By Sue O’Connell (@SueOConnellMath), author of Putting the Practices Into Action.
It seems ironic that the first Standard for Mathematical Practice focuses on problem solving. When I was a student, math problem solving seemed to be an afterthought. Problems were at the bottom of the textbook page or at the end of a lesson. My perception was that math was about computation. Most of my time was spent memorizing facts and algorithms and solving for the right answer. Thank goodness, then, that we have revised our goals for today’s students. While I do hope they can find the answers to math computations, I know they can accomplish so much more. With guidance from the Common Core Standards for Mathematical Practice, we focus on ways to help our students develop the skills needed to be proficient mathematicians who solve problems.
But how do you teach someone to be a problem solver? That is a particular challenge for most teachers today who, like me, were assigned problems to solve but not taught to be problem solvers by their own teachers. What can we do differently for our students? As we explore the expectations in Practice Standard #1, we gain insights into the complexity of the task. Are we considering process, strategies, and attitudes as we explore ways to teach our students to solve problems?
- Process: Do students understand the process for solving problems? What do they do first? Next? How should they approach and move through a problem-solving task?
- Strategies: Do students have a repertoire of strategies to solve different types of problems? Do they understand when to add, subtract, multiply, and divide? Do they have other strategies to rely on like drawing diagrams or organizing data to look for patterns that might lead them to solutions? Are they able to choose and adjust strategies as needed?
- Attitudes: Do they believe they can solve problems? Do they persevere when problems are complex? Do they have patience when a first approach doesn’t work?
While the teaching of problem solving is complex, there are many steps we can take to foster our students’ problem-solving skills.
- Integrate problem solving with computations. Begin lessons with a problem-solving task. Whether students are adding single-digit numbers, subtracting multi-digit numbers, multiplying fractions, or dividing decimals, the problem context helps students make sense of the computations that follow. The problem gives a context for the operations and helps students better understand when and why we use each operation. In this way, we are simultaneously teaching computations through problems and giving our students experience solving problems.
- Start with comprehension. Focus on first understanding the problem, then solving it. Ask students to retell the problem. Do they understand what is happening? Can they tell you the critical data? Can they justify why a particular operation would make sense for solving this problem?
- Encourage students to visualize the problem. One part of comprehension is the ability to visualize the situation. Can students draw a diagram or show the problem using manipulatives?
- Focus on building a strong understanding of operations. Emphasize the action of operations, not the key words. Are we comparing quantities? Are we splitting a total into groups? Are we combining quantities to find a total? Students who understand the action of each operation, through experiences with many contexts that show that action, are better able to select the operation—and build the equation—that makes sense for a given problem.
- Ask deep questions. Ask students to defend their choice of operations. Ask students to justify the data they choose to use or not use. Ask students to find other ways to solve the problem or to critique the methods of others. Ask students to explain their methods and prove the reasonableness of their answers. Pose questions that prompt students to talk about their thinking.
- Get students talking to one another. Set up opportunities for students to discuss the problem, from the initial interpretation of the task through to the final presentation of their method and solution. Have students debate their methods for both accuracy and efficiency.
- Pose problems that encourage perseverance. Pose problems with multiple steps or with multiple answers. Discuss ways students may have gotten “stuck” when solving a problem and how they got “unstuck.” Praise patience, reflection, and perseverance.
Problem solving is the starting point and ending point in meaningful math instruction. By beginning with problems, we set a context for the math skills students learn, and as students are able to apply varied math skills to solve problems they demonstrate their ability to use and apply mathematics. As our students make sense of problems and persevere in solving them, they are doing the work of a mathematician.
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Sue O’Connell is a nationally known speaker and education consultant who directs Quality Teacher Development. She is the coauthor of Putting the Practices Into Action and the Mastering the Basic Math Facts series. Her newest curricular resource, Math in Practice, will be available in early summer. Follow her on Twitter @SueOConnellMath.
Could you define what is meant by "reasonableness"? Does it mean "in the ballpark"? Does it mean "check one's work"?
