Wizard math… day 2

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WE’D LOOKED at the sequence generated when you take a hexagon and steadily add more hexagons. It gives a linear rule. Some students got this easily, others found it challenging.

For 20 minutes the task was to work through a handful of similar problems in the textbook.

The issue was how to give an extension challenge for those who could easily do this sort of problem. Jo Boaler in her What’s Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject advises open-ended problems are one solution. She also argues that so-called low-ability students benefit from doing hard problems, or, at least listening and eventually participating in finding solutions in mixed-ability group discussion. Talking math is learning maths. The higher ability students benefit by explaining their thinking to other students.

On the board I drew a square with diagonals drawn in red, a pentagon with diagonals drawn in red, a hexagon with diagonals, a heptagon and octagon, also with diagonals drawn in red. I also put up an incomplete polygon labelled n. I numbered the polygons 1, 2, 3…

I also wrote: Challenge question, spot patterns, how many red lines?

Breanna was out of her seat immediately counting the lines. Within seconds she announced the number of lines coming from each vertex was the same as the number above the polygon. Then she sat down. So? I asked.

“I can’t do any more,” she replied. I explained she needed to write down her discovery so she had something new to look at and work on to get the next step. No, she said.

Shane was busy counting lines. Rebecca stared at the diagonals in the heptagon and octagon and said it was too complicated. “What about this one,” I said pointing at the pentagon “start here… what you find out here will work for those.”

Breanna had started to build a table. “Breanna,” I said, “tell the others about how you counted the diagonals.”

Shane spotted the number of sides of each polygon was 3 more than the number above the polygon. Breanna explained the number of diagonals was the number of vertices multiplied by the number of lines coming out of one vertex. “And the polygon labelled n?” I coaxed. “The number of sides is n + 3,” said Jonathan from the other side of the room.

Robert who had earlier struggled with the basic exercises, sat watching the board and listening intently to the discussion on Breanna’s table .

As others finished the textbook exercises I drew on the board a 4×4x4 cube made up of 64 small cubes. Next to it I wrote, “if the cube is painted, what proportion of the small cubes have paint on them?”

Stopping the class, I focused everyone on the new problem. On our wall we have a list of Polya’s problem-solving strategies. I pointed to the list with my hand next to “Make it simpler”.

Shane was walking from desk to desk debating with other students the number of cubes. Rebecca asked, “you paint the back too?”

Jonathan and Nick were back and forth at each other, Jonathan slicing out invisible cubes in the air with flattened hands. Nick drew out the net of cube and cut it out: “Look,” he announced, beaming, “I’ve made a cube.”

Jonathan just wanted to explain how he had worked it out, how he got the total number of cubes, how he excluded the cubes inside the large cube, how he decided to not to double count cubes with paint on more than one side… all the time his hands slicing out cubes in the air.

What are the dimensions of the cube? I asked Jonathan. 4 he responded. 4 what? 4 times 4 times 4. How would you write that? 4 to the power three… 4 cubed… Oh! he exclaimed as a giant lightbulb flashed in his brain.

As the students left, Jonathan and Shane were still telling each other about how to solve the problem.

“Do you want to see my work?” said Robert showing me the textbook problem he had completed. “I saw you watching and listening Robert,” I said. Robert smiled.

The Calculus class tries the polygon problem to relax after an intense hour wrestling with implicit differentiation. OK, I say, tell me the number of lines in the 99^th polygon.

Becci runs to the board closely followed by Jared. Megan is shouting how to count the lines, but Becci and Jared are engrossed in mathematical disagreement about how to move forward. Josh, Jordan, Jesse and Nicole sketch out the pentagon and hexagon.

Soon the room of nearly 20 students is loudly split between those insisting the rule includes (n −3) and those who say it is (n + 3). As agreement settles on (n + 3), good-natured boasting and mocking ensue.

But, everyone was talking math. And talking math is learning math.

Wizard math

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THERE are 35 players standing in a circle. As the games wizard walks round the circle she kills every second player until only player survives. The players are numbered one through 35. Which player lives?

The guesses came fast. The first one. The last one. The one next to the last one. Then some students started to draw circles and cross off players.

