My son: Daddy, 1/3 is the same as 3/4.
Me: Why is that?
My son: The 3 in the denominator tells you how many quarters there are in the fraction.
Strictly speaking, this is not evidence that my son learned anything about fractions in that class. This is just a single performance. Many people would say that in this performance, my son failed to demonstrate an understanding of fraction notation.
The goal of teaching though is not to generate specific student performances. The goal of teaching is to produce long-term changes in what students know and can do. While we study performances in classes and use these to make short-term decisions about what to with our students, we should also systematically compare these short-term performances with the long-term changes in student performances that then correspond to their learning.
I asked my son a couple of weeks later if he thought 1/3 was equal to 3/4. He told me, “Oh no Daddy. 1/3 is definitely less than a half and 3/4 is definitely more than a half, so they can’t be the same.” As it turns out, the lesson in which he made the first statement was focused on being able to place fractions on a number line in order to be able to compare the relative size of different fractions. So he learned something about towards that objective, even though his performance during the lesson seems like he did not.
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The balance between conceptual understanding, procedural fluency, and application depends on your goals with your students and those goals should depend on what you know about your students. This is why I’m opposed to “Problem Solving Fridays” and “Practice Tuesdays” as these ways of deciding on goals over-simplify teaching to the detriment of student understanding.
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When I first started teaching, I was given a single sheet of paper by the district supervisor for mathematics with a list of twenty questions on it. He told me, “David, If you can get the kids to answer all of these problems by the end of the year, then you are set.” That was the entirety of the direction I was provided as a first year teacher in terms of curriculum and instruction. It took me weeks to find any textbooks in the school and I didn’t really learn what was on the end of year state assessment until I was scoring my students’ work on it. No first-year teacher should be provided so little direction and support.
In my second year of teaching, I was given a textbook and a pacing guide and told that my job was to “cover the curriculum” and make sure that my students got the same experience as other students in other classrooms. I was told that if my supervisor came and observed me and I was not on pace, then I would be automatically given a poor evaluation.
Neither of these approaches to curriculum support works. Either teachers are expected to each individually recreate the wheel or they are treated as completely incapable of making curriculum choices. Teachers must both have sufficient guidance on what to teach and simultaneously the autonomy to adapt and extend curriculum resources to meet their students’ needs.
There is a mindset across many in the teaching profession that teachers should be both be designers and implementers of curriculum. However as Robert Pondiscio notes in his article, “How We Make Teaching Too Hard for Mere Mortals” (2), this leads to a lot of teachers using Google or Pinterest as a primary lesson planning resource which results in an incoherent experience for children. It also is more likely to lead to lessons where the task hasn’t been fully thought through and so any classroom discussions and potential learning opportunities that are embedded within the lesson are more likely to fall flat.
Designing curriculum to support learning is a surprisingly challenging and time-consuming task. Suppose for example that you are designing curriculum for a unit on geometry at the elementary school level. It is highly likely that your curriculum will need to include, for example, pictures of triangles like these ones.
There are actually two issues with not being careful about how you use geometric representations with students. If every one of the triangles you draw has one side of the triangle parallel to the bottom of the piece of paper they are drawn on, then students may think that this is a property of triangles. If every triangle has somewhat similar length sides like the ones above then students may think that triangles that look longer and skinnier are not triangles. My own four year old son refers to long skinny triangles as lines, since these triangles have more in common, in his experience, with lines than with the shapes above he happily calls triangles.
Further curriculum is more useful when one lesson deliberately builds on the prior lessons. Teachers often focus on the day to day job of having things to do with students and rarely have the time to deliberately use what they have already done with students and build on their shared experiences. Some mathematical ideas are too big to contain within one lesson, and so planning day-to-day leads to an incoherent experience for children (3).
Curriculum has the potential to offer so much more than just resources for teachers to use in lessons. It can be a tool where they continue the learning about the content they teach that their education schools rarely have time to complete. Across the United States, many teachers now teach out of the content area for which they were trained, and so embedding opportunities to learn this content for teachers is especially important.
When the teachers’ manual, which usually contains somewhat repetitive suggestions for teaching, is only a click away from the resource a teacher is accessing, then learning about pedagogy can more easily be embedded within the curriculum. Deborah Ball is leading a project (4) to determine the highest leverage content (5) and pedagogy to be taught before teachers start teaching. What if curriculum authors were able to carry on this work with in-service teachers? What if the curriculum deliberately and explicitly offered multiple pathways through the curriculum?
