Prospective elementary teachers’ responses to unanticipated incorrect solutions to problem-solving tasks

Abstract

Eliciting and responding to student thinking are high-leverage practices that have powerful implications for student learning. However, they are difficult to enact effectively, particularly for novices, and more research is needed to understand how teacher education can support teachers in developing these skills. This study examined ways prospective elementary teachers (PSTs) responded to unanticipated incorrect student solutions to high-demand problem-solving tasks and how their responses changed over a 6-week field experience embedded in a practiced-based mathematics methods course. Data were collected in 6 weekly cycles of planning (written plans), enactment (video of problem-solving sessions with students), and reflection (written reflections on video). Problem-solving task implementations were analyzed using cognitive demand and math-talk frameworks. Of the three collaborating PST groups, two groups improved at responding to unanticipated incorrect solutions, but these two groups also developed a tendency to shut down anticipated solutions. The third group showed no patterns in their responses to unanticipated incorrect solutions, but did maintain the cognitive demand when responding to anticipated solutions. I present a case of one group of collaborating PSTs who made improvements in responding to unanticipated incorrect solutions in terms of the pedagogical moves they employed, cognitive demand associated with their responses, and how their responses changed over time. Implications for teacher education are discussed.

Appendices

Appendix 1: Focus tasks of each task set

Generalizing and explaining patterns

Task 1: Telephone club

Your class made telephones out of strings and juice cans. Each group of students has to work together to make a phone club that connects every person to every other person. If a group had four people, how many strings would be needed to connect every member of the group to every other member of the group? What if you used 28 strings, how many people would be in a group?

Task 2: 6 Numbers

Can you put the numbers 1–6 in the triangle shown so that each side adds up to the same amount?

Making organized lists

Task 1: 12 Pennies

Place 12 pennies in 3 piles with no two piles having the same number of pennies.

Task 2: Clock 6s

How many times in a 12-hour period does the sum of the digits on a digital clock equal 6?

Working backward

Task 1: Crayons

Mary has some crayons. Doug had 3 times as many as Mary. But Doug gave 4 to the teacher and now John has 2 more crayons that Doug. John has 7 crayons, how many does Mary have?

Task 2: Puppies

The pet store advertised that they had lots of new puppies on Monday. The owner took 1 puppy for his son. Then, on Tuesday he sold half of the rest of the puppies to a farmer with lots of land. On Wednesday, a mom took a half of the puppies that were left for her children. When you got to the pet store on Thursday, there were only 4 puppies left to choose from. How many puppies were there on Monday?

Reasoning algebraically

Task 1: Cupcakes

A baker makes chocolate and vanilla cupcakes. He packages the vanilla ones in boxes of 4 and the chocolate ones in boxes of 6. He made 38 cupcakes and used 8 boxes. How many boxes of vanilla and how many boxes of chocolate did he make? (alternate version: 58 cupcakes and 12 boxes).

Task 2: Tickets

Amy and Judy sold 19 play tickets altogether. Amy sold 5 more tickets than Judy. How many tickets did each girl sell?

Reasoning deductively

Task 1: Castle

Twenty men need to guard the castle below. For the castle to be safe there should be 7 men guarding each side. The men on the towers count as guards for both walls that connect to the tower. How would you place the 20 men?

Task 2: Women at the table

Five women are seated around a circular table. Mrs. Osborne is sitting between Mrs. Lewis and Mrs. Martin. Ellen is sitting between Cathy and Mrs. Norris. Mrs. Lewis is between Ellen and Alice. Cathy and Doris are sisters. Betty is seated with Mrs. Parks on her left and Mrs. Martin on her right. Match the ladies’ first names and last names.

