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If you have read my About Me page, you will know that I have researched ways to strategically approach word problem comprehension, but I wasn't sure what I was going to do with my research. I turned my research into a manuscript and submitted a chapter to six publishing companies. This was an arduous task considering publishing companies ask for a lot of information. Some of my submissions were 4 or more pages, without my chapter. Three companies sent me a kind rejection letter within a week of my submission. Two companies have yet to send me any news.

But one company did respond positively and informed me that they were interested in publishing my work. The email overwhelmed me. It took me a full day to calm down and respond. I signed a contract with Atmosphere Press and hope to be a published author this upcoming year. Thank you @atmospherepress!

For me, this is a decade-old dream come true. I cannot wait for my future.

I read an article titled Calculational and Conceptual Orientations in Teaching Mathematics by Thompson et. al. It blew my mind when I was a young teacher. For many of you this concept may not be new but for me it was career changing. When I visited classrooms, it was obvious which orientation the teacher practiced and it helped me relate to teachers from their perspective.

In a nutshell, calculational orientation means you may not know why the math is the way it is, but you can do it. There is a heavy focus on knowing the math and the process steps. Conceptual orientation means you understand why and how the math works and is applied. There is a heavy focus on the why and how. Below is a table to compare and contrast the two:

In a classroom setting with a limited mathematics block it might seem easier to take the calculational route. When I taught using calculational orientation, my lesson was very much like this: Here is the math. We will apply it to these 20 problems that you will do in class and what you don’t finish will become homework. I will meet with a small group for intervention while the rest of you work on these problems individually.

In a conceptual orientation lesson, a whole mathematics block might be used to build the concept without students solving any additional problems. For the teacher, it might appear or feel like you are behind. I hope that even with the pressure of teaching and getting so much math done in one class period, you still consider the conceptual approach, for the sake of how much it benefits your students. I spoke with my 17-year-old son, who at the time of our conversation was a junior in high school taking a college statistics course. On the topic of conceptual learning he said, “My world feels so much more open when teachers allow me to do things, like being told a question and told to figure it out. Quite plainly, I had no clue what a variable was until my ninth-grade teacher, who when she taught variables, asked us what a variable is. Before that moment, I always just solved for an equation. I never thought about what it meant. When a teacher skips steps, I skip those same steps. I have no idea what I missed because I was never taught it.”

Conceptual orientation empowers students with discovery and understanding that calculational orientation skips. It is time consuming on the front end but embeds conceptual foundations that do not have to be "retaught" throughout the year. In this sense, teaching with a conceptual orientation is actually the better investment. 

In my previous post, I wrote how taking away the math from a problem empowers students intimidated or less knowledgeable of mathematical concepts to equally engage in problem-solving scenarios. Taking away the math not only levels the field of intuition, it also prompts construction of ideas by the individual and group and is absorbed into their collection of knowledge. This is how Lost Souls and Fishes Out of Water are able to safely engage in a safe conversation that does not place them in a math-knowledge deficit. However, these conversations also benefit the Tunnel Visionaries  and Amnesiacs because as the group thinks collectively, they are also creating conceptual understanding.

To revisit the concept, here are two example questions that both ask for the same thing, but one gives the learner all information needed to solve the problem and the other, by taking away the math, requires the learner construct the knowledge (preferably within a group).

With the math:
Our class has eighteen students and one teacher. If each person eats three slices of pizza and each pizza has eight slices, how many pizzas does the teacher need to purchase?

Without the math:
​How many pizzas do I need to buy for our pizza party?

Without the math, students engage in making sense of their problem in order to build their knowledge of how to solve it. The active sense-making moments are where the teacher observes interaction and clarifies misconceptions and guides further application, and it is these teachable moments in which a connection to word problem comprehension happens. As students discuss, they might not always have the knowledge of academic vocabulary to communicate their ideas. Teachers do, and can model thinking in anchor charts that become references for academic discourse. While listening to students, jot down what they are saying. Recording their ideas respects their voices and their thinking and uses their baseline knowledge to build connections and verbalize their metacognition. Apply their thoughts in a mathematical setting so they can see how even if they did not know it, they can and do think like mathematicians.

I included an example anchor chart based on the question I asked my students, "How many pizzas do I need to buy for our pizza party?"


As my students and I review the anchor chart, I would point out what observations or comments students made and where it is on the chart. I would acknowledge their quantitative analysis and point out that this is something they do every time they are trying to understand the purpose of the numbers in their word problems. I would stress understanding the relationship between quantities and values is important because the problem could not be solved without knowing this information. Visually representing the students' metacognitive process helps the Mathmagicians because they already know how to solve but need help with explaining what they know. It also equips Junior Teachers with the right academic vocabulary.

Compare the two word problems, or compare them with similar word problems. Ask the students what the difference is and which one is easier to solve and why. This comparison serves as a reflection of their learning and will strengthen the math skills they acquired in the lesson(s).