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! 😀

Please!!! No More Timed Multiplication Tests!

These past few weeks have been some of my most enjoyable in my over thirty years in education. This semester I have the privilege of re-imagining the content and teaching methods used in a developmental mathematics course at MSU. With failures rates hovering around or above 50% in this course change is needed. 

 I have watched my students as they have become accustomed to my methods and been vulnerable enough with me to share their personal math journeys. Many feel like math has damaged them and one theme that emerged as they shared their stories is that it started with learning multiplication facts by taking timed multiplication tests. Their stories include frustrated parents, missed recess time, missed sundae parties, teacher scorn, and downright embarrassment because "other kids got it and I didn't!"

This past week I was sitting with a student helping her review for our first test. Despite showing me great thinking she began to cry. I asked her what was a matter. She said, "I am worried that if I do not pass this test, I might not pass this class,  I won't make it through school, and I won't get a good job! I have just struggled with math for so long I just don't know if I will ever get it!" As tears flowed down her face I asked her how long she had felt this way. She said, "Since we learned times tables." She shared, in great detail,  that she had been slow to learn them, could never do them as fast as the other kids, and missed out on going to the sundae party with the rest of her class. She said her parents would try to help her by doing flash cards with her, but it did not help. She said it made her feel stupid, hate math, and she stopped trying because she believed she would never figure it out.

I am not opposed to fluency with multiplication facts and practicing those facts, but why do we have to time the process? Do there need to be winners and losers based on speed? 

As we talked more I asked her if she could tell me what 8 * 6 was? She nodded her head that she couldn't. I asked, "How about 4 * 6?" She nodded again that she couldn't. I asked, "What about 2 * 6?" She said it was "12". I wrote the problem on a sheet and asked, "study 2 * 6 and tell me what 4 * 6 would be?" She said "24". I asked what did you notice? She said it "doubled". Then I asked, "What about 8 * 6?" She took a moment but said "48".  I did not do my best work to uncover her thinking, but I pressed her to think about what 16 * 6 would be. She wrote out 16 + 16 + 16 = 48 and doubled that to get an answer of 96. I knew she was starting to see something. I finished by asking her if she could use all her work to tell me what 15 * 6 would be? She paused a bit, and finally said "90".  She smiled from ear to ear in a way that made might heart grow "10 times larger!"

If you ask a mathematician, in general, what their work entails they would tell you it is about "looking for patterns and finding relationships." I believe there is a mathematician in every child, but it is slow thinking, not fast thinking, that wins the prize because often patterns and relationships take time to see.  Are we giving our students an opportunity to do this?  Think for a moment the power of the following sequence of multiplication problems. Oh the places you could go!

2 * 6

4 * 6

8 * 6

16 * 6

15 * 6

I think we are all wired to think this way. We just need the right sequence of problems, a little time, and the opportunity to explore. I know we want to do what is best for our children, but as this story exemplifies, and is repeated many times as I talk to other students, timed multiplication tests seem to be doing more harm than good.  If that is the case, why does it continue? 

 

Reflections on the Teaching and Learning of Algebra

Algebra teaching and learning has been on my mind lately.  I just came out of a semester in which I taught a development  algebra class at my university and I spend my evenings helping my 12-year-old daughter through her first foray into algebra. I have come to the conclusion that whatever we are doing in teaching this subject is just not sticking for many students--95% of the students in my developmental college class  had Algebra 2 or beyond in high school and 40% of those same students had at least one form of an accelerated mathematics course.  Interestingly, the algebraic reasoning capacities I see in my daughter just beginning her algebra journey are not much different than those college level algebra students who have extensive experiences with algebra. That is not a knock on them. They were fine folks that I do not think set out upon on their educational journey to struggle with algebra. I just do not think they have been equipped with the tools to engage in meaningful algebraic reasoning.

As I have worked through problems with my daughter, I have pondered the question, "What are the tools I would like to equip my daughter, and my other students, with so that they would be successful in algebra. I started to think about this through the lens of helping my daughter with her homework. 

The other night she was given the charge of "simplifying the following problems written as a single exponent". One of those problems was... 

She wrote an answer of 8h^8 and asked me, "Is this right?". I refrain from answering this type of question because I want her to be empowered with the mathematical authority to answer that question, but I did have to reflect on the tools she would need to do this. If it was incorrect, assuming that she knew what she was creating by simplifying was an equivalent expression, is a counterexample. However, if it was correct, an equivalent expression, there should be a series of definitions and algebraic properties that would lead her to that result.

