Multiplication and Arrays – Does order matter?

Happy New Year to everyone.  My resolution is to write more blog posts.  I am shooting for 2 per month.  I get asked questions through email frequently and I am going to start using those questions and responses as blogs.  I hope that helps keep this material relevant to all of you.   This first one is one about considering arrays as a mathematical tool and the ‘rules’ about  arrays and mathematical expressions.  And then in a couple of weeks I have one on Fractions ready to roll.

Let’s start here. This question is from a former student of mine who is now a first year teacher.  She is ‘shaking it up’ in her team meetings by asking questions such as this one:

Hi Dr. Faulkner,
 A question came up in our meeting today about the “order” of an array. Our workbook that we have for our kiddos explicitly states that the first number in a multiplication equation is the number of rows and the second number is the number of columns. For instance, 3X4 would be 3 rows of 4 columns.  Always.

I don’t know if there is a specific order to creating a multiplication array or not. I argued on the side of not (but I could be wrong!) The best reason I got from my team to explain why it was rows and then columns was because that’s how it will be when they do area (which made no sense and didn’t satisfy me).

Anyway, I was hoping you could shed some light on this problem for me. If I’m right, good for me, if not, I’ll teach it better next year 🙂

Thanks,   TM

I hear this question a lot and there are several issues involved.  My short answer is that I side with TM on this one – it is an artificial idea that an abstract multiplication problem MUST always represent a certain thing.  TM’s team does have one decent argument to make, but it is not that this is how we will rigidly address area problems, it instead relates to matrices in algebra. Let’s look at a few things here.

First let’s think about Practice Standard #2 – Reason Abstractly and Quantitatively  (* see below for full Practice Standard 2).  Creating rigid ‘rules’ ahead of time for students (3×4 always means rows x columns) makes it harder for you and the class to engage around Practice Standard #2.  If you have already told me what everything means ahead of time, why would I have to pause and consider what these abstractions mean in my communication to others?  In short, I wouldn’t.  I just have to memorize that the teacher said this is always rows and this is always columns.  The very idea of this practice standard is to understand that numerals by themselves ARE an abstraction.  They are abstract.  Only a context really brings them to life.  As I have heard several people say “numbers are adjectives”.  I think that helps.  So 3 and 4 are adjectives that describe something.  IF you want to convey that 3×4 means, to you, 3 rows x 4 columns, then write that.  This encourages you and the class to remember that ‘following your units’ is really important and helpful.  You might say something like “I am in the habit of writing my rows first so I know when I write 3 x 4 that I mean rows x columns  – If I know I am communicating to someone else, I would need to have my units in there.”    And something like this as well  “Remember 3 and 4 are just adjectives, descriptors.  What are they describing?  Well we don’t know that until we contextualize them. That is why units are so important, they help us connect these abstract descriptors to the things they describe.  Blue tells you, well, not very much. But a blue cow says a lot!  It is the same with numerals…”

We might also consider whether arrays themselves are merely abstractions and not really very concrete. An array is not exactly the same thing as a real context, right?  An array is a mathematical tool to make sense of a multiplication expression, but it is not actually that concrete.  What do the rows and columns represent?  Rows and Columns for rows and columns’ sake?  They are a tool to organize the thinking about what is multiplication.  If we want to think of them as more concrete, we need to make themIMG_20150407_135536058 more concrete through a context. A colleague in Asheville, Tima Williams, is currently thinking about the importance of remembering that many of our mathematical tools are not as concrete as they need to be for some students.  She sent me this picture to remind us that, for many students the Tool of the Array needs to be made more concrete and that this concrete level needs to be in place for a while for some students.  An easy interjection to support the thinking of all students.  So this leads us to considering Practice Standard #5 Use appropriate tools strategically.  Can you see that, by determining ahead of time what a given problem represents that we discourage students from considering what it means to use a mathematical tool such as an array (#5)?  Rather than thinking ‘okay so what does my 3 represent in my drawing and how does the array help me make sense of that’   students will be thinking ‘okay the teacher said this is always the rows – I don’t know why but it just always is’.  Do you see also how this directly affects the student’s sense of purpose and therefore impacts their ability to construct a viable argument for their drawing?

This then impacts Practice Standard #3 Construct Viable Arguments and Critique the Reasoning of Others. As we teach students to construct arguments we must remember that ‘because the teacher said so’ is not a viable argument.  On the other hand ‘because in my drawing you can see here that the 3 represents my columns and the 4 represents my rows’  is a viable argument that connects the students thinking to the numeric expression with which they are working. This argument (3 for my columns and 4 for my rows) is only ‘wrong’ in the context of the teachers arbitrary ‘rule’.  It is on the money with how we want students to be engaging with their models/drawings/mathematical tools and how we want them to be able to put into words what they represent as they learn to construct viable arguments.

