Archive | August, 2009

Z-minus 10,000 Meters, Mr. Spock

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Stu­dents come to our class­rooms with many assump­tions and mis­con­cep­tions, and it is the teacher’s job to antic­i­pate them, rec­og­nize them, and cor­rect them. Here are a few that I have seen or heard about:

  • When you add or sub­tract, always line up the num­bers on the right
  • When you mul­ti­ply, the answer is always bigger
  • Rock­ets work because the exhaust pushes against the Earth
  • Mag­nets stick to any­thing made of metal
  • Christo­pher Colum­bus was try­ing to prove the world was round
  • The Amer­i­can Rev­o­lu­tion was fought over high taxes

Many stu­dent mis­un­der­stand­ings are sim­ply a lack of expe­ri­ence. There is a scene in the 1982 movie, Star Trek II: The Wrath of Khan, where Khan, the vil­lain, is try­ing to hunt down our heroes. Kirk flies the Enter­prise into a neb­ula in order to obscure the ship from Khan’s scan­ners. After a few min­utes, Spock makes an obser­va­tion about Khan:

SPOCK: Spo­radic energy read­ings port side, aft. Could be an impulse turn.

KIRK: He won’t break off now. He fol­lowed me this far. He’ll be back. But from where…?

SPOCK: He’s intel­li­gent, but not expe­ri­enced. His pat­tern indicates…two-dimensional thinking…

Kirk looks at him, smiles.

KIRK: All stop.

SULU: All stop, sir.

KIRK: Z-​​minus ten thou­sand meters. Stand by pho­ton torpedoes.


Like Khan, our stu­dents are intel­li­gent but have lim­ited expe­ri­ence. I won­der, though, how often we rein­force mis­un­der­stand­ings instead of cor­rect­ing them?

Often in the name of mak­ing our lessons acces­si­ble or under­stand­able we sim­plify con­cepts and use stereo­typ­i­cal exam­ples. Con­sider geom­e­try, for instance. When we draw shapes, they always look essen­tially the same:

Standard pattern block shapes

Stan­dard pat­tern block shapes

Tri­an­gles are always equi­lat­eral and point up. Rec­tan­gles are always wider than they are long and are par­al­lel to the ground. At the extreme, we even refer to shapes by dif­fer­ent names depend­ing on their ori­en­ta­tion. I actu­ally heard this state­ment dur­ing a math les­son once:

And if you turn this dia­mond, it will become a square.


The shape was always a square; the direc­tion it faces doesn’t make any difference.

Try these sug­ges­tions to avoid rein­forc­ing the mis­con­cep­tions of your students:

  • Know your own mis­con­cep­tions. Begin with the assump­tion that you may have picked up your own wrong ideas in school or from pop­u­lar media. Review the mate­r­ial ahead of time and look for places where you your­self didn’t quite get it right. (Inci­den­tally, if you read any of the items in my orig­i­nal list and thought, “What’s wrong with that?” you may want to do a lit­tle research and find the sub­tle prob­lems with them.)
  • Plan ahead for stu­dent mis­un­der­stand­ing. Learn the places where your stu­dents are likely to get con­fused or have pre­con­ceived ideas about a topic. Many mis­con­cep­tions are com­mon and repeated, so it’s easy to pre­pare for them.
  • Use a wide vari­ety of exam­ples. Delib­er­ately choose exam­ples that stretch stu­dents’ think­ing. Use coun­terex­am­ples to help them bet­ter define con­cepts in their minds.
  • Let stu­dents con­struct their own def­i­n­i­tions. By let­ting stu­dents build def­i­n­i­tions and expla­na­tions around exam­ples you use, you are encour­ag­ing them to ana­lyze the exam­ples and under­stand the con­cept deeply instead of just mem­o­riz­ing a sen­tence some­one else has pro­vided them. After they attempt to build a student-​​friendly expla­na­tion, you can come in and pro­vide more pre­cise vocab­u­lary where nec­es­sary to give them a more con­cise way to express it.
  • Expect stu­dents to explain and jus­tify their rea­son­ing. Some­times stu­dents are able to apply a rote algo­rithm accu­rately and get a cor­rect answer to a prob­lem with­out really under­stand­ing what they are doing. Ask­ing them to explain, even when their process seems obvi­ous to you, will give you insight into whether their think­ing is accu­rate or has flaws that need to be corrected.

Soon after Kirk changed his tac­tics to account for Khan’s mis­con­cep­tion, he was able to sneak up behind Khan’s ship, ulti­mately win­ning the bat­tle. While it is unlikely that the mis­con­cep­tions our stu­dents carry through school will result in such life or death cir­cum­stances, we can make our own jobs eas­ier by pre­vent­ing them in the first place.

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SPOCK
                             Sporadic energy readings port side,
                             aft. Could be an impulse turn.

                                           KIRK
                             He won't break off now. If he
                             followed me this far he'll be back.
                             But from where...?

                                           SPOCK
                             He's intelligent, but not experienced.
                             His pattern indicates two dimensional
                             thinking...

                   Kirk looks at him, smiles.

                                           KIRK
                             Mr. Saavik, all stop.

