We all believe were good teachers, yet also we all believe we can improve. Our training and experience, and occasional validation by our peers helps us feel good at what we do. Students also give us comforting feedback, directly through their improved performance and “thank you” and indirectly as we hear from other students, friends and colleagues. We continuously share our ideas with our peers, discuss successes and failures, ask for guidance and help with a pedagogical idea. We teach with inspiration and concern, and treat students with respect, care and a nurturing attitude.
So, what’s the problem? How come we are distressed at the 4-5 students in a class of 20-some who don’t seem motivated, don’t do the work, sometimes don’t even show up for class? Are we to parent them? Are we to act as shepherds of their lives as well as math instructors? Are we to be motivational speakers to grab their attention and rouse them to dynamic learning?
Perhaps we believe, at the core, that everyone is capable of learning math, if only the learning environment were appropriate. If only we were able to be more adept at unraveling the history of their prior experiences and build from the math rabble they possess. And, paying attention to the importance of learning styles, we tend to adopt a host of teaching methods until we find the one that’s just right for each student. Well, at least, we try.
So, what’s the problem?
Is it them or us? This short question is the problem. Once ‘them’ and ‘us’ are separate entities, once we see ourselves as separate, the problem begins. Really? After all, we are individuals and it is pure happenstance that brings us together in the same classroom.
And that’s the point. We are a unit and theoretically, I’m the instructor and they the learners. There is no point to my teaching if there is no audience and there is no point of their sitting in a classroom if no activity is to occur. So, we are a unit; an educational unit.
And what is the dynamic of this unit? One way to view this is to allow that the instructor is to tell stories to the students. The stories will be about relationships between quantities. Usually these collective and cumulative stories will be called math; there are prescribed relationships that have been identified through the ages as essential and core to mankind, and these relationships are bundled together in a text and it is called mathematics.
Some of these relationships – some of these stories – don’t seem to make sense to the students because they have heard these stories before and the stories don’t seem to make sense or don’t seem to have any meaning, or further, don’t seem to have any meaning to that person’s unique story. Is this a critical event? Must all math – or for that matter – any content area have immediate and purposeful meaning to a person before they feel engaged with it and feel impelled to learn it because it has a visible connection to the daily lives of the student?
What about history, English, biology, psychology? Do these disciplines have value to the learner? Can the learner see how these can be used, integrated into their daily lives and become valued learning? What about math?
If math is taught as it traditionally is, then the likely answer is no, there is no apparent real-life immediate utility. This, of course, is based on the assumption that if the utility were made visible, the students would then be more engaged in learning it. But let’s return to the idea of ‘traditional’ for a moment. Traditional means that sequence of information presented in texts which math curriculum developers and teachers believe to be important for students to learn.
In basic math the four basic operations are learned, and then the order of operations is learned because a person confronted with a problem in which all the operations occur must have a ‘rule’ for which operations to do first. Why? So that not only will they get consistent answers but also so that all the people doing to problem will arrive at the same answer.
Is there something sacred about the order of operations? Not particularly, but it has been made to be so. Violate the rules and you will err. For example, in class the other day the problem was (2/5)^3x(5/8)^2.
The solution in the text showed (2/5)(2/5)(2/5)(5/8)(5/8) and then showed some canceling of common factors, resulting in an answer of 1/40. However, notice that in order to do this, the order of operations has been ignored. If one were to literally apply the order of operations the exponents would have been the first operation to do, giving 8/125 x 25/64. The next step would have been to do the multiplication of the fractions, and finally the answer would be simplified completely. Somewhere in the text, by the way, the way this problem was worked was consistent with what was described as a ‘shortcut’, meaning when you have multiplication of fractions you should look at the fractions to see if they can first be reduced, and then do the multiplication. This is a valuable and time-saving strategy, but in the example as worked out in this case, reference to the ‘shortcut’ wasn’t made.
The issue here is the story of how to do this kind of a problem. Is there more than one story that can be told? Yes. Be literal and follow the order of operations or alternatively, use the ‘shortcut’. Which one is correct? Both are correct. Which one should a student use? Whichever story resonates with them. What? Isn’t math a little more linear and logical than that? Allowing students to craft their own method of solution?
Well, isn’t crafting an individual and unique method for problem solution what students do in everyday life? Why not in math class?
A lot of students have the perception that math is very linear, very logical, and has one and only one path to the answer. Starting with the very basic idea, for example, that 2 + 2 always and only equals 4, this concept scaffolds into other math operations, like the above noted order of operations. Until of course, one has to do algebra, where the numbers don’t exist and letters representing them exist.
