This study examined the extent to which preservice teachers (PSTs) develop their capacity to attend to children’s strategies and interpret and respond on the basis of children’s mathematical understanding in the context of two welldesigned assignments: Inquiry into Student Thinking assignment and tutoring assignment. The two assignments were assigned after 6 and 10 weeks of instruction, respectively. The analysis revealed that PSTs attended to children’s strategies and interpreted children’s mathematical understanding but struggled with the component skill of responding to children’s mathematical understanding in the two assignments. Although the nature of tasks selected differed across the two assignments, generally PSTs focused on tasks that would develop children’s mathematical understanding. The findings have theoretical implications for a hypothesized trajectory of professional noticing of children’s mathematical understanding and the design of mathematics methods courses.
The question of what preservice teachers (PSTs) can learn in teacher preparation programs to ensure that they are wellprepared beginning teachers of mathematics has increasingly drawn attention in the United States and around the world [
Additionally, teacher educators have argued that lack of a common knowledge base, curriculum, or a common pedagogy in teacher preparation programs has made it challenging to study how content taught in teacher preparation courses supports PSTs to acquire the knowledge, practices, and skills required to effectively enact mathematics instruction as beginning teachers [
This study examined the extent to which PSTs’ capacity to respond on basis of children’s mathematical understanding using two scaffolded assignments in the context of a mathematics methods course developed. Teachers’ knowledge of children’s mathematical understanding is part of what Shulman [
In this article, the word “capacity” is used to refer to PSTs’ ability to attend to children’s strategies and interpret and respond based on children’s mathematical understanding as conceptualized by [
Furthermore, teachers’ capacity to use children’s mathematical understanding is fundamental and critical to effective teaching practice. A plethora of research findings have linked productive instructional environments to teachers’ knowledge of how to teach elementary mathematics in a way that develops and relates to the benefit of attending to children’s mathematical thinking [
However, although the studies reviewed above suggest that focusing on children’s mathematical thinking is a powerful mechanism for bringing pedagogy, mathematics, and students’ understanding together, other research studies have suggested that the expertise of noticing, understanding, and using children’s mathematical thinking to inform instructional decisions does not naturally develop [
More recent studies have documented teacher educators’ efforts to develop PSTs’ ability to attend and interpret and respond on the basis of children’s mathematical understanding [
The theoretical framework guiding this study is grounded in the notion of professional noticing of children’s mathematical thinking [
Summary of the hypothesized developmental trajectory of professional teacher noticing.
In addition, this study is informed by prior research that has focused on the extent to which teachers use what they notice to respond to children’s mathematical understanding [
This study examined PSTs’ capacity to respond based on children’s mathematical understanding using two scaffolded assignments in the context of a mathematics methods course, by investigating three research questions:
To what extent do PSTs develop the capacity to attend to children’s strategies, interpret, and respond based on children’s mathematical understanding in the context of scaffolded activities?
To what extent is the rationale of the PSTs’ choices of instructional tasks based on children’s mathematical understanding?
What type of tasks/problems do PSTs pose after assessing children’s mathematical understanding?
In this study, we used the nature of tasks/problems that PSTs selected and designed as they completed the two assignments as a lens to understand their ability to respond based on children’s mathematical understanding. We analyzed PSTs’ capacity to respond based on children’s mathematical understanding in the context of scaffolded activities in a mathematics methods course. Scaffolds have been described as structures, tools, and assistance from more knowledgeable others that allow learners to engage in practices beyond their independent capacity [
This is a qualitative interpretive study that was conducted in the context of a mathematics methods course in a large university in the United States. Two scaffolded assignments that were completed at two different times as the PSTs progressed through the course were used as a source of data. In the Inquiry into Student Thinking assignment, PSTs analyzed the mathematical understanding of one child from a case study of four children (Appendix A.1). In the tutoring assignment, PSTs interviewed two to three children from a nearby elementary school and then planned for a series of four tutoring sessions (Appendix A.2). We chose to explore the practices of attending to children’s strategies and interpreting and responding based on children’s mathematical understanding in the context of scaffolded assignment because these practices include activities that are essential to the work of teaching [
The elementary mathematics methods course met for three hours each week over a 14week semester. The course met in the university classroom for 6 weeks, followed by a clinical experience of 6 weeks in a K8 classroom. For the last two weeks of the semester, they met in a university classroom to debrief the practicum experience. The stated primary goal of the course was to develop PSTs’ capacity to design and implement mathematics instruction that is both reflective and mathematically significant. The course was organized around four learning goals:
How children’s thinking typically develops, including common understanding, misunderstanding, strategies, and errors
How to access and assess children’s mathematical thinking within different content areas
How to use children’s mathematical thinking to select and pose worthwhile mathematical tasks
How to use curriculum materials, family, and community resources and other supports to help the PSTs facilitate the development of children’s mathematical thinking
To meet these goals, the course was built around a variety of in class activities and homework assignments contained in a course packet developed by the mathematics education team at the university. These activities included using video clips, analyzing children’s written work, and developing and solving CGI problem types.
All the participants were enrolled in a twoyear teacher education program as part of their fouryear undergraduate degree. Fiftysix PSTs from three different sections of the methods course consented to participate in the study at the beginning of the semester, but only data from thirty participants were used because those were the participants for whom researchers had access to all their course work activities and reflections. While analyzing the data, we noted that some PSTs had not submitted some of the assignment or they submitted in a format that was not accessible. For every participant, copies of written work (the original work was returned to the students) with responses to the Inquiry into Student Thinking assignment, the tutoring assignment and any work related to the course across the semester was collected. In the next section, the expectations of these course assignments and their purpose are briefly elaborated.
For this assignment, the primary source of information was found in a case study of four second graders described in [
Summary of prompts used in the two assignments.
Component skill  Inquiry into Student Thinking  Tutoring assignment 