                                           SAAVIK
                             All stop, sir.

                                           KIRK
                             Descend ten thousand meters. Stand
                             by photon torpedoes.

Eradicating Busy Work

Crayola crayons, 24 pack, 2005.
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Last month some col­leagues and I ran a work­shop for teach­ers at my school on dif­fer­en­ti­a­tion. In prepar­ing for it, I came across the idea of anchor activ­i­ties. Unfor­tu­nately, many of the resources I found giv­ing exam­ples actu­ally list a lot of the tra­di­tional time-​​filler busy work (extra work­sheets, copy and define words from the dic­tio­nary, col­or­ing pages, etc.) and slap the “anchor activ­ity” label on them. In her book The Dif­fer­en­ti­ated Class­room, Carol Tom­lin­son defines anchor activ­i­ties as

mean­ing­ful work done indi­vid­u­ally and silently. This could be jour­nal writ­ing, free read­ing, for­eign lan­guage pat­tern drills, seat­work in math, or sketch­book assign­ments. It’s some­thing use­ful and impor­tant for stu­dents to do.… (p. 97)


The key words I see here are mean­ing­ful, use­ful, and impor­tant. We have to put as much thought into select­ing what we ask stu­dents to do in their unstruc­tured time so that it never actu­ally becomes down time.

At the same time, it’s impor­tant to keep in mind that stu­dents’ brains can­not stay in high aca­d­e­mic gear all day long. They need fre­quent short “brain breaks” (as Eric Jensen calls them) to be able to stay alert and focused through­out the school day. The real trick is find­ing the bal­ance and mak­ing sure that the breaks are built into our instruc­tion so that stu­dents are more able to con­tinue aca­d­e­mic work dur­ing their unstruc­tured time.

As with many dif­fer­en­ti­a­tion tech­niques, though, anchor activ­i­ties should be just a start­ing point. Tom­lin­son her­self explains that set­ting up anchor activ­i­ties as a rou­tine in your class­room should be a way to train stu­dents to expect that there will be times when dif­fer­ent peo­ple are doing dif­fer­ent things so that some stu­dents can break off from the group.

What do you do, then, when you have stu­dents who are ready to break off? Per­haps you have a few gifted stu­dents who have com­pacted out of part of a math unit. Or you have sev­eral stu­dents who rou­tinely fin­ish their work quickly and accu­rately. Here are a few ideas for ongo­ing, long-​​term activ­i­ties they can do that are mean­ing­ful, use­ful, and important:

  • Inde­pen­dent Study. This is of course the tried and true tra­di­tional approach, and much has been writ­ten about it. What I rec­om­mend is that you always give stu­dents a way to share their results or inte­grate it back into the class­room com­mu­nity. I had a stu­dent once who was fas­ci­nated with folk tales and fairy tales. Her fourth grade class was learn­ing about Africa that year, so her inde­pen­dent study project was to find and study some African folk tales and adapt one into a play (another inter­est of hers). She then selected stu­dent vol­un­teers and put on a very sim­ple (just a few masks and props) pro­duc­tion in the classroom.
  • Class­room year­book. Have your reg­u­lar early fin­ish­ers form a “year­book com­mit­tee.” Their job is to plan, design, and pre­pare a class­room year­book to go home with your stu­dents at the end of the year. They would need to inter­view each mem­ber of the class, pre­pare a page about each, take pho­tos, record impor­tant class­room events, and so on.
  • About Our School video. Have your kids take snap­shots of activ­i­ties around the class­room (and around the school if your sit­u­a­tion per­mits and your stu­dents are trust­wor­thy). Use Ani­moto to put together an intro­duc­tory music video that the prin­ci­pal could use dur­ing Back to School night pre­sen­ta­tions or post on the school website.
  • Unit recon­nais­sance. Enlist the aid of your bet­ter researchers to help you find good mate­ri­als for upcom­ing units. Tell the stu­dents what the next unit will be in one sub­ject area. Give them some guide­lines and some topic sug­ges­tions, then give them time to explore the library and the Inter­net for mate­ri­als that will sup­port what you will be doing. Use online tools like Diigo or a class­room wiki to gather the infor­ma­tion in one spot.

What are your ideas for keep­ing anchor activ­i­ties and big­ger projects con­nected and mean­ing­ful? How will you work to elim­i­nate busy work from your class­room and school this year?


Ref­er­ences:

Tom­lin­son, C. A. (1999). The dif­fer­en­ti­ated class­room: Respond­ing to the needs of all learn­ers. Alexan­dria, VAASCD.

http://​shop​.ascd​.org/​p​r​o​d​u​c​t​d​i​s​p​l​a​y​.​c​f​m​?​p​r​o​d​u​c​t​i​d​=​1​990
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Failure Is Not An Option...But It Should Be!

Eugene F.
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The ten­sion hangs in the air like wet snow on tree branches. Flight Direc­tor Gene Kranz lis­tens as his team tells him the com­mand mod­ule does not have enough air or power to return to Earth. In this scene from the movie Apollo 13, Ed Har­ris, play­ing Kranz, utters the now-​​famous line ver­bal­iz­ing what the team—and the audience—felt: “Fail­ure is not an option!” (Kranz, by the way, never actu­ally said those words, though he did bor­row them for the title of his memoir.)

