Imagining Possibilities: Algebraic Procedures and Algebraic Reasoning

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Is Mathematics Teaching in Need of "Retooling"?

 

In the movie Hidden Figures Dorothy Vaughn, played by Octavia Spencer, recognizes that with the advancements in technology her staff’s skills are going to become obsolete thus losing their jobs with NASA. She embarks on a journey to not only train herself but also to “retool” her employees so that when a different skillset was required, they were ready. "Retooling" is hard! Despite our best efforts  many of us have never been given the time, guidance and direction, or the resources to “retool” our teaching. In fact the constraints of assessment practices and the general public's reluctance to accept and embrace the need for change has left large swaths of the mathematics teaching workforce struggling to move beyond a 1970s mindset. This is in spite of the fact that many of us would agree that our students need a different experience.  Parenting and teaching are no different in a lot of ways. We tend to parent and teach the way that we were taught. If we are not “seeing” a different perspective modeled by our mathematics teachers, it is difficult to imagine ourselves doing something different when we have our own classrooms. 

What is a 1970s mindset? In 1970, Fortune 500 ranked computational skills as the 2nd “Most Valued Skill” by employers. Problem solving was ranked 11th. Computational skills are still important, but they are important in a different way and current mathematics teaching needs to reflect this difference. In the “Most Valued Skills” ranking over the past ten years, computational skills are no longer a top 15 required skill. Today’s workforce requires problem-solving, team-work, and quantitative reasoning skills, not computational skills. The reason that computation skills is no longer at the top of the list is that all of us have in our pockets the technology to perform these same skills accurately and efficiently. 

For example, if you need to know the answer to 2/3 * 9/10 all you have to say is “Hey, Siri!” or “Hey, Alexa”. Before you can write out the steps the technology replies with an answer of “point six”. [See! Even Siri and Alexa do not like fractions!!! More on that at another time!] The proliferation of technology enables us to shift our instructional focus away from a significant emphasis on efficient computational strategies such as “multiply straight across” to obtain correct answers because the technology can now do that work more efficiently. This strategy as well as others (e.g., “keep, flip, change”) were borne out of an era in which quick, accurate, and efficient calculations were required. Many of these strategies involve a set of steps to perform an operation without the student knowing the mathematical underpinnings (e.g., axioms, definitions, and properties). There was not as much concern for the “why” behind these "tricks" because it was not a focus of learning. I am not saying these tricks do not have a purpose later in solving more complex algebraic expressions, but they definitely do not have value in elementary and middle school classrooms when students are beginning to make sense of numerical relationships.

As was the case in the movie Hidden Figures, advancements in technology have enabled us to shift the focus of our teaching goals. Students can quickly get answers, so it is no longer the answer that matters as much as the thinking, questioning, and reasoning that we engage them in along their mathematical journey. Now we need students who can mentally reason as to whether the technology is providing a correct answer and, if not, how to adjust the technology so that it does provide a correct answer. Our students need to know the “why” behind what they are doing so that they can reason about the validity of the result and, if necessary, troubleshoot. For example, what our students need to know is that an answer of 0.6 or 6/10 to the problem 2/3 of 9/10 is a reasonable answer. Our students need to be empowered with the capacity to reason that the answer is less than 9/10 because taking 2/3 of something will result in less of that something. They also need to see that the problem 2/3 of 9 and 2/3 of 9/10 are similar because both involve the same actions. That is, the amount, 9, is partitioned into 3 equal groups (3), and the required amount is 2 of those groups is 6, or 6 tenths, respectively. 

After some consultation with some of my respected colleagues there is also a point to be made about efficient computational choices. The choice to partition 9 into 3 equal groups before selecting 2 of those groups was because I was able to easily determine the size of the groups. I could have just as easily "stretched" to create 2 groups of 9 (18) and then partition into 3 equal groups (6). It is also about the efficiency of those choices. I would not make the same choices if the problem was 2/3 of 5/7. I think that understanding is part of what we want our students to learn. It just about the "when" to apply an understanding as much as it is about the "why". My point again is a student may be able to obtain an answer of 10/21 to 2/3 of 5/7 but is that all we want them to know or is there something more we want from them? 

Another quick example to further illustrate this point. Much of the internet has been ablaze with problems like 8÷2(2+2). These problems claim to “break the internet”! Of course, there is an axiomatic need for an order of operations, but problems like this are not where the mathematical value lies. One of the values of order of operations lies in entering a formula into a spreadsheet. For example, assuming you have entered a numerical value in cell a2 you will get a different output if you enter =4 * a2 + 2 as compared to =4*(a2 + 2) into different cells.  That is because the parentheses create a different order of the operations. In the first example the order of operations is to multiply by 4 and add 2 while in the second the order is to add 2 and then multiply by 4. With quick access to spreadsheets, we can also quickly compute monthly payments using a complex formula that we can enter into a spreadsheet program.

This requires the correct placement of parentheses to perform the order of operations so that the output value correctly represents the formulaic computation. 

I think one of the most powerful thought experiment that all of us can engage is to reflect on the places in our own curriculum that might still be cultivating a 1970s mindset. We can then imagine what we can do to tweak it so that we can engage our students in the type of mathematical reasoning that will not only support current technology-driven workforce demands, but also give them the tools to meaningfully reason and think more deeply about mathematics. We have come so far with technology that we might even be able to have ChatGPT tell us where those areas in our curriculum might be! 😀