| JackieB | Thinking about what math students *really* need to know. How much do they need to be able to do by hand? |
Jul 04, 2008 -
01:38AM
| JackieB | Thinking about what math students *really* need to know. How much do they need to be able to do by hand? |
| mindelei | thinks | they need to learn how to do it all by hand...then the shortcuts. (I know, not making your job any easier.) |
| JackieB | Why do they need to solve complex equations by hand? |
| budtheteacher | Yeah, I'm curious, too. |
| budtheteacher | How many of them will actually use that later? Like, for really real? |
| mindelei | says | I just think you should know how to go about doing it without the tech. |
| mindelei | says | It's the same way when I learn how to use a program, I want to know the long way around through it before learning the shortcuts. |
| mindelei | says | Then I know how to use it better. But, hey - that's just me. |
| JackieB | Why? How is it "better" to be able to solve it by hand than using a graphing calculator or a CAS program? |
| mindelei | says | It's not really about "using the equations" it's about learning to think through a problem. |
| mindelei | says | I'm not an advocate for memorizing equations - they're in reference books. But I think kids need to learn to think their way through math. |
| JackieB | How does using tools (CAS, graphing calc) prohibit problem solving? I think it makes higher math more accessible to more students. |
| JackieB | I'm not trying to be contrary here, I really want to know. |
| mindelei | says | It's about building on the basic knowledge that can be expanded by the use of tools. Plus, these skills transfer to... |
| mindelei | says | higher thinking and reasoning skills. |
| mindelei | says | I'm not saying you shouldn't use the tools, I'm just saying they need to know how to think things through which will allow them to use... |
| mindelei | says | the tools to their advantage. |
| budtheteacher | mindelei - Do they? I had someone push back at me on that lately, and it's put me in a weird place. |
| milobo | says | Students need to know basic skills as building blocks so they can understand higher concepts. |
| budtheteacher | I'm all for lots of thinking - but there're plenty of practical and real problems requiring solutions. |
| milobo | says | We wouldn't say that because we have books on CD (or iPod) that kids don't need to have reading skills, right? |
| budtheteacher | Math problems in a book that have already been answered might not be the best problems for our kids to be looking at. |
| mindelei | asks | budtheteacher to clarify "pushing back". |
| budtheteacher | Would anybody here disagree that kids need "basics"? The trick is - what're the basic tools/concepts? |
| JackieB | But if your "goal" is to solve an equation, why not just use a graph or CAS? If your goal is problem solving with math, then use the tools? |
| milobo | says | math needs to be a mix of fact based skills as well as understanding the how and why those skills are important. |
| mindelei | just wants to say what a great conversation this is...this is the kind of stuff I have been waiting for! |
| budtheteacher | mindelei - by "push back" I mean they asked me if that was really true, to think through it some more. |
| JackieB | So, a student who has not "mastered" the basics can't have access to higher math? |
| mindelei | thinks | that purpose is very important. Kids need to see how these ideas relate to what is around them. |
| budtheteacher | Push back meaning that there wasn't automatic agreements and head nodding. |
| budtheteacher | jackieb - Uh oh, I smell Papert. |
| JackieB | And wouldn't use of available tools help in that? |
| budtheteacher | Must math be a linear progression from the basic to the higher order? |
| mindelei | asks | if someone hasn't mastered those basic skills how can they appreciate/learn the higher skills? |
| JackieB | Papert has been reinforcing a lot of what I already do.  |
| mindelei | asks | What is papert? |
| budtheteacher | or can kids, when they see a need, explore the fundamentals after tackling, or attempting to tackle, a particular tough problem? |
| JackieB | And extending my thinking. Yay! |
| milobo | says | it's funny to me (a math teacher by trade) that people argue against basic skills in math and not reading! |
| budtheteacher | jackieb - Score. |
| mindelei | wishes | milobo would further clarify... |
| budtheteacher | milobo - I'll push back about basic skills in reading. Kids need to be immersed in language. That's all. |
| budtheteacher | Okay - it's more complicated than that - but learning to read isn't a linear process. |
| budtheteacher | We just want it to be. |
| budtheteacher | mindelei - Seymour Papert. MIT Professor, mathematician. Smart dude. |
| milobo | says | but if they don't have the skills to interpret and appreciate the language, can they truly grow fully in that appreciation? |
| JackieB | mindelei www.papert.org/ I'm reading Mindstorms now. |
| budtheteacher | milobo - It's a recursive process. It's why kids read Hamlet six or seven times in school. |
| JackieB | Yes, but you teach the skills in context, right? |