One of the major areas I knew almost nothing about when I finished my degree in education was the ways students typically understand and misunderstand mathematics (6). In order to be best positioned to remediate common errors and misconceptions students make while learning mathematics, I need to know both strategies for this remediation and to know the ideas to be remediated. When you look across just the high school mathematics curriculum, the number of different ways students typically understand each mathematical idea varies greatly, and so the sheer amount of information about student understanding that a teacher would need to know is overwhelming. No one has successfully captured all of these ways of knowing yet, but some efforts exist, at least at the elementary school level (7). An ambitious curriculum would keep track of at least some of these common ways of understanding mathematics so that teachers who are planning units can reference these resources on demand.
Further when I learned more about actively supporting students in developing productive dispositions towards the mathematics I taught, I learned that framing student ideas as either ‘knowing the math’ or as a misconception isn’t all that helpful. I now think of students as making sense of the world around them and try to figure out how their ideas are logical, given what the children know, rather than try to find the mistakes children make (8). What if the curriculum teachers accessed explicitly offered suggestions on how to improve students’ self-conceptions of themselves as mathematicians?
Teaching in response to what children know or do while working on authentic mathematics tasks (9) is difficult yet most curriculum resources offer almost no support for teachers in doing this. The most common understanding of formative assessment is that it is a type of assessment teachers give in order to determine what children know, but as Dylan Wiliam’s book Embedded Formative Assessment (10) shows, this is a limited definition and perhaps not helpful definition of formative assessment. Given the importance of formative assessment in teaching, strategies for formative assessment, like the 5 Practices for Orchestrating Productive Mathematics Discussions (11), could be embedded right into the planning materials for the curriculum.
Another major area of my own learning was about the use of mathematical representations to support student learning. I knew mathematics and I knew many mathematical representations of that mathematics, but I had not deliberately been exposed to mathematical representations, such as tape diagrams or area models, during my teacher training. An excellent curriculum would support teachers in making explicit connections between different representations and as a result, being better prepared to select mathematical representations to use with their students. Mathematical representations are an excellent tool for students (and their teachers) to make connections between different mathematical ideas and as such should be forefronted in curricular resources. Further, a cohesive curriculum should be written such that the representations that start their use early in a child’s school career make it easier for the same child to make mathematical connections in their future mathematics classes.
When we can provide curriculum electronically, the resources that can be included for any given lesson or activity are limitless, yet we still design most curriculum with the limitations of print. A truly ambitious curriculum would have typical student approaches, useful mathematical representations, suggestions for pedagogy, specific tasks to support a mathematical idea, and other resources all within a click. Why not supply slides and templates for teachers to use so that teachers can focus on other more important aspects of their craft, such as anticipating student thinking for the upcoming lesson?
And it goes without saying that the strategic use of technology should be embedded within the curriculum with the caveat that given the current state of technology across the United States, that curricular resources should work in a variety of contexts. Some schools have one computer per classroom which might be hooked up to a projector, other schools have devices for each child; an ideal curriculum supports all likely arrangements of technology.
In Todd Rose’s The End of Average (12) he recounts a story about the design of the cockpit of an airplane, which I believe should inform curriculum writing:
…
After multiple inquiries ended with no answers, officials turned their attention to the design of the cockpit itself. Back in 1926, when the army was designing its first-ever cockpit, engineers had measured the physical dimensions of hundreds of male pilots (the possibility of female pilots was never a serious consideration), and used this data to standardize the dimensions of the cockpit. For the next three decades, the size and shape of the seat, the distance to the pedals and stick, the height of the windshield, even the shape of the flight helmets were all built to conform to the average dimensions of a 1926 pilot.
…
Out of 4,063 pilots, not a single airman fit within the average range on all 10 dimensions. One pilot might have a longer-than-average arm length, but a shorter-than-average leg length. Another pilot might have a big chest but small hips. Even more astonishing, Daniels discovered that if you picked out just three of the ten dimensions of size — say, neck circumference, thigh circumference and wrist circumference — less than 3.5 per cent of pilots would be average sized on all three dimensions. Daniels’s findings were clear and incontrovertible. There was no such thing as an average pilot. If you’ve designed a cockpit to fit the average pilot, you’ve actually designed it to fit no one.
What about curriculum? We all know that children enter classrooms in many different shapes and sizes, and that their understanding of the content we intend to teach is as varied, and yet we design curricular resources that mostly aim to support an average child. What if there is no such average child? The curriculum that aims to best support the average child may in fact support no one best.
It is well-known, for example, that students need multiple opportunities to both learn a mathematical idea and to access their memory of the idea in order to strengthen their memories (13). Almost no curricula deliberately interleave practice or offer opportunities for spaced retrieval practice. What if a curriculum deliberately included ideas from cognitive science into its construction?