The Apollo 13 astro­nauts of course made it back safely, and the intense search for solu­tions to impos­si­ble prob­lems still makes a riv­et­ing story.

In pub­lic edu­ca­tion today it often seems like we’re liv­ing this Hol­ly­wood scene. Fed­eral pol­icy man­dates that five years from now, one hun­dred per­cent of our chil­dren will meet grade level stan­dards in read­ing and math. Fail­ure is not an option. And why not? If you believe that all chil­dren can learn (and I do), what’s wrong with set­ting high expec­ta­tions for achieve­ment and doing every­thing we can to see that stu­dents meet those expectations?

There is plenty of debate about that very ques­tion, but that isn’t my focus here. What con­cerns me is how the expec­ta­tion of suc­cess and achieve­ment can get trans­lated at the class­room level. I fre­quently see this idea that fail­ure is not an option applied to daily assign­ments and tests. Teach­ers have no-​​tolerance poli­cies about missed home­work, for exam­ple, or grad­ing scales that doom stu­dents who do not pass every test.

This is par­tic­u­larly evi­dent with gifted stu­dents. I often hear both par­ents and stu­dents say that since these stu­dents are so capa­ble, any grade below a cer­tain level is unac­cept­able and likely means the stu­dent is sim­ply being lazy. The response is often puni­tive, requir­ing extra “make up” work or retests for par­tial credit.

It’s easy to for­get that gifted stu­dents may have the abil­ity to learn quickly and com­pre­hend at a deep, sophis­ti­cated level that other stu­dents don’t, but this doesn’t mean they already know every­thing or can do every­thing with­out instruc­tion and guidance.

It also doesn’t mean that a lack of suc­cess auto­mat­i­cally means a lack of effort. Young gifted chil­dren are used to suc­cess. Things come eas­ily to them, often auto­mat­i­cally, and they learn rapidly with­out even real­iz­ing they are learn­ing. With­out fail, though, every child hits a point where con­tent is beyond their abil­ity to absorb instantly, and they need to begin apply­ing con­scious thought and sys­tem­atic effort to their learning.

Most chil­dren reach this point early in life, often before school starts. They find out that some­times things don’t go right the first time, and they develop ways to cope with it, per­sist, and grow.

But gifted stu­dents may not reach that point until later, some­times not until mid­dle or even high school. When they finally do hit the wall, they often have no con­cept of what has hap­pened, and they don’t know how to respond.

It’s impor­tant for teach­ers to teach all stu­dents, and espe­cially the highly able ones, how to fail suc­cess­fully.

If you have made mis­takes, there is always another chance for you. You may have a fresh start any moment you choose, for this thing we call “fail­ure” is not the falling down, but the stay­ing down. (Mary Pickford)

I’ve missed more than 9,000 shots in my career. I’ve lost almost 300 games. 26 times I’ve been trusted to take the game win­ning shot and missed. I’ve failed over and over and over again in my life and that is why I suc­ceed. (Michael Jordan)


We can’t afford any longer to treat fail­ure like an end. Instead, we need to rethink it and con­sider it a begin­ning. I’ve writ­ten before about how to deal with per­fec­tion­ism, and those sug­ges­tions apply here as well. Here are a few other spe­cific things that teach­ers can do to cre­ate an envi­ron­ment that nur­tures learn­ing instead of sti­fling it:

  • Rede­fine the word “mis­take.” In your class­room, a mis­take should always be an oppor­tu­nity for growth and learn­ing, never a fail­ure. Which nat­u­rally leads to
  • Give sec­ond (and third and fourth) chances. Any stu­dent who does not achieve at the expected level should not be labeled as lazy or a fail­ure. Instead, give them the oppor­tu­nity to relearn and try again. School should be the one place where it is com­pletely safe to mess up over and over until you can get it right.
  • Cel­e­brate growth. Instead of focus­ing only on accom­plish­ment, give every stu­dent the oppor­tu­nity to expe­ri­ence the plea­sure of suc­cess by redefin­ing it. Progress should be con­sid­ered suc­cess, not just ris­ing above a tar­get level.
  • Reward effort. Giv­ing as much (if not more) atten­tion to stu­dents who work hard and take risks as to those who demon­strate more tra­di­tional types of suc­cess, we send the mes­sage that our class­rooms are a place for work­ing and try­ing, not just for accomplishment.
  • Model fail­ure. By show­ing stu­dents how to deal with times they don’t meet their goals or expec­ta­tions, we give them tools to cope when it hap­pens to them. We also let them see that mis­takes and fail­ure are a nor­mal part of life.
  • Set stu­dents up to fail some­times. Espe­cially with gifted chil­dren, there will be times that the more impor­tant les­son is how to recover from fail­ure rather than to expe­ri­ence suc­cess. Set stu­dents up to fail by giv­ing them a task they do not have the skill or knowl­edge to com­plete. Then help them pick them­selves up, think about what hap­pened, deter­mine what they need to do to suc­ceed, and walk them through that recov­ery process.