Answers varied. Musbah smiled and said, “number 7, I did it in my head.” Musbah never fails to delight. But what if the number of players changes I challenged him.

Emma, Brittany and Julia worked together drawing circles and carefully working, talking and comparing results. They confirmed Musbah’s answer.

Jason, at the back of the room, asked do the dead ones get dragged away?

Emma was quick to add, “it must be a prime number.” Then Jeremiah tried a circle that gave the answer nine. Haley added, “you have to be an odd number to survive.”

Heaven, Jessica and Celeste raced as they took up the suggestion to try some smaller numbers in the circle. They quickly listed the survivor for a page full of circles, busily crossing off imaginary players. “There’s no pattern,” Heaven declared, giving me an accusatory look.

“You’ve destroyed your evidence,” I commented. “What if you write out the numbers as you cross them off,” I suggested.

Taylor was sufficiently intrigued to not need to be reminded to put away her lines for the school play… each of her players was reprsented by an open circle. As each player was killed off by the game wizard she filled in the circle. While others students were crossing off dots, Taylor’s method was simple and clear and a guard against confusion.

“Taylor, show your method at the board.” She responded, “I’ll do 35.” I resisted suggesting she did a small circle, beginning to think the pattern might reveal itself easier with larger numbers of players.

Emma and friends urged her on and soon Taylor was also listing the numbers as she filled in circles. In a stroke of pure genius Julia suggested Taylor colored the circles a different color on the second revolution by the wizard. Soon the numbers were color coded too.

By the sixth revolution we’d exhausted the supply of colors.

On the second revolution “it’s every fourth player… the next every eighth… then every 16,” Brittany pointed out.

Hadassah and Amanda, on the far side of the room, pointing to their list with numbers outlined in boxes said there was a “doubling” as the wizard circled.

As the 50 minutes ended, Brittany asked, “you do know the answer, right?” No, I confessed.

+++

Not everyone in class was as enthused by the problem. Everyone attempted to answer the straight question, but a significant number then tried to take the question further. A pleasingly large minority were still working on the problem as the lesson ended.

Significantly the students who persevered were mostly girls, and not girls who always stay so focused in class.

❏ The use of an open-ended question for a whole lesson was inspired by The Elephant in the Classroom: Helping Children Learn and Love Maths by Jo Boaler, to be published next March, and Boaler’s US classic What’s Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject

Prof Boaler, who has done research on effective math teaching involving thousands of students in schools in both the USA and UK, argues for open-ended projects, mixed-ability group work and students talking math.

❏ The problem has a name: the Josephus problem.

The UK’s Royal Institution has used it as the basis for one of its master-class series. I got the idea to use the problem from the November 2009 issue of Mathematics Teaching, the journal of the UK Association of Teachers of Mathematics (MT216).

The “best” solution is to use binary numbers.

Grading gets an F

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THE PRINCIPAL of my school has bravely initiated a discussion about whether or not D and F grades should be used by teachers. The argument goes along the lines if D and F grades are ditched then teachers will need to work with their pupils to find the success within every student.

Ds and Fs don’t motivate or promote learning. Finding success does.

Ds and Fs have enshrined the out-dated pedagogy that grading is about sorting and ranking, that students need to be judged in comparison with one another.

The result of such alpha-numeric grading is that the traditional idea of the “good” student — the students who most closely resemble the aspirations of their teachers — is the scale against which students are judged; learning is demonstrated by turning homework in on time, doing well on quizzes and tests and putting your hand up to answer questions while not talking out of turn.

The result of such behavior is turned into a mathematically nonsensical percentage inside a computer, which then spews out the grade. And for many teachers that’s it.

If a student does not fit into this rigid mould, or cannot demonstrate learning by these criteria, then the result is F for Failure.

Yet the research should be pointing us to question this approach: Not only does this traditional way of measuring learning not reveal the learning going on among many students, it is actually an obstacle to learning for all students… the achievers as well as those who appear not to be getting it.

As part of the discussion in our school I was challenged in a meeting to summarize the case against alpha-numeric grading. I mumbled a few sentences as best I could for as long as it seemed polite to do so.

