Four Surprising Takeaways From the MDC Classroom
Why are my students able to answer questions correctly in class but unable to succeed on the assessment?
During SREB training for Mathematics Design Collaborative, Jillian Watson of Bottenfield Middle School in Adamsville, Alabama, immediately saw the potential for her classroom. In her fourth year of teaching math in 2016, she teaches in Jefferson County School District in the Birmingham area.
Here is her story, in her own words.
My first response when asked to participate in the Mathematics Design Collaborative was a skeptical, “Will this truly benefit my students?”
Teachers are asked to be a part of many initiatives, and unfortunately that leads many teachers like myself to approach any new initiative with caution and hesitancy. We come to the table with views about how students learn and a strong desire to do what is best for our students. Anything that takes us away from this desire and commitment conflicts with the very reason we get up early every morning to go to school, stay late to support our students in sporting events and plan engaging lessons on our weekends.
A teacher’s professional learning time is precious and should be spent on high-yield, research-based, proven practices to improve student learning. Though research and data are essential in determining best practices, the most convincing factor in proving to me the effectiveness of MDC was the immediate response I saw in my own students.
Here are four surprising takeaways from my own classroom after implementing MDC practices and formative assessment lessons (FALs):
1 My students didn’t know what I thought they knew
Figuring out what your students know is an issue constantly on teachers’ minds. I am often puzzled by the question: Why are my students able to answer questions correctly in class but unable to succeed on the assessment? MDC uncovered two reasons why teachers might experience this frustration that helped me analyze what often happens in my own classroom:
The Three-Legged Stool
The first of these is the three-legged stool approach. Mathematics instruction must balance procedural skill, conceptual teaching and application. If you are missing or overemphasizing one leg of the stool, your students may not make the mathematical connections. This principle became apparent to me when teaching the concept of slope to my eighth-grade students.
While my students were able to correctly subtract x and y values and divide rise by run to calculate the slope when given two coordinates, I over- emphasized the procedure, and this calculation had no connection to my students’ understanding of rate of change.
Thus, when my assessment contained an application question about finding the price per pizza from two coordinates that represented (number of pizzas, price), many students made procedural errors that gave them negative numbers or values that made no practical sense.
Finding a Balance
To avoid a frustrating grading session at the end of the unit, MDC emphasizes that instruction should be consistently balanced among the procedural, the conceptual and relevant applications of the mathematics being taught. Your students may be able to recite a three-step method for finding slope from a given representation, but if they lack the deep conceptual knowledge of what the slope tells you about a linear relationship and are unable to apply this knowledge in a practical situation, their knowledge is shallow and disconnected from other important mathematical ideas.
2 Questions matter
The second principle that transformed my ability to find out what my students know is effective questioning. Even when asking high-level questions, I am often guilty of trying to draw out a certain response from my students. For example, when engaged in finding slope from tables, graphs and coordinates, I asked my students, “What are the connections among strategies for finding slope in different representations?” When one of my students volunteered, “It’s basically the same thing on a table and graph — you just subtract,” I responded, “You mean the difference between the x-values on a table gives you the run, just like subtracting x1 from x2 gives you the run from a set of points?”
Filling in the gaps
This is a clear picture of what we as teachers are often guilty of in wanting our students to experience success in mathematics — filling in the gaps for them. Did my student truly understand the connection between seeing a progression in x-values on the table and subtracting the x-values from two coordinate points? Possibly.
If I really wanted to know what my student understood, I should
followed up with a simple question such as, “Can you tell me more?” or “What do you mean by ‘you just subtract’?” or “Can you give us an example?”
These responses let the student guide the discussion and truly reveal what he knows and doesn’t know. Though this can be uncomfortable in a class discussion setting, if you have created a safe space for your students to share their thinking and make mistakes, they will be willing to take risks.
Use open-ended questions
Implementing FALs taught me the importance of asking questions
that don’t have a specific desired response. When students work
on a task in pairs, simply start by asking, “Can you tell me
about how you are approaching this problem?” This type of
question helps my students get started on a task when they were
previously unsure how to start or simply unmotivated to
In the past, my approach with groups that have trouble starting was to give suggestions or repeat the directions.
Often, while I moved on to another group, the pair was still not motivated or equipped to begin the task. Having students talk out their approach to a problem often reveals misunderstandings or specific questions students have that guide them toward the real math involved in the task and help me see what understandings they have surrounding the concept at hand.