Try these in your class­room this year. Cre­ate a dif­fer­ent atmos­phere and see what hap­pens to atti­tudes and learning.


Update: Thanks to Kevin Wash­burn who pointed me via Twit­ter to this post he wrote recently which sum­ma­rizes research sup­port­ing these ideas.

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Racing to Catch Up With the Past

Melrose-Saugus Middle School Track Meet 111-5x7
Image by Paul-​​W via Flickr

When I was in col­lege earn­ing my edu­ca­tion degree, most of the research on learn­ing came out of behav­ioral psy­chol­ogy: Pavlov, Thorndike, and Skin­ner. We learned how to mold our stu­dents’ skills and behav­iors through drill and prac­tice, rewards, and pun­ish­ments. Instruc­tional tech­niques were built around how to train stu­dents to become flu­ent in the read­ing and com­pu­ta­tion skills they would need to be suc­cess­ful in life.

Then, about ten years later, when I was doing advanced grad­u­ate work and earn­ing my cer­tifi­cate in cur­ricu­lum design, we learned that learn­ing wasn’t quite as cut and dried as that. Cur­ricu­lum shouldn’t be com­part­men­tal­ized, it should be inte­grated. Instruc­tion shouldn’t be skill-​​driven, it should incor­po­rate higher level think­ing. Assess­ments shouldn’t be designed around dis­crete facts, they should be authen­tic. At the time we read an inter­view with Lau­ren Resnick, a major researcher into learn­ing and how it works. (The arti­cle is avail­able free to ASCD members.)

I recently came across the arti­cle again, and although it is now twenty years old, Resnick’s com­ments are thought-​​provoking, not the least because much of what she said then still has not become wide­spread in the field.

If knowl­edge con­sists of small bits of infor­ma­tion to be accu­mu­lated, then we know how it is learned and there­fore how to teach it. In that case the ped­a­gogy has to do with how you orga­nize prac­tice, how you struc­ture and sequence the mate­r­ial, and how you man­age moti­va­tion.… But if you view knowl­edge as some­thing more than an accu­mu­la­tion of lit­tle bits, if you want stu­dents to under­stand and be able to use knowl­edge reflec­tively, that’s dif­fer­ent. (Brandt 1988/​1989, p. 13)


If you read the pro­fes­sional lit­er­a­ture and lis­ten to what is said in train­ing sem­i­nars and work­shops, you might think this belief that there is more to learn­ing than dis­crete facts has per­vaded our school sys­tems. Resnick talked about how math­e­mat­ics, for exam­ple, is not a col­lec­tion of skills, but is an “orga­nized sys­tem of thought” (p. 14). But the cur­ricu­lum has yet to catch up with the past. Even twenty years later, text­books still look essen­tially the same as they did then. They still are struc­tured around the accu­mu­la­tion of facts and dis­crete skills, though they often fill them with lots of the lat­est terminology.

Teach­ing prac­tices really haven’t caught up either, because our schools aren’t struc­tured to facil­i­tate it.

What peo­ple learn is vir­tu­ally never a direct replica of what they have read or been told or even of what they have been drilled on. We know that to under­stand some­thing is to inter­pret it.… It is not enough to focus on mak­ing an excel­lent pre­sen­ta­tion, because you can­not assume that your ele­gant expla­na­tion will be heard and under­stood in its entirety. In fact, you can be almost 99 per­cent sure that no child in your class­room will get it the way you said it. (p. 15)


And yet what do we see in many class­rooms? Teacher at the front, telling stu­dents what to do, how to do it, and what to remem­ber. I’m guilty of it myself, and I think on reflec­tion it is a func­tion of time. Plan­ning and imple­ment­ing the most effec­tive forms of learn­ing expe­ri­ences take far more time than most teach­ers can spare. So we fall back on what is effi­cient, even if it is not as effective.

It’s not enough to stay com­fort­able with what we know how to do. If I keep teach­ing the way I’ve always taught, I can’t bring my prac­tice up to date with 1980’s research, let alone what is hap­pen­ing in 2009. I don’t think we can even afford to say, “the sys­tem isn’t set up for it, so why bother?” Find ways inside the struc­ture to start mak­ing changes toward a more student-​​centered, think­ing– and problem-​​solving-​​oriented approach.

How are you mak­ing this hap­pen in your class­room? What are the strug­gles you’re fac­ing? How can we work together to over­come the challenges?


Ref­er­ence

Brandt, R. (Decem­ber 1988/​January 1989). On learn­ing research: A con­ver­sa­tion with Lau­ren Resnick. Edu­ca­tional Lead­er­ship, 46 (4), 12–16.

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Messy Learning from Tidy Teaching?

Paperwork
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As I was reread­ing Wig­gins and McTighe’s Under­stand­ing by Design recently, it occurred to me that there is a dis­con­nect between authen­tic learn­ing and the way we are required to teach today. Teach­ing is increas­ingly focused into neat lit­tle pack­ages that are eas­ily assessed and can be boiled down into a sin­gle test score for account­abil­ity and record keep­ing. Cur­ricu­lum and unit plans are struc­tured and pretty doc­u­ments, hav­ing a well-​​defined begin­ning, mid­dle and end. Lessons are lit­tle self-​​contained deals, 45-​​minutes or less, with a clear struc­ture and clo­sure and don’t nec­es­sar­ily con­nect to any­thing else.