Then, later, I kicked myself for forgetting the key reason grading does not work. So, I decided to summarize in short sentence bites the best case I could muster for a two-minute contribution:

❏ Grades tell students nothing about what they need to do to improve.

❏ Grades tell students nothing about what they have achieved.

❏ Grades focus students on grades and collecting points, not on what they are learning.

❏ Grades introspectively focus students on ability, or their feelings of lack of ability, not on how they can work to improve.

❏ Grades destroy intrinsic motivation.

❏ Grades don’t measure learning: grades measure obedience, compliance and how well a student can jump through a teacher’s grading-policy hoop.

❏ Grades discourage intellectual risk taking.

❏ Grades divert the attention of teachers and parents as well as students.

❏ Grades encourage rote learning, memorization not reflection.

❏ Grades pit student against student, ranking and sorting.

❏ A grades require F grades. Grades force teachers to give Fs to justify the As. Grades work against finding the success in every student.

❏ Grades increase stress. Stress is bad for learning.

❏ Grades don’t describe learning.

❏ Grades throw students off the back of the boat.

❏ Grades discourage student collaboration.

❏ Grades reward skills not valued in later life, such as memorization.

❏ Grades demoralize and demotivate.

❏ Grades label and stigmatize.

❏ Grades are part of an out-dated carrot and stick, rewards and punishment behaviorist approach to education.

❏ Grades lower the self-esteem of low achieving students and discourage risk taking among higher achieving students.

Readers will find plenty of links elsewhere in this blog on the research behind these statements. But a good start would be From degrading to de-grading by Alfie Kohn.

Take it nice and slow

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AS KIDS move into adolescence they need to become less impulsive and more reflective. So the brain’s output controls in an adolescent are pushing the teenager to take her time and not do the first thing that comes into her head.

“This is ironic,” says Mel Levine in his A Mind at a Time, “since our high schools force our kids to do everything as fast as possible.

“They have to write quickly, think fast, remember on the spot, sprint through timed tests, and meet tight deadlines,” says the professor of pediatrics and director of the Clinical Center for the Study of Development and Learning at the University of North Carolina medical school.

“This frenzied pedagogical rhythm is totally contrary to what the students’ brains are striving to become. The output controls are crying out, declaring that they exist to promote thoughtful, slowly executed work, which should be one of the principal missions of adolescence and the high school years.

“I think we should reward adolescents for taking as much time as they need to do a good job.

“The output controls are doing what they can to decelerate thinking, decision making, and output, to make kids thoughtful rather than impulsive. Secondary education, therefore, ought to incorporate as one of its principal objectives teaching kids how to work slowly.

“That’s what the developing brains are trying to tell us.”

Mel Levine MD is also the author of The Myth of Laziness and is a co-founder of All Kinds of Minds, a nonprofit institute for for the understanding of differences in learning.

No gain from the pain of testing

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HIGH_STAKES testing on the rise since 2002 and No Child Left Behind, may have lead to more hours spent on reading and math in schools, but there have been no increases in learning.

And the curriculum, particularly in the humanities and liberal arts, has narrowed. “Today may actually be worse for poor children in the US than at any time in the last half century. This is because the lower classes are being kept from the liberal arts and humanities curricula by design,” a respected thinker on pedagogy told a key conference this summer.

“The newest difficulty in promoting the arts and humanities in the curriculum is due to the use of high-stakes testing,” Prof David Berliner told an international conference on redesigning pedagogy in Singapore.

“We need to remember that when administrators and teachers concentrate their efforts on raising only a few skills, they detract from the talent pool for individual and national success in an economy that will demand adaptability.”

In his paper the Regents’ Professor in the College Of Education at Arizona State University argues the result of high-stakes testing has been to increasingly narrow the curriculum, at a time when the challenges of the future demand the broadest possible liberal arts curriculum.

“The decrease in exposure to certain curricula is a rational response to high-stakes testing. But this decrease in exposure to a varied curriculum is of great concern as we contemplate what the 21^st century might have in store for our youth.

“Compared to the past, the future is likely to be more Volatile, Uncertain, Complex, and Ambiguous — A VUCA world for our children to face. I think adaptation to such a world requires a citizenry with the broadest possible curriculum, not a narrow one that constricts the skills of the youth because of a need to demonstrate accomplishments on a small set of assessments.”