Knowing that it was not correct I asked her to substitute h = 1 into each expression. This resulted in an output of 64 for the initial problem and 8 for her result. We talked about the fact that if we have two equivalent expressions any value that we substitute in for h into either will yield the same result. And, that we could be certain that they are not equivalent, but only get a "feel" as to whether the two expressions are equivalent. 

Based on her answer I sensed that she handled the operations involving the variable h but I was not convinced that she knew why, "I added the exponents, then multiplied." We talked about the definition of exponentiation, at least as it applies to whole number. That is, the exponent 2 signifies that we should multiply together two quantities of whatever is under the exponent [1] and that the exponent we should multiply together three quantities of whatever is under the exponent (i.e., h*h*h) [2] Because I wanted her to see why you add the exponents when multiplying I asked her to write out the expanded notation of each. It was a little messy but I wanted her to see this at least once. We then talked about the fact that since the only operation involved was multiplication that there are properties that allow you to reorder (commutativity) and regroup (associativity) to create equivalent expressions. [3][4].  Those properties and definitions are what guide the steps we take to create equivalent expressions. 

Substituting h = 1 into the original problem and final result [5] she obtained the same result, 64. I emphasized that while the substituting process would give us a "feel" for whether the simplifying work we had done was correct it was the proper application of definitions (i.e., exponentiation) and properties (i.e., associativity, commutativity) that determined whether the expressions were equivalent. 

This is part of a problem set my daughter was working on the other night.  She came to me asking if she had done the problems correct. Take a moment to see if you can find the error in her reasoning. 

Let me be clear I have no problem with her making mistakes. Mistakes are merely opportunities to learn, but you have to have the tools to learn from those mistakes. I knew she did not have any tools which means she has to rely on someone else to determine the correctness of her work. 

This is one of the areas in which I think the system is breaking down. We are not equipping students with the tools to determine the correctness of their work. What are those tools? At least in this case, a case that happens extensively in algebra, is that one of those tools is the concept that underpins the action of  "simplifying". That is the creation of  an equivalent expression. Using a functional approach, an equivalent expression means that for any value (except where the value of x results in an undefined function) that is substituted in for the variable should yield the same result. We can get a "feel" as to whether two expressions are equivalent by looking at the output after substituting a value or two into each expression.  If the outputs are the not the same then we are certain they are not equivalent. In other words, there is something amiss in our simplifying process. However, if the outputs are the same then we can feel a little more confident that we are on the right track. Let's try this for #5. 

What do we need to be certain? We need mathematical reasons in the form of definitions and properties. For example, by definition of exponentiation we know that r^3 is r * r * r. The problem  (3r^3)/(2r) is can be written in terms of multiplication  3 * r^3 * (1/2)*(1/r) or 3 * r * r * r * (1/2) * (1/r) and then using associativity and commutativity of multiplication this expression can be rewritten as 3 * (1/2) * r * r * r * r * (1/r) which, using the identity of multiplication, is rewritten as (3/2)*r* r or (3/2)*r^2.  Is this cumbersome and maybe mundane, possibly? But, what is the alternative? If these definitions and properties are not emphasized then the symbolic work becomes about "feel". In the end,  I just see my daughter struggling at times because she is unsure what to do. I do not want to have to answer the question, "Is this correct?". I want her to have tools to make those decisions on her own. 

Drugs and "Keep, Change, Flip"--Just Say No!

I am a product of 80s schooling.  A popular public service announcement during that time was a man with a perturbed look on his face, arms crossed, asking the viewer, “Is there still anyone out there who still doesn’t understand what doing drugs does?” Holding an egg, he exclaims “this is your brain,” cracking it up against the side of a heated cast iron skillet he says, “This is your brain on drugs” as the sound of the sizzling egg plays in the background. The ad finishes by him asking, “Any questions?”. The motto we rehearsed in school, thank you Nancy Reagan, was “Just Say No!”.

I didn’t like eggs or drugs, so I am not sure the ad had much impact on me--I had already said “No”! However, I did get the gist. If you take drugs you are going to do bad things to your brain like impede mental growth, stunt brain development, lower cognitive functioning, etc. As a thirty-year teacher I often reflect on the question, “Are their current practices in math classrooms, as well as my own, that are deadening minds and stunting development?  Yeah, I know, drugs and mathematics teaching practices, a tough analogy. I believe one such of those practices that we need to “Just Say No!” is tricks to teach mathematics. A prevalent one is the use of the phrase “Keep, Change, Flip” to teach division of fractions. Okay, all I ask is that you hear me out.  