It’s teach­ing in a Twit­ter and YouTube world where sig­nif­i­cance boils down to a 140-​​character sum­mary or a 30-​​second video clip.

But learn­ing in the real world, or at least in my real world, is messy, lumpy, and long-​​term. I was think­ing about how I per­son­ally learn almost every­thing I’ve learned in the last few years: web design, writ­ing inter­ac­tive fic­tion, cur­ricu­lum com­pact­ing, even IEP writ­ing. In most cases, I learned most of what I know sim­ply by jump­ing in with both feet, get­ting dirty, and muck­ing around with things. In a lot of cases, I learned some of the “basics” after I learned more advanced tech­niques. I learned things as I needed them. When I wanted to make a web page do what I wanted it to do, I just went in and fig­ured it out. There was very lit­tle sys­tem­atic about the process. When I ran into a road­block, I’d go look­ing for help, either from those more struc­tured resources or from my net­work of friends and colleagues.

Not that I didn’t have some struc­ture to my learn­ing. In most cases I did take the time to read tuto­ri­als, or intro­duc­tory level books about what I was learn­ing, and I tried some struc­tured activ­i­ties designed to walk me through what I needed to know. But often I didn’t know what I needed to know until I was in the midst of my own real project.

I think this is what Wig­gins and McTighe are inter­ested in get­ting at with more authen­tic ways of assess­ing stu­dents. But how to fit it into the struc­tured world of school? My own teach­ing the last few years has tended towards the messy, unstruc­tured vari­ety. Often, I’ll teach a unit by hav­ing an idea of a project I want my stu­dents to com­plete, and some spe­cific goals I want them to get out of it, and we just sort of dive in and work out most of the details as we go along. There’s some value in this, I think, and as much as I’ve crit­i­cized myself for not being orga­nized enough or plan­ning enough, when I look back I can see a lot of good learn­ing that has taken place in my stu­dents over the years. The feed­back I get from the stu­dents and their par­ents has also rein­forced this.

But to an out­sider (or an admin­is­tra­tor) look­ing on, it’s hard to explain. I don’t always have finely-​​detailed unit plans, and less often do I have well-​​structured daily les­son plans. I don’t always have the clear­est idea where some­thing is going to take us, and often the stu­dents push a project in direc­tions I couldn’t have imag­ined it going when I con­ceived it in the first place. More often than not, too, these learn­ing expe­ri­ences don’t always wrap them­selves up into a tidy pack­age with a bow that I can send home at the end of the mark­ing period. I can’t tell you how many times I’ve set up our annual end of the year open house dis­play with mul­ti­ple signs indi­cat­ing “works-​​in-​​progress”.

As a teacher of the gifted, I have much more free­dom to try these messy projects with my stu­dents. But there has to be a way to tighten things up, too. As much as authen­tic learn­ing is messy, I do want my stu­dents to be able to walk away from the year with a sense of accom­plish­ment and com­ple­tion, and I want to be able to help main­tain an appro­pri­ate focus.

So where’s the bal­ance? How do we keep things “authen­tic” (and there­fore poten­tially messy) and still have the neat, account­able pack­age that the school sys­tem demands? What are the con­flict­ing forces that pull you in two dif­fer­ent direc­tions as you teach and how do you rec­on­cile them?

(This arti­cle orig­i­nally appeared in a slightly dif­fer­ent form at Grandé With Room)

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Teachers Can't Read Minds!

Future predicters
Image by Freekz0r via Flickr

Every year I stand in front of a group of new fourth or fifth grade stu­dents and face the most chal­leng­ing teach­ing task I’ve ever had: train­ing them to be telepathic.

I always begin with a magic trick. Each stu­dent chooses a two-​​digit num­ber. Then I walk them through a series of sim­ple cal­cu­la­tions result­ing in a new num­ber. On that page in their math book, they choose a pic­ture and mem­o­rize it.

Ear­lier in the day, a mys­te­ri­ous enve­lope had arrived in the class­room, marked “DO NOT OPENTOP SECRET.” I now open that enve­lope, reveal­ing a dupli­cate of the photo they all have mem­o­rized. I can read minds!

Of course, it doesn’t take long for the class to real­ize it was a trick, and I don’t deny it. In fact, I remind them that I began the exer­cise by telling them I was going to do a magic trick. The point is why I had to do a trick: teacher’s can’t read minds.

So what does this have to do with math?” they ask me.

Ah, excel­lent ques­tion,” I reply. “When you put an answer down on a math test or a home­work prob­lem, how does your teacher know what you were think­ing when you solved it?”

Uh…she doesn’t?”

Pre­cisely. But for us to teach you, we need to know how you’re think­ing so we can help you learn how to solve prob­lems bet­ter. Since we can’t read minds, what’s the only way for us to know what’s going on in your head as you’re solv­ing a math problem?”