Certainly we would love our students to check their work (computational fluency), but I was referring to the reasonableness of their answer based on the problem situation and data. Does it make sense?
And what does "making sense" mean in the context of the problem situation. If the problem is: "There is a 2 mile relay race track and every 1/5 mile team members change places. How many members are on each team?"
If a person says "11" that's wrong but it's reasonable given the context of the problem is it not?
Hello Sue,
thank you for sharing your thoughts on mathematics education. I feel as though your approach to teaching mathematics is bit backwards however! Textbooks used to be carefully planned so that students would spend much time practicing the basics to the point of automatic recall (to me a good 80% basic facts versus 20% problem solving seems to be quite nice). After the basics were down, then it made sense to ask a problem solving question (hence them being at the end of the question list). Students who had practiced significantly with the material were then able to deal with the high cognitive load associated with problem solving.
Does it make sense to drop students into the middle of a problem solving question when they haven't mastered the basic computations? Not really. So much of their working memory will be spent searching for the answer to a part of the problem that they will get quickly overloaded, and it is highly likely that they will have learned nothing as a result (no change in long-term memory storage). We haven't even touched on the possible psychological effects that might occur!
I think it is fantastic that you are promoting problem solving, as students definitely do need to see problem solving as a part of their mathematics course. However, it makes more sense to keep the problem solving more to a minimum and use it later, after the students have dealt with mastering the basic computations.
Bryan,
I couldn't disagree more. Basic facts, number sense, and computation ARE essential to one's mathematical development. However, they are not the starting place. They can and should be developed in context, through connections, and grounded in problem solving situations. Fluency with facts is developed through experience and practice after foundational understanding is developed. Computational skills and basic problem solving should be developed in a mutual fashion. I have not seen any research that suggests otherwise.
Hi Sue. I think it is unreasonable to expect small children to understand an instruction that asks them whether their answer to a problem is "reasonable" when adults generally can't agree on what that means.
Instructions for children ought to lay out expectations as to what one would like to see. For example, if you mean you'd like to see a ballpark estimate, why not ask children to give a reason — without repeating the calculation — why they think their answer is "close"?
Often in mathematics we want students to justify their answer and that is what we mean by "show it is reasonable"– that is, it stands to reason, under scrutiny. However, this is an unfair instruction to small children as they do not have the logical sophistication to understand the rigors of analytic reasoning and argumentation. As brain scientists tell us the pre-frontal cortex does not fully develop until late in one's teens — this is, therefore, an appropriate kind of exercise for high school students, but not for primary grades.
"Explain your steps" is closer, and more age-appropriate, but what one is most likely to get is a *description* of those steps rather than a justification. But I'm not sure it is appropriate to ask for a justification of EVERY step. And after the first couple of times, what is the point in asking them to repeatedly, ad nauseum, always explain every step?
After all, this standard calls for "perserverance". I can think of no better way to squash perserverance than to mire students — after having successfully completed a problem — in pedantic and tiresome exersises of "showing reasonable". The children I know are easily exasperated by adults who, rather than saying, "Good work, Johnny, now let's move on to something else" says "We're not done yet — show me how/what/why — what is the meaning of this? Where did you get that formula? Can you do it in reverse? Show me in pictures now" etc. I know I hate that. What? why wasn't my first solution good enough? I thought I was done — you want me to do it … again?! Differently?!
Superior to all of the above approaches is "show your work".
A properly laid out working of a problem ought to function as a justification (though a child may not think of it in terms of "proof" or logical argumentation). It has the added benefit of giving you the opportunity to diagnose where a child's thinking or practice is in need of remediation. And both child and teacher can revisit the precise point in the solution where things went awry.
For this good mathematical hygiene ought to be taught. Children need to know how to properly compose mathematics on paper, and how to express themselves clearly in that form. It is not a hard skill, although it is abstract. In the past it has been taught effectively and students we received in our university classes could present work effectively. It seems a lost art, and probably the NCTM standards are at least partly to blame, for they have seriously degraded any emphasis on paper-and-pencil skills.
