In order to consider whether the Mathematical Mindset approach changed teachers’ ideas about mathematics learners, about mindset, and about mathematics teaching and learning, we gathered a range of data sources that have been used to triangulate findings. The different sources of data give a consistent picture of teacher change.
3.1.1. Observation of Lessons
On the 5 measures in the mathematical mindset observation tool, the teachers improved in every area, with three of the areas reaching statistical significance. To perform an analysis of the observation tool scores, the three scores (Emerging, Accelerating, and Expanding) were converted into numerical scores (1, 2, and 3). We did not assign numerical values for Not Observed
rating. For example, under Mathematical Mindset Practice 1 (Teachers fosters a positive culture), there are 3 sub-categories: mindset messages, praising effort, and students’ mindset. Thus, three sub-scores were generated for this practice and summed to construct a total score for that practice. Averages of scores were taken for each practice across the 27 teachers for whom we had complete observation records. The 5 areas of growth are shown in Figure 1
shows the instructional shifts in terms of subcategories and total counts, while Figure 2
shows the mean scores of the 5 teaching practices.
To measure statistical significance, a matched-pair t
-test was conducted for each of the 5 measures using mean score on the indicators within the measure (due to a county office error, only 2 indicators were used for practice 3, instead of the 3 on the observation tool). Practice 2 had a statistically significant difference from beginning to middle of the year, indicated by a t-statistic = 3.03 and p
-value = 0.005 with a Cohen’s d
effect size of 0.883. Practice 3 had a statistically significant difference from beginning to middle of the year, indicated by a t-statistic = 2.75 and p
-value = 0.012 with a Cohen’s d
effect size of 0.758. Practice 4 had a statistically significant difference from the beginning to the middle of the year, indicated by a t-statistic = 2.40 and p
-value = 0.024 with a Cohen’s d
effect size of 0.678. The 3 areas that showed significant growth (Practices 2, 3, and 4) included a number of sub-areas that are shown in Figure 3
, Figure 4
and Figure 5
. Sub-areas are reported to help illuminate, with standard error bars, where the significant changes in observable practices occurred in each practice.
The areas reflected in practices 2, 3, and 4, i.e., using more open tasks that encourage reasoning, depth, and multiple perspectives, maintaining high challenge and cognitive demand, and encouraging risk-taking and valuing mistakes, respectively, are critically important teacher practices. It could be argued that the 2 areas that improved but did not reach significant levels are more difficult to achieve, particularly in the first year of change. Practice 1 is focused upon providing a teaching environment in which all students can be successful, and Practice 5 is focused on the facilitation of productive student to student interactions. These two areas reflect high-level teaching practices.
3.1.2. Teacher Reflections (Online Course Responses and Surveys)
Online course reflection prompts and teacher surveys that asked teachers which of the ideas in the course were more salient to them and would be the basis for change, produced findings consistent with the observation tool. Teachers were asked, through an anonymous survey, to share how their teaching practice had changed from the previous year due to the professional development they had received. The following shows the coded responses to the question: “In what ways has your teaching of math changed this year?” Table 3
provides a sample of how surveys were coded. For all responses, we provide pseudonyms for all teachers we quote, as per our consent agreement.
Before the Mathematical Mindset PD Approach, the teachers talked about both themselves and the students “dreading” math time, and the time being taken up with strict teacher control and students working through textbook questions.
As a first year teacher, I never.... It was like every time we did math was like, “Open up a book.” And it was just this negative connotation to it.
—Maurice V (Online Course Reflection)
When reflecting on their previous practice, teachers shared that they relied heavily on direct instruction. When it became time for mathematics, teachers felt that they needed to control the environment in the classroom by being the sole provider of knowledge.
What I feel has made the most difference in my teaching is, quite simply, the change of mindset. I used to think that direct instruction was the only way. I felt I had to always be in control in order for learning to occur.
