Student-Invented Strategies: When the Right Answer Isn't Right Enough

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.

Math Talk Moves: Five Steps to Student Success!

Having interned in elementary schools for the past three semesters, I’ve seen--all too often—how very reluctant and unhappy some students are when the time of day arrives when they must leave their general education peers to receive “special help” in the resource room.

Compounding the already prevalent social stigma associated with requiring services in the resource room is the factiuyuyt that the students leaving their general education classrooms may be leaving behind much more interactive and enjoyable learning environments.  In the hands of a teacher who doesn’t recognize that his/her responsibility in a resource room setting should be to provide instruction that meets the individual student’s interests as well as his/her instructional needs, learning, for students with learning disabilities, can be a painful experience.    

This, however, is far from the case in the math lessons I’ve observed in the resource room at my placement school!  The supportive, collaborative atmosphere that the math resource teacher I observe has created makes it hard for students not to feel empowered and respected regardless of the difficulty of the tasks which they’re presented. During most of the lessons this resource room teacher implements, the small student-to-teacher ratio (usually ranging from one-on-one instruction to teaching groups of three students) allows instruction to be highly individualized. Consequently, throughout the lessons, each student is able to enjoy the opportunity to work directly with the teacher when solving some problems, to share his/her thinking trajectories, and to answer probing questions designed to elicit deeper reflection about the processes in which he/she is engaged.  It is during this time, that Chapin’s “Five Productive Talk Moves” often become the focus of instruction.

While reading about the “Talk Moves,” I could imagine their value in the classroom, but, really, it wasn’t until I saw this teacher actually using them during instructional time that I really understood just how empowering these questions can be to students.  During a particular lesson, I observed the teacher working one-on-one with a student who for the past few months has experienced a great amount of difficulty with the concept of a “fact family.”  According to the teacher, while the student is able to solve the math problems using manipulatives, the relationship between these numbers seems transitory to him, and, as such, while he night sometimes say four plus three equals seven, a few moments later he might give a completely inappropriate numerical response to seven minus four—almost as if he has lost track of the fact family.  In an effort to communicate the permanence of the relationships between numbers in a “fact family,” I observed the teacher first relating the three numbers that make up a “fact family” to the three members of the student’s own family. After creating a link between the number four and the student’s mother, the number three and the student’s father, and finally, the number seven and the student himself, the teacher explained to the student “You wouldn’t have some stranger come, join your family, and kick another member of your family out, would you?  Well, the same is true with these ‘fact families.’  Four and three and seven are always going to be in a ‘fact family’ together.”  She then followed this explanation with an example of the problem illustrated with manipulatives actually shaped like people—a nice touch, I thought!

With manipulatives in front of him and the answer in his mind, the teacher then introduced the first “Talk Move” in the lesson.  Asking the student what another formula in the “fact family” looked like, the teacher asked him to explain what he had been thinking when he initially said that “seven minus four must equal five.”  The student began to justify his incorrect answer when--suddenly understanding-- he turned to his manipulatives instead!  Counting out seven manipulatives, the student then used the “Counting Down” strategy to arrive at the correct answer, three.  The teacher then used another “Talk Move” when she re-voiced his strategy, saying something similar to “So, let me see if I get this right.  You first counted seven manipulatives, and, then, you looked at the problem and saw that you needed to take away four of the manipulatives.  But I didn’t see you take those manipulatives away!  How did you end up getting the answer?”  Re-voicing his strategy in a way that still asked him to explain the procedure he applied to eventually get to the answer made the student more aware of the strategy, perhaps increasing the likelihood that he will use the “Count Down” approach in the future.

 Finally, throughout the lesson, the teacher expertly employed the strategy of “Wait Time.”  As I watched the student process and then progress throughout the lesson, I came to see how critical this “Talk Move” is when working with a student with a learning disability.  So often in general education placements, it’s not unusual to see those students who take longer to answer questions being ignored in classroom discussions and allowed to fall behind in their coursework.  As I watched the teacher today wait, in complete (but comfortable) silence, I was able to witness her student really thinking about the problem, analyzing the possible solutions and offering a well-reasoned answer.

Reflecting upon the lesson I observed today, the apparent effortlessness of the teacher’s implementation of these “Talk Moves” almost obscured my ability to observe them!  While reading about the different questioning practices Chapin suggests, I had pictured a classroom, at times, more like an interrogation room, with the students’ thinking constantly being challenged.  Seeing the comfortable environment that the teacher was able to create and the way she actually used the “Talk Moves,” not to question student thought, but rather, to elicit reflective thinking, has made me anxious to try some of these strategies myself! 

In a classroom where Talk Moves are expertly employed, getting through lessons without understanding the content isn't an option. :)

Of all the strategies I saw this teacher demonstrate, I think the most valuable one to use when teaching students with learning disabilities might also be the most difficult to employ: “Wait Time.”  For students in resource room, as valuable as remediation of academic skills might be, instilling in them confidence in their ability to succeed with subjects they find difficult might be equally, if not more, important.  Despite all the potential benefits of the “Wait Time” strategy, I think that this strategy might be more difficult to implement effectively, at least initially, with students who require a lot of reinforcement and scaffolding.  For a student for whom experiencing silence causes worry and uncertainty, I think that teacher gestures such as nodding your head, smiling , and even using such phrases as “you’re doing a good job, keep going!” might be helpful as he/she tries to answer the question.  Over time, the frequency of these prompts could be reduced and the duration of  “Wait Time” could be increased. 

Without Wait Time, just being able to give an answer becomes a challenge!

Two other “Talk Moves” that I want to employ in my own lessons are “Re-voicing” and asking students to restate someone else’s reasoning.  As I watched the lesson today, I really liked how the teacher’s use of “Re-voicing” enabled the student to give voice to his own procedures and thought process, enabling him to critically consider the veracity of the thoughts which shaped his actions.  As Chapin suggested in her article, when teaching groups of students, asking students to restate someone else’s reasoning can dfprove to be an effective way to have other students restate the thinking processes of their peers.  For students who might be English Language Learners (ELLs), this technique, where ELL students would repeat what their native English-speaking peers stated, would provide both practice in considering other math thought processes and also valuable practice speaking English.