But that is precisely where presentation of mathematics is learned and appropriate communication of results must be honed. Mathematics is a language. It is a precise language, with rules, grammar, conventions and — yes — aesthetic considerations as well. It is the necessary language of science. It is the indispensible medium of communication in Engineering.
Advances in our society are dependent on children being fluent in that language. Yet year by year we are seeing our incoming students display weaker and weaker grasp of that language. Today half my students enter my classes fresh from high school and can express themselves mathematically on paper only in the equivalent of grunts and squawks.
And fewer and fewer of them are being successful in the technical disciplines that require mathematical fluency. Instead these spots are being filled by foreign students who come from countries where mathematical profiency and proper form in expression oneself mathematically are still bedrock principles in public schools. And well over half of the students successful at applying for our graduate program … come from overseas.
So … to "show an answer to be reasonable", please demand of students that they show their work. From a mathematical perspective, that suffices. And then skilled teachers who are cognizant of the correct form for expressing oneself mathematically ought to provide them continually, immediate feedback on the presented work as their ongoing formative assessment.
Yes, Barry, I would consider 11 a reasonable answer, because it is close to the correct answer, although in the case of that problem, number sense about fractions should easily tell us that 5 runners would be needed for each miles since five 1/5ths are 5/5 which is a mile. But many students come up with answers like 2/5 which is not reasonable. When our students aren't sure how to approach problems they often pluck numbers out of the problem and try an operation, so they see 2 and 1/5 and think 2/5. It doesn't make sense with what the problem is asking. And how can there be 2/5 runners? Or students might combine the data to come up with 2 1/5, which again makes no sense.
I think it makes a lot more sense to contextualize this problem on the number line first. We can show the students a distance of 2 and break it up into pieces of size 1/5. How many pieces of size 1/5 do we have? 10. Okay, now let's do a couple more to see a pattern. Now that we see the pattern, we can develop a rule: "flip and multiply."
Mathematics does not have to be contextualized within problems for it to be "understandable." If a teacher cannot produce a simple and clean picture to explain basic arithmetic, then I have questions as to whether or not they should be teaching in the first place. Think about how cognitively heavy just this number line is to our students! Certainly the word problem is much heavier for them: there are teams of people, a track, units of measurement, it's sunny outside, the birds are chirping, etc.
Bryan – I think we may have a different idea of the goals for math instruction. You mention practicing the basics to the point of automatic recall, then solving problems. First, I believe that problem solving is part of the "basics". Without an understanding of when and why students do math computations, they are simply memorizing procedures. In life, we don't do computations in isolation. We do them when we need them to solve problems. I am not sure why the problems need to be at a level to frustrate and overwhelm students. Isn't it possible to pose problems that give a context to the computation without overwhelming students?
In fifth grade, students learn to perform operations with fractions. They learn the procedure for subtracting fractions – but just consider if instead of practicing for 80% of the time with equations like 3/4-1/2 = n, finding common denominators, making equivalent fractions, and subtracting, they simply began with a situation like: Jan ran 3/4 mile and John ran 1/2 mile. How much farther did John run? That simple problem brings context to the equations. It helps thm see those numbers as real quantities and when they get an answer it helps them figure out if it makes sense (which may not happen when students are just manipulating numbers). Did they do the computations incorrectly and get 1/2. That wouldn't make sense because Jan didn't run a whole mile, so it has to be less than that! The problem is not overwhelming, but guides the computations. The equation makes sense through the problem.
I believe teachers are able to select problems that guide and enhance lessons so students not only learn how to perform computations, but see examples of when and why they do those calculations. And the goal is to teach problem solving, not just assign problems. Like you, we want to minimize the overwhelming feeling associated with problem solving, which many, many adults have based on their experiences in the classroom with problem solving as an after thought and little time spent actually teaching it.
I want more for our students than what I had in the classroom. Change is not always a bad thing. I have watched children become more engaged in lessons because problems were posed, catch errors in their computations because the context helped them realize the error, be able to express why they were choosing certain operations, and gain confidence in their math abilities.
