However true that characterization of elementary teachers is, we think it\u2019s a distraction. There is no kindergarten teacher anywhere<\/em> who doesn\u2019t know how to count and add and subtract, which is most of what her children will encounter during the year. And if the teacher isn\u2019t sure of the name of some particular geometric shape, that\u2019s way down in the noise of what will matter for teaching. Lack of mathematical information<\/em>\u2014even a lack of understanding<\/em> of why particular algorithms work\u2014is not the biggest roadblock in the earliest grades. The remedy might involve more courses in mathematics, especially mathematics they will teach, but we think that the key issue is not more<\/em> but different<\/em>, even for secondary teachers.<\/p>\n As we see it, what hurts elementary mathematics teaching most\u2014and hurts secondary teaching as well\u2014is some of the ways in which teachers know \u201ctoo much\u201d math without a tempering sense of what the mathematical enterprise is<\/em> and what not<\/em> to teach. We will give two examples, one from fourth grade and one from first year algebra, to illustrate what we mean.<\/p>\n <\/p>\n Fourth grade. <\/strong>Over a span of years and in several schools, we\u2019ve watched many fourth-grade teachers as they present And yet, all<\/em> of the teachers felt a need not only to point that out right away, before the unit started, but also to dive into vocabulary and other formalities. Referring to the The last statement is wrong, of course, but the big problem, in our view, is not the teacher\u2019s error, but the teacher\u2019s apparent feeling, common to all of the observations we made on this puzzle (and a vast number of other observations of other teaching) that everything the teacher knows about the situation is relevant now<\/em>.<\/p>\n Teachers need to know what (and when) not<\/em> to teach.<\/p>\n Of course, good teaching practice does involve looking for learning opportunities, sometimes milking a problem for more than what appears on the surface, but part of the mathematical preparation of teachers at all levels<\/em> must include ways to decide when not<\/em> to do that\u2014what is not<\/em> relevant at a particular time, or is essentially a diversion that will, at best, dilute the focus on an important idea and, at worst, mean nothing at all to the students. \u201cA letter, like x<\/em>, that stands for a number\u201d was certainly one of those; it wasn\u2019t needed and didn\u2019t clarify anything. The purpose<\/em> of the puzzle was to set the stage for a key mathematical idea\u2014a property of multiplication that the nine-year-olds would first explore, then apply, and only after that formalize to extend their ability to multiply. The distraction subverts (perverts!) the mathematics. Worse, because it became nonsense, it can convince students that understanding is neither necessary nor, perhaps, even possible for them. By contrast, when students are given time to solidify an idea first, naming the idea becomes useful, helping them talk about it and even indicating that the idea is important enough to warrant a name.<\/p>\n Starting mathematics lessons with vocabulary and notation seems nearly universal, even among teachers who know from their language arts instructional methods that vocabulary is best learned in context. And elementary school (and often secondary) mathematics teachers seem not to distinguish conventions and vocabulary from what can be reasoned out or understood. We\u2019ve seen that showing In our view, the fix for this<\/em> particular problem with elementary teaching is not for teachers to learn more mathematical content, but to change teachers\u2019 perception of what mathematics is<\/em>\u2014their sense of how the discipline works\u2014staying mostly within content they already know or once knew.<\/p>\n For example, how many fourth-grade teachers have students do age-appropriate research to find patterns in multiplication facts? Here is a particularly striking pattern that most teachers have never even seen. Presenting to students can be entirely silent\u2014no \u201cexplaining<\/em>.