Appendix 2: Sample task dialogue assignment

Task set: Make organized lists
Task 2: How many times in a 12-hour period does the sum of the digits on a digital clock equal 6?
Dialogue 1	Rationale for teacher moves
S: Things that add to six are 6 and 0, 1 and 5, 2 and 4, and 3 and 3. So to make times I get 6:00, 3:30, 4:20, 5:10 T:
Dialogue 2	Rationale for teacher moves
S: So 6:00, 3:30, switching the threes doesn’t change the time, 4:20, 2:40, I can’t have 0 o’clock 5:10, 1:50 T:
Dialogue 3	Rationale for teacher moves
S: So 6:00, if I put 6 anywhere else, it doesn’t make a time. 3:30, 3:03, I can’t put 0 in the hour place. 4:20, 4:02 2:40, 2:04. 5:10, 5:01, 1:50, 1:05 T:
Dialogue 4	Rationale for teacher moves
S: Student has all the times with 0’s (6:00, 4:20, 4:02 2:40, 2:04. 5:10, 5:01, 1:50, 1:05). Then 4:11, 3:21, 2:22. “Oh and changing them around. If I rearrange these too that should be all.” T:

Appendix 3: Example of one (of four) completed dialogues from a task dialogue dssignments

Task set: Make organized lists
Task 2: How many times in a 12-hour period does the sum of the digits on a digital clock equal 6?
Dialogue 3	Rationale for teacher moves
S: So 6:00, if I put 6 anywhere else, it doesn’t make a time. 3:30, 3:03, I can’t put 0 in the hour place. 4:20, 4:02 2:40, 2:04. 5:10, 5:01, 1:50, 1:05
T: How did you get those answers first of all? Explain your thinking	← Makes the student put words to their thought processes so that the teacher can identify what areas are needing help and what the child does understand
S: Well I thought of ways to make 6 (0 and 6, 1 and 5, 2 and 4, & 3 and 3). Then I put them in order of the numbers on a digital clock and put a zero in so they would stay at just 6. Then I moved the numbers around so that the numbers got a chance to be in all the places
T: Ok I see. But can I ask you something? I’m wondering if you can make 6 out of three numbers instead of just pairs of numbers and a zero?	← Leads the child to think about an option they didn’t realize existed
S: You mean like 1 + 2+3 is 6?
T: Yes! How can you put that into terms of time?	← Ensures that the student understands the concept of putting the numbers into a time frame
S: It could be 1:23. Or 1:32! Or 2:13 or 2:31. And there is also 3:12 or 3:21! Wow! I forgot about all of those!
T: Great job! That is exactly what I meant. Are those the only ones that were forgotten about or could there be even more ways to do this? Why don’t you work some more to figure it out? Is there a way you can organize all of this information so that it is neater and easier to see and understand?	← Acknowledges that the student is on the right track/encourages them. Then asks them to portray this new knowledge by putting it to work for the other numbers (being sure to give enough wait time for the concept to better sink in and be understood). Also, the last question gets the child thinking deeper about how it would best be organized so that no answers were skipped

Appendix 4: Revised math-talk framework (Hufferd-Ackles et al. 2004)

