What about the related equation that determines the kids we want in math? To move students toward successful math learners, we must first understand where they are mathematically. e = evaluate. Diagnosing math understanding, though, is complicated. A general math achievement test, like a state test or typical progress monitoring assessment, only provides an indicator of general health. Is the kid about where he or she is supposed to be? If not, you don’t know why. Since a general assessment covers a broad range of material, you typically only have one or two items related to a specific area of content. How much can you tell from that little bit of information?
Let’s look at some items related to fractions to dig into this question. Here’s one from the grade 8 NAEP test in 2007. Only 49% of students answered it correctly. The most common incorrect response was E, which, oddly is the only sequence in which both the numerator and denominator go from greatest to least, the opposite of what the question requires. However clever our error analysis, we will never know if the selection of E represented a common misconception (that smaller numbers, in this case numerators, means bigger quantities when it comes to fractions) or just a futile guess with the last available choice.
Even individual correct responses can be deceiving. We did some field research for a fraction program Tom Snyder Productions is releasing this fall. We found students who consistently exhibited their understanding of adding fractions with problems like this one. Note how the numerators are added while the denominator
remains constant. The student even shows the answer in simplified form. You might conclude from this single item that the student “understands” adding fractions. However, the same student suddenly forgot this understanding when confronted with the task of adding fractions of unlike denominators. Where did the fifteenths come from? Analyzing this second item in isolation might lead one to conclude that the student is applying whole number concepts to fractions. But that wasn’t the case with
the previous problem. To diagnose what’s really going on with this student we need to dig deeper.
What we really need, I think, are layered assessments. The high level ones -- periodic achievement tests -- act as triage. They let us know which students need more in-depth diagnosis. But what troubles should we diagnose for? I suggest targeted key concepts, like those identified by the NCTM Curriculum Focal Points or the National Math Advisory Panel’s algebra foundations. Focus on the foundations, like fluency with whole numbers and fractions. Employ targeted, deep assessments to get a true window into what’s happening inside our struggling students’ heads.
The final variable in the formula for producing the students we want is v for value. (It's tough not repeating coming up with all these variables without repeating a letter.) To get students who are engaged learners in math, we must value their effort and reinforce that their effort has value. Let's start with the valuing their effort idea. The research on praise suggests that we focus on the work not the person. It isn't that a successful kid is "smart." We tend to associate intelligence, being smart, with innate ability. If you don't have it, why do it. Fortunately, intelligence changes over time. Our brains are plastic; they change as we learn. That's the message we need to send our kids. It takes effort, but you can do it. Even natural abilities can be squandered without practice. It was great hearing Lebron James talk about his work ethic as the key to his basketball success during this year's NBA playoffs. Being tall is no guarantee that you'll be a good basketball player. It was also gratifying when my son's advisor extolled our kid's work ethic at his high school graduation. The poor kid is a victim of the research I've been following. He used to be smart. Now he's effortful. Some students will take more effort than others, but with focus and attention, much can be achieved.
The other part of the value proposition, that student effort has value, leads us to help students see that math is relevant. Mathematical and algebraic thinking are things we do all the time (check an earlier blog). And some particular mathematical skills -- arithmetic, data analysis, and statistics, among others -- have become part of being a 21st century citizen using 21st century tools, like spreadsheets. It's worth learning math, not for the grade, but to satisfy intellectual curiosity, to improve analytical thinking, and to be an informed consumer. Math has value and the effort to learn has value too.
There you go. b+i+f+a=the kids we get in our math classrooms (born+informal+formal+affect). And e+c+d+v=the kids we want (evaluate+connect+differentiate+value). Like most models and metaphors, these formulas are not the truth. But they do help us think about our students and how to improve their learning experiences.
Each student is different. Some will panic when faced with a math task. Some just need a little help. Some will move slowly. Some will move fast. Technology allows us to adjust both cognitive demand and cognitive resources. Based on performance we can dynamically make tasks easier or more complex, more concrete or more abstract. Keeping each student in his or her zone of proximal development, just on the edge of what is doable independently, is a beautiful thing. Few things are more satisfying than successfully completing a challenging task. Too easy is boring. Too hard is frustrating. But just hard enough is very rewarding (chemically, in the brain!). We like it.
At the same time, we can also adjust the resources, or supports, that a student has available to complete a task. We can offer alternative visualizations, definitions, and worked examples. Think back to my Verizon Wireless analogy a few entries ago. We can give students “the network” so that those scary math dead zones aren’t so daunting.
Having a notion of what’s inside our students’ heads can help us provide instruction that connects old knowledge with new, informal sensibilities with formal understanding. Educators sometimes use the swiss cheese metaphor when describing student knowledge. It’s full of holes. We simply need to find the holes and plug them. Unfortunately, this plug-the-holes approach only works if the new material bonds with what’s already in place. Otherwise, the plug just falls out.
Sometimes building on to the existing knowledge base will work, but it’s difficult to know what you’re building onto with every child. Instead, it can be more efficient and effective in the long-term to rebuild the structure from scratch. Provide students with the common concrete experiences that can provide the tangible foundation out of which you abstract the desired mathematical concepts. In any case the connection is key. If students see the knowledge as isolated bits of information to remember, you’ve got a cognitive overload situation on your hands. And you don’t want that.