Attending to children’s strategy  (i) Choose one of the four students from the case study and trace their responses and learning through the study.  What does each student know, think, and understand about number, operations, and problem solving? 
(ii) Summarize what you think they knew or understood at the end of the study that they did not know or understand in the beginning  
(iii) Choose 2 examples of this student’s responses as evidence for your claim from part (1) and explain how they support your claim about this student’s learning.  


Interpreting children’s mathematical understanding  (i) Summarize what you think they knew or understood at the end of the study that they did not know or understand in the beginning  How will what you learned in the interviews influence how you work with the children during the next sessions? 
(ii) Choose 2 examples of this student’s responses as evidence for your claim from part and explain how they support your claim about this student’s learning.  


Respond based on children’s mathematical understanding  (i) If you were to teach the next lesson to this group of students, write one problem that you could give them and explain why you think this would be a good problem for this group  How are these tasks intended to build on what you know about your students’ understanding and misunderstanding? 
The tutoring assignment is a fieldbased assignment completed in three K4^{th} grade elementary classrooms. PSTs were expected to plan for instruction based upon knowledge of subject matter, students, community, curriculum goals, and state curriculum models. To complete the assignment, PSTs interviewed elementary level students about their understanding of number and operations and problem solving. Based on their interpretation of the interview responses, PSTs planned and implemented three tutoring sessions using tasks that were challenging and yet accessible to the students. During the final session, they interviewed the student(s) again to assess their mathematical understanding after the three tutoring sessions. The activity was completed in four 50minute sessions over a period of one month. For each session, PSTs responded to the prompts in Table
Data analysis used quantitative and a qualitative interpretive research approach [
For the component skill of attending to children’s strategy, we considered the extent to which PSTs’ explanation demonstrated evidence of providing mathematical details. A code of
Examples of responses coded as most, some, and lack of mathematical details.
Code  Inquiry into Student Thinking  Tutoring assignment 

Most mathematical details  He solved it with a direct modeling procedure and drew each individual soccer ball in the designated three bags. Emilio worked on the problem: Dr. E has 4 rolls of candy and 11 loose candies. How many candies does she have altogether? He initially spat out the number 40 and explained on his own that there were candies in each of the four rolls. He had trouble counting up from 40 to 51 for the 11 single candies, but this is an issue he had the first day as well and shows that may be another issue. However, because he knew to count up from 40 by 1 single candies shows that he is able to distinguish groups of 51 from single units which is very significant in using base ten problemsolving strategies.  On the JRU problems, Tyler would use a breakingthenumberapart strategy. He would like to get the numbers into base 10 so that they would be easier to add together. For example, on the first set of numbers (42, 36) for the apples problem, Tyler told me that the answer was 78. When I asked how he knew that he wrote out that 42 + 30 = 72 and then wrote 72 + 6 = 78. I was really excited that he knew a shortcut for how to do the problem. He also used this same strategy for the SRU problems. 


Some mathematical details  Jack throughout the case study counted up by ones to find his answer. From the very first day, Jack miscounted the total number of soccer balls because he had the wrong number of soccer balls in one bag, even though all of the bags had simply 10 balls in each. In his first few sessions, Jack tended to write tall marks to keep track of whatever he was counting, no matter how big the number was.  Apple problem: The student started with the original amount of blocks (3) then found the number of picked apples (12). After this the student started counting the blocks starting with 3, counting up to 12 on, starting with 3, 4, 5, 6, 7, 8, 8, 10, … This led me to doing a problem that would include counting since my goal for the lesson was to get my student to be able to count in sequence starting from a given number in the known sequence. 