There are also strategies for supporting students with special needs and emergent bilingual students which offer additional avenues for students to access the same mathematical content as their peers and which when implemented effectively, support all students. Many curricula offer no deliberate pedagogical suggestions for different populations of students, but in New York City alone there are more than 170,000 students with special needs and over 140,000 (14) students who are learning English as a second language. Given that access to education is a fundamental human right (15), our curriculum resources should offer the greatest possibility that all students have access rather than being a potential limiting factor.
One suggestion for this curriculum is to embed the use of instructional routines (16) throughout. In our work (17) on curriculum we have discovered that instructional routines offer support for teachers and students in multiple ways. Since they routinize the “steps” to a lesson in predictable ways, they allow teachers to focus on the parts of the lesson that change in response to the students and the mathematical ideas presented within the routine structure. Similarly, curriculum developers are able to develop tasks for instructional routines more rapidly and with more confidence that they will be used in the way intended when the possible ways the task will be enacted with children are more narrowly defined. Since students and teachers know what to expect next when the routine unfolds, they can more completely focus on each other and their mathematical reasoning. These routines also allow teachers to learn from their enactment as they “temporarily hold some [parts of their teaching] constant while working on others.” (18) Finally, well designed routines embed formative assessment into the structure of the routine (19) so that the challenging work of responsive teaching becomes more manageable to learn.
Finally, the license and formatting of curriculum resources should encourage both thoughtful revisions to the curriculum and the sharing of those revisions back to the greater community. In the United States, we have a million mathematics teachers, each basically writing lessons and altering resources on their own to support their students’ understanding, and no shared pool of resources from which to iterate. While online communities (20) have sprung up around different instructional routines, there is no policy in place nor organization that has stepped up to help organize those routines and their associated resources into a cohesive collection that can be built upon. Further, access to curricular resources is highly uneven across the many thousands of school districts across the United States, and a more open access model may help mitigate education inequities.
It is not likely that such an ambitious curriculum will emerge on its own. It will require the combined efforts of many hundreds of teachers and curriculum specialists working over a decade or two to design, test, and iterate to create a coherent collection of resources. Educational researchers should be included from the beginning of the curriculum construction so that they can rigorously test the impact of the curriculum on student learning. Professional development providers should be included so that the professional learning experiences of teachers who volunteer to be part of this effort can be aligned to the aims and beliefs about learning embedded within the curriculum structures.
Given that funding for education appears to be already stretched, this additional effort may need to be funded by outside partners. However the potential for such a project to have a long-lasting impact on student learning is great. We already know that curriculum impacts student learning (21); what might be the impact of an ambitious curriculum?
References:
Ball, D., Hill, H., and Bass, H. (2006), Knowing Mathematics for Teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), p. 14-17, 20-22, 43-46., Retrieved from https://deepblue.lib.umich.edu/bitstream/handle/2027.42/65072/Ball_F05.pdf on September 6th, 2016
Across the United States, there is a continued focus on the use of formative assessment to improve the conditions for students’ learning. One common theme is that teachers want more support in implementing formative assessment strategies in their classroom and that they want the use of formative assessment to be instructionally efficient.
What is formative assessment?
While formative assessment is defined very differently in different places, the framework I’ll assume for this article is the one proposed by Dylan Wiliam (Embedded Formative Assessment, 2011) wherein formative assessment is built up of five key strategies:
1. Clarifying, sharing, and understanding learning intentions and criteria for success,
2. Engineering effective classroom discussions, activities, and learning tasks that elicit evidence of learning,
3. Providing feedback that moves learning forward,
4. Activating learners as instructional resources for one another, and
5. Activating learners as owners of their own learning.
Not only are there different definitions for formative assessment there are also different time-spans over which eliciting evidence of student achievement, a key aspect of formative assessment operates: over the course of a unit, day to day, or in the moment.
One very typical model of formative assessment involves school districts focusing on common unit assessments and common data tools in order to focus teachers’ attention on long-range planning in response to student performance data. Another shorter range focus is on modifying day to day lessons based on exit tickets and quizzes. However where formative assessment often falls short in terms of implementation is in the moment to moment decisions made by students and teachers in the classroom.
Further, much of what passes for formative assessment focuses on students making mistakes and teachers responding to those mistakes with corrective feedback. In many cases, choosing the feedback to give to students is time-consuming and actually ineffective. I propose that it is less important to focus on the mistakes that students make and more important to focus everyone’s attention, students included, on the reasoning that students do.
Instructional Routines as an answer
What would a routine that offers the rich features of formative assessment but allows teachers to be responsive in the moment look like? Can we design a routine which does not take an enormous amount of time for teachers to plan given the real time constraints faced by teachers? Can the instructional routine focus teachers and students more on understanding student reasoning rather than just identifying mistakes?
