I love language almost as much as I love numbers, so I tend to get excited about vocabulary. We talk about Latin roots in my class, and why we use the terms we do in math class. I have always stressed “constant” and “variable,” but I really got excited when I saw how contrasting the variety of possible scale factors in a table with the constancy of the “constant of proportionality” highlighted the concept of a mathematical constant. By talking about proportional tables and asking questions like “What’s the same throughout the table and what keeps changing?” my students have already heard me say the word “constant” (with something I could point to) more times than all of last year. When we start to write equations, it will be the word “variable,” and I expect that the idea of a variable will make perfect sense in the context of the proportional relationships and tables. And by spending as much time as we are going to on proportional y = ax equations before we move on to y = ax + b, students will have a deeper understanding of the subtler differences between constants, constant terms, and coefficients.
In my prealgebra classes I have used my own curriculum for a number of years, but it seemed that I was always re-writing or re-sequencing. This year I decided to try out the Illustrative Mathematics curriculum for Grade 7, and let someone else make the decisions for me. So far I think I put myself in the right hands. These are good problem-based units, they stretch the students in ways that I realize I have been shying away from, and they develop multiple threads of skills at the same time. Sometimes that sequencing is obvious, sometimes more subtle.
The first decision I made for myself was in fact a mistake: I was concerned about time for some of my own topics and activities that I wanted to keep, and I thought I could do without Unit 1. Scale drawings and Proportional Relationships seemed like they would be covering similar ground, and scale drawings weren’t as critical a topic as proportions. But the two units together develop the two kinds of relationships you find in proportional relations:
Scale factors are the relationships between pairs of ratios and the constant of proportionality is the relationship across each equivalent ratio … which stays constant for each row of the table. Unit 1 develops the idea of scale factors and Unit 2 applies scale factors to tables and adds the constant of proportionality. So they really build nicely on each other.
Fraction skills are also a concern of mine (more on that later) and Unit 1 is an important part of that sequence as well. In Unit 1, the students multiply by unit fractions and are reminded of the equivalence of dividing by 4 and multiplying by ¼. They also find 3/5 of a number or 2.5 of a number, and are reminded of reciprocals. In both Unit 1 and 2 IM ventures into fractions gently, introducing a skill with whole numbers, then using numbers that the students can make sense of mentally or with relatively easy calculations. I have tried to do that myself in the past, but it is nice to have the details thought through for me, and so far it has worked well for my students.
I have been able to make things work this year with a little extra time and attention to scale factors, but I will be using Unit 1 next year.
I stumbled across some notes I made years ago - easy to say, hard to implement...
I am thinking hard about this, and took some time to write down my current thoughts, some of them arriving just now.
Many years ago
Students can’t remember (or use flexibly) math procedures because they don’t understand them. If they explore and learn the concepts, they will remember the procedures that come from understanding.
Problem: they still forget the procedures
A couple of years ago
Memory and recall of facts and procedures happens in a different part of the brain than theory and conceptual thinking. We should be talking to the two parts in different ways. Once we arrive at the procedure through conceptual work, discussion and synthesis, then we follow that up with practice to engrain the procedure. The cognitive knowledge has been established and should remain, because it “makes sense.” Maintaining the fact/procedure part just requires repetition. Revisiting the theory (through reminders and mini-lecture, not re-engagement in thoughtful questions and problems) during practice uses up time and cognitive load, making practice harder.
Problem: That doesn’t seem to move knowledge forward overall. Eventually the procedural knowledge becomes disjoint from the conceptual knowledge. Also, I suspect that the way I revisited the theory, in the form of reminders from my knowledge, also tended to nurture “facts” (chunks of conceptual knowledge) that were disjoint from real understanding.
Maybe both procedural* and conceptual knowledge need active maintenance. And they should be integrated. My scales practice certainly benefits from thinking about the theory behind the scales – not at the same time, but consecutively… what is the major Ionian scale? If I don’t know that, that becomes my lesson for today... then I run a few scales based on that. A few minutes thinking, a few minutes practicing. If I do remember the scale, then I can go a little further… practice in C, now figure out how to practice in D… then practice in D.