This does not mean that students do not have time to practice computation skills, of course they do, it simply means that problems and computations are connected. The equations are the symbolic representations of the problems. Why wouldn't we want students to see them as connected?
R. Craigen –
Thanks for your comments.
I agree that "show your work" is a very powerful way for students to express what they have done. It allows us to see the the path they took to get to the answer, although I have found that it often requires interpretation from the students as parts of their work may be missing and understanding why a student chose to take a certain approach may be hard to identify. Still, it is incredibly valuable and should be nurtured and developed. Asking students to explain the steps they used to solve a problem is also a wonderful way to see into their heads to know why they solved a problem in a specific way. There are many questions that can yield great insights. "Is it reasonable?" is just one of them.
With that said, I do believe even very young students can explain reasonableness, although it is at their developmental level and certainly not with the same reasoning as older students. When a kindergarten student is posed with a problem like 3 cookies on the plate and I ate 2. How many are on the plate now? They can tell you that 5 doesn't make sense because you ate 2! They can show you with their fingers or cubes, that 5 is more than 3 and you ate some so it shouldn't be more. They will count for you to show you 5 comes after 3 or draw you a picture and show 5 is more. They are using their primary reasoning skills to justify the unreasonableness of that answer.
While you describe students who become frustrated by having to justify, I have seen the energy it has brought to many classrooms. Students enjoy explaining, debating, and justifying. It is up to the teacher to make the process productive and engaging, The goal is never to go on and on ad nauseam.
Stay tuned for Practice Standard #3 for more discussions about this.
"In fifth grade, students learn to perform operations with fractions. They learn the procedure for subtracting fractions – but just consider if instead of practicing for 80% of the time with equations like 3/4-1/2 = n, finding common denominators, making equivalent fractions, and subtracting, they simply began with a situation like: Jan ran 3/4 mile and John ran 1/2 mile. How much farther did John run?"
You seem to be advocating zero percent rather than 80%. All proper textbooks, if you can find them in K-6 anymore, always start with mastering basic operations before complicating things with words. However, the things you complain about when you were growing up haven't been around for at least two decades. When my son started second grade 11 years ago, they switched to Everyday Math. Before that for many, many years was MathLand. Neither of these are anything like your 80% description. In fact, the problem with these curricula are low expectations and the idea that math is some sort of generic thinking process rather than mastery of skills. You have to understand and master the skills and understand how to apply those skills to problem solving. You don't properly learn skills via problem solving that require those skills.
Another modern issue is an over-emphasis of what goes on in the classroom and not the day-to-day overall process of building and validating skills with homework. While group work might be exciting (for some), a process has to be followed to ensure individual mastery of skills. Motivating why one needs to learn the material in each new unit is important, but that should not drive mastery of skills in a top-down fashion.
BTW, when I was in school in the 50's and 60's, it was NOT just rote learning. One cannot be successful in math with rote learning. One can have partial or inflexible understandings, but that's not rote. One can attempt to do new problems by following exactly what was done on another problem, but that rarely worked. Mathematical understanding comes from mastery of skills, and understanding of problem solving comes from applying mastered skills.
CCSS now institutionalizes K-6 as a NO-STEM zone and talk of understanding and problem solving will not make that go away. Those properly prepared for algebra in 8th grade will be those who have parents or tutors who ensure mastery of basic skills outside of school. What is currently done in K-6 schools is a full inclusion process, not a feedback loop that ensures mastery at any particular level, especially a STEM level.
Hi Steve. Your No-STEM zone comment made me chuckle. It sounds like it belongs on a Fox News segment. I don't know of a district (CCSS or not) that doesn't teach math it everyday for extensive long amounts of time and since the M in STEM is math… I can't agree that STEM is being avoided.
You're right that some students can be successful with rote memorization of procedures. Many cannot. My only evidence is the large number of adults who can't balance a checkbook, make change, compute with fractions, calculate a tip, figure the interest on the mortgage, ….. (this can go on for some time).