I have learned so much about my students by asking fewer pointed questions, letting them talk more and listening more carefully.
Be ready for growing pains when you try this questioning approach. Many students will search for a quick answer to satisfy you because of their past experiences with what teachers want from them. The more you push your students with open questions and follow-up questions that ask them to explain their thinking, the richer your class discussions become and your knowledge of your students’ understanding expands.
3 Homogeneous pairing works
One important facet of FALs is homogeneous pairing for the collaborative activity. This means pairing your student who was unable to complete any part of the pre-assessment with another student who was similarly unsuccessful. Often these are your students with learning disabilities or a perceived lack of motivation. As teachers, our solution to a child who struggles academically or behaviorally is often to put them with a student who can help them, or who will be so motivated to complete the task that the assignment will still get done. If the latter is the case (as it often is), has this student really learned? Or, have you frustrated one student and enabled another to continue to be a fringe member of your classroom community?
Despite my reservations at pairing students with a history of academic and behavioral struggles, I gave homogeneous pairing a try during my first FAL assignment and was amazed at the results. One pair that surprised me was Ben and Chris, who both looked at me with disbelief when they saw their names together on the board. They were used to being separated any time a seating chart or partner assignment was given, but they both gave me pre-assessments showing very little understanding of the math concept in the FAL.
Masters of their own learning
Ben and Chris were not able to match all of the cards in the card sort that day, but I was amazed to see them start the task without any extra prompting from me. Since neither of them felt like the other was smarter or more able to complete the task, they were able to engage in it without fear of being wrong or unnecessary to their team. Though they did not get to the deep level of discussion and learning that some other pairs experienced in that FAL, they were engaged in a worthy task and were truly owners of their own learning on that day. As teachers we may feel better when we explain or “cover” a lot of material at the end of class, but we must look at what work our students have actually done to assess whether learning has occurred.
Since experiencing success with student grouping in FALs, I
use homogeneous pairing when pairing students for tasks, and I have seen my students’ confidence to engage in tasks and willingness to participate greatly increase.
While there is a time for “peer tutoring” or heterogeneous grouping, we must think critically before assigning pairs or teams about the goals of the activity and how grouping might affect a students’ ability to learn from and participate in the task.
4 Students actually enjoy the struggle
Seeing what my students are truly able to do with FALs is eye opening. The level of rigor and depth of the mathematics in a FAL pushes your students to a point that will undoubtedly frustrate some of them.
My students’ reaction to the first FAL I taught left me realizing that the depth I pushed students to on a daily basis was not enough, and I was leading them rather than letting them explore and struggle.
This became apparent to me when I did my first FAL and had students unable to move past the instructions because the task was unlike what they had seen in class. I saw then that I had allowed my students to become too comfortable.
Even though I was engaging them in what I thought were rich questions and applications, I was too quick to give them a prescribed method for solving problems, or give them a first step when they didn’t know how to start.
Students take ownership
Though you will get some pushback from students when you respond to their question with another question rather than a suggestion for where to go next, the ownership students take when they accomplish something at the end of the task is well worth the moments of frustration.
In my first FAL, I found out the hard way that helping my students through the collaborative activity meant they were less invested in the task and unequipped to sustain a class discussion at the end of the period.
When students truly struggle and find their own way to a solution
incomplete one), they are agents of their own learning.
The richness of discussions in subsequent FALs that I implemented exemplified this principle. When students struggle, they are excited to share what they learn and are better able to explain their thoughts.
And, they love it. I will never forget the day Jordan, a student
teachers might call disruptive or disengaged, left my class after a FAL and
said to his friend, without sarcasm or an awareness that I was listening,
“That was fun!”
Those are the moments you treasure as a teacher — students actively engaged in mathematics, gaining confidence as learners and mathematicians, and having fun in the process. Through engaging in MDC practices, I have seen and hope to see continued growth in my students’ deep mathematical understandings, as well as growth in myself as a facilitator of this type of learning.
While I only taught three FALs from January 2015 to the end of the school year, at my end-of-course assessment, there was a 13 percent overall growth in eighth-grade math. I credit much of this to the change in classroom practices the last half of the school year. I am convinced that as I embed these changes more deeply into my instructional program and continue to refine my own knowledge and understanding of this more balanced approach to math instruction, gains will be even greater this year. If you would like to know more about my experience with MDC, feel free to contact me at firstname.lastname@example.org.