In their 2009 article entitled “Instructional Activities as a Tool for Teachers’ and Teacher Educators’ Learning,” Magdalene Lampert and Filippo Graziani introduced the idea of instructional routines for ambitious teaching. They define an instructional routine as “designs for interaction that organize classroom instruction”. These instructional routines could become the standard operating procedures that every other profession, aside from teaching, has developed to standardize and professionalize their work. I offer that these instructional routines are also the rich formative assessment-embedded activities that we have all been waiting for.
The greatest benefit of these instructional routines is their routine-ness. Instead of having to invent a different lesson plan or activity for every different mathematical idea, a teacher can select an instructional routine appropriate to the mathematics at task. Rather than having to re-invent teaching each day, planning teaching becomes more about choosing mathematical goals, selecting tasks to support those goals, and then embedding these tasks within an activity.
But why are these instructional routines useful for formative assessment? Let’s look at one instructional routine, Contemplate then Calculate, to see how it aligns with Dylan Wiliam’s definition of formative assessment introduced earlier. (Here are other examples of instructional routines)
Here are some key elements that teachers need to consider when implementing instructional routines in their classrooms and their relationship to formative assessment.
Lesson preparation
In enacting this kind of instructional routine, a teacher needs to select a mathematical problem, ideally one rich enough for students to have something to talk about and one that will focus students on a particular mathematical goal. Once a task is selected, in order to make the best use of the instructional time with their students, this teacher anticipates the different approaches her students might make.
For example the task selected might ask students to identify the number of white squares in the following image without counting all of them. In this case, the goal is to support students using mathematical structure (Common Core Math Practice 7) to identify ways of chunking the figure in order to make it easier to work with and to allow students to connect what they know about multiplication to counting squares. (Here are more sample tasks, each of which could be the mathematics that is inserted into the instructional routine).
Before giving the task to students, it is useful to anticipate strategies students might use. For this task, one potential strategy is to chunk the diagram and notice that there are four identical triangle-like shapes. One can then zoom into one of these triangles to see that it is formed of four rows, each of which is two squares longer than the row above it, so since we start with two in the first row, the triangle-like shape has 2 + 4 + 6 + 8 = 20 total white squares for one triangle-like shape. Since there are four of these shapes, there are 20 × 4 = 80 total small white squares.
By identifying potential strategies in advance, the teacher is better equipped to circulate around the classroom during the portion of an instructional routine where students are working together and listen to student conversations and observe them work. During the portion of an instructional activity where students are presenting their ideas, anticipating strategies makes it easier for the teacher to focus students on a particular mathematical goal and make the mathematics evident in the solution explicit for everyone through careful annotation.
Here is an example of three annotations and three descriptions of student strategies. Which annotation goes with which strategy? How do you know?
Description 1
“I chunked the shape into four triangle-like shapes and counted the number of squares in each row, and then multiplied this by four.”
Description 2
“I noticed five groups of four grey squares, which is 20 squares and I subtracted that from the big 10 by10 square to get the remaining white squares.”
Description 3
“I noticed that each row has 2 grey squares, so there are 20 grey squares because there are 10 rows. I then subtracted these from the big 10 by 10 square.”
By deciding in advance on various ways to annotate the task when students describe their strategies, teachers can shift their cognitive load while teaching from remembering what they want to say next to really listening to what students are saying and support other students in the class in doing the same.
The Parts of One Instructional Routine: Contemplate then Calculate
Lesson launch
This instructional routine starts with a launch designed to make it more likely that students understand both the purpose of the activity and their role within the activity. This script allows students to orient toward the goal themselves and aligns most directly with ‘clarifying the learning intentions’ formative assessment strategy. The meta-reflection activity at the end of the instructional activities helps students see progress towards those goals themselves as well.
Orientation to the task
Students are then given an independent opportunity to notice key features of a problem, followed by time with a partner to share what each noticed and briefly discuss how that could be mathematically useful. During this time, the their teacher circulates around the room and gathers information on what students are thinking about. Students are then gathered together for a whole classroom sharing of the mathematical structures, patterns, and/or quantities noticed. The teacher records these noticings so that they become the shared knowledge of all the students in the classroom. Because the cognitive demand of “notice” is relatively low, all students can participate in this orientation; because the noticings are shared, all students develop ownership of the task–long before any struggles would prevent their investment.
Partner work
Next, students use the gathered information to build and share a solution strategy while teachers have amply opportunity to circulate around the classroom and listen to what students think and use this as evidence of what they understand and more importantly, how they understand it. Since each instructional activity includes preparation work ahead of time where teachers anticipate the kind of thinking they expect, teachers are now better able to contrast their expectations of student thinking against the actual live student thinking they encounter.
By working within a structured routine, students also have less extraneous information to pay attention to in the mathematics classroom and are expending less of their total cognitive load on remembering what’s happening next and what their role is now; therefore, they are better equipped to pay attention to each other and to the mathematical ideas surfacing in the classroom. When students are able to pay closer attention to how each shared mathematical idea works, students get better feedback about their own ideas. Both of these observations are also true for teachers.