“A 21st century workplace is likely to value such social skills as active and tolerant listening, helping each another to define problems and suggesting courses of action, giving and receiving constructive criticism, and managing disagreements. But in today’s high-stakes school environments, collaborative work where such skills can be developed is seen less frequently than ever because such work always means a loss of time that could be used for preparation to take high-stakes reading and mathematics tests.”

The narrowing curriculum is particularly undermining the education of the poor he argues. “America apparently has developed an apartheid-like system of education.”

“Using the argument that we must get their test scores up, we in the US are designing curriculum for poor children, often poor children of color but certainly, numerically, for poor white children, that will keep them ignorant and provide them with vocational training, at best. Their chances of entrance to college and middle class lives are being diminished, and this is all being done under the banner of “closing the gap,” a laudable goal, but one that has produced educational policies with severe and negative side effects.

Focussing on research by Hong and Youngs (2008) the response to high-stakes testing in Chicago and Texas, Prof Berliner says:

“In Chicago the researchers found that high-stakes testing seemed to narrow the curriculum and make it harder for students to acquire higher-order thinking, writing, and problem-solving skills. In Texas, it was found that schooling changed in ways that emphasized rote learning, not broad intellectual skills.”

A study by Lipman (2004) of Chicago schools found that the more affluent students in Chicago received a much richer and more intellectually challenging curriculum than did the poor children in Chicago. Poor minority children, in particular, were required to memorize fragmented facts and information, and they were constantly taught simple test-taking techniques.

“Lipman is probably quite right when she says that this differential access to high-quality curriculum will have significant consequences in terms of the social inequalities we will observe in the future. White students who possess a great deal of the cultural capital valued by schools are going to be much more likely to get to college and thus more likely to attain higher status through higher paying jobs. But low SES and minority students in Chicago’s schools are much more likely to end up in lower-skilled and lower-paying jobs. The decisions about curriculum and instruction in Chicago and other urban districts results in access to rigorous curriculum for some, but not for others, thus allowing for the continuation of the current unequal social structure.”

What to do?

“Change the tests used for school accountability under NCLB. Currently almost all the tests used to comply with NCLB make heavy use of multiple-choice items and thus are designed to reward memory of decontextualized bits of knowledge. But we know that tests with high-stakes attached to them drive curriculum and instruction. So the construction of tests that measure things like creativity and critical thinking need to be designed so teachers have tests worth teaching to.

Simply using tests with open-ended items has also been found to change teacher’s instructional behavior. Under those conditions teachers more frequently required their students to explain their answers in the classroom, and the teachers used more open-ended tests in their own classrooms as they tried to give students experience that would help them on the end-of-year tests.”

In conclusion Prof Berliner argues: “The same politicians and business persons that want high-stakes testing to be the cornerstone of a school accountability system also want 21st century skills developed. They do not yet understand that they cannot have both at the same time. These are incompatible goals.

“It seems to me that all but the most privileged students come into public schools where the pedagogy may actually be closer to that of the 19th rather than the 21st century. In schools for the poor, Dickens’s (1854/1868) wonderfully written caricature of a teacher, Mr. Gradgrind, still lives. Gradgrind said:

Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!

“But it is not just pedagogy that needs improvement. Many of our students receive too limited a curriculum for dealing with what the eminent psychologist Howard Gardner (1999) reminds us are always the most important questions facing humankind: what is true, rather than false; what is beautiful, rather than ugly or Kitschy, and what is good rather than compromised, or evil.

“A broad liberal arts curriculum is needed to deal with these eternal questions. But we in the US are far from providing that now, and moving further away from that model as high-stakes testing changes what and how we teach.

“No one really knows what 21st century skills are needed to foster success for individuals and nations. But developing critical thinking, engaging in activities that require problem solving and creativity, and doing individual and collaborative projects of complexity and duration, are all good candidates for helping each child and both of our nations to thrive” told the teachers and educationalists gathered in Singapore.

Prof Berliner’s complete paper can be read at www.susanohanian.org

In this review Prof Berliner’s citations have been removed for readability.