John, I don't think you read what Steve wrote at all. He said that it was not JUST rote learning, and that rote learning all by its lonesome will not be beneficial. You have set up a straw man argument, and have backed it up with no significant data (just "feelings"… hey kinda looks like the math we are teaching our kids nowadays). If we are comparing useless data: most of the adults over 30 that I know are quite good at all of the above topics and it's the students we are beginning to see at the university level (16-20 y.o.) that are not well-versed in these topics. This is likely due to the fact that their mathematics education is failing them by not allowing them to master the subject.
The highest level ("distinguished") in the PARCC implementation of CCSS only means that one will "likely" pass a college algebra course. "Highest." This level of expectation starts in Kindergarten. This is nothing really new since NCLB, but it does clearly define and institutionalize the low expectations. I was able to get to calculus with absolutely no help from my parents. That is impossible these days even with "math brain" kids. I know. I had to specifically help my son with basic skills in K-6. Incredibly, his school sent home notes telling us parents to work on "math facts." The rich get mastery and the poor get platitudes about understanding. Educators don't like tracking in K-6 so it's hidden at home and with tutors, and what they work on is mastery of skills. Meanwhile, K-6 educators never ask the parents of the best students what they did at home. Go ahead, ask us. We are working on basic skills.
How do educators feel when so many content experts tell them that they are completely wrong? I wanted more in math for my son than what I had when I was growing up. Then I found out that our schools used MathLand, and now Everyday Math. They got it completely wrong.
Hi, Sue,
Thanks for these good reinforcers for optimal teaching. I especially agree with the tenet to embed problem solving within skills. To a first grader, “There are 10 balloons at a party. 2 popped. How many are there now?” is a much better way to begin subtraction than a number line or fingers. And to a fifth grader, “There is a 2 mile relay race track and every 1/5 mile the team members switch. How many members are on each team?” is a much better way to help students visualize division of fractions than the standard algorithm. Conceptual understanding is important, and many adults have no idea exactly why we invert and multiply to divide fractions.
One way that I build reasoning into lessons at the 8th grade level is to give a different word problem to each small group of 2 or 3 students. They have to solve the problem with different representations (graph, equation, table, words) and then present their poster (old desk calendars) to the class. The teacher can then ask “Why did you…” or “Explain why your graph…” Other kids aren’t bored b/c the problem is new to them. This works well at many levels; I am a math specialist for gr. 1-8.
Sue, I want to weigh in on this discussion, having read your post and comments of others.
You say that you want to begin teaching skills by putting them in a problem context. There is nothing wrong with a sneak preview of the kinds of problems to be solved, such as often appears at the beginning of chapters of math textbooks at the secondary level. It provides a brief example of where the skills to be learned will be applied. But make no mistake, the skills need to exist before they can be applied! I am about to begin a chapter on solving systems of equations using matrix row reduction. It is completely unreasonable to expect my students to do this when they have never been instructed on how it works. There are other ways to solve systems of equations, but the algebra of matrices is quite fascinating, and it mirrors the solving of a regular equation (such as 3x=7) very nicely. If that skill isn't there in the first place, then the algebra of matrices will make no sense at all — hence, no understanding on the part of students. Calculators are programmed to do this sort of thing. However, if students don't understand what it is to begin with, there is little hope for them to develop deep understanding without owning the skills that lead to it.
I agree that comprehension is critical. I sometimes find that students do not comprehend what they read, and often cannot produce an appropriate diagram to go with the problem. For example, when using indirect measurement, if one is using the length of a shadow and an angle of elevation to determine the height of a tree, it is not a good thing when students draw the right triangle with the shadow on the hypotenuse! Sometimes students are unable to take what they have experienced in life (e.g. walking on a sunny day and noticing that one's shadow is on the ground and not floating in the air) and transposing it into a diagram on their paper.
