The article is the second of a two-part series looking at the challenges faced by teachers and parents in teaching kids math in the Common Core era. Dr. Mahovsky spoke with several teachers and observed the actual teaching of math in a K-12 school in her exploration of this topic. The articles highlights the challenges teachers currently face teaching math while offering insights into how we might do it more effectively in school and at home.
Part II: The Challenge of Teaching Math in the Common Core Era for Parents and Teachers
The following vignette highlights the frustration felt by a third-grade teacher as she tries to balance procedural and conceptual teaching of mathematics in a highly diverse urban classroom:
As Joyce prepares for her math lesson, the students hustle around the room putting away the residual from their literacy lesson and proceed to get out their math notebooks. The students have their math notebooks and pencils at the ready as Joyce displays a subtraction problem. The hope for Joyce is to see the students’ heads move in formation following each stroke of her pen as she writes on the paper displayed on the doc camera much like sparrows flying south for the winter as they follow the lead sparrow swooping left to right, up towards the sapphire blue sky swiftly diving down again towards Earth in almost a figure eight formation. However, the students do not mimic this sparrow dance at all, but there is a mixture of random movements that indicates to Joyce that their interest is disparaging during this activity.
Joyce sits at a student desk next to the doc camera displaying the problem 149 – 132 = ___ which has half of the class scrambling to copy it down into their notebook. “So, we can change the 132 into a friendly number like we did when we learned rounding. If we change 132 to 130, we can subtract. What would that give us?”
Without hesitation, she gives the students the answer to the problem. “We would now have 121, but we would need to add back the two taken away when we rounded initially. The final answer would be 21.”
Several of the students write feverishly into their notebooks as others gaze around the room totally unaware of their surroundings. Suddenly, the teacher scans the room and sees mass chaos; not physically, but as if she could see their minds in motion. Joyce instinctively visualizes each student’s mind like gears churning in varying directions. Some of the gears are rusted and brittle as they spit flecks of amber rust around the room. Other gears are shiny silver illuminating her reflection. It saddens Joyce to see the flakes of rusted metal littering the classroom floor. Although, she remains optimistic since it seems as if some of the gears are lodged with something foreign which can be removed regaining their mobility.
Abruptly, it dawns on Joyce that there is mass confusion, and her lesson has suddenly come to a screeching standstill. She calms their fears by adding, “Guys, this is just another strategy. If this does not make sense to you than don’t subtract this way. Some methods are a lot faster than others. To be good mathematicians, you need to think about the numbers.”
The sound of metal on metal of the gears brings shivers down Joyce’s spine. She suddenly realizes that a preponderance of her students are lost entirely, and she must try a different strategy to get through to them. Joyce decides that it is best to assign five subtraction problems that require regrouping to her students while she works with a small group that is totally lost.
After working with this group using the same exact method she presented to the entire class, she scans the student work that has already been turned in. There are some students that have shown the traditional regrouping while others show no work at all. There are a few problems where the students subtracted each place value without regrouping even when the smaller number was on top. Those students whose gears mirrored her reflection turned in their work with very few errors. As she tries to trudge through the piles of rust littering the carpeted floor, she glances over her shoulder and sees the containers of base-10 blocks piled up in the corner of the classroom.
With Joyce being out the next two days attending district meetings, she decides to plan for the substitute to use the base-10 blocks to review subtraction with regrouping. When she returns on Monday, she administers a quiz on subtraction of three-digit numbers. To her surprise, a majority of the students failed. She decides that she is going to reteach subtracting with regrouping, but procedurally completing problems on the doc camera.
After following this monotonous regiment with a handful of problems, she assigns her class four word problems to complete independently while she works with five students that struggle daily. As she begins to call names, a student saunters up to turn in her paper. “You are done? I have already had two people come up to me telling me they are done. This means that you were working while we were reading the problems.”
Joyce’s frustration oozes from every orifice on her body. Sadly, she does not understand that these students are probably advanced and would benefit from differentiation. After working with the small group using the traditional subtraction method of borrowing and regrouping on a miniature whiteboard, Joyce recognizes that she will need to retrieve the place value blocks to help her students visualize the regrouping. She displays the problem 590 minus 204 on the whiteboard. Joyce takes out 5 flats, 9 longs and 0 ones and places them on the place-value mat. Her intention is to show how the base-10 blocks replicate what she is doing on the whiteboard. Although, Joyce is the only one manipulating the blocks while the students scrutinize in envy. Their eyes flit back and forth from the place value mat, to their worksheet, and to the loud rumble of students behind them. Joyce is distracted herself as the students clamor to turn in their paper.