Relax… and watch the kids shoot some squid or crash a car

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THREE weeks from the end of the semester I decided to bring back the laptops for my three Algebra 2 classes and see where we would go. We would go further than I could possibly have hoped. And it would be me who would learn the most… the more the teacher pulls back, the further the kids go on their own.

The students would be using Geometry Expressions, a symbolic geometry system that reveals the geometry behind algebra and the the algebra behind geometry¹. To get us going we would be using problems from Geometry Expressions activities for Algebra 2 and Pre-calculus, written by high school math teachers Tim Brown an Jim Wiechmann.

For a start, I photocopied some of Tim’s and Jim’s problems and hints on parametric functions which took the students through the basic steps of using Gx and launched in. I could model the steps using Geometry Expressions on the interactive whiteboard.

First impressions were positive as the first lesson logs record:

“Positive response to the program from students (girls + boys). Most problems around familiarity with the program, but no complaints about impossibility… Several said liked the way it draws straight lines, can see numbers, etc. (p2)”

“Three girl students who normally do nothing working well on task and asking mathematical questions… Higher level of engagement than normal for this class… Girls who are rarely engage in the lesson comment, “it’s fun… we should use it more. (p4)”

But not entirely: “This class (p5) has high rate of deliberate helplessness and unwillingness to figure stuff out. Ditto here, many students get stuck, do little to get unstuck and then blame me. It’s par for the course… Least successful class to date… as expected… Need re-think for next class.” Classic grumpy teacher response. It’s the kids fault! Bad teacher!

And so it continued for several lessons. Some ups and quite a few downs. The lesson logs record:

“The first part of the investigation seems to be useful and keeps most students absorbed. (p2)”

“TAG boy resists experimentation… Explain the aim is to get him to push assignment to one side and to take over under his own initiative and inquiry. Try to explain a method of changing one thing at a time in the equations and spotting/writing the change. Look for pattern. Explain he’s used to being more spoon fed… this will seem slower, but will be more effective. He does not look convinced. (p4)”

“Better than usual focus (still not brilliant). (p5)”

But I’m still nervous. I’m off the curriculum, ignoring the upcoming finals and can see a lot of students being much better at toggling in/out of their web browser as I walk the room (we need internet access for the students to logon and the laptops to work). “Move students on… seems to indicate that need to put in more traditional lesson somewhere to check some links are established. I seem to be more phased by this than the students. (p5)”

Focus on trying to lighten up the atmosphere and take one problem very slow. Enter the powder-puff kicker trying to kick a 33-yard field goal over a 10-foot high goal.

For starters, we looked at investigating how to find one possible path for our powder-puff kicker’s shot. We decided the path would be a parabola, a quadratic.

Then by guess and check we started to use Geometry Expressions to find some possible paths.

❏ First, the parabola opens upwards, which does not fit the problem. Why?
❏ Then how to get it to start at the origin (where we had drawn a powder-puff kicker using the interactive whiteboard ability to draw over the Geometry Expressions projection).
❏ Where might the ball land if the kick had cleared the goal? Would picking a landing spot help us find a parabola and its quadratic function? Why? Is there more than one?
❏ But the highest point of the ball’s flight seemed to be at more than 2,000 feet! What would give us a reasonable curve?

Most of the classes — working in pairs — get curves that show how the ball could fly over the goal.

This gave a good review of quardratics, and particularly how we could use the kick spot and the landing spot for the x-intercepts, the zeros, and find a function in factor form.

But we still don’t know how the speed of the ball at kick-off or the angle of the kick affects the curve.

The lesson logs: “Apart from a few rough edges and reluctance of students to actively participate in much discussion, lesson feels good. Most pairs follow along and are still working using Gx at end of lesson. (p2)”

Well, most of the kids did 70 minutes of work. But, notice, they followed along, they were reluctant to participate. I ran the lesson, not the kids.

Same exercise for the next class, but with the aim: loosen up teacher-man.

“Explain we’re going to be looking at parametric equations, but we’re not likely to get there this lesson, it’ll take two. I like the idea of encouraging students to think that we’re not in a rush to get an answer, that the process is more open-ended… Second time around, much smoother. Also class more actively engaged in discussion back and forth. Keep almost all students actively engaged and on task for all lesson… I found it interesting to teach by finding the curve that fits the problem with guided guess and check approach. More students found the final curve, more quickly and more independently in this lesson.”