For example, what happens to the order of operations in algebra vs. in arithmetic? In arithmetic, students are taught to do the work inside any grouping symbols first, if there is work to do. So, 2(5 + 6), is done by adding 5 and 6 = 11 and then multiplying by 2 = 22. What happens if a student knows the distributive law and uses it? Is that student to be corrected?
But back to algebra. In algebra a statement like 2(5 + 6) can be seen as a(b + c) and of course, the order of operations still applies. ‘Do the work inside the grouping symbol first’, so the student is to add ‘b’ and ‘c’., Well, algebraically, it can’t be done. All one can do is indicate it by leaving the expression as written, b + c. Then multiply this expression by ‘a’. Is this consistent with order of operations? Absolutely, so why does this bedevil some students?
There are some students who contend that b + c = bc. We now have to address issues of notation. In algebra, ‘bc’ means to multiply ‘b’ times ‘c’, once we have actually substituted numerical values for ‘b’ and ‘c’. But if we were to be very literal, if ‘b’ = 7 and ‘c’ = 8, then what would be written is 78. But this isn’t done. By the way, 78 means 7(10) + 8(1), a whole other notation issue, and certainly doesn’t mean 7 times 8.
But when we write 78 because of having substituted ‘b’ = 7 and ‘c’ = 8, different notation is employed. To keep the arithmetic consistent with the algebra; we use one of the available notations for multiplication such as (7)(8). Why not use (b)(c)? Because b + c doesn’t result in (b)(c).
I think you get the point. Math stories seem to have inconsistencies. We seem to lurch around and come to realize that some of the arithmetic rules work with algebra and some of them don’t. And, what’s more, we have to know both domains, because while doing algebra, the arithmetic rules still apply, but the converse isn’t necessarily true. So, although 2 + 2 = 4 is true in arithmetic and in algebra, students must understand that although the order of operations is the same in both cases, it ‘feels’ different.
So, why don’t we as math instructors, teach students the algebraic story of grouping symbols to bring consistency? Why not allow that if one sees 2(3 + 4), use the distributive law? Is it because of exponents? A student would have to see that 2(3 + 4) and 2(3 + 4)2 have to be handled differently. And, by the way, look at what happens – and it happens regularly with some students – when a(b+c) and a(b+c)2 are presented. Have you ever seen (3 + 4)2 = 9 + 16?
Part of the issue here is that sometimes, the rules that apply when the values or expressions have an exponent of 1 are different from the rules that apply when a value or an expression has an exponent greater than 1. Wow, what a story.
For example, when trying to make the point of the difference between (-4)2 and –42, one very direct way is to say ‘follow the order of operations’. In (-4)2, a student would do the exponent first, thus 16. If the student were to do –42 according to the order of operations, the ‘-‘ has to be seen as subtraction, not the sign of the number, and the order of operations says to do exponents before subtraction, so the outcome is –16. Is this too subtle for students to see? Should this understanding of it be presented? And, why-oh-why does math allow the same symbol to be used for two different things: ‘+’ can mean addition or be the sign of the number and the same for the symbol ‘-‘.
So, look at (-4)2 and –42 again. Using grouping symbols makes it clear that the student is to apply the exponent to everything inside the grouping symbol. This works readily. But what if students had been taught to read the ‘-‘ not as subtraction but as the sign of the number? Or, that the symbol ‘-‘ can mean either? Given this understanding of ‘-‘, then (-4)2 and –42 could mean the same thing. The symbol ‘-‘ is the sign of the number, therefore understand this to mean ‘negative four times negative four’.
Here again, however, the idea of an exponent of 1 versus an exponent greater than 1 can come into play. If a value has an exponent of 1, the ‘-‘ can be either the sign of the number or the operation of subtraction, but if the exponent is greater than 1, the ‘-‘ means subtraction and not the sign of the number. Given this, should it ever be taught that ‘-‘ can mean both, or more critically, knowing that this distinction between the forms (-4)2 and –42 forces different meanings, perhaps the point should be made about how the rules differ when exponents are equal to 1 or greater than 1. Or, ignoring all this, simply point out that the order of operations applies.
