For many, the appeal of the Investigations program is that students are encouraged to discover their own mathematical connections, problem solving strategies and computation algorithms; teachers are encouraged to value the diversity of student thinking; and teachers and students can learn from each other. This is all well and good, but what do you as a teacher do if you have diligently tried to make all this happen and you are just not seeing what you think of as addition, multiplication, division, or subtraction happening in your classroom?
Here's the scenario: It is March, you've waited months for your students to show some understanding (as you are used to seeing it) of double digit addition and subtraction (or multiplication and division), you have been patient because you think Investigations is a good program, but you do not see students using "the algorithm" or even strategies that you view as efficient. You are frustrated and you want to know that you have taught these operations before you send your students on to the next grade.
How can you calm this March Madness and not let it get into April Anxiety and May Mayhem? You may say that the solution is to teach your students a way that you know works and make sure they know how to do it consistently. A little direct teaching and your fears will subside. But you may then say that Investigations is about discovering and investigating and you're not allowed to teach "the right way". You may also wonder whether teaching your way may somehow devalue their attempts at inventing their own algorithms. Is there perhaps a middle road?
I believe the middle road is created through making time for students and teachers to be explicit about their thinking, their discoveries and the strategies they are trying. What do I mean by being explicit? Just as a friend may always stick to a familiar, yet excessively long, route to get to your house unless someone explicitly shows him/her a shorter one, it is comfortable for most students to do what works for them no matter how laborious. To help them take the next step or find comfort in a more efficient strategy, it is the job of the teacher to explicitly set up situations for them to encounter, understand and evaluate other strategies.
Take some time to become explicit with students about what they already know about addition and subtraction (or multiplication and division), what strategies they are currently using, why they are choosing specific strategies, and when some strategies are more efficient than others. Spend time focusing on the strategies that the individuals in the class are actually using in their games and problems and already trying to make sense of. If the traditionally taught algorithm is one of these, which I suspect it is, then it too must be addressed.
As an example, I have watched many students play Close to 100 (or 1000). As they play, many students adopt a strategy of finding numbers to add to 9 in the tens (or tens and hundreds) column(s) and 10 in the ones column. However, many other students continue to try to make 10 in all columns or try to use their largest numbers in the left most column believing that they will get a big number which must be close to 100 (or 1000). Some students add the numbers in their heads, some on a calculator. Some students add only in columns and some add whole numbers. This is a situation in which it is up to the teacher to draw out the key mathematical concepts s/he sees emerging (or not emerging) in the game strategies currently in use and make them explicit to all students.
In this case there are a variety of ways in which I might go about making the strategies explicit. I may start by floating around the room during a game session and asking questions, or specifically describing to individuals, or listening to them describe, the process being used. "I notice that with your cards you can get exactly 100. How could you do that? How did you know?"
During a sharing time I may choose specific students to talk about their strategies. "I looked at my cards and saw that I could have sixty something and forty something and six and four is ten so I did that but then I had too much so I tried sixty and thirty." I may then ask the rest of the students to decide which strategy was most like theirs, ask others to explain the strategies, ask students to discuss which made most sense to them and which seemed best for the game.
The next time the game was to be played, I may introduce the session with students deciding on a strategy they wanted to try and then pair up students so as to match them with a partner using a different strategy than they had previously used.
Throughout all these methods it is important that I keep my own mathematical standards in mind. In this case I want students to come to an understanding that making 9 tens and 10 ones (or 9 hundreds, 9 tens and 10 ones) is an efficient strategy because 90 and 10 makes 100. I want them to see the connection between 90+10 and carrying the one. I want them to see that if they make 10 tens and 10 ones they will have 110 which is not as close to 100 as 98 or 94. I could just tell them that if you make 9 in the tens place and 10 in the ones place you will be sure to get 100 but what if they can't make these combinations? Will they have any other strategies to fall back on, not to mention number sense?
As Hollee Freeman says in the Spotlight article, "Equity in Math Cooperative Groups," "students need to critically look at (other) procedures juxtaposed with procedures that they have invented themselves" in order to find a procedure which can be used with "efficiency, accuracy, confidence and understanding of number relationships." By using students' own thinking as the basis for explicit teaching a teacher strives to meet the students where they are and stretch the limits of their current understanding without pushing them beyond what makes sense or what can be done comfortably and consistently. If I instead put forth what I find to be the best strategy it may hit a few students where they are and provide a strategy which meets the above criteria. However for many other students it very well may be interpreted as the right strategy to use but may then be used in a rote way, without sufficient understanding of number relationships or an ability to modify it to changing situations.
As Investigations users we should value the discoveries happening in the classroom and the many interesting mathematical connections students are making as they play games, work through problems and discuss their thinking with each other. By making these discoveries and connections explicit for students we demand that students work to understand them, evaluate them, and use them to develop more and more efficient strategies. Just as we value, support, and guide the successive approximations of verbal fluency that young children must go through when learning to speak, we must also value, support and explicitly guide children's attempts at mathematical fluency and efficiency. So hang in there, focus on the approximations and work to guide your students to the next level of fluency. You may well be surprised by the efficiency of their invented algorithms.
Katy O'Reilly McGraw, Math Specialist,
Westwood, MA
February 2001
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.