| milobo | says | agrees and thinks that kids that are kept out of higher math and reading because of a preceived lack of skills are being disadvantaged. |
| budtheteacher | Louise Rosenblatt would be someone else for folks to look up here - she's a theorist who talks a lot about Reader Response theory. |
| mindelei | Thanks for the link - I'll be sure to check him out! |
| milobo | says | (and having this conversation while also watching "History of the World" on TV isn't easy!) |
| mindelei | Rosenblattt sounds good too! |
| budtheteacher | It's way more complex than 140 characters - but basically she and others argue that reading a text is a conversation between a writer's . . |
| milobo | says | RE recursive - YES! and that should be the beauty of math - it's a spiral curriculum, not a linear one. |
| budtheteacher | brain and the reader's brain. What's learned is what the reader takes away and blends with prior experiences. |
| budtheteacher | So different readers will have different readings of the same text. |
| budtheteacher | Depending on their backgrounds. |
| budtheteacher | I think you could make a similar argument for math problems. |
| budtheteacher | real problems. |
| mindelei | says | budtheteacher She sounds right up my alley! |
| mindelei | feels | that each person does come away with something different - particularly in relation to symbolism. |
| budtheteacher | How interesting it would be to take a look at reader response theory and twitter or plurk - talk about different experiences. |
| budtheteacher | Each network is its own text. |
| budtheteacher | but anyway - back to math. |
| mindelei | says | That would be interesting! |
| mindelei | is | Impressed that there are now 72 responses to this plurk. |
| JackieB | No, this is exactly what I'm looking for. Keep it coming! budtheteacher, thanks for the new addition to my reading list. |
| JackieB | And regarding the intimidation factor, if we keep trying to "drill" the basics, isn't that more intimidating/frustrating than ... |
| mindelei | says | It goes back to dividing fractions. Rather than learning how to actually cut things into smaller pieces, we simply flipped and multiplied. |
| JackieB | here, let's explore this problem. |
| JackieB | So, if there are tools that help them explore this (in the context of a larger problem) why not use them? |
| mindelei | I'm not really into "drilling." I do agree that we do to much of that - no real learning. But you do need to get basic understanding. |
| milobo | says | if you have to "drill" for understanding, then the students don't really understand and then it is frustrating for students. |
| mindelei | But do they understand what the tools are doing? I've been in countless science labs where there is no real learning of why/how. |
| JackieB | What is "basic understanding"? Estimation? Ability to use the skill in context? Ability to do 10 of the same problem? To extend the problem? |
| mindelei | Ability to apply the skill to a new problem. |
| mindelei | Maybe not new...but different. |
| mindelei | Ability to determine how to use what I already know on something I don't know. |
| mindelei | Or at least move in that general direction. |
| mindelei | I should have a general understanding to think my way through things that aren't completely familiar. |
| JackieB | Can't we use larger problems to figure out smaller ones we need to figure out first? Gives us a real reason to learn the concept? |
| budtheteacher | What about "an ability to see what it is you need to know that you don't know - and seek it out." |
| mindelei | likes | what budtheteacher has to say! |
| JackieB | Thanks Bud, that's what I was *trying* to say. |
| mindelei | thinks | that's a better explanation of what I was trying to say. |
| mindelei | asks | Hey - how do you get italics? |
| JackieB | One * on either side gives you italics, **two** on either side gives you bold. |
| mindelei | thinks | these little plurk tricks are very cool. |
| milobo | says | budtheteacher As long as students know enough to know what they don't know...  and that's where some of my students struggled. |
| budtheteacher | Okay - so we agree on what we want kids to be able to do. The argument comes when we try to figure out *just how much* context is needed. |
| JackieB | We agreed? What do we want them to be able to do? |
| mindelei | thinks | she agrees. But also believes it comes down to problem solving skills. |
| JackieB | Ah, so problem solving. Then why do they need to be able to do it by hand? |
| budtheteacher | mindelei - That's a bit circular - we learn problem solving by learning problem solving skills. |
| budtheteacher | Okay - how do we learn those skills? |
| budtheteacher | *By actually engaging real problems with a skilled facilitator in our presence.* |
| mindelei | feels | that by knowing it by hand they will have a better understanding of what types of shortcuts work best and when to use them. |
| milobo | says | the best way to learn any skill is to apply it authentically. |
| mindelei | feels | This also helps to identify which tools are of most assistance when solving similar problems. |