Lack of any mathematical details  In the beginning, Jack did not recognize ten as a numerical unit. It seemed that, to him 10 was no different than 4 or 9. Because of this, he often counted up to the answer. As the study went on, he began to develop an understanding of ten, first by using a representation of 10 (rather than tally marks or other such onetoone representations) in session 5 and later by solving number sentences by counting tens rather than counting up by ones (seen in session 9, but also hinted at from session 5 on.  Second, any straightforward problem (e.g., 23 + 57 = __) was not difficult for them. It did not seem to matter whether a task was JSU, SCU, SIU or SRU; those sorts of problems were simply too easy for these three students unless the numbers were sufficiently large enough to require them to use paper just to keep track of their carrying … 
Note: Join Result Unknown (JRU) Join Change Unknown (JCU) Join Initial Unknown (JIU) Separate Result Unknown (SRU) Separate Change Unknown (SCU) Separate Initial Unknown (SIU).
Due to the nature of the prompts given in the two assignments, the first glimpse of data analysis revealed that most PSTs took an interpretive stance rather than descriptive or evaluative stance [
Examples of responses coded as robust, limited, and lack of evidence.
Code  Inquiry into Student Thinking—interpretation  Tutoring assignment—interpretation 

Robust evidence  I believe that Jack, by the end of the study, had a much better understanding of how to use his knowledge of base ten in solving problems  Brian had a good understanding of base 10 and basic problems. He was able to count very well and almost never stumbles when switching decades, e.g., 97, 98, 99, 100, 101, … He also demonstrated that he was capable of counting by 10s both forward and backward. I was especially happy to see him easily counting backwards, 204, 195, 184, 174, etc. Brian also understand the use of the equal sign … 
One of the problems was this: (JCU) 22 pennies, how many more to have 50. Jack solved this problem by counting up by ones from 22 using tallies to keep track, which solves the problem but shows evidence that he does not fully understand how to use his base ten knowledge to help solve problems. In session ten they did a problem that was as follows: 30 pencils, 29 more. This is a somewhat similar problem from the one in example 1. For both he needed to count up by about 30 to get the answer; but this time to solve the problem he drew a picture that represented groups of tens and then ones. This time he did use his knowledge of base ten to help make this problem easier to solve.  


Limited evidence  10 have now become a unit for Jack instead of just the 1’s unit. His thinking for base 10 is fragile though, and he will need more practice. For the last problem, he couldn’t decide between if 45 beads could make 4 or 5 necklaces.  I think both Calvin and Karl seemed to have a good understanding of numbers when counting forward and backward by rote memory. They were able to answer all the questions with ease. I even tried using some of the secondgrade questions and they were able to answer them without even thinking. 


Lack of any evidence  Jack didn’t have an understanding of base ten at the beginning of the case study but, by the end, he had a concrete understanding of the base ten process. At first Jack got confused with the terminology of loose and thought that he should subtract the balls instead of adding the balls together and he misunderstood the problem type. By the end of the case study, Jack had a better understanding of how to decode problems more properly.  Overall, I was surprised by how much my students know along different strategies to solve the problems. I was also surprised how the students knew how to solve the CGI but, when I presented them with the true or false and opennumber questions, they struggles 
Similar to Jacobs et al. [
Examples of responses coded as robust, limited, and lack of evidence.
Code  Inquiry into Student Thinking—selected task  Rationale  Tutoring assignment—selected task  Rationale 

Robust evidence  I would create JointResultUnknown (JRU) or SeparateResultUnknown (SRU) problems for the students. A JRU example would be “Sunny has ____ fish, and then she buys ____ more. How many fish does she have now?” Number choices would include {(10, 50) (20, 30) (10, 41) (15, 25)}.  This type of problem would be good for all of the students. Sunny and Daniella struggle to count by tens past the numbers 20 and 30, and this problem challenges them to do so. Emilio would be challenged to count by tens and keep track of the ‘one’ in 41. Both Jack and Emilio would be challenged by the last number choice, as both understand the concept of counting by tens, but they would have to extend their understanding to non‐zero ending numbers.  Cornor has ___Wii games in his cupboard. He found ___more Wii games under his bed. How many Wii games does Cornor have? 
After my initial interview with the students, I knew they did not have a clear understanding on how to count on from a number other than one. When I gave each student that question in the interview none could count on from the number I had given them … 


Limited evidence  If Quinn had 89 pieces of pizza, and 10 pieces of pizza made a whole pizza, how many whole pizzas can Quinn make?  This is a Separate Result Unknown problem. I chose 89 because the students have the concept of base 10 down; they are able to do it with the easy numbers, now I want to challenge them with bigger numbers, hoping they would use the manipulative[s] and not their fingers. I would hope students could lay out the manipulative[s] and see easily that they can make 8 pizzas. If students understand this concept they should have no problem with this problem.  The student will be given visual balance with numbers in blocks. One black on the right side will be blank. My number choices are 6 and 2 on the left sides and a blank and 4 on the right side  I plan to work on commutative property and relational thinking to help with those problems. Latter I plan to focus on his subtraction skills so that he will be willing to use them in other problems … 