—Jolie (Online Course Reflection)
Because for years it was I teach, we practice, you practice alone. That type of directed instruction, and now it’s given me the confidence as a teacher to just sit back and watch them explore, and it’s okay. Even though they’re making mistakes as they work through it it’s okay, because now they’re learning the conversation that they need to have with each other as groups, as partners.
—Anonymous 5th Grade Teacher (Survey)
The survey responses indicate that teachers had moved far from direct instruction, instead valuing student ideas and strategies. As Table 3
shows, the 2 most commonly cited changes the teachers reported making were asking students to share their ideas, and students having the space to explore mathematical ideas. Teachers also talked about valuing different student strategies, and students owning their ideas. There was a clear contrast between the teacher-led environment the teachers describe as dominating their practice before the Mathematical Mindset PD, i.e., working through textbooks and pacing guides, to an environment they describe in which students explored and played with ideas and then discussed them with each other. The teacher online responses and surveys give teachers’ self-reports of their changes, which are consistent with the observations of the coaches who saw the teachers using more open tasks that encouraged student reasoning. One of the teachers wrote in an anonymous survey about the “complete shift” s/he had gone through:
I’ve gone from direct instruction (DII) with little exploration of tasks to a complete shift of teacher role and I’ve become a facilitator. I anticipate their answers to math tasks, but they share their thinking and ask each other questions. They clear up their own misunderstandings most of the time with little direction from me. My students are thoroughly enjoying math.
—Anonymous 5th Grade Teacher (Survey)
The vast shift in teaching practices that emerged from the coaches’ observations and the teacher survey are further elucidated by the teacher interviews.
The coaches’ observations, teacher online course response, and surveys reveal considerable teacher change. Interviews with 12 of the teachers who volunteered to share their perspectives add depth and nuance to the other data. The coding of the interviews produced 8 different themes that were important to the teachers and reflected their biggest areas of change: Mindset, Using Open Tasks and Questions, Encouraging Struggle, Celebrating Multiple Ways of Seeing and Doing Math, Scaffolding Less, Valuing Slower Thinking, Student Role in Change, and Own Relationship with Mathematics. Each of these themes will be discussed below.
In the online class, teachers were introduced to research showing that brains grow and change, and that self-belief—having a growth mindset—is important for learning. When asked in their second teacher survey about the ideas they were implementing from the course, over half, 25 of the 40 teachers, chose to write about mindset, which was the single biggest idea chosen. In the interviews, teachers expanded on the ways they had integrated mindset ideas into their teaching:
I will definitely change my way of thinking about all of my student abilities. All of the data given to me regarding my students will not be looked at the same way as it had been in previous years. I will now see the student first rather than see a test score and then a student. My Students will be viewed as individuals who no matter what level of math skills should be attended to as a student with mindset that needs some TLC.
During the interviews, teachers also shared the ways in which they had integrated mindset messages into their classrooms. They also reported that students were not only internalizing the messages, but also taking the initiative to set the tone of the classroom, and encouraging each other to have growth mindsets.
I’ve really seen an impact on my students and how they view themselves, and it’s changed their level of perseverance in the classroom. And everything, really. And even during art, one kid was saying, “I can’t. I can’t do this,” and another kid said, “That’s a fixed mindset.” And I’m getting positive feedback from parents about this. Their children that really didn’t believe that they could do math, or really just saw themselves as deficient in some way, that their attitudes are changing.
Teachers also shared that they brought the research they learned in the online course back to their classroom to share with the students. The teachers talked with the students about brain growth, neuroplasticity and synapses firing in the brain, which became part of the classroom conversations:
From the get go we talked about and continue to talk about the Fixed and Growth mindset. I continue to remind my class that we must change our old ways of thinking about math to a new way, which is to always approach math with the attitude of “yes, if I try and keep on trying, I can” explaining that this is the Growth Mindset!! The information on the brain and how its plasticity is affected by the learning or lack of learning/struggle. In my class we always refer to our brain synapses firing which is a good thing especially when we are challenged.