Understanding of operations is big, particularly of inverse operations, if students are to be able to learn algebra. They need to know that one can undo adding 3 by subtracting 3. They need to think of fractions like 3/5 as multiples of 1/5, so that the numerator is a counter and the 1/5 is the unit fraction. In other words, 3/5 is three 1/5 quantities. When I see students do 3 times 1/5 as 3/1 times 1/5, I know that they do not comprehend what it means to have 3 times 1/5. They can count 1/5 quantities, so that 1/5+1/5+1/5=3/5, but multiplication is another thing. Repeated addition is just not always appropriate.
Moreover, if a student is trying to factor the trinomial x^2-2x-24, there is not time to take a bunch of manipulatives and line them up into rows of 2, rows of 3, rows of 4, etc. Students MUST get away from the use of manipulatives for this purpose quickly, and move on to the abstraction of factoring 24 in several ways, quickly. Otherwise, they are crippled for learning algebra.
My students will sometimes say that their answer is right "because the calculator says so." I cannot think of a poorer excuse for declaring something to be right, even when it clearly makes no sense. For example, when students need to evaluate x^2 for x=-2, they will enter into the calculator -2 and then the squared key, and they will get -4 (because that's what they told the calculator to do!!). The can look me in the eye and say that when you multiply two negative numbers together, you get a nonnegative result, and yet when the calculator suggests otherwise, they become believers in the -4 result. Knowledge of reasonability comes with years of practice with computation so that students have confidence. You mentioned student beliefs — when students believe in their skills over the result from a calculator (in which a calculation may have been entered in error), I will feel more certain that they have what it takes to be problem solvers.
I grew up in the 1960s, and was taught math by explicit instruction all the way through, including some of the "New Math." My K-6 teachers taught clearly and with understanding of the operations, and every student in my class knew how to compute. They all had their basic facts down and they all could add, subtract, multiply, and divide without drawing pictures or a lattice or any other kind of aid. I went to a rural school and there were 88 in my graduating class, so this was not an affluent community with lots of tutors available. We all learned math at school without any outside help, the internet, tutors, Kumon, or Kahn Academy — and even without much help from parents. It was all accomplished at school. In 8th grade, we spent an entire year on problem solving — percents, ratio and proportion, etc. By 9th grade, algebra 1 was a breeze, because of the strong preparation that occurred ahead of it. We owned the skills that led to learning algebra!
I think that math education is moving in a wrong direction, and would love to see a return to strong skills. If we keep passing students forward to junior high and high school with weak computation skills, we are slamming the door in their faces to learning algebra and thus, to going to college. And we are also barring access to the coveted STEM fields.
I think you bring up a very valid example of linear algebra Joye. And I like how you explain that operations with matrices and matrix equations do make sense if the foundations in algebra are strong. The major problem I keep seeing in mathematics education is "well our calculator can do it, so it is not important to teach." This is further from the truth! This statement is an excuse to remove math content from the curriculum to introduce more time for "wishy-washy" and "fuzzy" math where students are expected to solve with multiple strategies or waste copious amounts of time problem solving an open-ended question. Thus, students are walked through yet another grade level not learning anything of substance that can be used to solve a real mathematical problem!.
Hi Sue,
As an Elementary Mathematics Instructional Leader and mother of three, I couldn't agree more with your comments on Problem Solving. Just today I was working with a group of second grade teachers on posing and solving problems that align with the Problem Solving Structures in CCSS. We discussed, at great length, the importance of students understanding the problems they would solve as Mathematical Comprehension. The teachers were excited that their students were gaining an understanding of addition & subtraction through the purposeful use of problems. We worked on representing the problems mathematically so all students could access the math involved and we planned follow up questions to support student learning. Your comments would have been the perfect closure to our Professional Development!
As a mother, I have seen the benefits of my children learning to solve problems independently. They have confidence in themselves as learners and are risk takers as they work. When they struggle with a problem, I encourage them to stop and think about what they know or understand about the problem. Most times it just takes a gentle prompt to get them thinking and on the road to a solution. They are able to communicate about the math they are doing because they understand the problems and the computation required.