I witness on a regular basis the intellectual residue of students who have been raised on math tricks such as “Keep, Change, Flip” to learn mathematics. I teach mathematics content classes to future elementary, middle, and secondary mathematics teachers. I also teach the lowest college-level developmental mathematics course at our university--a course in which 95% of students have successfully completed Algebra 2 or above in high school. I also have conducted research in 4th through 7th grade classrooms related to understanding of fractions. I often ask these students, including preservice teachers,  to explain the meaning of 3/4÷1/3 or solve a word problem such as, “A cookie recipe calls for 3/4 of a cup of sugar and you have 6 cups of sugar, how many batches, including partial batches, of the cookie recipe, can you make?”.  Questions like these throw many students into a panic. They often reach for a calculator, or simply state, “Is this where I use, ‘Keep, Change, Flip’?” I have also had the opportunity to teach students from South Korea, Taiwan, China, Oman, Germany, Greece, as well as many others. The one constant is that most of the students from these countries have never heard of “Keep, Change, Flip”. Many U.S. students know “Keep, Change, Flip”, but have no idea mathematically why it makes sense to even perform that operation. This is an American-based shortcut that yields correct answers to division number sentences which are prevalent in our assessment-driven culture. However, a shortcut, like the 80s claim about drugs,  that promotes intellectual neuropathy. 

Now there might be a place for “Keep, Change, Flip” in simplifying an algebraic expression such as 

 but, I am becoming more and more convinced that it does not have a place in elementary classrooms. You are welcome to try to convince me otherwise, but at least give me a chance to make my argument.  I myself have done this and I agree that “Keep, Change, Flip” is an easy, efficient way to get an answer to a school-based problem. I am now sitting on the other end witnessing the unintended consequences of this narrow view of mathematics. Our educational goals should push students beyond the “quick high” of just getting a correct answer.

What is the alternative to “Keep, Change, Flip”? Provide students with the tools that they will need to reason about division problems in the real-world, the language necessary to make meaningful interpretations about any division problem and learn an algorithm that connects to whole number division. The book Extending Children’s Mathematics: Fractions and Decimals by Sue Empson and Linda Levi is a great resource for anyone desiring to learn more about a different view of division of fractions. 

Take a moment to think about the solving the problem, Gwen has 6 yards of ribbon to make bows. If each bow takes 3/4 of a yard of ribbon, how many bows can Gwen makes before she runs out of ribbon. In fact, can you create a picture that would represent the situation? Before we formalize the notation that cues dividing fractions it is important that students intuitively experience the actions associated with the process of division. I would venture that in modeling this solution you created 6 whole objects to represent the 6 yards of ribbon, partitioned each whole object into 4 equal pieces (i.e., fourths), and counted sets of 3 of those pieces (i.e., 3 fourths). In general, we want students to connect the process of partitioning a whole unit into n equal parts with the size of a part, 1/n, that results. And, that they can combine unit fractions to make fractions that are multiples of unit fractions (p. 74, Empson & Levi, 2011). In the ribbon problem, the answer of 8 means there are 8 groups of 3/4 of a yard in 6 whole yards.

Students must also develop the language associated with a symbolic division problem. For example, the above problem 6÷3/4 is asking, consistent with the visual model, “How many groups of 3/4 can be created from 6 whole units?” The first number in a division number sentence signifies the whole and the second number, using a measurement perspective, signifies the size of the group. Many of the college level students I teach do not have a language beyond “Keep, Change, Flip”. This is very limiting to their intellectual development because it does not provide them with the tools to use mental arithmetic to solve problems nor the means to estimate or check the reasonableness of their answer. Consider how a student empowered with this language might reason about problems such as 4 1/2÷1/4  or 6÷0.01.  For example, 4 1/2÷1/4 is asking how many 1/4s are in 41/2?, and, 6÷0.01 could be interpreted as, how many pennies in $6?  And, how they might estimate the answer to a problem such as 1/4÷1/3. That is, that the answer is less than one because the whole is smaller than the size of the group. 