If the les­son were out­side at night, this ques­tion would nor­mally be answered by the sound of crick­ets chirp­ing. One brave soul usu­ally raises a cau­tious hand: “Uh…we tell you?”

A sim­ple con­cept. A dif­fi­cult task. Actu­ally get­ting the thoughts from their heads into words—and even­tu­ally onto paper—is some­thing that takes much prac­tice and many exam­ples. Yes­ter­day I talked about one of the ways to begin this process by teach­ing and using the cor­rect vocabulary.

We need to teach stu­dents that math is not about rote manip­u­la­tion of abstract sym­bols. Those sym­bols, and the ter­mi­nol­ogy that goes along with them, are tools with two pur­poses: solv­ing prob­lems, and com­mu­ni­cat­ing ideas.

I’ve devel­oped a struc­ture that helps stu­dents orga­nize their think­ing and chunk the way they com­mu­ni­cate it. I tell them, “Wear Your C.A.P.E.”:

C Cal­cu­la­tions Show all of your math work and computations
A Answer Be sure to answer the ques­tion or ques­tions that the prob­lem asks!
P Pro­ce­dure or Plan Show each step of how you solve the prob­lem, includ­ing draw­ings, tables, etc.
E Expla­na­tion Explain your math reasoning—tell why you did what you did

The most dif­fi­cult aspect of this, of course, is the explanation—describing the why, not just the what. In order to help with this, I teach them the Magic Words. Just like using clue words to iden­tify the oper­a­tion in a word prob­lem (like “all together” sig­ni­fies addi­tion), these words can help to sig­nify their math­e­mat­i­cal rea­son­ing when they talk or write. (This list is based on an arti­cle by Diane Hurst pub­lished sev­eral years ago in the PA Math Assess­ment Hand­book, but no longer appears to be available):

to get because
to find since
to fig­ure out there­fore
to show so that

Stu­dents who learn to use these words cor­rectly will begin to unpack the rea­son­ing that is going on in their heads.

How could you adapt this to your sit­u­a­tion? What other sub­ject areas might it work for? Do you have other ideas about teach­ing stu­dents to be “tele­pathic” and com­mu­ni­cate their think­ing to other people?

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Vocabulary for Developing Math Reasoning

Tyrannosaurus Rex
Image via Wikipedia

Teach­ers of math­e­mat­ics need to rec­og­nize that there is a strong link between lan­guage, writ­ing, and prob­lem solv­ing. In most of the assess­ments that states use to deter­mine stu­dent and school suc­cess, a stu­dent must demon­strate math rea­son­ing abil­i­ties through writ­ing. This skill is not auto­matic, though. It devel­ops through a recur­sive process:

Vocab­u­lary & Lan­guage <—> Rea­son­ing <—> Talk <—> Writing

Begin­ning with vocab­u­lary and lan­guage, a stu­dent learns to rea­son, then to com­mu­ni­cate those thoughts ver­bally, and finally to write. Each of the lev­els feeds back to the pre­vi­ous one, rein­forc­ing and fur­ther devel­op­ing it.

Thus if we’re going to teach rea­son­ing skills effec­tively, it fol­lows we need to care­fully con­sider the vocab­u­lary we use.

It isn’t uncom­mon, espe­cially in the pri­mary grades, for teach­ers to sim­plify the lan­guage we use with chil­dren to explain com­plex con­cepts. Although this is use­ful, it can also lead to sloppy lan­guage if we aren’t care­ful. It is par­tic­u­larly impor­tant that we don’t per­mit stu­dents to use pre­cise math terms improp­erly and that we teach the “real” terms as quickly as pos­si­ble. Even if stu­dents don’t use them right away, they should be hear­ing the cor­rect ter­mi­nol­ogy in con­text from the beginning.

Here are a few exam­ples of sloppy math lan­guage that I often hear from older stu­dents. If these go uncor­rected, stu­dents will have a very dif­fi­cult time com­mu­ni­cat­ing well when they need to explain their thought process–a skill that is essen­tial to upper level math.

Instead of these… Use these…
take-​​away minus
“plussed” added
“minused” sub­tracted
“timesed” mul­ti­plied
answer sum, dif­fer­ence, prod­uct, quotient
amount length, height, vol­ume, num­ber, etc.
num­ber digit, addend, fac­tor, div­i­dend, etc.
big­ger, smaller greater than, less than

I believe it’s essen­tial to require stu­dents to be pre­cise when they com­mu­ni­cate. Often when stu­dents don’t use the cor­rect term, or use a valid term improp­erly, it is a sign they just don’t have the right words.

I’ve heard teach­ers argue that young chil­dren just aren’t capa­ble of such sophis­ti­cated lan­guage yet. My father, a retired pro­fes­sor of speech/​language pathol­ogy, has often said, how­ever, that if sec­ond graders can learn and cor­rectly use terms like “Tyran­nosaurus Rex” and “Diplodocus”, why on earth can’t we teach them to say “sub­tracted” instead of “minused”? Vocab­u­lary instruc­tion should be as much an inte­gral part of math­e­mat­ics as it is of read­ing, writ­ing, and other con­tent areas.