Lack of any evidence  If I were to teach the next lesson to these students, one problem I could give them would be a: Jack has 45 crackers. Sunny gives him 10 more. How many crackers does Jack have?  I choose this Separate Result Unknown problem because I wanted the students to continue using addition. These are the problems they have been used to and need to keep getting trying to understand. I chose the numbers 45 and 10 because the students need to continue using large numbers so they can’t just count by ones and learn to use going by 5’s or 10’s as a first choice.  The student will be presented with these problems one at a time and they determine whether the problem is true or not. 
These equations allow the student to look at the two different equations and see that although the numbers are different they equal the same thing. 
After coding the PSTs’ responses for each component skill, we quantified the data in order to foster more meaningful comparisons and allow patterns to be identified and further explored [
Description of the coding scheme.
Component skill  Code  Subcode  Score 

Attending to children’s strategies  Considers the extent to which PSTs’ explanation demonstrated evidence of providing mathematical details  Most mathematical details  2 
Some mathematical details  1  
Lack of mathematical details  0  


Interpreting children’s mathematical understanding  Considers the extent to which PSTs’ explanations demonstrated evidence that the interpretation was based on children’s mathematical understanding  Robust evidence  2 
Limited evidence  1  
Lack of evidence  0  


Responding based on children’s mathematical understanding  Considers the extent to which PSTs’ rationale demonstrated evidence that it was based on children’s mathematical understanding  Robust evidence  2 
Limited evidence  1  
Lack of evidence  0 
Using the coded values, calculations of the number of PSTs’ responses that had most, some, or lack of any mathematical details as they attended to children’s strategies were performed. Similarly, calculations of PSTs’ responses with robust, limited, or lack of any evidence that they interpreted and/or responded on the basis of children’s mathematical understanding were performed during the two assignments. Further, we conducted a paired sample
To examine the nature of tasks that PSTs posed after assessing children’s mathematical understanding, data from PSTs’ responses to the last two prompts given in the Inquiry into Student Thinking assignment and the tutoring assignment were used:
In this stage of data analysis, open coding was performed to make meaning of the PSTs’ conceptions and rationale of tasks that would engage students with either highlevel or lowlevel thinking using the levels of cognitive demand described by work done by Stein and Smith [
To measure reliability and validity, an independent member check was used [
This section presents the results organized in the following subtopics: (a) extent to which PSTs attended to children’s strategies, (b) extent to which PSTs interpreted children’s mathematical understanding, (c) extent to which PSTs responded based on children’s mathematical understanding, (d) outcome of the paired
As shown in Table
Summary of PSTs’ responses in the two assignments.
Component skill  Inquiry into Student Thinking  % # of PSTs responses  Tutoring assignment  % # of PSTs’ responses 

Attending to children’s strategies  Most mathematical details  40% ( 
Most mathematical details  73.3% ( 
Some mathematical details  56.7% ( 
Some mathematical details  23.3% ( 

Lack of any mathematical details  3.3% ( 
Lack of any mathematical details  3.3% ( 



Interpreting Children’s mathematical understanding  Robust evidence  60% ( 
Robust evidence  70% ( 
Limited evidence  33.3% ( 
Limited evidence  23.3% ( 

Lack of evidence  6.7% ( 
Lack of evidence  6.7% ( 



Responding based on children’s mathematical understanding  Robust evidence  13.3% ( 
Robust evidence  36.7% ( 
Limited evidence  60% ( 
Limited evidence  33.3% ( 