It seems noteworthy that the teachers were sufficiently excited about the research they learned that they returned immediately to share it with their students, who in turn used it to encourage each other.
Using Open Tasks and Questions.
As mentioned in the introduction to this paper, encouraging a growth mindset means more than teachers using different words with students. Narrow textbook questions, with one method and one solution, often give students the idea that mathematics is a fixed subject. Considerable portions of the online course and the network meetings were devoted to mathematics, and the value of more open tasks that encourage different student ideas to be shared and discussed. Additionally, all of the teachers in the network used the “Week of Inspirational Math” tasks from youcubed: a pre-planned week of low floor, high ceiling mathematics lessons, and accompanying lesson plans encouraging student discussion of the ideas. The teachers spoke about the impact of the youcubed lessons (that they referred to as imaths), which were very different from their previously used textbook questions:
The week of imaths was incredible for kids, I’ve never seen a classroom more fired up for math in my life.
One of the teachers reflected that the mathematics that students learned through open tasks did not “look” like math to the students, as they were so used to narrow closed questions:
I think it’s because it doesn’t look like math, a typical practice sheet with 50 problems on it. … So, they’re more excited, because it’s more like an activity. It’s more like they get to discover on their own and do something that is different than what they’ve done before. So, I think that’s why they’re enthused about it.
The results from the coaches’ observations tool suggest that teachers continued with open tasks throughout the year, and in interviews the teachers talked about the ways they managed to do this: finding tasks on line and creating their own:
The whole week was wonderful. At first we pulled different youcubed problems so we didn’t have to do GoMath on Fridays, now we come up with our own open ended questions. We just create. The other day on the board I wrote: The answer is 17 how many different ways can you get me to the answer? I thought they might just say 1 plus 16 but they were doing orders of operations and they got really fancy and I was really impressed with them.
Janice’s reflection reveals a high degree of teacher agency, as she describes first looking for new tasks online, then creating her own, using the principles she had learned through the network. One teacher shared that teacher collaboration time had changed since taking the course, and that he and his colleagues now work to develop rich tasks together:
What we’ll do is we’ll plan an enrichment task. We’ll kind of go through the process of how we think the kids are gonna solve it, maybe how we think what we’re gonna see. … So that’s been our biggest goal, kind of setting up a rich task…
As the teachers describe their creation and use of different tasks, they suggest a different relationship with mathematics, which is more enjoyable and playful, and leads to deeper learning of their own.
A key message of the online course concerned the value of mistakes and struggle for brain growth. Many mathematics learners come to feel ashamed if they make mistakes and do not realize that times of making mistakes and struggle are very important for brain growth. In the online class, participants were asked this question:
Think back to some time when you were learning maths and you made a mistake. How did that make you feel?
I’m not good at this.
Slightly irritated—I need to do the problem again.
Great, I can think about what I did wrong and learn from it.
Happy, I feel challenged and more determined to learn.
58% of respondents answered “I’m not good at this”, while 42% answered “Slightly irritated—I need to do the problem again”. Teachers talked in interviews about the ways they integrated their new knowledge of the benefits of struggle and mistakes into their classrooms, creating, as one teacher said “safe, risk-free environments” where students were “more willing to share openly without worrying about right/wrong because they know as we work on they will have (the) opportunity to modify their answer/thinking without any stigma being attached.”
The teachers shared that the creation of mistakes-friendly environments had deeply impacted their students:
The kids were thrilled, going “Oh my gosh, he’s doing it like that? It’s OK that we struggle? It’s OK we think differently?”