Tomor­row I will tackle a more chal­leng­ing vocabulary-​​related issue in math­e­mat­ics: ver­bal and writ­ten expla­na­tions of a student’s cog­ni­tive process.

(This arti­cle is based on mate­r­ial I orig­i­nally posted in Grandé With Room.)

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Quick Classroom Activity about Authors

Have desk, will write
Image by Bright Meadow via Flickr

Here’s an inter­est­ing idea for a quick class­room activ­ity that has poten­tial for many dis­cus­sions. This could cer­tainly be applied in many dif­fer­ent ways to stu­dents at all levels.

Begin by tak­ing kids to this site: http://​wherei​write​.org. It is a small site with one pur­pose: to show­case por­traits of authors (they all hap­pen to be in the sci­ence fic­tion genre) in the spaces where they do their writing.

A few thoughts come to my mind as I scan through the pictures:

  • Nearly every space is a work space. Cre­ativ­ity isn’t about flashes of inspi­ra­tion. It’s about doing. And effort.
  • Almost every writer sur­rounds him– or her­self with books. Dozens or hun­dreds of them. Writ­ers read. A lot.
  • Writ­ers are ordi­nary peo­ple. They have pets. They even have stained glass thin­gies hang­ing in their windows.

I think there’s a great les­son for stu­dents, espe­cially reluc­tant writers.

Some other ideas for fol­low up activities:

  • Have stu­dents share pho­tos of their writ­ing spaces and talk about them
  • If you could cre­ate a bet­ter space to write, what would it look like? Why not cre­ate it?
  • How could we design our class­room space to make it bet­ter for doing our work?

What do you see in these pho­tos? What ques­tions would you ask of your stu­dents about these pic­tures? What else do they tell you about what writ­ers do and how to be one?

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Kitchen Sink Link Buckets

The kitchen sink
Image by alancleaver_​2000 via Flickr

I had a con­ver­sa­tion the other day with Kelly Hines on Twit­ter about resources for teach­ers. There are a num­ber of sites out there (like this one and this one) that col­lect links to resources for teach­ers in one place for easy ref­er­ence. Some peo­ple find them use­ful, and there are undoubt­edly some great resources there.

But I tend to find them dif­fi­cult to use, at best. Though they fre­quently are cat­e­go­rized by topic, or grade level, or some other sys­tem, in prac­tice, it is sort of like hand­ing some­one a list of all the book titles in the library. I think these kinds of kitchen sink link buck­ets have value, and I’m not say­ing there’s no point in hav­ing them or using them. I just per­son­ally find their value to be lim­ited. For some­one who plans around resources, they are, I’m sure, invalu­able. Browse the list, gather a few good sites, and then build your instruc­tion around them.

But I tend to plan in the oppo­site direc­tion. I’ll select my objec­tives and projects and develop a gen­eral out­line of where I want to go, then look for resources that will plug into the out­line. Link buck­ets are not the best way to track down this sort of resource.

Prob­lem is, I’m not sure what really would work. I’m a mem­ber of Diigo and Deli­cious and do search through the socially tagged links there. I do use the link buck­ets from time to time. But what I think we need is some­thing that’s a blend of a wiki and a social book­mark­ing site. The prob­lem I see with cur­rent social book­mark­ing is that each person’s links are sep­a­rate. When I search, I get an uncat­e­go­rized, unsorted list of links which may well con­tain dupli­cates. I can see if mul­ti­ple peo­ple have saved the same link, but a dif­fer­ent link on the same domain will show up as a sep­a­rate item.

Here are a few fea­tures (in absolutely no par­tic­u­lar order) I’d ide­ally like to see in the per­fect resource site:

  • Search within search results to nar­row the focus
  • Group­ing and sort­ing within results
  • If I save a link, I will imme­di­ately see who else has saved the same page or domain so I can make an intel­li­gent deci­sion about how to save the link
  • A way to dis­play anno­tated links (not just titles or sum­maries) for a par­tic­u­lar tag or search term
  • A broad but struc­tured way of iden­ti­fy­ing links as appro­pri­ate for var­i­ous cat­e­gories (e.g. K-​​2, math, geol­ogy, etc.)
  • A way of edit­ing the cat­e­go­riza­tion made by oth­ers and mod­i­fy­ing the struc­ture itself (like a wiki)

There has to be some way of bal­anc­ing vol­ume (lots of worth­while links) with find­abil­ity (lim­it­ing links to a few strong can­di­dates so it is more brows­able, like this list I put together for a work­shop I did recently).

What am I miss­ing? What other fea­tures should it have? How would it look in prac­tice? Is this even a pos­si­bil­ity? Maybe it already exists and I just don’t know it yet. I’d love to hear what other peo­ple have to say.

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Lessons in Responsibility from Spider-Man, Part 3

Wordle: Leadership Day: The Pace of Change
Lead­er­ship Day: The Pace of Change, from Prac­ti­cal The­ory by Chris Lehmann

In my last two posts, I wrote about the respon­si­bil­i­ties that go along with using pow­er­ful tech­nol­ogy tools, both for stu­dents and for teach­ers. Today I will con­sider a third group: administrators.