If students are to learn an algorithm it should be an algorithm that connects with what they already know about whole number division. That algorithm is called the Common Denominator Algorithm. The visual action of the earlier ribbon problem illustrates this algorithm. The 6 whole units was recomposed into 24 fourths which was used to form groups of 3 fourths resulting in an answer of 8. In other words, the fraction division problem 6÷3/4 is equivalent to the fraction division problem 24/4÷3/4  which is equivalent to the whole number division problem 24÷3 because it is asking, “How many groups of 3 fourths can be made from 24 fourths?”.

Consider another problem introduced earlier 3/4÷1/3. The problem is asking, “How many 1/3s can be created from 3/4?” or making the units explicit, “How many 1 thirds can be created from 3 fourths?” Using what we know about equivalent fractions, an equivalent problem can be created, “How many 4 twelfths can be created from 9 twelfths?”. In other words, the fraction division problem 3/4÷1/3 is equivalent to another fraction division problem  9/12÷4/12 which is equivalent to the whole number division problem 9÷4. 

If we START with, “Keep, Change, Flip”, we are promoting intellectual neuropathy, a numbing of the brain, because our students feel they have all they need to get answers. As a colleague of mine said, “If we choose to share with students ‘Keep, Change, Flip’ we are making the decision that our students no longer need to think about ideas related to division of fractions.” It has been a challenge to develop a deeper understanding of division of fractions because students believe they have all they need if they can get correct answers. I have seen this first-hand in my college level classes. Recently, I asked a class of 50 students why, “Keep, Change, Flip” works. Not one student was able to provide a mathematically valid response (e.g., dividing is the same operation as multiplying by the reciprocal). There is so much more that we want them to understand about division of fractions than getting a correct answer. Let’s, “Just Say No” to “Keep, Change, Flip” and hope that what we replace it with is more effective than those 80s drug ads! I love to engage other teachers in ways to improve mathematics teaching. Feel free to reach out for further dialogue at patricksullivan@missouristate.edu.

Adding Fractions: Making the Connection to Whole Numbers

One of the five recommendations put forth by the Institute for Educational Studies in Developing Effective Fractions Instruction for Kindergarten Through 8^th Grade (2010) is, “to help students understand why procedures for computations with fractions makes sense” (p. 5). One way to help students make sense of procedures for computations with fractions is to leverage procedures they may already understanding for computations with whole numbers. The argument is that fractions represent numbers and the reasoning to solve problems involving operations with fractions is closely related to the reasoning to solve problems involving operations with whole numbers because both problems involve numbers. For example, “addition is addition” whether the problem involves whole numbers or fractions. That is, addition is a process of defining numerically the amount of objects in two or more sets of like objects with same unit (e.g 9 apples and 4 apples is 13 apples). The purpose of this first blog is to make a clear link with addition. The necessary understandings of fractions as well as important terminology will be highlighted.

 Depending on the operation and the specific situation it is important to understand the different possible meanings of a fraction. In some instances it is important to interpret the fraction 3/4 as a scalar. For example, is 3 copies of 1/4 or 3(1/4), in which the 1/4, or fourth, is the iterating unit—that is 1/4 is repeated 3 times. In other instances the fraction 3/4 is interpreted as an operation. For example, in the abstract problem 3/4 * 16, the fraction 3/4 can be thought of through the lens of partitioning 16 into 4 equal groups {4} and choosing 3 of those equal groups                                     {3 equal groups of 4 = 12}.

Consider the following addition problems.

Jenny has 4 whole apples and her friend Sue has 9 whole apples. Between the two of them                 how many total whole apples do they have?

 Jenny has 1/3 of a whole pound of cheese and her friend Sally has  3/4 of a whole pound of cheese. Between the two of them how much cheese do they have?

The operation of addition is about describing numerically the sum of the number of elements in two or more sets of the same objects sharing the same iterative unit. For example, to answer the question “How many total whole apples do they have?,” the iterative unit is 1 whole and apple is the object. In performing the computation the action of addition involves combining 4 copies of 1 whole apple with 9 copies of 1 whole apple for a total of 13 copies of 1 whole apple or 13 apples.

 The critical aspect of addition in contextual situations is that not only is the object is the same, but so is the iterative unit. These ideas are lost sometimes when discussing de-contextualized problems such as 1 + 1/2.  The answer 1 and 1/2  is describing the numerically the combining action of 1 copy of 1 whole unit with 1/2 copy of 1 whole unit yielding 1 and 1/2 copies of 1 whole unit or 1 1/2 whole units. It seems important to mention that 1 + 1/2 can also be thought of as the sum of 2 copies of 1/2 whole unit and 1 copy of 1/2 whole unit which is 3 copies of 1/2 whole units or 3(1/2) or 3 halves.