However, texts present the following: “When subtracting, add the opposite” and demonstrate this algebraically as a-b = a + (-b). So, in a problem such as 25 – 42, if a student were to follow the subtraction algorithm, the outcome would be 25 + (-4)2 and thus again showing that (-4)2 and –42 mean the same thing. And it is again true that if a student were to be literal about the order of operations, the 42 has to be done before the subtraction. But where is it ever designated that changing the notation form isn’t allowed? After all, changing the notation form is not doing the operation. But, is rewriting 25 – 42 as 25 + (-4)2 actually doing the operation of subtraction or simply changing notation? Again, if it’s understood to be doing the operation, then again the order of operations prevails, and thus a student couldn’t apply the subtraction algorithm before doing the exponentiation. But, how is a student to know this? Well, maybe again, this defaults to distinguishing between the operations if the exponent is 1 versus greater than 1. If it’s greater than one, suspend transforming subtraction to addition. So, it seems that raising numbers to exponents greater than one has a subtle but critical meaning which isn’t addressed in texts or by instructors.
Perhaps all the above is irrelevant since the rules and algorithms that we present are presented in a very direct, concise, logical way. We need not presume that any student would ever get caught up in the tangle of symbols, words and relationships as written above.
But these little inconsistencies, which never reach consciousness necessarily, can generate cognitive dissonance. It may well be that some part of the students ‘math brain’ is saying, “Huh…this doesn’t make sense. We were taught that 6-2 means 6 + (-2) is true in one circumstance, but not true in another circumstance, 6-22 ≠ 6 + (-2)2.”
Perhaps this gives too much importance to the thinking students bring to math classes. Have they ever been bedazzled by other such seeming anomalies in math notation and math meaning? Have they every sensed something about the rules not always being applicable and no instructor has ever pointed this out? Should I, a student, ask about this if I see it?
Fundamentally, I believe teaching math is telling stories and listening to students’ math stories, the stories that they carry around in their heads, perhaps fantasies, hardly realities of math.
Getting people to reflect on what they know – actually, reflect on the fragments that they carry around that they call math – is important to the story telling. Again, these are the student’s stories. For example, writing on the board that 2/4 = and asking them what the answer is usually gets a collective ½. If it is then asked “What math operations did you do?” the answers will vary but be things like, cancelled, or divided, or reduced, simplified, got rid of. The point is that students know how to do, metaphorically, math. The words they use aren’t math operations but symbolically represent several math operations. In order to get 2/4 to 1/2 involves several math operations. These are sequential events, algorithmically organized, compartmentalized, and further trivialized by summarizing and compressing them into a word like ‘cancel’.
Getting students to reflect on what they know gives them a starting point from which understanding, not just doing, math can occur. Puzzling them, disabling their usual routes to answers, and making them appear as deer in the headlights. But not doing it in a threatening or punitive environment.
For example, we were talking about dividing by powers of 10, and used the example of 300 divided by 40. I then wrote it as a fraction, and brought it back to the same notation form as 2/4. I got a wow response, which was nice but also something which I could get only if students had been misperceiving what was really being expressed. How did they generate this misperception? They’ve been in math classes before mine and have come ‘armed’ with these fragments and pieces of math-like words.
I do believe that if they can gain an understanding of the basics, then they can always reconstruct in their own way with their own logic a pathway from the problem given to the set of possible solutions available.
Keeping it basic and visible is helpful. For example, most textbooks give one page to exponents and square root. It is written in ‘mathese’, dry presentation of what the notation is and what it ‘means’. However, what it means can also be demonstrated.
Put on the board the following: (2)(2)(2)(2)(2)(2)(2)(2)(2)
Then talk about notation; no mystery here, simply a notation issue.
Try this. Put on the board a cluster of 9 “postits” and a separate cluster of 16 “postits”. Ask for a volunteer to come to the board and arrange these in a square. Once done, they will be able to see that the square made by 9 is 3 on a side, and 4 on a side for the 16. Then ask for a volunteer to put them together in a square, and this gives a square of 5 on a side for the total of 25. This demonstrates square root.
Students have been trained to believe that if you do math well, you must also do it fast. This I find to be one of the perceptions that students bring which interferes with their being able to learn math. The part that isn’t seen is the thinking, the reflection, the connecting the ‘new’ stuff with stuff they already know. Their stories of math typically aren’t complete. Rather than identifying what they do as wrong, I try to identify what they do as incomplete. Sometimes I prompt them with the idea that if they have a strategy that works, they need to know if it works all the time. If it doesn’t, let’s talk about it and find out how to make it work all the time.
To be continued … meaning there is always more stories to tell – a lot of stories by a great number of teacher and students. I would hope that framing teaching math as a way that teachers and students can share math stories will reframe some of the classroom conversation.