| budtheteacher | mindelei: What's "it"? Basic math? Is that a problem solving skill? |
| milobo | wishes | I could stay up and hear the rest of this conversation! I'll check in again tomorrow. |
| budtheteacher | Solving real problems isn't a linear process. |
| mindelei | says | It's like only knowing how to use Word. If I have the skills to use multiple word processing programs, I won't have issues when Word isn't |
| mindelei | says | available. |
| budtheteacher | So why is math instruction? |
| mizminh | mathematics is beautiful, engrossing & challenging - having a firm grasp of the basics promotes appreciation, engagement & creativity |
| mindelei | says | she never said that. |
| mindelei | says | that understanding how to go about things by hand helps with the use of tools. Only knowing how to use the tools doesn't help the process. |
| mindelei | says | No problemo. Clarification is always good! |
| mindelei | says | |
| JackieB | Can't it work in reverse? Knowing how the tools work can help in understanding? |
| budtheteacher | Wish I could stick around. Solve this one for me, y'all. I need some answers. |
| mindelei | feels | that it could - but what about the ones who don't really understand the tools - just know how to enter data? |
| mindelei | thinks | that many kids who have math disabilities go undiagnosed which is partially why tools seem to help so much more. |
| mindelei | thinks | that tools shouldn't be substituted for knowledge. They help the process, but don't take the place of the process. |
| mindelei | agrees... |
| JackieB | I agree. A tool without understanding is useless. I'm just questioning what the process is. (symbolic manipulation or problem solving?) |
| mindelei | I think you need to know multiple routes to come to a solution. Providing the knowledge of how to do it by hand creates an additional route |
| mindelei | I also think that allows you to have a better idea of what other routes are most worthwhile. |
| JackieB | Okay. I agree it is a route. However should that route be a gatekeeper to other routes? |
| budtheteacher | What about for problems that we don't know how to solve? Should students engage in those? Makes "by hand" or "with tools" irrelevant. |
| mindelei | feels | that learning "by hand" helps in solving unfamiliar problems. |
| JackieB | Why would we work on problems we already know how to solve. That's boring. Unless it is to find a better/different way - then it's fun. |
| mindelei | has | a goal to get to that - using knowledge to engage in solving unknown problems. |
| mindelei | thinks | finding better/different way is where tools come in handy. |
| colleenk | thinks | that basic skilss need to be taught alongside authentic problem solving rather than before. (first plurk) |
| colleenk | says | skills not skilss |
| colleenk | thinks | a certain amount of fluency can make it easier to focus on the problem solving task especially in middle school and above. |
| JackieB | I agree that fluency is a goal. But should lack of fluency stop further mathematical work? |
| colleenk | thinks | how we teach math should be revised from the ground up. We don't teach math with any consideration of the way children (people) learn. |
| JackieB | I agree. However I'm working on change with the students I have (h.s.) It's tough to change their (and their parents) idea of what math is. |
| colleenk | thinks | fluency should NOT get in the way. Fluency is not an indication of a students' ability to do higher level math and should not be a barrier. |
| colleenk | shares | that she is having the very same problem where she is. |
| JackieB | Yay! I just had a conversation with a friend (different school) who said students can't get into alg. until they've mastered basics, so they |
| colleenk | wishes | Mindstorms was required reading for parents. |
| JackieB | put them in a year long "basics" course. I asked if they were that much better after a year, she said no. |
| JackieB | Heck, I wish it had been required reading in my math ed courses. |
| colleenk | Are you trying to integrate higher level problem solving into you classes? Are people questioning your curriculum? |
| JackieB | We use a "reform" curriculum (next year is our 4th year, so first year of offering all 4 years). Lots of solving using graphing calculator, |
| JackieB | we're incorporating more Sketchpad. Everything is "taught" in context of a larger problem. Some staff still entrenched in old ways, some ... |
| JackieB | parent confusion (though less now), still various levels of staff buy-in. I'm still questioning my own thinking (hard to undo they way I was |
| JackieB | taught to think about what it means to "do math". |
| JackieB | I'm also questioning what kids need to know to be "ready" for college. (I'm teaching the senior course - 1st time we're offering it). |
| mizminh | the "by hand" usage is interesting. Mathematical skills are best developed by moving from the concrete to abstract -graspable to symbol |
| JackieB | Hmm. What if the tool allows them to "grasp" the understanding. Isn't looking at a graph or a list or computer output more concrete than |
| JackieB | symbolic manipuplation? |