  The same argument holds when adding fractions. In order to perform the combining action associated with addition we must have the same iterative unit and like objects or referent units.  We cannot combine 1/3 of a whole pound of cheese and 3/4 of a whole pound of cheese because although their referent unit (1 pound of cheese) is the same the iterative units are not. The fraction 1/3 can be interpreted as “1 third” or “1 copy of the iterative unit 1/3.” Similarly, the fraction 3/4 can be interpreted as “3 fourths” or 3 copies of the iterative unit 1/4.”

While thirds and fourths are different iterative unit preventing us from completing the combining action of addition it is possible to re-define each of those fractions in terms of the same iterative unit. That is, 1 third is the same as 4 twelfths or 1 copy of 1/3 is the same as 4 copies of 1/12 . Likewise 3 fourths is the same as 9 twelfths or 3 copies of 1/4 is the same as 9 copies of 1/12. 

Now that both fractions are expressed in terms of the same iterative unit the combining action of addition showing the sum of 4 twelfths and 9 twelfths as 13 twelfths i.e. --An analogous solution to the whole number addition problem.

Making Sense of Two-Digit Multiplication and Much More!

Making Sense of Two-digit Multiplication and Beyond

            24

      x    35

Take a moment to think about how you would perform the computation. How you probably learned to perform this computation involved using an algorithm driven by speed and accuracy. Most of us never were asked by our teachers or considered why the algorithm worked. We simply computed. Our goals are different now. If we want students to perform the computation with speed and accuracy we can simply hand them a calculator. Our desire now is for something more from our students than performing steps to a memorized algorithm.  We want them to understand the purpose behind the steps and the meaning associated with the steps of the algorithm. We want them to understand the structure of multiplication. We also want students to understand the relationship between place value and the computations performed to complete a two-digit (or more) by two-digit multiplication algorithm. The goal is to make a clear connection between the computations performed with a visual, concrete representation and the abstract representation. Base-Ten Blocks are an effective visual, concrete representation for modeling a two-digit multiplication problem such as 35 * 24. Making a clear link to place value ideas, the numeral ‘35’ represents 3 tens and 5 ones and the numeral 24 represents 2 tens and 4 ones. These two numerals are represented by the outer boundaries shown in red. The computation shown on the right is the Partial Product Algorithm whose results are also shown in the Base-Ten Block representation.

The top Partial Product Algorithm computations are written in terms of the units shown in the four quadrants of the Base 10 representation {20 ones, 10 tens, 12 tens, and 6 hundreds}. Three of the four computations can be re-expressed (i.e. 20 ones is also 2 tens, 10 tens is also 1 hundred, and 12 tens is also 1 hundred and 2 tens) yielding an answer of 8 hundreds and 4 tens (840). Interestingly, when you write the numerals representing each of these partial products as shown in the bottom Partial Product Algorithm (20, 100, 120, 600) both interpretations are visible. For example, the numeral ‘20’ can be interpreted as either 20 ones or 2 tens and the numeral '120' can be interpreted as either 1 hundred, 2 tens, or 12 tens.

A less concrete, visual model than Base 10 Blocks is an area model of multiplication. Each numeral is separated by the place value of each of the numbers. The area model representing 35 * 24 is shown with 35 as 30 (3 tens) and 5 (5 ones) and 24 as 20 (2 tens) and 4 (4 ones).

One of the key points of making sense of multiplication through visual representations is the reasoning and sense-making potential these representations afford. Although not an exhaustive list of the potential reasoning and sense-making possibilities the three examples shared exemplify the potential of these models to help students prove conjectures generated from analyzing number pattern relationships, to explain why a mental arithmetic strategy works and to make sense of algebraic relationships.

Proving conjectures generated from analyzing number pattern relationships

Suppose that you introduced the following pattern to students and asked them to reason about how one could use the given information to determine the value of 85².

15² = 225

25² = 625

35² = 1225

45² = 2025

A typical response from students is that 85² is 7225 because you take the tens digit {8} and multiply it by the next consecutive tens digit {8*9} to find the last two digits and drop 25 down to represent the first two digits {7225}. Reasonable follow-up questions are 1) Does the algorithm always work when the units digit is a 5? 2) Are there other multiplication facts in which the algorithm will work?, and 3) Why does the algorithm work? Understanding the structure of the multiplication in terms of visual representations might enhance students’ ability to reason about each of these questions. Using an area model one of the instances in the pattern, 25², is shown.