*[I think we need to distinguish procedures (how to multiply fractions or get the area of a triangle) from facts. And maybe distinguish particular types of facts: multiplication facts, labels for mathematical objects (angles, denominators), labels for mathematical concepts or more abstract objects (integers, linear equations, commutative property)… worth thinking more about.]
So what are best practices for maintaining procedural knowledge? This goes back to the fact that symbols and quantity are processed in different parts of the brain. We want the symbols to engage thinking, not shut if off. How to do we practice procedures with the concepts still engaged, feeding the retention and integration of both?
For conceptual knowledge? Do we practice conceptual knowledge? Or are we really practicing a third type or a hybrid sort of knowledge, factual knowledge about concepts/relationships (multiplying by numbers between 1 and 0 have a common effect – making the other factor smaller)? Should practice of procedures be included when we practice concepts? Or how about the same type of short burst – re-engage the concept in a thoughtful way, then have students summarize in a way that is meaningful to them. Now what we are demonstrating and practicing is generalization, the existence of conceptual knowledge itself, which the students probably understand less than I think, and the variety of ways to do that: with language, with pictures and diagrams, with algebraic symbols, with examples, etc.
Do slogans and posters have a role in helping students to chunk and integrate new conceptual knowledge and help, or do they tend to divorce the fact from the understanding? Answer: they won’t be divorced from the facts if the students make them up. Give the students a chance to make their own procedure cards, posters, slogans, reminders, etc. (Embarrassed to be just now seeing the actual value in that, but there you go. I guess I had to figure that out for myself - meta; and isn't it cool that "meta" is a cultural concept now?).
Appendix A - Ideas for maintaining procedural knowledge-
I like the idea of little chunks… pick an idea behind a procedure, revisit that in a thought-engaging way, with a clothesline or what do you notice or open middle/open beginning (create the problem)… then quick chunk of repetitive practice of that idea… if possible with reference to the idea… maybe the procedure is written on the board next to the work that we just did.
Represent a concept visually and have students write the symbols for the problem and the answer.
Practically, what happens when some students already have learned the procedure and understand the knowledge? Perhaps just tweaking my attitude a bit: rather than asking them to explain it or go further with a little bit of chip on my shoulder, saying out loud “You can always go deeper” while really thinking “Why can’t they all just be in the same place for once?!” or wanting to prove that the student doesn’t really know it all – I can get really excited about what they know, and maybe figure out a better type of extension than representing it a different way, or explaining their thinking…
What is a multiplication problem you can’t do? What is a multiplication problem you think I can’t do? What do you notice, what do you wonder? … Is there some category of activity (3 or 4) that they might look forward to doing after a quick session of practice that would extend the concept while I help other students who need conceptual engagement earlier in the process? (I have decided this year to try to avoid the term "struggling").
BECAUSE… procedural practice should be concept driven… students should only be practicing the procedures that come out of the concepts they understand.
This is connected to this Twitter stream: twitter.com/crstn85/status/1016730819912708096
We accept statements like “If something is understood … it can’t be forgotten” at face value, but I think it is a much more nuanced relationship. For starters, the “something to be understood” is typically actually different than the thing we want to remember.
If we are studying the area of triangles, I want my students to understand that area means counting squares, and that seeing triangles as half of parallelograms and rectangles can help find the area of triangles more precisely. What I want them to remember is b·h÷2. If we are exploring exponents less than 1, the thing I want them to understand is what the notation means and how we can extend, generalize and apply it. What I want them to remember is that anything to the zero power is 1 and negative powers give reciprocals of positive powers.