\u201d<\/em> On a number line, choose a single number like 4, draw two arrows up from it and write 16, then two arrows from its neighbors 3 and 5, and the product 15. Then start<\/em> the process from one other number (e.g., 3), writing the square 9, draw the neighbor arrows and let students call out their product 8. If students need another example to \u201cget\u201d what you\u2019re doing, give the 8 and start a new pair (e.g., at 6) leaving the numbers to the students. Keep going until kids are bouncing up and down dying to describe the pattern they see. For teachers in pre-service preparation, this is one example of what it means to do<\/em> mathematics within a territory they already know. There are many others. For teachers, this does<\/em> offer opportunities to develop new mathematical ideas, terms and notation, but if the preservice goal<\/em> is treated as \u201cmore math to know,\u201d rather than how to do<\/em> mathematics (research, problem posing, puzzling through to find results), it stamps in the very problems we see so often in classrooms. Teacher preparation cannot ignore content, but it cannot be about<\/em> content; it must be about mathematical ways of thinking, using content as the opportunity to do<\/em> that thinking. Students come to view mathematics the ways their teachers view it. That, in turn, is influenced by the mathematical experience teachers have in their preparation. Though classroom curricula also influence students\u2019 image of mathematics, teachers are key.<\/p>\n High school. <\/strong>We observed a class on graphing linear equations using the \u201cslope-intercept\u201d method. For readers outside the culture of middle and high school, this means that you transform whatever equation you have into the form y <\/em>= mx<\/em>+b<\/em>, and, from this, you produce the graph. There is, of course, a sensible and simple method for graphing an equation like 2x<\/em>+3y<\/em>=9 but on this day, the teacher\u2019s goal was the slope-intercept method.<\/p>\n So, students transformed the equation into y <\/em>= (\u20132\/3)x <\/em>+ 3. Then a 3-step procedure is used: (a) go up 3 units on the -axis and put a point; (b) from here, go to the right 3 units and down 2, and put a point; (c) connect the two points.<\/p>\n Most kids followed the procedure and produced the correct graph. Almost as an afterthought, the observer asked one student if the point (1, 2.5) was on the graph. The kid looked baffled, plotted it, and said that it looked as if (1, 2.5) was on the graph. When asked if (300, \u2013595) was on the graph, the kid had no idea how to tell\u2014it was off the paper.<\/p>\n We\u2019ve seen this phenomenon in most classes. For many students, y <\/em>= (\u20132\/3)x <\/em>+ 3 is a kind of code; from it, one obtains three numbers (\u20132, 3, and 3) and uses them to produce a picture. Completely missing was the idea of determining if a point is on a graph by testing to see if its coordinates satisfy the graph\u2019s equation. Assessments didn\u2019t detect this deficit because, given an equation, students could transform it to slope-intercept form and produce a correct graph. The goal was about procedure, so the gaping hole in students\u2019 understanding remained hidden.<\/p>\n This example might seem just plain weird to many readers, but this kind of thing happens often in secondary classrooms. There\u2019s the \u201cbox method\u201d for setting up equations to model word problems, a different box method for factoring quadratics, the \u201cswitch x<\/em>\u00a0and y<\/em>\u00a0and solve for y<\/em>\u201d method for inverting functions, and a host of other special purpose methods and terminology that have no existence or purpose outside of school.<\/p>\n Note the parallels to the fourth-grade example. In both, the teaching emphasis was on form<\/em>, one particular way<\/em> of writing and doing the problem, not on what the problem meant, which the fourth-graders naturally and instantly gravitated to and which the high-schoolers could have, too.<\/p>\n What can we learn from this?<\/strong> Part of teachers\u2019 mathematical preparation must<\/em> include an understanding of fundamental results and methods\u2014content specified in state standards. Missing, though, is the aspect of mathematics that involves research, play, experimentation, sense making, and reasoning. Mathematics is not about how much you know but about how much you can figure out with what you know.<\/p>\n The problem these stories illustrate is not just what\u2019s missing, but what\u2019s there<\/em>\u2013\u2013a view of mathematics that most mathematics professionals would not recognize. Wu[1]<\/a> has written about \u201ctextbook school mathematics\u201d as a dialect of the discipline that lives in precollege curricula. Wu\u2019s main criticism is lack of precision, sloppy (or missing) definitions, absence of logical sequencing, and missing distinctions between assumptions (again, ill-formulated) and results that follow from those assumptions. But how best to mend those flaws? In many classrooms that attempt to remediate these deficiencies, the current practice is to put instruction in vocabulary and memorizing forms and formulas first, to teach without first (or perhaps ever) allowing students to build the mathematical sense of the underlying logic. This practice has failed. Wu calls for reducing \u201cteachers\u2019 content knowledge deficit,\u201d remaining \u201cconsistent with the fundamental principles of mathematics (FPM).