Sharing and studying strategies
Now the teacher orchestrates a whole class discussion around the mathematical focus for the day. The objective here is to use the thinking students have done and explicitly make this thinking public for everyone to understand by probing students for more detail, asking students to listen to and restate each other’s ideas . As a group, the role of students is to work as mathematicians do, by constructing and assessing arguments for the validity of their mathematical approaches and looking for connections between the various ways the class solved the problem for today.
During the partner work and sharing out portions of the activity, students are oriented not to the teacher’s talk but to each other’s speaking and thinking, which activates them as resources for each other. When the strategies conjectured by students are unpacked and made explicit for students through gesturing, annotation, and restatement, students can make clear sense of what mathematics is being discussed and why it works (or in some cases, why it does not work).
Meta-reflection
Since the goal of mathematics classes is not to learn how to solve individual problems, but to use problems to learn mathematical ideas, many of the instructional activities include a meta-reflection portion of the activity. Here students respond to a prompt or write about what they learned during the lesson. Ideally students should be able to describe what they learned in relation to the goal articulated at the beginning of the lesson.
Conclusion
“Instructional routines support teachers in studying the impact of teaching practice on student learning by focusing attention on the outcomes of a small, well defined, common set of practices that are repeated for a given period of time.” (Deeper Learning, Magdalene Lampert, 2015) The goal of these activities is to allow teachers to better study their own impact and to allow students to better see their progress in learning mathematics over time–to see themselves as doers of mathematics. Simultaneously, the same features that make instructional routines useful tools for developing teacher practice support teachers in integrating formative assessment practices as part of their teaching.
Formative assessment is not an exit ticket at the end of a lesson or a quiz every Friday. It is a framework for teaching and as such an instructional routine which aligns well to the framework is more likely to be useful for teaching than any attempt to layer formative assessment strategies on top of an existing set of routines.
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What they found was startling, especially for younger kids. Students who had been randomly selected as being likely to show potential based on an imaginary test and then this information communicated to their teachers, showed tremendous gains in the intellectual ability compared to the control group.
The reverse effect, called the Golem Effect, has also found to have an impact through randomized-control-sample experimental design. This means that one factor into student performance is the expectations teachers have, either high or low, on that student performance.
Every time you refer to your students as “low kids” you are re-enforcing expectations of future achievement that limit children. #NCSM16
— David Wees (@davidwees) April 12, 2016
This is why I get incredibly frustrated when teachers use labels like “high” or “low” to describe their students. Invariably these labels are like self-fulfilling prophesies that lead to strengthening or weakening student beliefs about their own ability to be successful and lead to teachers, subconsciously, taking actions that lead to either higher or lower student performance, and as performance is related to student learning, this can be devastating.
Of course, this pernicious labeling of students is built right into our systems of education so we can hardly consider individual teachers to be at fault. We have the ELL students, the SpEd students, the “lower third”, the students who only achieved a 1 or a 2 on the NY state exam, the students who failed algebra last semester, and other unhelpful labels for students ad nauseam.
So what do we do? I propose a good starting place is to eliminate unhelpful labels from our own vocabulary. Another step is to talk to our colleagues about our shared use of labels for students. Find ways to look for and talk about strengths in students rather than perceived deficits.
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Empower math communities to improve the teaching and learning of math through the use of dynamic tools in a Web 2.0 world
Last night members of CLIME and other interested people attended a meeting of CLIME to discuss the its future.
In order to understand the role of CLIME in promoting the use of technology in math education, one has to understand a bit of the history, so Ihor Charischak (the long-time President of CLIME) started us off with a brief recap.
We then discussed some ideas for how we could better support the meaningful and productive use of technology through the NCTM annual meeting. Note that for this meeting our focus was on improving the NCTM conferences rather than all of the other ways we can support technology use. We brainstormed the following list of ideas.
1. We could find people doing interesting work with technology and invite them to submit proposals on that use.
2. We could set up an area in the exhibit hall and run mini-technology based sessions where educators could come to learn about how to use dynamic geometry software, learn how to get started with blogging, how to set up a Twitter account, etc… One benefit of this arrangement is that we could offer to help people install software (or find and bookmark websites) so that people who wanted to run workshops on the same technology later would be more likely to have a group of attendees with the software already ready to go.
3. We could suggest the labeling of sessions on technology as beginner versus advanced so that people who need help installing software, finding the menus in that software, and getting started with their initial exploration of the technology can have support and that people who are already experts in the use of technology can share ideas back and forth.