Lack of evidence  26.7% ( 
Lack of evidence  30.0% ( 
Similarly, analysis of PSTs’ responses in the tutoring assignment revealed that 73.3% of PSTs’ responses demonstrated evidence of attending to children’s strategies by providing most mathematical details as compared to 26.3% PSTs’ responses that demonstrated some or no mathematical details.
A closer look at PSTs’ responses revealed that out of the twentytwo PSTs whose responses demonstrated evidence of providing most mathematical details during the tutoring assignment, nine had previously provided most mathematical details during the Inquiry into Student Thinking assignment. The remaining eleven participants provided some mathematical details in the Inquiry into Student Thinking assignment, and their responses during the tutoring assignment had evidence that they provided most mathematical details. These results indicate that the eleven PSTs’ performance shifted from general descriptions of children’s strategies to providing details of children’s strategies and including details on how the children interacted with the mathematical ideas.
Analysis shows that 60% of participants’ responses demonstrated robust evidence, while 33.3% demonstrated limited evidence that they interpreted children’s mathematical understanding in the Inquiry into Student Thinking responses. Only 6.7% of participants’ responses demonstrated no evidence that they made sense of details of children’s strategies when interpreting children’s mathematical understanding. In the tutoring assignment, analysis revealed that 70% of the PSTs’ responses demonstrated robust evidence and 23.3 % of the PSTs’ responses demonstrated limited evidence that they made sense of details of children’s strategies as they interpreted children’s mathematical understanding. Only 6.7% of the responses demonstrated no evidence that the participants made sense of children’s mathematical strategies.
Furthermore, detailed analysis of individual PST responses revealed patterns that represented their performance in the component skill of interpreting based on children’s mathematical understanding. For example, the number of PSTs (18 out of 30) whose responses had robust evidence in the Inquiry into Student Thinking assignment that they interpreted children’s mathematical understanding was larger than the number of PSTs (12 out of 30) whose responses had most mathematical details. Specifically, nine out of the 18 PSTs’ responses demonstrated evidence of providing most mathematical details and demonstrated robust evidence that their interpretation was based on children’s mathematical understanding. The remaining nine participants’ demonstrated evidence of providing some mathematical details in the component skill of attending to children’s strategies but their responses had robust evidence that their interpretation was based on children’s mathematical understanding.
Similarly, the ten responses that had limited evidence that the PSTs’ interpretation was based on children’s mathematical understanding in the Inquiry into Student Thinking varied in their performance in the component skill of attending to children’s strategies. Three of them had provided most mathematical details in the component skill of attending to children’s strategy, but their responses had limited evidence that their interpretation was based on children’s mathematical understanding. Seven of the participants only provided some mathematical details in the component skill of attending to children’s strategies, but their responses had limited evidence that their interpretation was based on children’s mathematical understanding.
For the component skill of responding based on children’s mathematical understanding, only 13.3% of the participants’ responses demonstrated robust evidence that the responses were based on children’s mathematical understanding in the Inquiry into Student Thinking. In comparison with the other component skills, few PSTs seemed to have demonstrated robust evidence of responding based on children’s mathematical understanding. Most PSTs (60%) demonstrated limited evidence that their responses were based on children’s mathematical understanding while 26.7% of participants’ responses lacked any evidence that they were based on children’s mathematical understanding. In the tutoring assignment, only 36.7% of the responses demonstrated robust evidence that PSTs’ responses were based on children’s mathematical understanding and 33.3% of the responses demonstrated limited evidence that PSTs’ responses were based on children’s mathematical understanding. Further analysis revealed that 30% of PSTs’ responses demonstrated no evidence that their responses were based on children’s mathematical understanding. The results indicate that PSTs struggled more with the component skill of responding during the two assignments than with the component skills of attending and interpreting children’s mathematical understanding.
Finally, a paired sample
The results presented in this section inform the third research question:
What type of tasks/problems do PSTs pose after assessing children’s mathematical understanding?