When a student asks if “It’s OK that we think differently?”, it suggests that they had held damaging ideas about mathematics learning, including the idea that they should all think in the same way. The teachers were quick to notice how powerful the new ideas were for the students, particularly around the permission to struggle:
I liked watching them struggle, and I know if the teacher says that it always sounds kind of odd, but it was a really good struggle because it really got them to think much differently about what math was, and that math is not just going to come out of a textbook, I am not just gonna sit there and look at you and go, “You’re gonna do this many problems.” We don’t do that, and the kids have now accepted that going, “Okay, so things are definitely gonna be different.” And they all have math notebooks and keeping ideas of their journals, and what they need to do.
Allowing and acknowledging the value of struggle helped teachers redefine what mathematics was for their students. Instead of a finished product that is laid out in a textbook, the teachers helped students see mathematics as the product of student thinking and purposeful struggle. The teachers were grateful for the changes they saw in their students:
I just want you to know this [the online course] has meant a lot. Seeing how positive the kids are about their learning now has made a world of difference. The confidence they have is unlike anything I have ever seen.
Through their new-found appreciation of struggle, the teachers became more relaxed when teaching mathematics, and were less tense when students made mistakes:
But now it’s easier for me to accept the wrong answer from a student. Usually it’s, “Come on, you guys can do this.” Now it’s, “Okay. Let’s do it this way. Let’s see what else we can come up. Talk to your partner.” So it’s more engagement for me, and it’s a lot easier for me. It’s a lot more relaxing.
One of the teachers described the frustration students could fall back into when they struggled, but when she reminded them of the importance of struggle, she saw a “wave” of change with students developing growth mindsets and appreciating mathematics more:
But I still had a lot of kids that were incredibly frustrated, “This isn’t gonna work.” The next day we talked about those norms again, and those mindsets that in this class you’re going to be expected to think, and it’s okay, you’re gonna be challenged, and you are gonna struggle. And struggle is not a bad thing. And then all the sudden, you could almost see kind of like this wave of kids, and the whole reaction started to change, so their mindset totally completely began to change. And then they got more of them pulled in.
Celebrating Multiple ways of Seeing and Doing Math.
An important area of change that was discussed in the online course and the teachers’ network meetings concerns the need to make mathematics more multi-dimensional, encouraging students to see mathematics visually and to make connections between areas. Many of the teachers discussed the change in the way they and students saw mathematics:
Oh, the visuals.... They love that too, cause with their ideas of how it would form, and how they would build. I do it periodically. Just throw up a visual with different things and say, “Okay, what do you see? What don’t you see? What might you see? What could be the next thing?”
Well, the big take-aways for me are the connections they make. We were doing a paper-folding activity where we were using it for fractions, but the paper-folding activity, they started folding the paper with triangles from a square. And they on their own discovered that there was an exponential relationship, so when they folded it once they had two pieces. And then when they folded it twice, they had four. And they started seeing the exponents of two with every fold. And they made that connection all by themselves, cause we’ve been doing base 10, and powers of 10. So I see those connections happening during these iMath lessons, and that’s huge for me.
Another key message of the online course is the importance of allowing students to struggle and the need to step back on scaffolding and rescuing students. The reduction in scaffolding was one of the themes that emerged from the interview:
Last year I probably would’ve been more of the one to help jump in a little bit more and give them more scaffolding. This year, I’ve been working really hard at not jumping in and scaffolding quite so much, but the kids are helping each other, so it’s not relying on me. It’s where the kids, when we’ve been doing these different tasks from the five practices, where it’s me kind of stepping back and letting them, and doing maybe more of the question asking.
As the teachers developed an appreciation of struggle they became more comfortable leaving the students to engage in productive struggle and to use each other as resources.
Valuing Slower Thinking:
One of the online course lessons focused on the value of depth in mathematics, and the need to counter the damaging idea that to be good at mathematics, students need to be fast with numbers. Many of the teachers talked about the ways they had slowed down to allow students to go into more depth and think more conceptually about math:
And even individually, I can tell that they sit and they think about it when it’s time to think. Before, they just tried to work through it and be done, and that’s a lot of what happened last year, was just the race to be done. And so I’m trying to expel that, I just wanna do away with the whole race. There’s no value in that.
