Inquiry in a Course for Future Teachers

Once upon a time…
In a galaxy far far away…
In a world…

I can never seem to find an opening line that I really like when I write, so I will just jump in and I ask you to please pretend that I started with a very compelling and gripping opening. My goal for this blog post is for you to get a sense of my philosophy and approach to one of the most enjoyable courses I teach; as well as to highlight how an IBL (inquiry-based learning) approach to the course is fundamental for the learning goals that I and my institution have for the course. Notice that you can download some of my explorations below.

I think it’s fair to say that I am a card-carrying member of the IBL club. Here at Westfield State University (where I teach) I get the opportunity to teach a broad variety of courses, from service courses to various departments, to courses we offer for students to satisfy their general education requirements, to both lower and upper-level courses for majors, and inquiry is the foundation for all of them. I can tell that IBL is really the teaching method I’m most naturally suited to because even though I’m pretty good at lecturing (I think), I very rarely use direct instruction in the classroom.

Courses for future elementary school teachers at WSU

The courses I teach most frequently and consistently are the service courses that my department offers for WSU students pursuing an elementary school level education degree. These are the students earning a teaching license in either elementary education (licensed for grades 1-6), early childhood education (licensed for PreK-6)), or special education (we have two such programs, one licensed PreK-8 and the other 5-12). Here at WSU, following the curricular insights of some past faculty with impressive vision for where teacher preparation would go in the future, we have four separate mathematics content courses for these future teachers, each of which is aligned with the primary mathematics content strands prescribed by the Commonwealth of Massachusetts for teacher candidates at these levels. These four courses are Foundations: Number Systems; Foundations: Patterns, Reasoning and Algebra; Foundations: Geometry; and Foundations: Data Analysis and Probability. The course I am going to focus on in this blog post is the second of these, Foundations: Patterns, Reasoning and Algebra. This course is the second that students take in the sequence of these courses, has the first (Foundations: Number Systems) as a prerequisite, and is required of all students in the ELED, ECED, and SPED programs. If you’ll humor me, the most natural abbreviation for this course for me to use is it’s number in the WSU course catalog, so from henceforth I will refer to Foundations: Patterns, Reasoning and Algebra as MATH 250. Here is the course description for MATH 250 :
An introductory course in the foundations of mathematics. Topics include: finding, analyzing, and describing patterns; sets and classification; functions and relations; inductive and deductive reasoning; problem solving; and logic. Students will develop a conceptual understanding of the course material in a learning environment that models the pedagogical foundations of the Massachusetts Curriculum Frameworks for Mathematics and the National Council of Teachers of Mathematics (NCTM) Standards.

I’m including this to give a perspective of not just the types of mathematical content we cover in this course, but also to showcase how the learning environment is so fundamental to the experience we want students to have (in this course and all of the others in the sequence) that we have written into the course description what they should experience. My own interpretation of how to organize the course in this way is having students experiencing the Standards of Mathematical Practice (SMP) as a central element of the course, see Common Core State Standards Initiative

The Standards of Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

There are eight Standards of Mathematical Practice:

  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tool strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

I don’t have a proof that it’s impossible to have a direct instruction course that provides a meaningful opportunity to engage with and experience the SMP, but I will say that such a course would miss valuable opportunities to engage in them during the time that students are in the classroom. A direct instruction classroom in a mathematics course for future teachers would also fly in the face of where the NCTM and other professional organizations have been trying to take the elementary school classroom for a very long time. When I read between the lines of the SMP I see a group-work style IBL classroom where the students are engaging in rich mathematical tasks. (For the flavor of what I mean by a rich mathematical task, see the article Rich Tasks and Contexts.)

As anyone that has taught future elementary level teachers will tell you, many of these students come to college with more than a few negative experiences with mathematics in the K-12 system; and though I want them to have a completely different (and more meaningful) experience with mathematics, by this point in their academic career the direct instruction classroom dynamic, along with a sense that evaluation of mathematical understanding means fluently mimicking procedures, is deeply ingrained into what they perceive as the nature of learning the subject of mathematics. For these students I have not found it to be very meaningful to throw something like the SMP or the NCTM process standards at them, and assume they’ll understand what these standards mean for the classroom they are in and, more importantly, for the classroom they will one day lead. I’ve heard and seen too many very well credentialed career educators talk past each other, and confuse themselves over what these standards mean, thus to ask that of my undergraduates seems a bit unfair. In my courses, to start the conversation about all of these standards and how they provide a blueprint for how teaching and learning mathematics differs from the (still) far too typical K-12 experience, I boil down everything to three non-equalities:

Teaching ≠Telling,

Learning ≠Listening,

Understanding ≠Repeating.