\u201d We would concur, but his FPM seems easy to misread, allowing undergraduate instructors to conclude that it supports what they\u2019ve always done. Wu\u2019s FPM starts<\/em> with \u201cevery concept has a definition,\u201d which is not a claim that teaching<\/em> must start that way. But it is easy to interpret as such.<\/p>\n For us, the classroom stories above illustrate something deeper and more fundamental than the \u201cflatness\u201d that raises convention to the same level of importance as matters of mathematical substance. And they are only partly about deficits in content knowledge. What they illustrate is a lacks of the perspective that learning mathematics means developing a collection of practices that help you figure out<\/em> what to do when you don\u2019t know<\/em> what to do\u2014developing the habits of mind that underlie flexible proficiency in the discipline. These classroom examples treat mathematics as a collection of special-purpose methods that allow one to perform specific tasks that are the calisthenics\u2013\u2013the finger exercises\u2013\u2013of mathematics. Practice is valuable for mastery in any field, but exercise as an end in itself produces muscle-bound results that can impede performance. Knowing how to transform an equation to some canonical form is an important skill, developed best through orchestrated exercise. But knowing when<\/em> to use a particular form is much closer to what mathematics is about. More generally, it\u2019s the doing<\/em> of mathematics that gives people an understanding of the discipline. Learning mathematical facts and methods is absolutely essential but, by itself, builds a view of the subject that emphasizes getting to a particular form, like , rather than understanding the connection between an equation and its graph. By itself, it elevates what you know over what you can figure out.<\/p>\n Is it a stretch to trace the roots of such stories back to teacher preparation? We don\u2019t think so. Yes, other forces are at play\u2013\u2013curricula, pressures from high-stakes exams, oppressive working conditions, school lore. But a mathematical preparation that focuses on the doing as well as the learning of mathematics would give prospective teachers some tools to overcome the schoolish nonsense common in commercial curricula, to prepare students for state tests while immersing them in real mathematics, and to downplay the clutter in district syllabi so that there\u2019s time to concentrate on what\u2019s really important. That doesn\u2019t mean ignoring<\/em> the district syllabi\u2014often a teacher can\u2019t. Instead, one might seek a mathematical context, topic or activity of genuine intellectual worth as a venue for presenting the lightweight clutter. Teacher educators could look seriously at school curricula, think hard about how to prepare teachers to find the mathematics within or behind the school-only terms or methods such as \u201cthe box method\u201d for whatever, or idiosyncratic or curriculum-specific terms like \u201cnumber buddies,\u201d or terms like \u201cfriendly numbers\u201d that have<\/em> a mathematical definition but appear in school with completely unrelated meanings. Children and teachers will<\/em> hear these in school, and they may sometimes even be useful in school, but they are school-only, and will never be used outside of school. Educators could help teachers learn how to craft age-appropriate research activities that respect time constraints and content requirements but help kids experience the doing<\/em> of mathematics. One way is by giving prospective teachers<\/em> such experiences of doing mathematics.<\/p>\n Teachers know that what they value is communicated to their students. When teachers come to understand and value the heart of mathematics, they communicate this focus to students even when a particular day\u2019s lesson must be about \u201cwhat you have to know for the test.\u201d<\/p>\n [1]<\/a> E.g., Wu, Hung-Hsi. 2015. Textbook School Mathematics and the preparation of mathematics teachers. https:\/\/math.berkeley.edu\/~wu\/Stony_Brook_2014.pdf<\/a> Retrieved September 15, 2019.<\/p>\n![\"7](\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2019\/09\/7xsqc56-1.png\")
<\/a>, <\/a>which their curriculum uses as a two-minute warm-up puzzle in advance of a unit that introduces the distributive property that students will apply to multi-digit multiplication. Every time we\u2019ve watched, many children blurted out that ![\"\"](\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2019\/09\/sqc-1.png\")