PSTs’ responses to the following prompts in the Inquiry into Student Thinking and tutoring assignment were used:
If you were to teach the next lesson to this group of students, write one problem that you could give them and explain why you think this would be a good problem for this group (
What was your plan for this week? Describe the activities, problems, literature etc. that you planned to use during this week’s tutoring session and explain your rationale for this plan. Included in your rationale should be: What makes these tasks high cognitive demand for your students? (
The nature of instructional tasks selected and rationale of PSTs’ choices, both in the Inquiry into Student Thinking and tutoring assignments, were examined.
As shown in Table
“Sunny has ____ fish, and then she buys ____ more. How many fish does she have now?” Number choices would include {(10, 50) (20, 30) (10, 41) (15, 25)}
Nature of tasks selected during the Inquiry into Student Thinking assignment.
Problem type  # of tasks  PSTs example  Number choices 

Join Change Unknown  5  “Joshua collects rocks. He likes to keep them in bags of ten. Last week he had a total of ____ bags. After finding some more this week, he now has ____ bags. How many rocks did Joshua find this week?”  Number Choices: 5, 6) (5, 8) (8, 12) (9, 15) 
Join Result Unknown  8  I would create JointResultUnknown (JRU) or SeparateResultUnknown (SRU) problems for the students. A JRU example would be “Sunny has ____ fish, and then she buys ____ more. How many fish does she have now?” Number choices would include  Number Choices: {(10, 50) (20, 30) (10, 41) (15, 25)}. 
Compare change unknown  1  Emilio had 64 soccer cards. Sunny had 28 soccer cards. How many more soccer cards does Emilio have than sunny?  
Separate result unknown  4  Emilio has __soccer balls. __ Soccer balls roll away. How many Soccer balls are left?  Number Choices: (15, 5) (26, 16) (57, 27) (81, 51) 
Group size unknown  2  Jack has 65 pencils. He wants his 5 classmates to each have an equal amount of pencils. How many would each classmate get?”  No number Choices 
Number of groups unknown  3  Sunny has 94 chocolate chips. She needs 10 chocolate chips to make a cookie. How many cookies can Sunny make?  No number Choices 
Both product unknown & Joint Result Unknown  4  Sunny has 8 rolls of candy. Each package has 11 candies in it. She also has 12 extra candies. How many candies does she have in all?  No number Choices 
Product unknown and compare result unknown  1  Mary had __ bags of cookies with __ cookies in each bag and Amy had __ bags of cookies with __ cookies in each bag. Which one had more cookies?  No number Choices 
Relational thinking problems  1  40 + 60 =  +   No number Choices 
True/false sentences  1  10 + 2 = 6.  No number Choices 
10 + 5 = 5 + 10  
10 + 10+10 + 9 = 20 + 19 
This type of problem would be good for all of the students. Sunny and Daniella struggle to count by tens past the numbers 20 and 30, and this problem challenges them to do so. Emilio would be challenged to count by tens and keep track of the ‘one’ in 41. Both Jack and Emilio would be challenged by the last number choice, as both understand the concept of counting by tens, but they would.
Notable in this response is the fact that this PST selected and used number choices that would advance the children’s mathematical understanding. The PST focused on how the task would support Sunny and Daniella’s understanding because “they struggled to count by 10” and both Jack and Emilio would be challenged by the last number choice because they “understand the concept of counting by 10.” This type of reasoning seemed to be a tendency for PSTs even when they did not select multiple number choices. For example, one PST selected the task, “Danielle has 55 beads. She wants to make as many necklaces as she can, but she must have 10 beads on each necklace. How many complete necklaces can she make? How many beads will she have left over?” Explaining why she selected those numbers, she remarked:
I chose 55 because the students already demonstrated knowledge of knowing 50 is 5 groups of 10 and 5 is an easy number to work with as a remainder. I chose 10 because the goal is to get the students to develop and use the idea of ten as one and use it to problem solve
The examples also provide persuasive evidence that the PSTs had some capacity to choose tasks based on children’s mathematical understanding by the time they did the Inquiry into Students’ Thinking assignment.
To examine the nature of tasks that the PSTs selected in the tutoring assignment, PSTs’ lesson plans for the first tutoring sessions were examined. Although we used PSTs’ responses for the interview and the first tutoring session, PSTs had an opportunity to tutor the same child or children for three 50minute sessions. Therefore, they got an opportunity to interpret children’s mathematical understanding and select and pose tasks for three times. PSTs had been instructed to start with an opening number routine followed by the main activity, which they used to tutor the children based on the interpretation of children’s mathematical understanding. As shown in Table
Nature of tasks selected and/or generated during the tutoring assignment.
Type 