The small space in the mobile flashes its ugly fangs exhibiting the horrible acoustics that make it almost impossible for students to concentrate during small group work. Now that she has the base-10 blocks out to replicate 590. She asks the students if they can take 4 away from 0. The students sit quietly waiting for the teacher to answer her own question. It is what they have become accustomed to since it allows them to sit idly by letting the teacher do all the thinking. “If the number on top is smaller than the bottom you cannot subtract.”
She swiftly takes a long (10 block) from the nine currently in the tens place and places it in the one’s place to illustrate borrowing. “Joseˊ, could you count out 10 ones and place them here (she points to the ones place on the place value mat). “Can we take away 4 now?”
The students just stares at her blankly as she takes four ones away. Do you see what I did there? This is what it looks like on paper.” Joyce quickly crosses out the nine to make it an eight and crosses out the zero and writes 10 above it. The students’ eyes dash back and forth from their paper to their view of the whiteboard in order to copy down what the teacher has written as quickly as possible. “Now you can subtract the rest of the numbers without regrouping. I want you to complete the last three problems on your own. You can use the base-10 blocks to help you”.
It is as if their confidence is instantly drawn out of them as they look at their blank papers with the exception of the first problem Joyce modeled for them. The students desperately work together with the base-10 blocks in hopes of making sense of the problems. However, the small group becomes discouraged and writes random answers down unable to show any work.
Joyce sits down at her desk to inhale her lunch as she looks over the math worksheets turned in. All she can think about is how she cannot spend any longer on this subtraction skill due to how far behind she is in preparing the students for the high-stakes test. In her mind, she would rather touch on all the standards that will be tested rather than miss teaching various standards in order for her students to master only a few. It is such an ugly game of tag the teachers play with the standards and the high-stakes testing.
So What Can We Learn?
This vignette illustrates how teachers are convinced that teaching procedurally is the only way to get through the content in an efficient way ensuring the students at least memorize a method to use for a particular mathematical skill. Unfortunately, they will not remember that procedure when the high-stakes testing is administered. The reality is that many teachers find themselves switching to telling and explaining when things are not going as planned during a mathematics lesson. This shift is mainly due to a teacher’s disciplinary knowledge of mathematics . It is essential for students to feel the struggle and persevere in solving real-world tasks not just equations. Students are motivated to solve problems that directly reflect their lives. For example, the vignette above comes from a classroom in an urban school district with an exceptionally high Hispanic population. A real-world problem for this classroom could be:
Isabella is planning her Quinceañera where she wants to serve cookies. Isabella originally bought 38 cookies, but after inviting more friends, she went back to the panadería (bakery) to buy more. She ended up having 81 cookies. How many cookies did Isabella end up buying at the bakery the second time? Use manipulatives to solve this problem.
Problems, like the example above, can be given to groups of students and varied based on ability level. A group’s progress can be monitored using a rubric to score problem solving, reasoning, communication, connections, and representations ensuring accountability. The above problem allows students to enter and exit the problem in multiple ways. Several equations can be used to solve this problem along with varying interpretations. Persevering in problem solving and arguing constructively about their strategies allows students the opportunity to understand the focused skill using their own background knowledge.
As parents, if your students are using a conceptual curriculum, you are probably dumb-founded by all the bizarre strategies being taught. What teachers forget to stress to their students and parents are that these strategies are optionsfor solving problems and not mandatory. Student should be free to solve the problem using any strategy that makes sense to them. Strategies are shared and taught in class so that students can see different ways to look at a problem.
Unfortunately, equations are presented instead of real-world tasks which encourage students to think procedurally versus conceptually. Having teachers acknowledge they have a procedural mindset is the first step towards embracing the conceptual world of mathematics. If teachers cannot embrace teaching conceptually in mathematics, nothing will change for our students and their ability to problem solve. As Albert Einstein so eloquently stated, “The definition of insanity is doing the same thing over and over again, but expecting different results.” For teachers, those results are gauged by high-stakes testing.
 J. Mason & B Davis, “The Importance of Teachers’ Mathematical Awareness for the In-the-Moment Pedagogy” 2013