4. We offered that the program NCTM has started where presenters add additional information about their sessions and invite participants to comment on and ask questions about sessions could be extended. This way the 50 words or so presenters have to describe their work could be increased without dramatically changing the experience of conference organizers (who have to read all of those descriptions and make decisions about who gets to present at the conference).
5. We could continue to review the existing program after it is published and offer feedback to the NCTM program organizers to use with the next conference.
6. We could run our own technology in math education conference. We noted the importance of a face to face conference for encouraging networking between math educators but we still considered a hybrid or entirely online conference as well.
7. We wondered about ways we could encourage the younger generation of math teachers to participate in NCTM’s conference.
8. We could form a technology study group with the aim of cataloging and reviewing different technologies in use in math education and then potentially presenting our findings at an NCTM conference.
If you were tasked with promoting the meaningful use of technology in math education through a conference experience, what else would you do?
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Every time you or your students make logical leaps when explaining mathematical ideas, your students must fill those logical leaps with what they understand about mathematics or invent their own logic to fill the gaps.
Consider this task:
Make the task explicit
This task as currently written is actually ambiguous. There are a lot of vertical and horizontal line segments in this picture; are they meant to be included, or not? Are we only supposed to focus on the bold line segments?
We could change the prompt to something like “Which of the bold line segments is steepest?” This of course assumes that students understand what a line segment is and interpret bold to mean the same thing their teacher means. (It’s fine if students don’t completely understand what steepest means though so since the goal of a task like this is to come to a common definition of steepest.)
Use Gestures
Another approach is when asking students to solve the task, having it projected on a screen, and using one’s hand to trace and emphasize the line segments in the image, while asking the question.
Push for clarity
Now consider these (simulated) student strategies for solving this task and imagine students are describing their strategies out loud to share with the class.
If you have taught students how to interpret lines or line segments on a graph (or remember the mathematics associated with the task from when you learned it), you can probably figure out what these strategies mean. But there are gaps or missing steps in each explanation and since the explanations are out loud, there are ambiguities in each explanations as well.
With respect to the first strategy presented, what does it look like to extend all of these line segments so that they are the same length? How does the first student know that just because the line segments are now the same length, that one of them is steeper than another? And which line segment did they actually find to be steeper anyway?
In the second strategy, where are those little triangles drawn? Are they connected to the line segments in some way? And even if they are, one of the line segments is horizontal; how do I draw a triangle under that? Why does the largest rise over run correspond to the steepest line? Is that always true?
In the third strategy, where exactly are the angles between the line segments and the x-line? And what is an x-line? And why does the largest angle correspond to the steepest line segment?
Use questioning
One strategy is to ask clarifying questions about the strategy or to prompt students to ask clarifying questions of each other. In order to be able to ask critical questions “on the fly” it is extremely helpful to have anticipated the approaches students will use and at least some of the possible leaps in reasoning they may make, so that you can prepare questions in advance.
Use annotation
If you or students talk about mathematical ideas with no public written record of what was discussed, chances are high some students will either not be able to follow the argument being made or will quickly forget the argument. You can, and should, keep this record for students during discussions and use color and symbols to make the connections between mathematical ideas clear.
Here are some examples of annotations related to the student strategies above. Do these annotations make the ideas being discussed more clear? Is it more obvious why these strategies work?
Keep a public record
Here is a record of what participants noticed and what their meta-reflections were when I used this task with them.
Having this public record means that if a student’s attention wanders, they can get back into the flow of the class. It also means the information you want students to take away with them remains up for as long as possible. Further, when you move to prompting students to consider why the mathematical strategy works, students’ cognitive load around what the actual strategy being considered is decreased while there is a public record that they can access.
Prompt students to consider each others’ ideas
Unfortunately, usually when people (our students included) listen to each other they listen for what they expect to hear rather than what was actually said.
While it is worth saying that you should actively listen to what students say rather than changing its meaning or filling in the gaps in logic yourself, you will also want to use talk moves (like revoicing, restating, asking a student to restate, asking questions, wait time, etc…) to push your students to actively listen to each other as well.
Use independent think time
When we run professional development sessions, we virtually always incorporate a section on doing the math as this increases the odds that our teachers are able to have meaningful discussions about how to teach the mathematics. If students have already thought about a problem themselves, they will find it easier to understand someone else’s approach more easily.
What are other ways we can make the mathematics in our students’ strategies explicit while simultaneously respecting the thinking that students put into these strategies?
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Instructional [routines] are tasks enacted in classrooms that structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.
Here is an example of a task intended to be used in an instructional routine called Contemplate then Calculate.
The high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem. A critical aspect of this and other instructional routines is that they embody a routine in which one makes teaching decisions, rather than scripting out all of the work a teacher is to do with her students.