The final piece of the formula reminds me of Verizon Wireless commercials where a cell phone user is confronted with the prospect of getting lost in a “dead zone.” The road ahead is scary. It might overwhelm most cell phone users, but the Verizon Wireless customer has a network of resources with him. He can prevail. That balance between the demands of the task and the resources we believe we can bring to it is a critical part of students’ affect, the state of mind they bring to learning math (or anything for that matter). I’ve written about affect, the work of Carol Dweck, and neuroeconomics in other blog posts, so I’ll keep this one short. Emerging research continues to show that students’ willingness to participate in the learning contract in the classroom along with their belief that their effort will matter pays huge dividends in their performance. It’s tough to do well if you don’t try. Do kids feel daunted at the task of learning math? Do they believe that it’s pointless because they’re just not good at it? Or do kids feel that they can meet the challenge? And do they think that meeting the challenge will bring benefits, immediate in the satisfaction that comes with meeting any challenge and long-term in their future academic and occupational success? This variable, often overlooked in when we assess our students, is a huge part of the formula that determines the kids that enter math classrooms everyday.
Unless you’re teaching preschool or kindergarten, your students already have a history of formal instruction in math when they enter your classroom. Did the prior instruction build on the child’s innate mathematical capacities and informal experiences? Did it treat each state standard as a separate, isolated and atomized bit of content, divorced from the other objectives. Or did it provide a coherent and connected progression of instruction with new instruction explicitly building on what students already had learned? Did the prior teaching protect students from cognitive overload by constraining new learning demands and providing time for practice and mastery before moving forward? Did the students fall victim to the math wars with overemphasis on either mindless procedural mastery or unformalized invented algorithms and approaches?
I could ask more questions, but the point is that past instruction matters. Many of the struggles students have with math come from what and how they were taught. Confused ideas about equivalence and the equal sign (=), overgeneralization of whole number algorithms into fractions, the lack of a unified number system across integers and rational numbers, and so on are really instructional issues. Without a good understanding of the models and approaches students have accumulated in school, it’s tough to make the kind of connections that can move them forward sensibly.
The variable i stands for our informal experiences and actually has two parts, i(sub1) and i(sub 2). i(sub1) refers to our informal mathematical experiences. Children have variable informal opportunities to count place settings, divide up Halloween candy, play numerical board games like Candyland, or share continuous quantities like pudding or juice.
Sadly, the variation often falls along socio-economic lines, with kids from poorer homes experiencing fewer early number experiences. It matters, just as it does in reading. Children who grow up in a home full of books, who are read to and enjoy rich language exchanges with family members come to school familiar with the alphabetic principle and the structure of books. They have a growing vocabulary and a head start in learning to read. Similarly, kids who play board games (the research is very strong here), read thermometers, tell time, count anything, share equally, and do all kinds of other informal numerical stuff have a richer number sense when they start school. They are ready to learn math.
In addition, children have all kinds of informal experiences that have nothing directly to do with math but a lot to do with their attitudes and abilities as learners. Chronic stress (like hunger or fear) early in life, for instance, may contribute to a reduced working memory capacity that in turn hinders the acquisition of certain math skills, like math fact automaticity. Early responsibilities, actions, and interactions can influence the development of self-regulation and executive function, the ability to control and manage one’s actions. Children who can monitor their own behavior are highly correlated to academic success. Put simply, students with the skills and attitudes tuned to school culture and formal learning increase their likelihood of classroom success across the content areas.
Even though we frequently hear people complain (or maybe apologize is a better word) that they just aren’t good at math, we are all, in fact, born to do math. Studies with infant humans (and many other animals) show that they can recognize small quantities, like one, two, or three, without counting, an ability known as subitizing. Google the term and you’ll likely find a little subitizing game that let’s you compare your ability to subitize versus a chimpanzee’s skill.
We’re also born with the ability to make comparisons. We can tell the difference between a lot and a few. From an evolutionary point of few, it’s not surprising that we (and, again, other animals) can quickly make quantitative comparisons. Knowing whether the odds favor running away or staying to fight helps improve the chances of survival. We’re generally not as good when the ratio gets closer to 1 to 1, but some of us are better than others. In fact, you can once again poke around on the web and find a game to test the edges of your ability to determine which of two sets is bigger at a glance. Some researchers found that students who performed better at comparing sets as the ratio got closer to 1 to 1 had histories of better performance in math in school. Interesting.
Using puppets and watching how long babies stare at something unexpected, several research groups have found that we also seem to be born with the ability to add and subtract, at least up to the number 3. Show 3 puppets and then show 1, and the baby looks puzzled. Show 2 puppets, and the baby is still puzzled. Where’s the third one? We’re pretty amazing even before we’ve had any formal education.
There’s even some research suggesting that we have some innate ability to recognize fractions and ratios. It’s tough to explore this ability, but very young children do look surprised when something like a book is hanging more than halfway over the edge of a counter, and it doesn’t fall. Maybe all the times that my son pushed his sippy cup or food bowl off the edge of the table, he was actually exploring his concept of proportionality (and gravity).
These born-with-it math abilities are variables. They are not the same for each of us. Some of us have more robust spatial awareness than others. Some can subitize larger quantities. But the natural math capacity we bring into the world is part of the equation for our mathematical identities.
I gave a brief breakfast talk at the NCSM (National Council of Supervisors of Mathematics) national conference this spring in Washington, DC, and I was gratified to receive numerous requests for the slides. The slides by themselves, though, are not that useful. I just use them as prompts for what I want to talk about. I figured I’d give a shot at trying to capture the talk in my blog. We’ll see how it goes.
I figured that since I was at a math conference, I’d work with a math metaphor. Here are 2 equations: one captures the variables that determine the students we get in our math classes; the other captures the variables we want to control to turn them into the math learners we want. Let’s take the variables one a time.