Common Core Standards: 8.NS.A.2 . Disclaimer: This post includes data from estimating square roots, but focuses on the concept of a square root. More on how to estimate square roots in a future blog post.
I was working with a teacher who had just given a Smarter Balanced Interim Assessment Block on Expressions and Equations. She had not realized that the assessment included estimating square roots as one of the types of expressions. I used the opportunity to explore the idea of student preconceptions & misconceptions and how they could inform first, best instruction.
It was clear that students were making the common mistake of dividing the radicand by two like finding the square root of 16 as 8 rather than 4. We discussed how we might confront this error directly during the initial instruction knowing the importance of students’ first experience with a new concept.
A week later, I performed a professional development event for a middle school site in our district focused on using assessment data to reengage students more deeply while dealing with their documented preconceptions & misconceptions. I used the experience above as an example.
Incidentally, this misconception seems to be related to number sense as it becomes more pronounced as the size of the radicand increases. Here is some actual data to illustrate this point and where we were in our discussion:
You might notice the highlighted spread of incorrect responses: 22%, 33% and 17%, which seems to indicate a gap in student understanding. The teacher’s own report that she had not yet done any instruction in this area supports this fair assumption. Notice what happens to incorrect responses though as the size of the radicand increases:
The incorrect responses are now more unbalanced with 33% choosing C compared to 6% and 11% for the other two distractors. This becomes even more pronounced, 50%, if you take into consideration that distractors C & D are both related to the common misconception of dividing by two, which leads to identifying a misconception or in this case a preconception.
I offered the following as an example of using this information on preconceptions to guide first, best instruction:
I have taught this content before via direct instruction and had some students somehow come away with the same misconception; therefore, I suggested making it a student task. Let me stop to define my terms here. I am using task in this context to denote the shift of responsibility for sense-making from the teacher to the student. You could save time and literally tell students, “Do not divide by two to find a square root;” however, now you are relying on innate procedural memory rather than developing mathematical understanding. Furthermore, the teacher acts as the provider of knowledge rather than the students developing mathematical agency, authority and identity. This leads to two points:
student tasks do not have to be large events & taking the time to do them initially will save you time in the long run by decreasing the need for reteaching later.
Let’s examine the task itself. Rather than starting with the square root of 16 and breaking it down into its factors demonstrating how to get the answer of 4,
what if we provided the mathematics already completed and asked students to make sense of it?
In this case, the square root of 16, 4, is provided along with a different representation of a square with a side length of 4 and an area of 16. By the way, do students even know that square roots are related to actual squares from our instruction? This representation of an actual square has been chosen over the more abstract form of area, (4)(4), while students are just beginning to form an understanding of the concept in order to avoid the error of the presence of two factors indicating that one can divide by 2. It is hard to get an answer of 8 looking at the picture; however, students lack of number sense surrounding multiplicative thinking or misapplication of additive thinking while looking at two 4’s may lead to an answer of 8. Debriefing this task or prompt, lays a conceptual foundation for the rest of the instruction while incorporating the standards for mathematical practice, specifically elements of MP1, MP2, MP3, MP6 & MP7, which also makes it a valuable use of time. While we have laid a conceptual foundation to help avoid the mistake, we have not attacked the misconception head on.
The following images represent my attempt to attack the misconception head on:
Please excuse the incomplete radical symbol above. Research supports that at some point more examples fail to have an effect, but error analysis has been demonstrated to have a profound effect. Not only are we addressing the misconception head on in just the second example presented to students, we have kept the responsibility for sense-making with the student and increased the level of rigor by asking if Kayla’s shortcut works all of the time, some of the time, or none of the time, which is an effect strategy to add to almost any instruction. While the prompt focuses on the shortcut, the discussion should focus on the difference between the correct conceptual understanding as opposed to a shortcut happening to arrive at the same answer. By the way reader, does her shortcut work all of the time, some of the time or none of the time?
Notice also the presence of the actual 4×4 square to activate knowledge just acquired and provide a structure for students to test their theories on. The square root of 4 happens to work on Kayla’s method, because 2+2=4 and (2)(2) = 4. Looking at a drawing of the perfect square of 4, we can see that it is an example of a time where half of the product is equal to its square root or side length:
OK, but how did all of this lead to Sticky Math being born?
As I related the story to Derek Rouch, a fellow math aficionado, I grabbed two stickies to record the two representations. I often grab the nearest scratch paper or whiteboard as I am explaining mathematics to establish a visual connection to what I am saying.
I explained, “Tell me what each of these symbols mean [on the right sticky], using only this picture [on the left sticky].”:
When I finished, Derek exclaimed, “Oh, Sticky Math!” We both stopped in our tracks considering the implications of what he had just said. We both looked at one another with the realization that we were on to something so simple yet so profound.
You see, with Sticky Math, understanding becomes the task at hand rather than a goal I hope to reach through some long winded instruction to which I am often prone. Connections necessary for the retention of information are the task rather than a goal I hope students internalize. When used as a warmup, Sticky Math can be a valuable source of interleaved practice, which has been proven very effective in aiding retention.
We believe using Sticky Math will improve procedural fluency as students learn what the procedure does as opposed to only what the procedure is. As mentioned previously, we also believe it has the chance to build students’ mathematical agency, authority, and identity as they begin to interpret mathematics for themselves.
By often providing the mathematics, students’ working memories are freed up to make meaning and connections. It also acts as formative assessment for teacher and student, providing a source of self-assessment and reflection for students.
Sticky Math provides opportunities for students to participate in mathematical discourse and use content specific, academic vocabulary. It provides students with an opportunity to construct viable arguments and critique the reasoning of others along with many other Standards for Mathematical Practice.
Sticky Math can be a quick easy way to bring conceptual understanding into the classroom as students connect mathematical representations, one of NCTM’s eight Effective Mathematics Teaching Practices. In fact, Sticky Math fulfills elements of all eight of the Effective Mathematics Teaching Practices.
Finally, Sticky Math can be a source of professional development as teachers learn to teach conceptually through multiple representations. It is short, incremental professional development that can be implemented immediately. Likewise, this immediacy applies to results as teachers can assess the understanding of their students and reflect on their own implementation & facilitation of the process.
We hope you join the Sticky Math movement, and maybe even consider becoming a contributor. You can do so by clicking here. We would love to see what you are doing with Sticky Math and hear what connections your students are making. Please feel free to submit a comment or question below.
Common Core Standards: 8.NS.A.2