This year we noticed a number of benefits to using instructional routines that lead us to plan continuing using them next year as well.
1. Instructional routines allow teachers to communicate about classroom practice with each other using a common language and common understanding of what kinds of instructional strategies are being implemented.
Usually conversations about classroom practice are extremely difficult because each teacher’s context is so different and because teachers visit each other so infrequently. My experience suggests that these conversations often devolve from talking about specific decisions that were made and the rationales behind those decisions and into discussions about mathematical topics and what order they should be taught. With a common instructional routine, teachers’ conversations can shift to a more granular level of discussion since so much more of the context can be assumed.
2. Instructional routines can support teachers and students in having access to high cognitive demand tasks by reducing the cognitive demand needed to attend to “what am I doing next”. Since the activity is routine and well-defined, the steps to doing the activity can fade into the background over time for both teachers and students. This allows more of the cognitive load for teachers and students to be potentially focused on making sense of each others’ reasoning and the mathematics of the task at hand.
Teachers already have many routines they use in their classrooms but those routines may or may not be used by other teachers (see point #1) or they may have too many different routines that they enact for each type of mathematical task they use. In order for the cognitive benefits of an instructional routine to occur, the instructional routines must in fact become routine.
3. Instructional routines have allowed our curriculum team to rapidly develop mathematical tasks to fit into these instructional routines because we don’t need to communicate the routine separately for every task. The routine stays the same (but see point #4) over time while different tasks are enacted within the routine.
4. Since an instructional routine keeps much of the classroom interaction the same, it becomes possible for individual teachers or groups of teachers to iterate on their practice more rapidly. If every day a teacher has to re-invent her practice, then it becomes more difficult to figure out what teaching strategies work in her context, when those teaching strategies work, and why she might choose a different teaching strategy.
I remember my first year teaching. I was unprepared. I didn’t know how to structure lessons. Each day I was floundering. I kept experimenting and trying different activities, different ways of communicating with students, etc… I would have benefited from more support in planning lessons.Note that this benefit supports newer and experienced teachers differently. A new teacher needs a starting place to iterate on their practice from. An experienced teacher who wants to refine her practice needs a tool with which to do so.
5. In professional development settings, we and teachers in our project can model teaching strategies more easily (this is really a combination of point #1 and #4). Since the routine is well-established, when someone does something different within the routine when modeling it with a group of teachers, it becomes easier to focus on the something different.
For example, we used the routine Contemplate then Calculate to model instructional moves intended to facilitate student discussions. We then, as a group, unpacked just that aspect of the routine. This was enabled because instead of everyone participating having to keep all of the teaching occurring in one’s head at one time, the routine aspects of the teaching could be ignored to focus on the non-routine aspects for that day.
6. The instructional routines all include built in opportunities for formative assessment and responsive teaching. We struggled to find ways to staple formative assessment practices on top of existing teaching and mostly failed. Instead the different aspects of formative assessment (as described by Dylan Wiliam) are embedded within an instructional routines, which as it turns out, makes them easier to learn how to use.
There are other more mathematical benefits of these instructional routines, but those depend largely on the specific routine being used.
We intend to continue supporting the two instructional routines we used this year (Contemplate then Calculate and Connecting Representations) and to add one or two more routines to support different mathematical goals teachers may have, because we have seen each of the benefits of these routines listed above play out in various ways across our project.
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On the day before I first started teaching, the district coordinator came to me and handed me a piece of paper with twenty questions on it. “Here’s what you have to teach, David. If your students can answer all twenty of these questions by the end of the year, you will be fine.”
Needless to say that this was insufficient curriculum. But what kind of curriculum would have been useful for me at that stage in my career?
A place many math educators, especially in our accountability-driven system, seem to start when teaching mathematics is teaching children how to solve questions. “Here’s how you solve this kind of question.” The connections between different problems are left for students to discover on their own.
The advantage to this system is that you can look at the state assessment and check off of all of the question types and feel like you have done your job. For the old NY state exam, this approach works in the sense that students were able to sit down at the exam and feel like they had been prepared for every question since the assessment was so predictable, even if they didn’t always know how to solve the problems .
The problem with this approach is that students now have to remember each question archetype and each solution to each type of problem separately, leading to a relatively unorganized and over-whelming set of problem-solving schema for students. This leads to students forgetting how to solve individual problems, forgetting which solution strategy they should use when, or misapplying strategies to solve problems for which the strategy is not appropriate. Even when students do master all of the problem types, knowing how to solve problem x, y, or z doesn’t help students make connections when they start studying further areas of mathematics.
To their credit, every textbook author I’ve ever read takes a different approach (to some degree). Textbooks start by dividing the year’s worth of mathematics to be learned into units of study and apportioning mathematical principles into those units. Within each unit of study, specific problems are used to illuminate mathematical ideas and ideally students at the end of a unit can articulate the mathematical ideas they have learned, rather than just the problems to which they apply. Where textbooks often fall down is in making the connections between units and ideas explicit.