My experience in the classroom tells me that conceptual learning is necessary but not sufficient for procedural fluency. Helping students understand a concept doesn’t automatically produce that procedure or algorithm that is useful to remember. And my experience with math and music tells me there should be a blend of understanding and procedural fluency. Algorithms are not the unfortunate byproduct of math education. They are a valuable part of it. But to bring them about from and connect them to learning and understanding (and not unintentionally overvalue either learning or fluency), I have to want it to happen and I have to do some intentional stuff to make that happen. What? How do the two interact with and support each other? Or how might they work at odds (an algorithm introduced too early… or is it just so clearly established as the “real” objective, since that gets students the right answer)?
More acutely, for a significant number of students, there seems to be a divide between mathematical insights and algorithms: no real connection ever arises naturally between them in their mind. Unintentionally clumsy attempts to force the issue and create connections for them just results in the confusion that has hindered the attempt to make understanding central to math classrooms – the model made so much sense, we had such a great discussion, how come they still keep asking me to tell them whether or not they should add the denominators? Or we just knowingly or unknowingly proceed along parallel but separate tracks in math class: problem solving and memorizing. It seems to me there is a lot to clarify here to be more effective teachers. What is the nature of that divide? Is it bridgeable? How?
AND when it comes to remembering, again I think we have dangerously conflated two things. Students may remember what they understood for a lot longer than they will remember the procedure. Or at least the memory is constructed and reinforced in different ways. Back to the area of triangles, after we have discussed and justified the formula, the next day fully half the class will go back to drawing rectangles around triangles and cutting them in half. If you ask them the formula for the area of triangle, they have no idea. No judgment here about which is more valuable, just an important fact: remembering one does not lead to remembering the other. We want them to know/remember/be fluent/demonstrate expertise with both. So we need to understand how to promote the memory of concepts we learn (intuitively this seems to be automatic, which may be the source of some of the confusion) and the memory of generalized algorithms and rules that come from those concepts.
I would imagine there is some research out there, I just don’t know where it is. For instance, part of the organizing principles for Illustrative Math discusses using practice to develop procedural fluency. I wonder if they used recent brain and learning research to help design their practice structures, or based it more on just common sense and typical practices. If so, I think we can be more deliberate than that about what makes practice effective with lasting effects, ie. how much? how often? in what ways is it useful to tie it to the conceptual learning and what are the challenges there?
Sometimes we are so deep in a particular worldview that we can’t possibly imagine there is any different way to look at things. Or even if we believe there is another way, what we are imagining is really just derivative of what we already see. You have probably heard about the differences between a U.S. mathematics classroom and a Japanese mathematics classroom – in particular, the amount of time students in Japan might be left to struggle with a problem on their own. And while the following might seem obvious, this morning I really got them (anyone remember “grok” from Stranger in a Strange Land?) – I really grokked it.
I was doing tai chi. And I was thinking that there probably things that my tai chi teacher wouldn’t want to teach us directly, because when he saw us doing them, he would know that we had figured something out, that we were experiencing the positions correctly. And it occurred to me that I do the same thing in my classroom. There are things about math problems that I am always pleased to see students figure out on their own. When they do, it tells me something about what they understand. And there is a huge difference between knowledge that students gain on their own and knowledge that I, or other students, give them. The former directly indicates growth in the student’s capacity to think and problem-solve. The latter may also contribute to that, but the connection is less direct. And, I think, only the former contributes to the confidence and growth mindset that we know is an important factor for success in anything.
So in math class, there are times when I am deliberately not explaining something to students that I know would be useful to them, in order to allow them to figure it out on their own.
Duh. So what’s the point? I think lots of our students' parents nod their heads when we say at Back to School Night that we aren’t going to explain everything to their children, that the students need to figure things out on their own. But in practice, I think many of those same parents would be confused or even stunned to hear a teacher say “Steve is still really challenged by finding the areas of triangles. I am working on some good problems that I think will stretch his thinking.” We know in the U.S. that figuring things out for yourself is a good idea, but we still have the expectation that teachers explain things, and the best teachers explain things well, meaning quickly and painlessly. This is a deeply rooted priority, and what appears to be a superficial gap between teacher as explainer and teacher as challenge-giver and coach is really more of a gaping chasm.