Example  Number choices 

CGIword problems  17  Student 1(use their names in the real setting) has ___race cars. Student 2 gave student 1___more race cars. How many race cars does student 1 have in total?  Student 1 adding to 10_(5, 5) 7, 3) (4, 6) (15, 5) student 2 adding to 100 (50, 50) (80, 20) (35, 65) (42, 58) 


True or false sentences  3  The student will be presented with these problems one at a time and they determine whether the problem is true or not. 
Those numbers are chosen because they are within the range of 1–10 and they are familiar 10s fact for the student. The values of the equations are slightly higher but the sum allows for more differing equations to be used. I chose this equations and numbers using low numbers in value while presenting anew concept in to make her more comfortable and confident in the use of those numbers … 


Number sentences and/or equations  3  5 + 8 = 8 + 5 
5 + 8 = 8 + 5I chose this number because I want to see if my students understand that the number to the right is the same with the number to the left 


Counting  5  After the students complete the number 10 worksheet the teacher will pass out the dottodot worksheet. Students will need to complete both worksheets by drawing lines from the numbers 1–30 and 5–500 first by counting up by ones and then by 5’s. This will give the students the bases for counting so that they will be able to count the “how many objects” worksheet.  No number choices 


Place value  2  I will start by writing a two digit number on my scratch paper for both students to see e.g., 76.I will ask them to say the number and then I will point at the different digits and then ask them what this number represents (prompting students to point out the place). After discussing the twodigit number, I will add to digit to the end of the number making it a 3 digit three digit number.  I chose some two digit and three digits because at this grade level students know three digit numbers and breaking them into place values is a good task, but I also choose two digits so that they can see the difference … 
Further analysis revealed four emerging patterns that characterized the rationale for the number choices and nature of problems created: (1) selected number choices that started with easier numbers to more challenging numbers, (2) considered the strategy that the children would use to solve the task, (3) considered children’s understanding or how the children will make sense of the problem, and (4) no rationale to the number choices. A brief explanation of each pattern is given in the next section.
Twelve of the participants selected number choices that started with easier numbers to more challenging numbers. In this approach, the PSTs’ rationales revealed a tendency to start with easier numbers that the children could manipulate with ease, followed by larger numbers that the children could not manipulate using mental strategies alone. The PSTs either referred to those first numbers as “easier numbers” or “familiar numbers.” From their reasoning, “the easier” or “familiar numbers” could make the children more comfortable with the concept before working with more challenging numbers. For example, one PST selected the following task for kindergarten children:
James has ___Wii games at his house. Cornor let James borrow ___more Wii games. How many Wii games does James have at his house now?
From the above rationale, the PST was cognizant of the number patterns and the connections that she wanted the children to make. One PST considered numbers that were easier to challenging but paid specific attention to the way the numbers would advance children’s mathematical understanding. She chose the following task for two kindergarten children and selected the number choices based on the strategies she anticipated the children to use.
Student 1(use their names in the real setting) has__race cars. Student 2 gave student 1 ___more race cars. How many race cars does student 1 have in total?
Number choices:
Student 1 adding to 10 (5, 5) (7, 3) (4, 6) (15, 5)
Student 2 adding to 100 (50, 50) (80, 20) (35, 65) (42, 58)
To justify her number choices, the PST anticipated that children will use doubling strategy, use the “make a 10 strategy,” counting from a bigger number to get to the answer, counting by 5’s, and using decade and nondecade numbers. The rationale of the number choices revealed that they were purposefully selected to progress individual students’ thinking. She also considered different numbers that would sum up to 10 and 100 respectively. Although the multiple number choices could be characterized from easier to more challenging, the PST paid careful attention to each number.
Some PSTs selected tasks with number choices that would support the children to identify patterns and use specific strategies. For example, one PST created a multiplication story problem with two sets of number choices as shown below:
“Sara has __bags of candy bars with __in each bag. How many candy bars does Sara have?”
Number choices:
(2, 5) (2, 6) (2, 3) (4, 5)
(2, 10) 2, 12) (4, 3) (4, 10)
Rationale:
The numbers that I chose in the main activity start by doubling one number then only one of the numbers in the next set would double. I was looking to see if Hilary would see that when one number is doubled in the number set then the answer for the sets would double.
The above explanation shows that the PST paid attention to the number choices, focusing on the strategy that the students would use and anticipating that the children will recognize the pattern and use it.
Three participants considered children’s understanding and how the children would make sense of the problem. For example, one participant selected the following problem:
You had __ cookies. Your brother ate __of your cookies. Now how many cookies do you have?
(20, 4) (30, 12) (15, 9) (37, 11) (21, 49) (33, 24)
To justify her number choices, she explained:
I chose the numbers that I did because the girls could do decadetodecade problems but couldn’t do nondecadetonondecade all of the time and Betty couldn’t do all the decade to nondecade problems.
Again, it is notable that the PST put into consideration what the children knew as she selected the numbers (doing decadedecade) and wanted to extend that to nondecade numbers. In addition to considering students’ understanding, the other two participants put into consideration how the children would make sense of the mathematical idea. For example, one PST selected the following word problem and provided the following explanation:
Mary has __grapes and ___oranges. How many grapes and oranges does she have altogether?
(3, 1) (3, 3) (3, 5)
Rationale:
Started with the same number of grapes every time so that they don’t have to start all over every time.
(3, 1)—Started off adding one since these are different problem types than in the opening number routine
(3, 3)—Stayed with the same first number to see if they can use prior knowledge from first problem to help them solve this problem. Also working on doubles.
(3, 5)—Want to see how to start the problem
This PST chose numbers that would support children to build connections and make sense of the mathematical idea.
Finally, fourteen PSTs did not provide an explanation or the reasoning behind their number choices. This group provided different number choices but there was no written rationale why they chose the numbers. For example, one student who chose the following task did not provide any rationale on the tasks used.
“After the students complete the number 10 worksheet, the teacher will pass out the dottodot worksheet. Students will need to complete both worksheets by drawing lines from the numbers 1–30 and 5–500 first by counting up by 1’s and then by 5’s. This will give the students the bases for counting so that they will be able to count the “how many objects” worksheet.
From the above example, the PST did not elaborate how the numbers selected might develop children’s mathematical understanding. As indicated in the Discussion section, the lack of a rationale might not necessary imply that the PSTs did not have a rationale for the number choices.
The need to design the content taught in teacher preparation programs to ensure that PSTs are equipped with knowledge, skills, and practices to become wellprepared beginning teachers is of prime importance. Although the mathematics education community have highlighted lingering challenges in teacher preparation programs, the need for highquality teachers in K12 classrooms has been and remains imperative [