The Inter­na­tional Soci­ety for Tech­nol­ogy in Edu­ca­tion (ISTE) rec­og­nizes the impor­tance of strong lead­er­ship to the effec­tive use of tech­nol­ogy in schools, shown by the fact that they have devel­oped national stan­dards for admin­is­tra­tors. It is not enough to sim­ply cre­ate poli­cies aimed at enforc­ing safety and pro­duc­tiv­ity for stu­dents and teach­ers. Admin­is­tra­tors must accept their respon­si­bil­ity for vision­ary lead­er­ship, which takes sev­eral forms.

Understand

Just as teach­ers and stu­dents must under­stand a new tool or tech­nique before they can use it prop­erly, admin­is­tra­tors must deeply under­stand what is hap­pen­ing in the realm of tech­nol­ogy if they are going to be able to lead effec­tively. This means learn­ing about research and best practices.

It also requires them to use the tech­nolo­gies them­selves. Con­sider what a “chalk­board pol­icy” might have looked like if designed by admin­is­tra­tors who had never used one:

  1. Chalk­boards by nature are open and acces­si­ble forums, and as such have inher­ent risks involved with their use. In the best inter­ests of stu­dent and employee safety, it is the pol­icy of this admin­is­tra­tion to restrict access to chalk­boards and to mon­i­tor their use at all times.
  2. Chalk­boards will be main­tained behind a locked panel when not in use. The key to this panel is avail­able in the main office of each school build­ing and must be signed out when needed.
  3. Only autho­rized brands of yel­low or white chalk may be used on Dis­trict chalk­boards. Other writ­ing imple­ments, includ­ing but not lim­ited to col­ored chalk, serve no dis­cernible edu­ca­tional pur­pose and are forbidden.
  4. Only those who have a signed “Chalk­board User Agree­ment” on file may write on the chalk­board at any time.
  5. The use of the chalk­board is a priv­i­lege, which may be revoked by the admin­is­tra­tors at any time for abu­sive con­duct or vio­la­tions of this agreement.


Of course, there are many places where the par­al­lels break down, and it is not my intent to make light of the real issues and risks involved with Inter­net use by stu­dents. I believe, how­ever, that if more admin­is­tra­tors had a thor­ough under­stand­ing of the tools affected by their poli­cies, those poli­cies would have a dif­fer­ent focus.

Plan

Many dis­tricts seem to think that the prin­ci­ple of reverse psychology–where doing the oppo­site of what is expected will have more pow­er­ful results–also applies to pol­icy imple­men­ta­tion in this three-​​step process:

  1. Imple­ment new policy
  2. Deter­mine how the pol­icy should work (usu­ally after 6–12 months of practice)
  3. Decide whether the pol­icy was war­ranted in the first place (often after a year or two of failure)

In real­ity there must be thor­ough plan­ning before a pol­icy can be put into effect, with con­sid­er­a­tion for how it will impact all areas of cur­ricu­lum and instruction.

Another, more sub­tle sort of plan­ning is required if the use of tech­nol­ogy tools is going to be any­thing other than just a tacked-​​on option to an already over­stuffed cur­ricu­lum. This is where a clear, long-​​range vision for the future is cru­cial. The higher in an orga­ni­za­tion an admin­is­tra­tor rises, the more that per­son needs to see the big pic­ture and proac­tively design, not man­age, what that pic­ture will look like as the dis­trict develops.

Model

Edu­ca­tors know that more pow­er­ful than telling is show­ing. Good teach­ers build mod­el­ing into their instruc­tion because it pro­vides stu­dents with an exam­ple of what skilled, expert use looks like.

Admin­is­tra­tors can­not expect teach­ers and stu­dents to sim­ply fol­low their vision with hav­ing an exam­ple to fol­low. If admin­is­tra­tors expect tech­nol­ogy tools to be used prop­erly, they must show what that proper use looks like. If they want to see more stu­dents and teach­ers blog­ging, they must blog. If they want to see Twit­ter used as a pro­fes­sional tool rather than sim­ply a social one, they must be on Twit­ter themselves.

Communicate

A clear, effec­tive vision will never become real­ity until it is com­mu­ni­cated with those who are ulti­mately required to put it into effect. Just as teach­ers must com­mu­ni­cate with stu­dents, giv­ing feed­back, shar­ing goals, set­ting expec­ta­tions, admin­is­tra­tors must com­mu­ni­cate in all the same ways with their constituents.

Conclusion

Imple­ment­ing tech­nol­ogy respon­si­bly and effec­tively is a com­plex thing. Because it has so much power, I believe it would be irre­spon­si­ble not to embrace these tools in schools today. Many stu­dents, teach­ers, and admin­is­tra­tors are under­stand­ably reluc­tant to take on the respon­si­bil­i­ties that come along with the power of the tools. But like Peter Parker, who received his pow­ers with­out ask­ing for them, we can­not ignore them. We have to dive in, accept the fact that we have been handed great respon­si­bil­i­ties, and use our pow­ers to become super­heros instead of villains.

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