<\/a> had to be 8 as soon as they saw the puzzle, pleased to show that they knew the multiplication fact 7\u00a0![\"\"](\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2019\/09\/times-sign-1.png\")
<\/a>8. The purpose of the puzzle, at this point, was merely to have kids recognize that the \u201c8\u201d could be the sum of two numbers and to have them come up with several possibilities for ![\"square\"](\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2019\/09\/square.png\")
<\/a> and ![\"circle\"](\"https:\/\/i0.wp.com\/blogs.ams.org\/matheducation\/files\/2019\/09\/circle-1.png\")
<\/a>. At this introductory moment, even checking that if <\/a>\u00a0is<\/em> 8, then <\/a>\u00a0+ <\/a>\u00a0also gave 56, would be overkill because that\u2019s exactly the work students would next do on their own.<\/p>\n<\/a> and <\/a>, several teachers asked questions like \u201cWhat do you think the word \u2018variable\u2019 means?\u201d even though the term isn\u2019t used anywhere<\/em> in the lesson. Children generally had no idea. Some teachers then defined variable as \u201ca letter, like x<\/em>, that stands for a number,\u201d even though <\/a> and <\/a>\u00a0aren\u2019t letters! Why?! The teachers recognized something they<\/em> knew and felt compelled to teach the children the \u201cright way.\u201d One teacher wrote out\u00a0<\/a> as a justification for the 8 that the kids had already shouted out. One teacher put up a table to show how values for <\/a> and <\/a> \u201cshould\u201d be recorded. And after kids had offered a few possibilities for <\/a> and <\/a>, that same teacher took extra time to say that 4 + 4 — which no kid had suggested — \u201cwould not be right because then the equation would have to be written with two squares or two circles.\u201d<\/p>\n<\/a>\u00a0and asking \u201cWhat can you say?,\u201d encourages lively thought and participation even though it\u2019s a piece of notation, because its form allows kids to make sensible guesses about it. On the other hand, asking \u201cWhat do you think variable<\/em> means?\u201d shuts down logical thought. There\u2019s no context. Kids who know vary<\/em> might come up with a reasonable thought, relevant or not, but to these fourth graders variable<\/em> could mean anything. Vocabulary and conventions are needed<\/em> for clarity and precision of communication, but the mathematics is something else: logic and the inclination to puzzle through a problem and figure it out rather than the disposition to treat each problem as something for which one must first be taught a rule or method. When learning vocabulary and formulas becomes the focus<\/em> of mathematics education children move away from the skills they need to be mathematicians and they don\u2019t develop confidence in their own mathematical abilities. That is because people can<\/em> puzzle through mathematics, but what things are called or how they are notated is convention and can\u2019t<\/em> be \u201cfigured out.\u201d Children who proclaim themselves to be \u201cbad at mathematics\u201d are likely not to have seen mathematics as an exercise in logic and reasoning, and have likely not had enough opportunity to see how good they can be at that. Readers of an AMS blog know that memorizing vocabulary and formulas, while it can be useful, has little to do with mathematical aptitude, but many teachers have been prepared to think otherwise and thus emphasize those at the beginning of every lesson, giving students the false and often destructive idea that those are<\/em> the math.<\/p>\n
\n<\/a>
\nThen suggest some new research projects for the children to try on their own. For example, what if the outside pair of arrows are drawn from neighbors that are two<\/em> spaces away from the original (squared) number? Or, what if the inside pair of arrows does not come from a single number (squaring it) but comes from adjacent numbers (e.g., 3 and 4) and the outer pair comes from their nearest outer neighbors (e.g., 2 and 5)? Do the patterns hold with negative numbers? What patterns do we see if the line is numbered with consecutive odds? Consecutive squares? Consecutive Fibonacci numbers? Students get plenty of \u201cfact drill\u201d doing research projects like this, and have opportunities to describe what they see.<\/p>\n
![, <\/a>which their curriculum uses as a two-minute warm-up puzzle in advance of a unit that introduces the distributive property that students will apply to multi-digit multiplication. Every time we\u2019ve watched, many children blurted out that](\"http:\/\/blogs.ams.org\/matheducation\/files\/2019\/09\/sqc.png\"){kind=link}