When teachers stopped valuing speed, it allowed them to focus more on mathematical concepts:
I’ve kinda slowed things way down, and focusing more on the concept for the day.
Teachers talked about the students who, previous to the Mathematical Mindset PD, had decided they were not “math people” because they were not fast, and how they had been able to feel included as competent mathematics students:
So, I have noticed, I have a couple of students that really have taken on that it’s okay to take their time to do their work. I have a girl who just raised her hand and shared this with the rest of the class which I thought was “Wow.” She always thought that she wasn’t good at math because it takes her a long time to do the work.
Many teachers do not feel that they are able to slow down in their teaching, even if that means moving quickly over important concepts, because of pacing guides from districts. The teachers in the network were greatly helped in bringing about change as the County leaders had worked with administrators to make sure that they were supportive of the change teachers were making. This next comment highlights the importance of administrator support, in letting the teachers embrace the important research they were learning without being hampered by the need to meet district pacing guides and benchmark tests:
Okay, why are these kids so much better at focusing? Is it because they’ve had a couple more years of Common Core and they understand it better? They’re used to the routine? They’re used to the collaboration? Or is it because they have a different mindset now? And that’s what I’m leaning more towards, because last year it was like if they can’t finish it in five minutes they don’t wanna do it. And this year they’re okay with taking time on it, and I’m okay on taking time. In fact, I mentioned to our superintendent this morning, she was here with us and I said, “I don’t know that I’m gonna be completely prepared for our benchmark in a couple of weeks.” And she goes, “Don’t worry about it. Just let the kids explore, it’s more important that they think about what they’re doing. That they grow this mindset, because it’ll work out for them better in the end.” And so that’s the major difference I see.
The different teachers appreciated the administrator support that is often lacking in teacher reform efforts and one of the reasons teachers are fearful of making changes.
The biggest freedom that we’ve had is our administrator has empowered us to not look at “This is the lesson I have to get done,” it’s “How can I take elements and resources?”, covering the standard… So I think hearing that as a teacher, it’s like you have a blank canvas. It’s your freedom to... You have the freedom to add to what you want with the structure of common core and standards. But hearing that, as a teacher, there’s no complaining. It’s taking from here, taking from there, and then blending it together.”
Student Role in Change:
One of the most encouraging and unexpected themes to emerge from the teacher interviews was the role of students in encouraging teachers to make changes. Many of the teachers talked about being afraid of some of the more open tasks, or not thinking that students would be able to cope with them, and then being surprised when the students did well. In one instance, a teacher did not want to use one of the recommended tasks and then was surprised when students took ownership of it. The task is called: Four 4’s, and students are invited to find every number between 1 and 20 using exactly four 4’s and any operation:
When it said on the week that I might have to teach this, like, “No, no, no, no, no that’s too much for fifth graders.” But they thought it was really cool.
And the very first day we did it, oh my gosh, the struggle and the upset and, “Oh my gosh, I’m gonna have to do this, this is hard.” The second day, they got more into it, they got more experience. And then I had two girls the following week come in on their own, and figured out the missing ones, and they thought that was the coolest thing, and they were just like, “It’s totally filled out!”
Others talked about how the students took the messages they had learned and used them to encourage the teachers to have a better mathematical relationship, in a reversal of the expected role of teachers encouraging students in their mathematical relationships:
The most powerful is the mindset, and that’s not only for my students but more for me because I didn’t think I was a math person. I’ve never enjoyed math, I’ve never felt that I was good at math… before when it was just that one right answer then that left a lot out. I like the mindset and the kids have really picked up on that and they’re really applying that. In class, we’ll say, we don’t say that something is hard, we say it’s challenging. I am the one that usually says about something being hard and then they’ll all tell me, “No, it’s not hard, it’s challenging.” So they’re doing really well with the growth mindset.