The results in this study indicate that PSTs’ reflections evidenced emerging understanding of what it means to attend to children’s strategies and interpret and respond based on children’s mathematical understanding, consistently in the context of the two scaffolded assignments. Specifically, there was significant difference in the evidence noticed in attending to children’s strategies during the two assignments. As Van Zoest and Stockero [
These findings are noteworthy and encouraging in light of other studies (e.g., [
In addition, the findings from this study provide insights to the growing body of research that have focused on identifying core practices that PSTs can learn and develop the curriculum around those practices [
Furthermore, this study provides important insights on the hypothesized theoretical trajectory of professional noticing of children’s mathematical understanding. Jacobs et al. [
Model for PSTs’ learning of the component skills of professional noticing of Children’s mathematical understanding.
Over time, researchers have emphasized that students learn from the kind of work that they do during class [
These findings extend what we know about PSTs’ problemposing strategies using the context of letter writing [
Based on the benefits of using children’s mathematical thinking in mathematics classroom, teacher educators should consider how to decompose the practice and carefully provide PSTs with opportunities to learn in the mathematics methods course [
We propose that teacher educators need to purposefully choose what to teach, how to teach, and the activities to use as they develop PSTs’ capacity to attend and interpret and respond based on children’s mathematical understanding. As Jacobs et al. [
However, we cautiously interpret the results in this study in case we underestimate or overestimate what teacher candidates can/or cannot do as beginning teachers. Although reflections have been used to evaluate PSTs’ learning, it is possible that PSTs might have responded based on children’s mathematical understanding but not necessarily documented their decisionmaking process in the analyzed reflections. On the other hand, we might be overestimating PSTs’ ability in case they mirrored the kind of tasks that they had discussed in the university classroom. Despite these limitations, many teacher educators use PSTs’ reflections as a measure of learning as they go through the teacher preparation program. Additionally, to complete Educators Teacher Performance Assessment (edTPA)
Undoubtedly, more research is needed to determine activities that develop the component skill of responding based on children’s mathematical understanding in the context of a methods course, given that the activities provided in this study did not significantly develop the component skill in all PSTs. Since we used thirty PSTs in this analysis, a future study conducted in multiple contexts and a larger sample might provide more insights into PSTs learning. Furthermore, a future study could also track how PSTs would respond as they progress to student teaching experience and as first year teachers to document how their capacity changes in different contexts. Although previous research has suggested that developing the skill of responding to children’s understanding takes long to develop, teacher educators cannot underestimate the need to equip PSTs with competencies to become well started off teachers. It is hoped that the knowledge generated by this study will promote further discussions about how PSTs learn, and more activities will be shared and considered in mathematics teacher education.
For this assignment, your primary source of information will be a case study of four second graders found at
Read through the entire case study first to get an idea what it is about. Then, answer the following questions in a total of 23 doublespaced pages:
Choose one of the four students from the case study and trace their responses and learning through the study.
Summarize what you think they knew or understood at the end of the study that they did not know or understand in the beginning.
Choose 2 examples of this student’s responses as evidence for your claim from part 1.a and explain how they support your claim about this student’s learning (you will probably want one example toward the end of the study and one example toward the beginning).
Choose 2 tasks or problems that were posed to the students that seemed particularly productive for advancing the thinking of the students
Choose one instance of teacher decisionmaking or reflection that was particularly interesting or surprising to you. Summarize in a paragraph what made that instance stand out for you and how you might use it to inform your own tutoring experience.
If you were to teach the next lesson to this group of students, write one problem that you could give them and explain why you think this would be a good problem for this group. Use what you have learned in class about problem types, number choices, and students’ solution strategies to support your decision.
For this assignment, you will first interview up to three students about their understanding of numbers and operations. You will be provided with gradeappropriate questionnaires for this interview. Based (at least in part) on what you learn in the interview, you will plan and implement three tutoring sessions with these students. You will use
Briefly describe the setting of the interview and the interviewed students. Remember to use
For each of the questions on the provided interview, describe what
What does your chosen student know/think/understand about number, operations, and problem solving? Think about what strategies or problem types they DO know and which they are on the verge of learning but have not yet learned. Be
Based on your answers to part a, what specific mathematics will you focus on with these students? How will what you learned in the interviews influence how you work with these children during the next sessions?
You will tutor your student(s) two times after the first interview. The tutoring sessions must include an opening number routine activity and CGItype problems.
There is a lot to think about while you are teaching/tutoring. First, you will be thinking about what you are doing and saying–you want to follow your lesson plan but also be flexible based on your formative assessments of what your students are and are not understanding. During each session, you will also want to pay close attention to your students’ thinking–their strategies, responses (correct and incorrect), questions, etc.
Remember that your students should be doing the bulk of the talking. They are the ones doing the mathematics. You will be unpacking the problem, asking questions, planning the sharing of solutions, and doing some rephrasing. If you find yourself explaining most of the time, you are doing it wrong.
After
What was your plan for this week?
What makes these tasks high cognitive demand? Use the criteria from the packet to justify your answer.
How are these tasks intended to build on what you already know about your student’s understandings and misunderstandings? Here
What actually happened during your tutoring session this week?
Describe
How effective were your plan and your teaching?
What did your assessment plan tell you about your students’ growth, and what did not they tell you about your students? Provide
What do you still wonder about your students and their mathematical thinking?
What will you focus on during the next session, and why?
Finally, spend some time reflecting on this experience.
What did you learn about learning, teaching, mathematics, and students during this experience?
How do your original goals for the tutoring sessions compare with what really happened?
Overall, how do you feel your interviews and tutoring sessions went?
What would you do next with these students?
How can you use this process in your own teaching?
Data used to support the findings of this study are included within the article. However, any supplemental tables and analyzed data from responses to the Inquiry into Student Thinking assignment and the tutoring assignment are available from the corresponding author upon request.
This manuscript is based on a larger unpublished graduate thesis and dissertation by authors, currently in the Iowa State University digital repository that examined elementary preservice teachers' capacity to use children's mathematical understanding to select and pose mathematical tasks at “
The authors declare that they have no conflicts of interest or funding sources to disclose.
The authors thank Corey Drake, Susan GallagherLepak, and Scott Ashmann for their feedback on different versions of this article. The authors also sincerely appreciate the preservice teachers who agreed to participate in this study. This work would not have been feasible without their contribution.