In observing students engaged in mathematical thinking, both during the math interview and during the math lessons I teach at my school, I have been able to see students applying different strategies based on their developmental stages and past experiences in math. Despite differences in age and grade level, many of the students with whom I have worked tend to show a clear affinity for solving problems using the direct modeling approach, an approach that Van de Walle describes in his book, Elementary and Middle School Mathematics, as “the use of manipulatives or drawings along with counting to represent directly the meaning of an operation or story problem” (p. 214). For some of the students I teach, the direct modeling approach is the only strategy they have available when solving problems. However, others among my students have been encouraged, throughout their mathematical development, to experiment with different counting strategies, and, as a consequence, they have come to see that math is not solely defined by a set of algorithms; they understand that solving problems is not restricted to formulas or drawings, but can actually be a process of invention. The ability to use student-invented strategies, which is described by Van de Walle as “any strategy other than the traditional algorithm or that does not involve the use of physical materials or counting by ones” (p. 215) represents a more complex approach to problem-solving that requires a deeper level of understanding about mathematics and number sense.
Using the Math Talk Moves during my math lessons has caused me to more closely analyze my students’ thinking as they solve problems, an analysis which draws my attention to their different levels of understanding and their familiarity with different problem-solving approaches. By using Talk Moves that require all of the students to listen to and actively think about the approaches used by their peers, I have observed that the students look forward to the opportunity to play the role of “teacher” during the lessons, scaffolding for their peers and exposing each other to different modes mathematical thinking. Their own explanations of their thinking (as they approach each problem) by my students who are using invented problem-solving strategies, gives my other students the chance to see the potential that exists when traditional algorithms and direct modeling are forgone in favor of what can be more efficient problem-solving strategies. In the future, perhaps such modeling will encourage other students to look at problems from this inventive perspective as well.
Last week, I had the opportunity to observe two of my students using an invented strategy as I introduced them to the concept of subtracting with regrouping. When the first student, whom I will call Zoe, was presented with the problem 23-19, her first step, after borrowing* 10 from the 20, which gave her 13 in the ones place, was to set up the addition problem, 9+_____=13.
For Zoe, thinking about the problem through the “lens” of addition, seemed more natural, and this decision enabled her to use her knowledge of the fact that 9+1=10 plus 3 more equals 13, giving her the answer 4. Using this “Add Ones to Make a Ten, Then Tens and Ones” strategy meant that, instead of counting on her fingers or taking the time to draw representations of each number, Zoe was able to quickly arrive at an answer in her head even though she did not have that particular math fact memorized!
The second student, whom I will call Erin, initially solved the problem by simply using her knowledge of the math fact that 9+4=13. When I asked her to explain another method that she could have used to get the answer, Erin responded, “Well, I know that 10+3=13. Ten is 1 more than 9. That means that 4 must be added to 9 in order to get 13.” Because Erin set the problem up in a way that made the subtraction problem into an addition problem, the strategy that she finally used when explaining her thinking to me most closely resembles that of the “Use a Nice Number and Compensate” strategy listed in the table of “Invented Strategies for Addition with Two-Digit Numbers” in Van de Walle’s book. Erin’s ability to see and then correctly implement different processes for solving individual problems suggests to me that her number sense and understanding of the addition and subtraction processes are quite developed and, perhaps, indicative of her readiness to move on to the more complex multiplication and division procedures.
As I consider the problem, 23-19, which these two students solved, I noted that both opted to break the problem into two smaller problems based on the ones and tens places (a decision probably influenced by the way I had modeled the process of subtracting two-digit numbers earlier in the lesson). Had they chosen to look at the problem as a whole, one way that they could have solved it using an invented strategy would have been to use the “Add Ones to Make a Ten, Then Tens and Ones” procedure. With this procedure, they could have added 1 to the 19, giving them 20. They then could have subtracted 20 from the 23, giving them 3, and making the answer 3+1, which is 4.
Although both of the students implemented “counting up” strategies, a “counting down” invented strategy they could have used that would have given them the correct answer is the “Take Tens from the Tens, Then Subtract Ones” strategy. With this strategy, they would have first taken 19 away from 20, giving them 1. They then would have added 3 to the 1, giving them 4 as the answer.
Reflecting upon what I was able to learn about my students’ math understanding based on their different problem solving approaches, I can really see the benefit of the Talk Moves strategies! Not only did using Chapin’s Talk Moves make it possible for the students to hear descriptions of different strategies that might influence their own problem-solving in the future, but also, getting to hear vocalizing of the students’ thought-processes when solving the problems, made me see and appreciate the complex thinking that my students are capable of employing on a regular basis! What once might have been a lesson with a quick check at the end to determine the correctness of students’ answer on a particular problem, became a lesson where I was able to assess, not only their correctness, but also, and more importantly, their degree of understanding and the quality of the cognitive processes they employ. With the help of the Talk Moves and the knowledge I have gained through the readings on Cognitively Guided Instruction, I’m able to see that what really matters in the math classroom is less about the answers students end up with, and more about the cognitive processes and approaches that the students employ to get to those answers. Encouraging students to think about math problems in ways that go beyond the commonly seen “plug and chug” methods of typical algorithms, not only creates an environment where student invented strategies are more likely to be seen, but also, as I’ve learned from my own observations, creates an environment where students feel more empowered and valued when they are engaged in the problem-solving process.
*Though Van de Walle encourages the use of the term, "trading,” rather than “borrowing,” the term borrowing is used in the students’ classroom, so I have adopted this term in my lessons as well so as not to confuse the students.