to try another student's strategy. If Jaime always uses the traditional method of adding two digit numbers, ask her to try Jaden's method or try another method that makes sense to her. This requires Jaime to go deeper into her number sense and how to manipulate calculations.
Communication when different strategies are discovered is crucial. Too many times we miss the opportunity to discuss how strategies are similar and different. The goal of looking for similarities and differences requires a much higher level of understanding and is often glossed over in classrooms. It is important for students to not only make sense of their strategies but to also be able to critique the reasoning of others. Creating Anchor Charts of different student strategies is a great way to keep this discussion forefront and center.
Cognitively Guided Instruction (CGI) is a research-based strategy that engages students. Bottom line - CGI is a great strategy that goes across many instructional strategies and is a wonderful way to get at students creating multiple strategies and persevering in solving problems.
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