When well done, problems become vehicles for teaching mathematical principles. Mathematical representations (like graphs, tables, etc…) are embedded across the units so that students get multiple exposures to these critical representations and can use them to make sense of similarities and differences between different mathematical ideas. Ideally students explicitly learn connections between different mathematical ideas so that they see, for example, how solving a linear equation is related to solving a quadratic equation and how the graph of an absolute function is related to the graph of a linear function.
One way to support students in making these connections is to ask them to answer reflection questions like: How is what we learned today related to what we learned yesterday? How is this problem we solved today like the problems we solved last week? What did we learn today that we can probably generalize and use to solve other problems? At the very least while planning, make sure you can answer these questions yourself.
A major disadvantage of the second approach to teaching mathematics is that it takes much more work to organize curricular resources into the general themes and to make the connections between ideas explicit. Rare is the classroom teacher who has time to do this all herself. This is one of the reasons why I think well-organized curriculum of some kind is always going to be a helpful resource.
For further reading on a related topic, check out this post on instrumental versus relational knowledge.
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At some point I decided that being very economical with my planning was the mark of good teaching and so my lesson “plans” ended up being really short. “Teach Pythagoras” was an actual lesson plan I wrote. Of course, “Teach Pythagoras” is not a lesson plan. It’s not even really a topic. It’s a short-hand for pick some examples, tell kids how to solve those examples in an ad hoc fashion, followed by make up some examples for kids to try and solve themselves.
When I started using longer-term projects, that meant planning lessons got even easier as I could assign a time-line to the completion of the project and in each day I would support students either with some examples for the whole class to move their work in the project forward or by circulating through the class to help students out.
Up until this stage, any student mathematical discussions that occurred were ad hoc and almost always initiated by students. Not once in my classroom teaching experiences did I plan for student discussion.
About three years ago, I started a new job as a formative assessment specialist. It was then that I first read Peg Smith’s Thinking Through a Lesson protocol. As I read the article, I realized right then that I had spent most of my career planning poorly.
Next I read Dan Willingham’s book on Why Don’t Students Like School? and realized that I had spent most of my career planning tasks for students to do and not planning the thinking I wanted students to do. Tasks prompt thinking but what thinking? Here’s an example of a lesson that could have been one my early lessons. What are children being asked to think about?
Fortunately, I had an opportunity to test out these new ideas around planning for myself. I started teaching my son and between 6 and 8 other kids close to his age in a Saturday class. I decided to plan the student thinking, to try out the Thinking Through a Lesson Protocol, and most importantly, to ask other people to comment on my plans. Here’s my first lesson plan for this class. It’s by no means perfect but it has far more detail on what I will do in response to what I expect students to do, and the first example I can recall of a lesson where I explicitly planned a whole group discussion.
But who has time to plan lessons at this level of detail five days a week, potentially 4 or 5 times each day? The level of planning linked above is unsustainable for classroom teachers.
Last summer we started introducing instructional activities to teachers across our project. Instructional activities, as defined by Magdalene Lampert and Filippo Graziani, are “designs for interaction that organize classroom instruction”. Essentially they define a set of moves a teacher makes to position students to talk to each other about mathematical ideas, surface student thinking about those mathematical ideas, and then orchestrate a classroom discussion around the ideas in order to focus students on a mathematical goal.
These instructional activities have the advantage of bounding the scope of decisions a teacher can potentially make when planning a lesson while focusing the decisions that are made ahead of the lesson on planning for the thinking students will do, and then in the lesson enactment, allowing teachers and students the space to think about and respond to each others’ thinking. The routineness of an instructional activity, if the same structure is used many times, allows thinking about roles, what’s coming up next, to fade into the background so that more thinking can be focused on the mathematical ideas.
When planning one of these instructional activities, I find myself choosing an appropriate task based on some understanding of anticipated student thinking, then imagining how students might approach the task and what they will think about, then considering how to sequence the different strategies student might use toward a big mathematical idea, and then creating the resources to enable me to use the instructional activity in the classroom. This level of planning is sustainable.
As I reflect on my own development of planning over the course of my career, it seems to me that I would have benefited from knowing about planning routines that other people use. I would have benefited from learning at least a few instructional activities so that I didn’t need to plan every aspect of my teaching. I would have benefited from access to tasks where student thinking was anticipated for me. I don’t think a highly scripted curriculum would have developed me as a professional (but maybe my students would have benefited). I would have benefited from seeing how other people sequenced mathematical ideas.
In short, I would have benefited from more explicit teaching of how to plan lessons.
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