I write this to warn young teachers about two possible results of this chasm, in the hope that making them explicit may make them more manageable. First, you may forget the difference. In an effort to maintain relationships with parents, administrators, and students who have the “teacher as explainer” model in their heads, you may lose sight of the value of letting students struggle. That does not mean you are deserting the cause, or that you are losing ground. The challenge of managing how much struggle and how much explaining you are going to do in your classroom is just part of the job. You will have to make choices about that every day, they won't be perfect, and you will get better at it.
Second, just as you are doing the very thing that you think is the most valuable for students, when you see yourself as teaching at your best, they may be resentful and disappointed, and see you as an inept, ineffectual teacher. That's the students themselves, their parents, even other teachers or administrators. The gap between the two approaches is both vast and nearly invisible. Everyone seems to be nodding their heads at the same time, but the pictures in their minds of what is good teaching are vastly different. You may want to think about how to talk to both parents and students about the value of struggle, and about why you are going to challenge the students with difficult problems without providing a clear explanation or walking them through the steps. Highlighting your instructional decision and outlining the carefully thought out reasons behind it may go a long way in bridging the gap. Find allies who are eager to support you at your school, conferences, or online (check out some of the My Favorite blogs or follow them on Twitter). Take time to reconnect with the big picture, however that works for you.
I speak from experience. I am always little embarrassed when this suddenly becomes clear to me ... again. It always feels like something I just figured out and something I just remembered at the same time. But that's how it works. We are all captains of our own ship. The course may not always be known, but we keep sailing and doing our best.
***Note: I am aware that the terminology for Common Core states that "sequences" are a high school topic, while "patterns" can be studied starting in elementary school. I use the term "sequence" in an informal way, interchangeably with "pattern".
Starting with the 7th grade, we ask students to develop an understand of the relationships behind linear expressions. We ask them to write linear expressions and solve linear equations from real-life situations. Whenever I did this work with my students, I found that their understanding was often very shallow and disconnected. For instance, with the table on the left, my students could easily write the relationship as 3x. The table on the right was more difficult.
I realized that the students were not seeing the numbers in the right hand column as a sequence, and that the kinds of questions I wanted to ask them required that they be able to imagine that sequence and reason about it.
I don't use a textbook for my prealgebra class. I tossed the textbook out because it finally dawned on me that the most important idea in the year was just a one-class two-page spread (back in the days of that kind of textbook). That was the lesson when students were asked to write a function for a linear table. Linear functions seemed to be the concept behind 70%+ of the book: evaluating and solving linear expressions, graphing equations, and ratios and proportions; and this seemed to me to be the essence: understanding linear functions as a pattern of steady growth or decrease. As I struggled to find a way to have students make the connection between the patterns in the numbers and the functions, I tried The Pattern and Function Connection by Fulton and Lombard, the Creative Publications' Algebraic Thinking: First Experiences, and other places where students were looking at visual patterns, making tables and writing functions. (This was long before Fawn Nguyen had created Visual Patterns).
The students could see the patterns in the visual sequences, and that made it easier to write functions from the shapes, but writing a function from a table was another matter. As I wrote above, they saw how the numbers in the right column (the range) were increasing or decreasing by a constant amount, but they couldn't understand why that appeared in the function as multiplication, not addition or subtraction. They didn't see how the function could both describe the pattern and predict other values when there were gaps in pattern, if x = 9, 32, or 105. And there was another trouble spot. When we got to graphing linear functions, the students could find the rise and the run, and they could even use slope and the y-intercept to write a function, or find those values from the table, but it seemed those connections were just correlations, and they weren't really seeing the pattern of growth behind them. They were just numbers.
That's when I figured out that the students did not see the numbers in the range as a sequence at all. They were focused on the (x,y) number pairs. Or if they did see the sequence, they had a difficult time extending it mentally, or filling in gaps. The students needed time to look at linear sequences and reason about them, find missing terms, and find connections between the generating function and the sequence. Using Dan Meyer's metaphor, if linear functions are the aspirin, then missing terms in a sequence should be the headache. I also realized that I should start with just sequences, lists not contained in a table, then introduce the concept of term numbers, then combine the term numbers and the sequence to make a horizontal table, so that the students could develop this way of seeing and understanding linear patterns and graphs as a pattern of growth in constant steps.