      

Recall that the student’s algorithm involved multiplying the tens digit {2} and multiplying by the next consecutive tens digit {2 * 3 = 6} and placing this value to determine the number of hundreds {6__ ___}. It is important to note that mathematically the student was actually multiplying 20 * 30 or 20 groups of 30, or 2 tens and 3 tens.

 Moving the rectangle representing 20 groups of 5 to the right of the figure it can be seen that the two rectangles together represent 20 groups of 10 which means there will be a total of 20 groups of 30 {600} and 5 groups 5 {25} or 625.

Modeling multiplication using an area model provides the means for students to reason and make sense of why not only this particular student’s algorithm works but also why other potential algorithms generated by students might work. The area model also provides the means to reason about why the last two digits are always a 2 and a 5 and why problems such as 23 * 27 {621} can be computed using the student’s algorithm but other problems such as squares not ending in 5 (e.g. 27²) or those in which the units digits do not add up to ten {e.g. 23 * 26} will not work.

There is also the potential to make a connection between the concrete representation shown with the area model and the abstract representation that shows in general why the student’s algorithm works. The abstract representation of the length of each side of any square can be represented by the expression 10n + 5 in which n represents the number of tens. For example, 25² can be written as 10(2) + 5 in which the 2 represents the number of tens. Squaring the expression 10n + 5 results in the expression (10n + 5)² = 100n² + 100n + 25 = 100(n(n+1)) + 25. The first term 100(n(n+1)) yields the number of hundreds and is analogous to the student’s step of taking the numeral representing the number of tens and multiplying it by the numeral representing the next consecutive tens digit (e.g. 100(2(2 +1) = 100(2 * 3)) = 600). The abstract representation 100(n(n+1)) + 25 also reveals why the last two digits are a 2 and a 5 since the first term reveals the number of hundreds and the last term reveals the number of tens and ones (e.g. 100(2*3) + 25 = 600 + 25 = 625).

Making sense of why a mental arithmetic strategy works

A mental arithmetic strategy for computing a multiplication problem such as 18 * 22 is to square the number that is in the middle of the two numbers {20² = 400}. Subtract that number {400} from the difference each number is from the middle number squared {2²}. In other words, or symbols, 18 * 22 can be computed by taking 20² and subtracting 2² {400 – 4 = 396}. Check it out! Pretty cool! Try it for 33 * 27! Remember in an earlier problem it was discussed that 25² = 625. We can apply that knowledge and the mental arithmetic strategy just discussed to also find 24 * 26 or 23 * 27. The big question is, why does this mental arithmetic strategy work? As before, refer to an area model for 18 * 22, but this time the area model is represented using a Cartesian plane.

The model is representing 18 groups of 22 or 18 * 22. Consider cutting the last two columns (18 groups of 2) and moving the piece to the location shown to make a larger square with a smaller square (shown in black) missing. The area of the larger square is 400 {20 * 20} and the area of the smaller square is 4 {2 * 2}. This results in an area of 396. 

This mental arithmetic strategy can be connected to an important algebraic idea, difference of squares. The generalized version of difference of squares is (x + y)(x – y) = x² – y².  Now think about this idea in terms of computing 18 * 22. We can represent 18 * 22 equivalently as (20 – 2)(20 + 2) = 20² – 2² = 400 – 4 = 396. Once again making a connection between what we are doing arithmetically with an important algebraic ideas.

Making sense of an algebraic relationship 

A mistake that many algebra students make in expanding the expression (x + y)² is to square each term resulting in the expression x² + y².  Using an area model to represent this action not only exemplifies the meaning of “squaring” but also why the expression  (x + y)² is not equivalent to x² + y² but is equivalent to x² + 2xy + y².

Findings from brain research suggests that vision is the primary way that most of us receive stimuli and that analogous thinking is a strong learning mechanism. (Almarode, 2016, personal conversation). It makes sense then that teachers can significantly impact student learning by providing opportunities for students to reason about important mathematical ideas, such as multiplication, using visual representations. It also makes sense that student learning can be greatly impacted by providing them opportunities to reason about the relationship between what he/she “sees” in the visual, concrete representations of those ideas and what he/she “sees” in the abstract symbols, as well as making connections between these two representational forms.