	Questioning	Explaining thinking	Source of ideas	Responsibility for learning
0 All activity is teacher-directed. Students passively receive information and their only input is to provide answers to yes/no or basic fact questions.	T asks many short direct questions and/or yes/no questions in a series; S give short answers	T does not elicit S thinking, but does elicit answers; T may answer own questions	T shows how to solve or tells correct answers or appropriate strategies	T verifies and shows correct solutions; T shows out to carry out most effective strategies
0.5 All activity is still teacher-directed, but there is some opportunity for student input, mostly in the form of factual recall. When teacher poses more complex questions, students are not prepared to answer, often because teacher did not provide adequate time for student work.	T asks about strategies before S has applied any strategies	T elicits answers about applying a T-suggested strategy	T suggests a strategy via questions for S to apply, T elicits her own strategy, no re-voicing	T verifies correctness of S’s answers as S carries out T’s strategies or T draws conclusions for S’s based on S’s work
1 Teacher shifts her attention to student thinking, but once student ideas are elicited she is unsure how to incorporate them in the discussion. Discussion still includes more teacher talk than student talk.	T asks about S’s strategies or methods, but does not follow up	T elicits strategies and Ss give brief descriptions of how to find solutions; T does not push for more details; T may fill in explanation	T elicits S ideas but does not explore them; T may revoice correct and more sophisticated explanation than is given by S	T gives feedback, but does not verify correctness; T encourages S to execute S’s ideas that will lead to correct solutions
1.5 Teacher elicits student thinking and tries to follow students’ ways of reasoning, though she may not always be successful in these attempts. Talk is more equally shared between teacher and student.	T asks for S’s strategies and follows up with open-ended questions; T asks Ss to justify ideas and strategies	T elicits explanation about why (part of) strategy works; T begins pushing for clarification though T may still accept confusing or unclear explanation; T may seek multiple strategies	T elicits and explore S ideas; T helps S articulate idea or reasoning behind a particular strategy S employs	T encourages S to execute S’s ideas; T sets up for Ss to evaluate their own errors, though T does not always explicitly follow up on this; S’s draw their own conclusions
2 Teacher follows student thinking and encourages students to listen to one another. She begins to prompt students to interact, though students may still direct dialogue to the teacher. Teacher may model interaction by speaking to one student on behalf of another student.	T prompts Ss to ask questions of each other, T often supplies the exact question to be asked or asks it for the student; Ss ask questions of T or asks T about another S’s work	T does not accept poor explanations; T probes Ss for elaboration and looks for multiple strategies; T prompts Ss to explain to each other though Ss still filter explanation through teacher; T may repeat one student’s explanation for another	T describes one S’s strategies or ideas for another; When asked to compare/contrast solutions or strategies, Ss check answers against each other	T holds Ss accountable for listening to others; T asks Ss to make sense of other’s ideas, though Ss may struggle with this; T is solely responsible for ensuring ideas are expressed clearly
2.5 Student-to-student talk occurs, and is maintained beyond initial teacher prompting. Students respond to one another, no longer using teacher as intermediary.	After initial T prompting, Ss ask each other low-level questions without T prompting, T still guides discussion with questions requiring explanation and justification	T probes for complete explanation and looks for multiple strategies; Ss give detailed descriptions of how solved and when prompted by teacher will describe how they solved to another student	When prompted by T, Ss can discuss similarities or differences in their strategies and solutions; Ss confer with one another about how to find solutions	T asks Ss about each other’s work, asks Ss to evaluate each other’s work (do they agree with another’s ideas); Ss hold each other accountable for expressing ideas clearly
3 High level student interaction with minimal teacher prompting. Discussion is guided by student ideas and reasoning	Ss ask each other questions without T prompting; Ss ask each other to explain and justify work; T monitors S questions and may help clarify Ss’ questions	T does not have to probe for complete explanation; Ss probe one another; when asked by T, Ss can explain one another’s ideas; Ss defend and justify to each other and T	Students can generate ideas together; Ss take up apply one another’s ideas; T uses S ideas as basis for extension problem	Ss take initiative for their learning; Ss ask for clarification/explanation of another’s ideas; Ss judge correctness of others’ ideas using their own reasoning; T is not evaluator of idea; T coordinates equitable S participation

Appendix 5: Guidelines for coding levels of cognitive demand

Cognitive demand level		Indicators	Example
0	Memorization	Teacher reduces task to answering a series of basic skill questions; students answer questions using only memorized facts and rules	T: Make a list of numbers that add to 6 S: 1 and 5, 2 and 4, and 3 and 3
1	Procedures without connections to understanding, meanings, or concepts	Teacher and students focus on students finding solutions; Teacher asks how student solved a problem and may suggest strategies for student to carry out	T: You could think of all the options that started with 9 [S nods.] T: Then what would you do? S: Keep adding one to each, like it would be 8 plus that plus that… T: You want to try that? So you’d make a list so you could keep it organized
2	Procedures with connections to understanding, meanings, or concepts	Students focus on finding solutions; teachers focus on having students explain observed patterns, how the solutions were obtained, and why students’ strategies worked	T: On the last problem we saw if there’s 4 numbers you can arrange them 6 different ways, remember? Why can’t we arrange these 6 different ways? S: I started with 2 and 4, but I can’t start with a 0 cause there’s no time like 0:40 or…
3	Doing mathematics	Teacher encourages students to make and test conjectures, apply reasoning to situations, generalize strategies across multiple similar problems, justify conclusions	S1: ….I’m predicting there will be 12 strings T: Why are you predicting 12? S1: Each string has to go to 3 people S2: [draws his solution] I think 6 T: Can you explain yours to S1? S1: I think since you have to do 3 strings to each person you spin it 4 times, so 4 times 3 is 12 T: Why do you think there are 6 strings? S2: There are four people and they all have to connect to each other. So that would mean this person would need to connect to this person and that person. Since they’re already connected you only have to draw two lines

1. Examples were edited for clarity and correct grammar