Proceeding in this linear fashion :), my students develop a much deeper understanding of linear relationships. I do a lot of work with situations that are linear - writing linear functions, exploring sequences, filling in tables, etc. - before we start writing linear equations and solving. You might say that gives the students a context for solving equations, but I think more importantly it gives the students a mathematical model and a tool that allows them to do independent thinking about all of the above.
I hope at a later date to post some of the materials I use with my students for exploring linear growth situations and functions. You can take a look at some of the sequence activities I use in the powerpoint I mentioned in my previous post. The powerpoint outlines a progression of increasing challenge that takes place over a couple of months for me. You can use it at any pace that works for your students. I use sequences the same way one might use Visual Patterns: present a couple of examples at the start of class once or twice a week, have a good discussion about what people did and noticed, then extend the ideas to a slightly more challenging problem the next time.
I have been thinking a lot about discipline recently, and I have also been watching a lot of baseball. I occurred to me that the outcome of a class is a lot like the outcome of a baseball game, or even the microcosm of a single at-bat. Success depends on the performance quality of the batter, the pitcher, and a good dose of luck. And in the classroom it’s the same – the outcome depends on whether I have “got my stuff”, how my students are doing, and some luck. It isn’t entirely in my control.
When I set my expectations and hold myself accountable as if it is entirely in my control, I lose my sense of humor, tend to take things personally, and can fall into making power struggles out of student confusion or misbehavior. When I remember that it’s only 1/3 me, I can ease up on myself and gain some distance. I am still accountable, but that shift in perspective makes it easier to stay calm and respond with my best stuff. And when things don’t go the way I want them to, it is sometimes comforting to know I get another chance tomorrow.
Two years ago Elizabeth Statmore (cheesemonkeysf) wrote a blog post reflecting on the book How People Learn. She shared the blog with us in her session at TMC17, and I have been thinking a lot about the post and some of the comments she got.
Blaw0013 wrote: “I cringe when the term "discovery" is used. It suggests to me that the teacher has a certain way of knowing or thinking in mind for the student to use. This seems to me to be a coercive way to interact with people. C. Kamii uses "re-invent" which seems better to me (don't know why). I most prefer invent--maybe because it is my way to remind myself to allow students to make the meaning that they do, it does not need to be mine.”
I have struggled with this question of how free the thinking should be in an ideal math class, and I think the first thing I realized is that I don’t teach in an ideal math class. There are time constraints, and there are people constraints. And I have a curriculum, ideas about what concepts I want the students to work with, a broad sequence that I think helps to build understanding in a productive way, and some goals for insights I want them to have.
I think of that as “mining student thinking.” I agree with Elizabeth: “I don't believe in playing ‘guess what I'm thinking’ because I find it psychologically and emotionally insulting.” I would add exhausting and usually discouraging, both for me and the students. I pose questions that highlight the ideas I am interested in, and then I accept whatever work they do in response, again highlighting the work I see as most pertinent. So it feels like a good balance of my leadership and their initiative. If something comes up that looks like an interesting detour, and we have time (that’s just the reality) then we will pursue it.
I also give the students opportunities to explore and investigate. We don’t spend nearly as much time in this open environment, partly because of the time, and partly because some students (I would say most) really struggle with the lack of structure. Clearly that is because they don’t get much practice with it, but there are also students who just aren’t that interested. Maybe that is a problem with schools, but as I said, I don’t teach in an ideal school. But students investigating is just one format for getting students to think about and discuss mathematics. Engaging the students in mathematical reasoning pretty much constantly is my broader goal, and that can occur in all sorts of ways.
I would caution beginning teachers from getting too caught up in trying to create an environment where the students follow a “pure” path of discovery. Students don’t have to discover all the math that they learn. But the process of discovering math should be a regular part of both what and how they learn.