| mizminh | we speak of levels of abstraction maybe levels of concrete or "graspability" would be useful |
| JackieB | How would you quantify that? Wouldn't it vary from student to student? |
| mizminh | of course & that is the eternal pedagogical challenge - differentiation- catering for the individual learner |
| mindelei | asks | What are the final goals? Different majors require much different math skills in college. |
| mindelei | says | Earlier it was mentioned that "by hand" wasn't necessary because most of this stuff wouldn't be used after HS anyway. |
| mindelei | wonders which aspects can be made more relevant. The process of thinking through the problems is what can be taken beyond the classroom. |
| mindelei | asks | Which other aspects can be taken beyond the classroom? |
| JackieB | I want them to take away problem solving skills, the ability to evaluate the solutions of others, to communicate solutions (and process) to |
| JackieB | others, to have the willingness to tackle big/new problems, to be able to make a mistake and keep going,... just to name a few. |
| JackieB | And that was part of my original question. What do students need to be successful in college? (not that that is my only goal, but ... |
| JackieB | it is a factor - preparing them for future studies). |
| mindelei | asks | Is this one of those instances where it's more helpful to work backwards from your end-goals? Or at least look at that in conjunction with |
| mindelei | asks | initial goals? |
| mindelei | likes | the goals that you're working with so far. |
| JackieB | Yes, but I'm trying to figure out exactly what the end goals are (I know mine, I don't know what colleges want. Want to make sure I'm taking |
| JackieB | all things into account when I'm setting my goals). |
| mindelei | has | a suggestion: Check w/ some prof friends or local unis to see what they feel their freshmen are lacking. |
| mindelei | That should be another good perspective. |
| mindelei | I think that would be an excellent plurk - an alert to math profs. |
| mindelei | I do have an opinion on what part of the problem is: not enough good math teachers at the elementary level. |
| mindelei | I was an Elem Ed major, A LOT of the ed students had difficulties with the math courses we had to take (they only go through 8th grade math) |
| mindelei | There are a few people I know who had to switch to Secondary Ed because they couldn't pass the math courses for Elem Ed. |
| mindelei | Many had to take extra prep courses before math to get them ready. |
| JackieB | And that is just indicative of the larger problem. |
| mindelei | I think if we had more teachers who had a better grasp of math in elem and could better understand the problems the students are having |
| mindelei | you would have fewer issues by the time those kids get to you. |
| mindelei | Agrees on that point. |
| JackieB | Yep, but blaming the el ed teachers isn't going to solve anything. We need to figure out how to teach the students we have now. |
| JackieB | (and fix the system) |
| mindelei | Oh - I don't want to blame them...I just think they need to be better prepared too. It's a vicious cycle. |
| mizminh | surely the goal is to have autonomous learners exploring their own meaningful learning environments. |
| JackieB | Well, that's one of my goals. Will that help them in a college math class? With college admissions tests? |
| mindelei | says | when you teach them well, the tests will take care of themselves. |
| JackieB | Ah, but what if they can't use the tools on the tests? I'm still trying to figure this all out. |
| mindelei | feels | like we need a big white board or something. Do you ever watch NUMB3RS? I feel like we need to be making a big chart. |
| JackieB | I thought this *was* the chart. |
| mindelei | You could say that....but at the moment it has no sense of organization (although there is a definite flow). |
| budtheteacher | To be continued, right? Not finished? |
| JackieB | Oh no, not finished by a long shot. Just my initial thoughts. |
| JackieB | And I'd love for others to keep adding tot he conversation. |
| milobo | thinks | part of the problem when students get to HS is they see "doing math |
| milobo | thinks | "doing math" as adding/subtracting/mult/div whole numbers, fractions, and decimals. |
| milobo | thinks | and they "do math" with no connection to anything in the real world. |
| milobo | thinks | that's why they come to hate math in MS and HS. We need them to see that math is a form of language used to communicate and create. |
| milobo | thinks | the hand calculations are less important than the facility with understanding why and how you use each calculation to answer a question. |
| mizminh | what milobo said" math is a form of language used to communicate and create." |
| JackieB | Thanks to everyone for their thoughts. I'm still processing what's here so far. |
| mindelei | says | I can't wait for the next conversation. That was a lot of fun for me and very interesting. |
| mindelei | thinks | JackieB needs to check out this link www.plurk.com/p/11ah0 |
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