SMP #2: Reason abstractly and quantitatively.

SMP #2: Reason abstractly and quantitatively.

As a math teacher, one of my goals is to teach students more than math. I’m not talking about diving into a history lesson or science experiment (although I love a great cross-content lesson). I am talking about teaching soft skills. Soft skills are those non-quantifiable skills that are absolutely necessary to career success. We are talking about communication, leadership, time management, problem solving, and critical thinking. In math, a common student question is “when am I going to need this [insert any topic here]”. In the moments where I am stretched too thin for an in depth discussion of mathematical application, my fall back is that “I am teaching you to think and that is a lifelong skill that applies everywhere.” I am not necessarily taking the easy way out, I really mean it! Teaching students to think is always important to me. Problem solving and critical thinking is a lifelong skill that I hope to instill in my students.
In math, an example of problem solving skills is the ability to deconstruct problems to analyze the situation, and reconstruct the information to produce an answer. These are the core values of Standard for Mathematical Practice (SMP) #2: Reason abstractly and quantitatively. When a teacher incorporates the SMP’s into their everyday routine, they are undoubtedly teaching their students to think and preparing them for life beyond mathematics. While several of the SMP’s foster skills that are applicable to real life, I believe this one is the most applicable!
Problem solving is such an important skill in math and life. Let’s dive in a little…here is a Ted Talk about problem solving in Chess: Working backward to solve problems | Maurice Ashley↗. Maurice Ashley proves how the mind leads to incorrect assumptions when it doesn’t take context into consideration. It’s a great watch. He proves that context is needed because the human mind is very logical and proceeds forward at the cost of accuracy. Here is my example.

On the first day, Sofia has ten mangoes. On the seventh day, Sofia’s amount of mangoes doubles. How many mangoes does she have on the seventh day? 

Without decontextualization, a student's brain may push forward with math and say the number 14, which is double the days, when the problem asked for the number of mangoes to be doubled.

To read more about working backwards, visit a previous Mango Math Blog - Problem Solving: Working Backwards↗. Working backwards is sometimes the best strategy and one of the most difficult for students to understand.  

"I was one of those that didn’t understand this strategy because rarely did I read all the information.  I saw numbers, key words and started putting things together but not always correctly.  Also being dyslexic it was hard enough reading the information let alone interpreting it and then working backwards and apply inverse operations."

Abstractly & Quantitatively

The common core initiative website states:

“Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.”

The two stand-out words are “decontextualize” and “contextualize”. To decontextualize means to take the abstract mathematical representation and put it into context. 

Isabela has 7 mangoes on the kitchen counter. She eats two mangoes. How many mangoes are left on the kitchen counter? 

For a student to decontextualize, they must understand that the object of the problem is the mangoes and understand there is a quantity associated with the mangoes: 7. There are seven mangoes, and a student must visualize, understand what 7 means in the sense of separate items. Next students must understand what to do with the 7 mangoes within the clues of the sentence. The words “eat” and “left” point toward subtraction. How many do we subtract? The two that were eaten. This first grade problem may seem simple and straightforward, but it really is amazing when you think about all of the problem solving involved. Our first graders are just beginning to develop this skill.
To decontextualize means to then take the context you make, and return to abstract mathematical representation. To put the problem into context, it is important students don’t forget units. To continue the problem above, students need to translate their understanding that five items, mangoes, will be left. How do we transcribe that amount? With the number 5 and the understanding that these are mangoes. So the answer would be 5 mangoes.
Students need to have versatility and be able to move fluidly between both stages as they explore the relationship and quantities within a problem. This will look different depending on the grade level so let's look at some higher grade-level examples of a similar problem.

3rd Grade

In third grade, mathematical proficient students recognize that a number represents a specific quantity. A symbolic number can be used to create a logical representation of a context. 

There are 7 crates of mangoes with 15 mangoes in each crate. How many mangoes are there in total?

To solve this problem, students must decontextualize that there are seven groups of 15 mangoes, 15 separate entities in each group. The mangoes are placed in crates (wooden boxes). They can reason through this problem as groups with repeated addition or as a multiplication algorithm to present the situation 7 * 15 = ___. To decontextualize, students take the number obtained by totaling the groups, and writing the result abstractly as a number in terms of the situation. 

5th Grade

Lastly, mathematical proficient 5th grade students can “generate two numerical patterns using two given rules and “identify apparent relationships between corresponding terms” (Core Standards↗). 

In its first year, a mango tree has 5 mangoes. In its second year, a mango tree has 8 mangoes. In its third year a mango tree has 11 mangoes. How many mangoes will be in the tree when it is in its fifth year.

As we can see, the level of rigor has gone up! There is so much to decontextualize here: fruit grows in a tree, a tree bears more fruit as it matures, each year there are more mangoes. This information produces a pattern. Each year the pattern grows by three. Grows means addition. Repeated addition can mean multiplication. There are so many ways to think about this problem!

By continuing the pattern, students can reason that there will be 17 mangoes on the three in its fifth year. To complete the problem, students must decontextualize their understanding of 17 total entities at the fifth stage of the pattern, turning the information into an answer of 17 mangoes. 
To wrap up this blog, decontextualizing a problem is an important skill for students to develop. As a teacher, work with students to break apart word problems. Decontextualizing helps students make sense of the problem. This SMP really compliments SMP #1 Make sense and persevere. As a caution, don’t over contextualize a problem just to have students decontextualized. It should be a natural part of the learning process within real world problems. 

Further Research: