In his commencement speech ( story shared by @monsoon0 ) to a Memphis graduating class, President Obama said:
Through education, you can also better yourselves in other ways. You learn how to learn – how to think critically and find solutions to unexpected challenges. I remember we used to ask our teachers, “When am I going to need algebra?” Well, you may not have to solve for x to get a good job or be a good parent. That’s true. But you will need to think through tough problems. You will need to think on your feet. So, math teachers, you can tell your students that the President says they need algebra.
I really don’t get how solving for x in 3x + 17 = 5x + 2 will "help students think through tough problems [in life]" and "think on their feet." I see algebra skills as useful, but not out of context of the types of problems they help us solve. Instead of students learning an algorithm for which almost none of our students will ever get to see a real application; what if we taught students areas of mathematics which had direct application in their lives, and which actually helped them think?
I think that we do need some people who learn algebra in a really deep way, but the type of algebra that people use in their day to day lives is fairly simple, and doesn’t take very long to teach, especially if students see the value in what they are learning. Too long people have learned math because someone said they should.
Well Mr. President, I don’t think that telling my students that just because you think it is useful will mean they will want to learn it. I’m going to keep focusing on presenting the mathematics I teach in the context of the lives of my students instead, thank you.
Salehen Shovon Rahman says:
Although Obama’s statement begs a lot of question, I don’t think, however, that he meant that you actually need algebra to solve problems in life, but I think what he meant was that the level of focus needed to solve tough problems is present in algebra.
It’s amazing the type of critical thinking skills you gain if you keep an open mind about math, and that you dedicate just a little bit of your time to at least solve just one challenging problem, such as 3x + 17 = 5x + 2.
I’ll give an example that is sort of off topic: at SFU (and maybe in other Universities as well), it’s mandatory for software engineering students to take an introductory course to circuit design. Will software engineers ever need to build circuit boards themselves? No, of course not. That’s not what they do. Software engineers work through a series of abstraction, and there absolutely no point in trying to dabble with hardware. Even the professor that taught the circuit design course said that you will never need to know all the nitty gritty details of circuit design. But it sure helps to think critically about how data and instructions are passed from the memory system to the CPU. That way, someone trying to come up with an efficient algorithm to solve a problem will be two steps ahead. Not only will the engineer come up with a solution that is algorithmically efficient (that is, having a very low big-order O), but one that is also physically efficient when it comes to CPU usage.
I’m not saying that you should coerce kids to learn algebra. I’m just saying that the level of thinking required to solve an algebraic problem to find x is at the same level of thinking in the real world. Sure by the time that kids grow up, they will forget a lot of the math stuff, but the one thing that they will retain forever is the ability to stay focused to solve problems, and I think that’s what counts, and I think that’s what Obama meant.
May 16, 2011 — 4:23 pm
David Wees says:
It probably is what he meant, but my question is, can we find something more engaging for kids to work on and end up with the same result, an adult who is willing to struggle through a problem in order to find a solution?
May 16, 2011 — 5:00 pm
Mike says:
Kids solve problems every day playing videogames, with a laser-like focus. They strategize, plot courses, engage in basic physics. Not much of that skill transfers to real life though. It takes a huge mental jump and a lot of initiative to bridge one domain to another, and most students won’t do it on their own.
The same is more or less true with algebra.
Teaching algebra to teach basic problem solving skill works on some level, but it’s about as efficient as trying air condition your back yard. Almost none of the effort that students spend learning algebra will go to anything else in life, even indirectly. And because it goes unused, even direct domain specific knowledge will fade away within a couple years.
Meanwhile, Alan Simpson, who the president appointed to reform Social Security, said recently that it was nonsense to say a phrase like “The average life expectancy of someone that reaches 65 is 78.” He literally did not understand why the average life expectancy at 65 would be different than at 12, or at birth.
I bet Alan could solve for x just fine. But, as the cognitive science literature will tell you, skills aren’t easily transferable/generalizable, and we never bothered to teach him stats, and because of that a lot of old people may go very cold and hungry. So it really does matter, and matter quite a bit what we choose to teach people — it really is a matter of life and death, and is that quite frequently.
May 16, 2011 — 11:29 pm
David Wees says:
I agree completely. I didn’t think of transfer being a problem, but I can definitely see that it is. Students are expected to learn problem solving skills from algebra and then transfer those skills to the ability to fix a flat in their car, or whatever other problems life throws their way. Not going to happen.
May 16, 2011 — 11:37 pm
George Haines says:
Hey David– another great post. I’d like to see you flesh out your ideas starting from scratch. If you could design a year-long curriculum that accomplished what Algebra advocates claim Algebra accomplishes, but has that missing ingredient of tangible application I’d be interested for sure.
Mike makes a huge point about domain knowledge. Anyone who has ever worked with smart but “non tech savvy” people know how vital specific knowledge is to problem solving. I can’t debug Java because I have no domain knowledge. I can’t correct spelling on a French test for the same reason. Deep knowledge is almost the whole game when it comes to problem solving. Without broadening the schema of our students, they will be cut-off from the ability to learn deeply and solve problems in most areas. Some educators claim to want to teach problem solving skills at the expense of a broad schema. Never understood why those beliefs still circulate. Maybe it’s because they never read a comment like yours! Valuable piece of the puzzle for sure.
Thanks for sharing this, David. I think you are exploring interesting regions regarding math education.
May 16, 2011 — 11:59 pm
David Wees says:
Your analogies are good, and I think they point to an important issue in mathematics education. Many math teachers teach nothing but domain knowledge, and produce kids that can’t solve real problems. So domain knowledge isn’t everything. Similarly, as you point out, student’s can’t solve any problems if they don’t have the tools. So we need to teach both domain knowledge and the problem solving tools students need to be able to move on in their lives. Since we are teaching hardly any of the problem solving skills in mathematics, and ending up with a largely innumerate public, I think we should teach more of the problem solving and less of the domain knowledge.
I’m not sure exactly what that looks like yet. I’m still thinking about it, and think I am not qualified to make that determination alone. I am working with a small team right now of other progressive mathematics educators interested in improving mathematics instruction in the same way.
May 17, 2011 — 12:30 am
George Haines says:
re: “Many math teachers teach nothing but domain knowledge, and produce kids that can’t solve real problems.” I think that is essentially what Mike was talking about. There is no real thing as “problem solving skills” there is just a certain amount of domain knowledge needed to solve a specific problem.
Understanding how my lawn mower works is all I really need to be able to know how to solve any problem with it. Knowing how my lawn mower works doesn’t help me figure out why my soccer team can’t breakdown a defense at midfield. In order to solve my soccer problem, I need to know a lot about soccer. Knowing Algebra doesn’t really help me fix that lawn mower either. Understanding Algebra might help me understand more about the relationships between numbers though, which might help me in other areas of math. Understanding how my lawn mower works might help me fix a go-cart or a motorcycle because there are similar patterns and vocabulary and materials involved, meaning, I already have *some* of that domain knowledge or I can more easily add new info to my schema because I have a solid foundation in “motors” or “combustion” or whatever.
“Problem solving skills” is a phrase that doesn’t have much real meaning. Asking people to use strategies like “take a break and look at it fresh tomorrow” or “examine your assumptions and see if your assuming something to be true that isn’t true” are certainly helpful reminders, but they are a miniscule part of problem solving. “Problem solving ability” essentially = “level of domain knowledge.”
I might add that confidence/freedom to act is an important part of the equation, but that *usually* comes with experience as well. Sometimes students are capable of solving a problem, they have the know-how, but they are afraid for some reason. In those cases, domain knowledge is insufficient. I am not saying domain knowledge is everything, but it is an overwhelmingly large piece of the puzzle.
To put it in numbers, maybe 95% of problem solving is simply domain knowledge coupled with freedom to act on that knowledge.
May 17, 2011 — 11:49 am
David Wees says:
Boy, I can’t say I agree with that. The problem with domain level knowledge is that it comes from someone else, and is tied to their understanding of the situation, not yours. The more domain knowledge you learn, the less able you are to think about that domain knowledge outside of the language constructed around it. In other words, being deeply entrenched in domain knowledge is to be entrenched in the ideas of others, which limits your ability to think creatively about the problem.
If we carefully instructed all of our students in the ideologies of environmentalism, for example, whatever lense we chose to share those ideologies would become the lens through which they would examine all new instances of environmental problems. Ask a typical politician about the environment for example, and because they’ve been so deeply rooted in the economic model of environmentalism, they can’t see solutions to problems outside of that domain.
May 17, 2011 — 12:24 pm
George Haines says:
I think your example proves the opposite point though– asking a politician how to solve problems with our global ecosystem is a bad idea because they have virtually zero domain knowledge. I think climate scientists have a much better chance of finding root causes to environmental problems because they understand the subject better. Plus, hey have a much more developed schema to weave new knowledge into.
The solution they come up with might be something like “we need some way to build taller buildings, or invent new drainage systems or something. They would be poor choices to pick to actually build those things though because they have no domain knowledge. Engineers, on the other hand, have that domain knowledge so they would be better to solve those problems. People can’t solve any problem without specific domain knowledge of the problem area. You don’t necessarily need to be a master to begin solving problems, but the more knowledge you have the better chance you have to solve the problem. Not only that, but the more knowledge you have, the better chance you have to learn more about the problem on your own in the future.
May 17, 2011 — 12:34 pm
David Wees says:
If someone actually was 95% steeped in the domain knowledge of others, how would they look at the problem from a fresh perspective? Further, achieving such a high level of domain knowledge in students is only possible through a standardization of the curriculum. Do we really want all kids to have the same understanding of the issues relevant to our world? How could we possibly find solutions?
Some of the freshest and best approaches to solving problems have come from amateurs and people outside of the field. Cross-pollination of fields has produced some of the most amazing ideas. The entire field of chaos theory, essentially invented by Lorenz, happened because a mathematician (and a nonstandard one at that) explored an area outside of his original expertise, meteorology.
May 17, 2011 — 12:41 pm
George Haines says:
Not sure what you mean by “95% steeped in the domain knowledge of others.”
Are you saying that 7×7=49 to you because you are being influenced by people who knew this before you were born? If someone with no knowledge of multiplication looked at 7×7, they might find a different, better answer?
Totally confused.
How about this– can you give me at least one example of a problem in math that people can solve without domain knowledge? If I couldn’t add, subtract multiply or divide and I didn’t know what the number values were or what any formulas were or anything in the domain, what problem could I solve with problem solving skills alone?
Also- can you give an example of a “problem solving skill?” What would you teach someone instead of building their schema? You are advocating sacrificing domain knowledge and increasing problem solving skills, but I don’t understand what you mean by that. Can you give me a concrete example?
May 17, 2011 — 1:01 pm
David Wees says:
Now you are being ridiculous, and not reading my post correctly. When did I say that domain knowledge was completely pointless? Obviously I can’t solve any problems without some domain knowledge, but to think that you need absolutely every piece of the puzzle in order to solve problems is also completely crazy in my mind. When in life do we actually have every piece of the puzzle?
As for your second question, there are a few things kids need to be able to do to solve problems.
May 17, 2011 — 2:04 pm
George Haines says:
David, not being ridiculous. Trying to understand the comment about “domain knowledge of others”. You seemed to imply that domain knowledge was something that hurts problem solving. I am guessing what you meant to imply is that people can teach other people misleading things and give them a false sense of having acquired knowledge. I agree that if someone has the wrong answers to questions, those answers will not help them solve problems. We agree on that, if that was what you meant.
If you were saying all knowledge we learn from books, teachers, TV, radio, friends, is somehow wrong . . . I don’t know how to respond to that.
re: “to think that you need absolutely every piece of the puzzle in order to solve problems is also completely crazy in my mind.” I agree. I don’t even think it is possible to know everything there is to know about a subject. The inescapable fact though is that the more we know about something, the better chance we have to solve problems that arise in that domain. If we know a little, we can solve some problems. If we know a lot we can solve many problems. I don’t think anyone would debate that. There is a direct relationship between domain knowledge and problem solving ability.
The Lorenz example is a perfect illustration of how important domain knowledge is to problem solving. Lorenz was a mathematician. If he was a janitor or a sportcaster or a basket weaver, the fact that he solved a math/weather problem would be unusual. The fact that he took his domain knowledge of math and then added to it an extensive set of really knowledge from the domain of meteorology, underscores the fact that he was possibly the *most likely* person on earth to uncover chaos theory. He was able to see what others couldn’t because he had more knowledge of the domains than others did. Mathematicians don’t know weather and meteorologists don’t know math well enough. He knew both. His domain knowledge separated him from others.
His story is a great example of why building not just a deep schema, but a *broad* schema is so valuable. When we know a lot about a few different subjects, we can make connections and see things others who lack our knowledge can’t see.
I think everything on your list is just another way to emphasize the value of domain knowledge. I think we are actually saying the same thing just using different words.
re: finding answers/building teams– I am saying we can’t solve problems without domain knowledge. You are saying we can, just find people that have that knowledge or find a resource where you can learn it. Same difference to me. Either way, the problem doesn’t get solved without domain knowledge. It is the only piece of the problem solving puzzle that is mandatory. I think we both agree on that, no?
re: variety of representations– not sure what sharing via multimedia has to do with solving a problem. If the specific problem they are solving is “how do I communicate X to others?” then having knowledge of multimedia tools will help them find a solution, but for most problems, multi-media presentation tools are not applicable. Fixing that lawnmower takes knowledge of lawnmowers, not multi-media presentation tools.
My understanding is that the goal is not to teach kids how to get other people to solve their problems, but to help build within them the ability to solve their own problems. teaching kids how to ask for help when they can’t solve a problem on their own is great. I lean on people all the time. It doesn’t help me learn to be a better problem solver though- it helps the person who solves my problem for me. What helps me become better is learning why the solution they proposed worked/didn’t work. That knowledge adds to my understanding in that domain and may help me solve my own problem in the future.
re: Cultural implications and consequences– these skills assume that the problem solver has or will need to have domain knowledge to come up with answers. Can’t know the repercussions of something you don’t understand. If I don’t understand gravity, the repercussions of space travel on my bones will be a terrible shock to me. If I know about gravity and bones, I will have a better understanding of the consequences.
re: research- Researching is defined as a way to increase your domain knowledge when you don’t have enough to solve the problem. I agree this is vital. The more knowledge we have, the more problems we can solve.
Is there anything in this you disagree with?
May 17, 2011 — 2:56 pm
David Wees says:
I agree with all of this information, excepting that multimedia representations is pretty broad and would include pictures and diagrams, so a lawnmower-fixing-person might absolutely have to be able to explain their problem using a picture, or a diagram to explain exactly where the problem is if they are unable to solve the problem themselves.
Where I think we differ is when this gathering of domain knowledge occurs. Should it occur during school, in an out of context setting, or should it occur when the person actually needs the knowledge? I’d prefer to see people who knew a lot about finding accurate, reliable, and useful information about an area of need, and perhaps a bit less than we do now of people expecting to know everything about a topic when they leave school. In other words, in our current system, I think we spend way too much time front-loading kids with information, in the expectation that they will find it useful, and not nearly enough time spending time actually finding useful information for themselves.
A consequence of this, is that sites like Answers.com and similar junky-user-entered-info-with-lots-of-ads profilerate through the web, and people have very little idea on how to actually find useful information. I was talking about this problem with our school librarian, and she was complaining that kids just do random Google searches to find information, and lack any kind of critical awareness of the reliability of that information. She said that the student at the very least should check to see if the article they are reading has references. So I opened up a textbook, and pointed out that traditional textbooks lack references as well.
I know what your counterargument will be, which is that people who have vast exposure to domain level knowledge can spot the BS more easily than people with little exposure to domain knowledge. However, it’s always a trade-off. We can’t possible train the kids in everything they could be exposed to, so we might be better off giving them a moderate amount of domain knowledge in a broader range of topics, and focusing on techniques they can use to establish the reliability of new knowledge.
However, there is also value when people make leaps and try to fill in gaps in their existing domain knowledge. Most of the time, they get it wrong, but once in a while they stumble across a different perspective in the gap, and it ends up re-arranging our existing knowledge in an area. It feels to me like your perspective on domain knowledge is that of a static representation of the world, when in fact I see our knowledge over time as having been dynamic. We know the world differently than we used to, our perspective on it has changed. How has that happened? At least occasionally, there are people muddling around and messing around and making assumptions about the world, and at least some of these assumptions change the nature of what we know. While domain knowledge is absolutely important for expanding areas of knowledge, too much of it is a hindrance in terms of finding flaws with existing knowledge.
May 17, 2011 — 3:44 pm
George Haines says:
I agree, it is a trade-off. The tug of war between depth and breadth is tricky.
The deeper the knowledge, the more higher-order thinking skills begin to emerge. Critical thinking, problem solving, etc., all emerge only after a certain tipping point. “Covering” a lot of different material (in a surface way) prevents these higher-order thinking skills from emerging. I think this is the real issue with increasing the importance of fact-based testing in many schools– standardized test designers value breadth over depth.
The broader the knowledge, the greater the student’s ability to learn new things in the future. We can only learn things that connect to our current schema, so leaving things out of the curriculum basically eliminates a student’s ability to learn those things later in life. It’s not impossible, but it makes it highly unlikely that a student will learn that thing from scratch later in life. This is the reason why millions of Mexican children speak Spanish, but millions of American (or Canadian) adults who want to learn Spanish never do.
In some ways it is a balancing act between higher-order thinking ability and the ability to be a life-long learner. Keeping both of those plates spinning is the difficult part.
May 17, 2011 — 4:13 pm
David Wees says:
Yes, we completely agree on this. Finding the right balance point is the tricky part. Right now, I think we are too far on the content area of the spectrum, but as you point out, we have to be careful not to go too far in the other direction.
May 17, 2011 — 7:39 pm