Though these are easy to say, and pretty easy for students to remember, the conversation as to what they mean, both philosophically and practically, goes on all semester long. The first two non-equalities are distillations of the foundation of a meaningful classroom dynamic when it comes to the interaction of the teacher and the students. The third non-equality is a distillation of the foundation of what constitutes meaningful understanding of mathematics, and also what constitutes meaningful assessment of that understanding.

Brian Jennings' Classroom

So, if my teaching doesn’t (or rarely will) involve telling, and the students aren’t spending their time listening to me, then it’s a natural question to wonder what my classroom look like. A typical day in class will involve students in groups of 3 or 4 on the exploration de jour. The explorations we work on (more on this later) tend to be pretty involved, and take anywhere from 2-4 weeks. Variance in the time spent on each exploration comes down to a combination of the explorations themselves (some are just more involved than others), the mathematical background and comfort level of the class as a whole, and explorations tend to take longer early on in the semester when students are still acclimating to the course. Students are distributed in groups more or less randomly, with some alterations to allow for some level of uniformity in ability. (I don’t try to make the groups as strictly uniform as possible, but I don’t want them to be too non-uniform.) During class I will move around the room visiting with each group (typically there are 6-8 groups total) following a few basic rules:

  • Check in with each group at least once during each class.
  • The primary purpose of visiting with a group is to provide them with the minimum possible nudge that will get them productively working
  • Beyond an initial check-in, as long as groups are focused and working productively, and don’t have a question, then don’t interrupt their thinking (this inevitably leads to a lot of eavesdropping on my part)
  • To force groups to work to fully utilize the nudges they get, don’t revisit a group for at least 10 minutes.

Whole-class discussions happen occasionally throughout each exploration, and at the end of an exploration. With enough consistency from me students will fairly quickly come to internalize the "rules of the game" and start to shed behaviors that have served them well in direct instruction classrooms, namely

  • Students realize that I will almost always answer their question with a question, and that they need to answer the question I posed as well as determine how that answer relates to their original question before they can understand the path to resolving their uncertainty or confusion;
  • Students quickly stop asking “Is this right?” (or the large N number of variants of it), and start to ask more meaningful questions;
  • Students come to expect that no matter why I am visiting their group (whether I came over unprompted, or they asked me over) that the first thing I will ask them to do is to explain where they are at, how they got there, and what they have been thinking about (if they want help from me then they need to demonstrate that they’ve been at least trying to help themselves);
  • Students will eventually come to focus on understanding the ideas and concepts behind the specifics of the mathematical tasks they are given, once they see that the types of tasks that they will be given on homework and exams will involve the same concepts, but will have different details.

Throughout the semester we have group discussions about my teaching philosophy and practice, and in those discussion I focus their attention on connections between the ways that our class operates and the SMP. Once the students internalize the connections between my pedagogical choices and the standards that they will be expected to bring into their future classrooms they tend to have much more buy-in to the class as a whole.

Brian Jennings' Explorations (including free download)

The backdrop of the classroom structure and the classroom dynamics I outlined above are the rich mathematical explorations (see some examples below) that we spend class time working through. The kind of mathematical work that will give students the opportunity to meaningfully engage with the SMP are ones that have some real depth to them, involve multiple mathematical structures that can be utilized to answer questions, and will require students to use both inductive and deductive reasoning skills. For the purposes of a class for future elementary teachers, the tasks should also involve mathematical content that’s found in the content standards of PreK-6. To give a flavor of the exploration I use are, consider the examples below:

If you are interested in more of my explorations, please email me at .

Notice that and the end of each of these explorations is a reflection exercise, which asks the students to answer the following five questions:

  1. Put yourself in my position, and consider what you think my goals for you were for this exploration? What did I want you to take away from this experience? 

  2. What struggles did you experience during this exploration that pushed you to grow in your understanding of mathematical content? 

  3. What experiences did you have that helped you to better understand and appreciate the Standard of Mathematical Practice, and more generally the process of doing mathematics? About which standard, or standards, did you learn the most? 

  4. Describe some experiences you had during this exploration with which you were impressed by how well you responded to struggle. (Examples of the sort of experiences I want you to consider could be that you found a creative way to approach a question, or that you recognized you were struggling and you actively worked past/around/through your negative feelings, or that you supported others in their work.) 

  5. Describe some experiences you had during this exploration where you recognize that you did not respond to struggle as well as you could. What lessons can you take away from this experience that will help you to grow into a better student, and ultimately a better educator? 

These reflection questions focus the students on seeing the bigger picture purpose for the mathematics that they have finished, specifically thinking about their experience from both a student and a teacher perspective, and also on thinking about how the SMP have manifested