One of the reasons I love TM’s question is that, to me,  she is grappling so clearly with these standards.  As a young teacher (or an old one like many of us, for that matter!) it is great to get in the habit of thinking about the math you are teaching in ways that line up with the practice standards. TM is figuring out what it means to use appropriate tools strategically (in this case, arrays) and she is grappling with how to construct a viable argument to bring to her team*.

Area and Matrices and Practice Standard # 4 Model with Mathematics – Now, what about the team’s argument?  Is there a viable argument for their position? I don’t buy their argument about ‘this is what we will do with area so get used to it now’.    Why does area need to be represented in this way always?  For some reason we have gotten caught up with some idea that length and width are different things.  They really are both lengths.  If you want to call the longer length a width, you can certainly do that–but that is a little bit of a non-mathematical artifice. Perhaps we can just say it is a daily usage that doesn’t serve us well in the world of mathematics.  In an area representation both 3 and 4 are lengths. We can see just by looking at it which side is longer, we don’t need to name that something different.  Where the 3 and 4 go in a certain context determines the orientation of the rectangle.  Here we are going to use our mathematical expression to model a patio we want to build (Practice Standard #4). With respect to your front door you could have a 3×4 patio or a 4×3 patio. Maybe one fits better and avoids a drain, maybe one looks better with the flow of the front yard. From there students could be asked to demonstrate their ideas for a patio and then explain which mathematical expression they want to use to describe it and why.  Oh, and by the way, here I am talking in meters or yards, not in feet.

Early investigations with PS #4 (Modeling Math) can be fairly straightforward: In early grades, this might be as simple as writing an addition equation to describe a situation (PS #4).  In our patio example we used a simple multiplication expression to model our patio.  If we make a ‘rule’ about these situations ahead of time (‘the ‘length’ always comes first and the ‘width’ second’) we discourage students from connecting meaning to their context and the mathematical expression they want to use to model that context.

For the record, I happened to be at a meeting at NCDPI and was able to consult with both the High School and Elementary Math representatives there on this question.  They both agree with the arguments I made above.  They also agree that it is artificial to say that this is how we should always address area.  They did, however, mention that there is one decent argument for this position.  That is that when we get to Algebra there is a formal understanding that the first numeral/adjective/factor in a multiplication expression does represent the row in a matrix and the second numeral/adjective/factor represents the column.   But this is still not reason enough to make this a ‘rule’ in an elementary class.  Let’s refer to Practice Standard #4 again.  In the explanation for this standard it says “Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.” The key here is that they are not to be generalized or made formal until later grades.  When you formalize ideas into rules too early, you detract from students’ abilities to have discussions and describe their thinking. You also injure their ability to think about the difference between an abstract expression (3 x4), a tool to represent it’s meaning (array – rows and columns), and the actual thing you might want it to represent (a patio).

We want to have these conversations with young students and commanding a rule-from-on-high hurts your efforts to have good math talk.  Saying to students “when you are in high school you will see that mathematicians have an understood habit of having the first product represent the rows, but here we are focusing on whether you can describe your model and connect it to the numerals that represent it”   is very different than saying “the first factor is always the row and the second factor is always the column”.

Feel free to add comments and thoughts to what I hope is a viable argument that we should NOT be in the habit of presenting 3 x 4 as an Always Rows Then Columns but instead present it as an opportunity to work on Practice Standards 2, 3, 4 & 5.

*Postscript – Just like TM, I am grappling with the standards as I write this as well. I sent this blog post to some colleagues and they suggested that I make a clearer distinction between practice standards 4 (models) and 5 (tools).

This is what my colleague Temple Walkowiak suggested I consider regarding my interpretation of Standard 4 (models) versus 5 (tools) (which I did and made changes to my blog above : ).

Temple argues that TMs question is more about Standard 5 than about 4:

Here’s why….as defined by the CCSS-M, modeling WITH mathematics is when we use mathematics to model real-world situations.   As written in your blog post, it sounds like they were grappling with just arrays absent of any real-world context.  So, I  might say, “TM is figuring out what it means to use appropriate tools strategically (in this case, arrays) in her classroom and she is grappling with how to construct a viable argument to bring to her team.” 

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

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. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

CCSS.Math.Practice.MP4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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I am a Teaching Assistant Professor in the Elementary Education department at NC State where I specialize in Math Methods. I went to Duke where I earned a B.A. in Anthropology and played on the Golf Team. I have an M.Ed. in Special Education and a Ph.D. in Curriculum and Instruction with a concentration in mathematics education, both from NC State. My interest in Math waned in middle and high school. But when I began to support high school students in math as part of my job as a special educator, I became interested in the cultural and instructional aspects of mathematics. And now, 20 years later, that's pretty much all I think about. But I do love to swim, garden and, with my partner Jenn, watch our backyard bees make honey which they store in their hexagonal spaces--see 'the isoperimetric problem' or take a look at this website:


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