Sue,
I wholeheartedly agree with your comments regarding problem solving and computation. When we think about the problems we encounter as adults in our jobs and beyond, usually the most challenging part is making sense of the problem and sticking with the problem long enough to actually solve it. Whether I'm analyzing data to find trends in website traffic, figuring out the least amount of expensive tiles I will need to redo a bathroom, or deciding on the best ways to save money in our family budget, computation is usually the least of our worries. Kids will have ample time to develop computational skills and fluency in elementary math classrooms – but the problems they solve will help students know why they are learning mathematical operations.
Sue,
Thank you for your blog post on MP 1 and problem solving. It's interesting to think about problem solving in mathematics compared to problem solving in the 'real world'. We are surrounded by problems every day. Before we attempt to solve them, nobody walks us through the procedures first to get there. We consider what we already know, use the tools that we have, and if it doesn't work, we try something different. My belief in the role of mathematics is not to create human calculators but to develop thinkers and problem solvers. By starting off with computational procedures, students are limited to 'inside the box' strategies. They practice a rote procedure and then apply that procedure to the word problems at the end. This does not develop the skills we need to solve problems that are isolated tasks.
The other day, after getting home, my 3rd grade son wanted to know how much free time he had until bed time – a real world problem. Instead of walking him through multiple examples of solving elapsed time problems, I asked him questions about what he already knew about time. "What time is it now? When is your bedtime? What do we need to do with this information?" He had all the components he needed to apply them without me telling him how to.
This also allows for students to develop their own strategies for solving problems. When we begin with procedural computation we unintentionally cause students to think that it is the only way to solve. Conversely, students need opportunities to see that there are multiple approaches for problem solving and arriving at the coveted correct answer.
Mathematics is not computation in isolation. Context provides us all with the foundation on why we are doing what we do. This is certainly a shift in how I learned. As a math specialist working in nine school districts, I've seen a significant increase in student success and perseverance when they are encouraged to solve problems and build computational fluency with context while applying the tools that already exist in their mathematical understandings.
It took the best (and brightest!) mathematicians in history standing on each other's shoulders to arrive at the body of mathematical knowledge we have today. How on earth can we expect a middle school student to do the same in 45 minutes? Are you essentially asking students to reinvent the wheel? If so, using solely this approach to mathematics in the early grades you will find students to be vastly under prepared for college level mathematics and any mathematics related field. Is that what the author would want? Less students in mathematics related fields?
I'm trying to understand how the kids are able to develop a sense of what is reasonable. At what age is there an expectation for this? Elementary students are not known for their ability to assess what is reasonable. They still believe Santa can travel fast enough to reach every house in the world. They don't even dress in reasonable attire with shorts on a -6 degree day in the middle of winter. Doesn't determining what is 'reasonable' come with experiences? Providing lots of repeated situations, or practice and repetition is what givens humans the ability to determine what is reasonable.
Hi Sue~
Everything that you wrote about reinforces Practice #2 in NCTM's Principles to Action. I fully agree that effective mathematics teaching incorporates tasks that engage students, allow for "productive struggle", and have them discussing various strategies. Problem solving is essential to build mathematical understanding.
As a student in the 70's, I agree that problem solving seemed to be an afterthought. It was that way for my own children (who are in their 20's) as well. Their experience was to learn computation and then solve a couple of problems at the bottom of the page.
As a Mathematics Support Teacher, I have focused on guiding teachers through the shifts in instruction and helping them to turn their classrooms into an environment of problem solving and rich mathematical classroom discussion. Teachers carefully select tasks that build on and extend students' mathematical understanding. They ensure that students comprehend the problem and use good questioning to support student thinking. They use various number routines to promote perseverance and reasoning.
It is essential for students to apply the computation and procedures to real world situations. Their careers will require "thinkers" and problem solvers. This cannot happen without developing understanding through problem based learning.
In Shift Happens, Karl Fisch and Scott McLeod point out “We are currently preparing students for jobs that don’t yet exist . . . using technologies that haven’t been invented . . . in order to solve problems we don’t even know are problems yet.” As educators we can only do this be teaching students to think and solve problems!