The students also took it upon themselves to use the mindset messages they had learned to encourage each other:
I am making more of an effort to help my students develop a math mindset of success. One example I have took place about two weeks ago during our Go Math lesson. A new student had arrived in our classroom, and he came with an IEP(Individualized Education Plan). We were doing a problem together whole group, and I noticed he was not writing down the problem. I asked him why, and he said “I am not good at math; it is hard.” Before I could respond, the entire class looked directly at him and said “don’t say that, everyone can be good at math.” We then gave him our class cheer which is “I can, you can, and we can.” It was totally spontaneous, and I am seeing more effort from him.
Eight of the teachers who took the teacher survey chose to talk about the ways students felt empowered by the new knowledge the teachers were sharing, which certainly seemed evident in the reflections of the teachers in the interviews.
Own Relationship with Math.
The seven themes reviewed speak to important changes in teaching, confirming the observational data and the survey data. The considerable teacher change that resulted from the Mathematical Mindset observation is noteworthy, particularly when considered against a backdrop of a general lack of interest in professional development by teachers [40
]. One of the codes that emerged from the interviews was different in nature from the other seven, although related to the teaching that was described in the seven codes; it concerned the teachers’ own relationships with mathematics and learning. As the teachers learned in the course that they could, themselves, learn mathematics well and there were no limits to their learning, they began to rethink their past experiences in school and as learners. The teachers, like many learners of the past, had believed the myth that they were not a “math person” and could not learn math well. They had internalized this idea and it had shaped their experiences, not only as a student but later as a teacher. While this theme emerged in the coding of the interviews, teachers also shared how they had been affected by the myth of the “math person” in their online responses and surveys.
In one of the reflection prompts inside the class, one of the teachers shared how his fear of failure had shaped his own personal experiences in education, in both English and mathematics.
The information has made me reflect on my fear in my English and literature courses. I have always loved my literature courses but I was always scared of failing and the subject not being for me as a Mexican-American and English learner. Applying it to math I had a similar thing going on where I liked math and even considered being a math major but decided against it because of the fear of failing.
—Miguel (Online Course Reflection)
This extract speaks both to the ways Miguel had been held up in his learning by myths around failure and the effectiveness of the spaces in the online class to reflect. The online reflection spaces seemed important for teachers to do the emotional work necessary to engage in personal sense-making of the information provided.
Many of the teachers in their interviews described feelings of being incapable of learning or teaching math well prior to the intervention, and were surprised by the impact the course had on their own relationship with mathematics:
I thought it was going to be great for the kids, I never expected it to change me, that’s been my greatest revelation in all of it.
The teachers talked about taking problems home and working on them in the evenings:
I have to defend math a lot and why I teach math the way I do because it goes against what parents have learned. It goes against what I have learned. I’ve never done it this way. I tell parents I go home now and I study and I practice and I figure it out. If I’d learned math this way I wouldn’t have cried every night in math going through school.
Another teacher shared that she had been in a network meeting working on a challenging problem when another teacher had looked up the answer on his phone. Rather than look at the answer she chose to keep working on the problem, not only in the meeting but through the next day and into the next week, until she solved it. The teachers started challenging themselves to look at mathematics visually and approach it differently, and struggle through hard problems until they had a solution.
The new relationship with mathematics that many of the teachers seemed to develop, based upon the knowledge they had learned about brain growth, seemed instrumental in their willingness to embrace new teaching approaches. They started to see mathematics and themselves differently, which allowed them to see learners differently.
shows the relationship between the themes we found across the teacher surveys and interviews, as they map onto the five key mathematical mindset practices of our observation protocol. Such a mapping provides information to researchers and network leaders about areas that appeared more salient to teachers. This mapping could also indicate practices that might need more emphasis or support. By mapping themes to key instructional practices, both researchers who design follow-up interventions and network leaders can maintain programmatic coherence.