And I’m learning from lesson to lesson:

“Spoonfed up to this point, but with input from class,” records the p5 log. “Now decide to leave and not intervene… ‘it doesn’t work’ insists best student in class… ‘we found out what we were doing wrong’… Announce remember we’re trying to do it like mathematicians have done it through history… a sort of guess and check&hellip ‘We got it… awesome’.”

Two steps forward, one step back: “Summarize where we were at… try to get responses. Little comes back… Front of room led lesson, spoon feeding all the way… but attempting to get feedback at every step.” The lesson moves on to open up the ideas of parametric functions.

It depressingly concludes: “As a didactic lesson ok. As a discovery lesson, useless. First lesson of the day always an issue with sleepiness, etc. Difficult to motivate students. (p2)”

One class goes out, the next comes in. Teacher is starting to think:

“Smoother and more straightforward explanation of trig relationship to the problem… More feedback from students and most of lesson achieved in shorter time… Tomorrow will try again, but will ponder on how to move in direction of less didactics and more student discovery. One key might be to slow the process and simply give less… the fear here is that the students simply don’t respond but just sit and talk! The issue-elephant in the room is that students who are used to being spoon-fed, see math education as getting an answer and who are not expected to be engaged on their own behalf are slow to rise to the challenge… My challenge, next year, is to work out how to build a new culture in the classroom specifically aimed at teaching the kids how to learn by themselves for their own sakes.” That’s what I thought I’d been trying to do this year.

Hands off with p5, “encouraging them to explore” using a table to record their results.

“The students in this class (p5) who are normally willing to engage, did so in this lesson. There was some engagement by those who are normally resistant to the lesson. I wandered around the room on several occasions and asked if students needed help and most said they were ok… Unfortunately could not get much for the table of results… Asked one of the more engaged students how it felt just being left to get on with it… ‘it was fun… I knew what to do and you gave us support… it was ok’.”

It’s getting close to the semester’s end and finals week. p2 “sleepy and non-responsive.” We get into a muddle with degrees and radians which throws off the results: “One of those problems that gives students the excuse to stop,” notes the lesson log.

I’m becoming resigned to spending the summer thinking about what went wrong. Last day for our seniors, the sun is out, carnival atmosphere, so try to keep things light. Let’s try to get some pirates to shoot a cannon ball at a giant squid: “What do we do first? What did we do the other day? Draw a silly picture. I want to see lots of pirates and squids… and a bit of math thinking.” A coloring-in lesson to get us through the 80 minutes!

After two weeks, some pretty drawings. “I hit enter and the point just disappeared,” says the most engaged girl in class with apparent irritation. Another frustrated girl: “The point just kept going up and down.”

Next class (p4)… “I want lots of drawings” I say with resignation. After ten minutes I decide to walk the room, for the sake of form, expecting I know not what, but not a lot. Here’s the lesson log (the numbers represent times, TAG = talented and gifted):

09:55 TAG girl trying to reconstruct a Quadratic to plot the curve. Two TAG boys appear to have drawn a giant quid in the computer (at the correct coordinate) plus a pirate ship complete with sails and a cannon (at sea level!). I go up to suggest they now… when they fire off two cannon balls complete with parametric coordinates, both of which directly hit the squid! I point out the cannon balls are traveling at different speeds… also why not put the cannon on the deck of the pirate ship?

10:00 TAG boys have shifted their cannon and can still shoot the squid.

A plethora of pirate ships drawn in Gx now appearing on laptop screens. TAG girls importing pictures of pirate ships off the internet and placing inside Gx.

10:05 TAG girls who plotted quadratic have stopped work… explain the need for them to drive a point using parametric functions as the coords.

10:17 Lots of experimentation now going on… one boy suggests putting less or more powder into the cannon to change the velocity.

TAG girls wrestling with working out the parametic functions in the coordinates… what’s theta?  Why t? “We’ve forgot”

TAG girls(2) have animated Jack Sparrow firing the cannon… alas not so much math

Remind need to find two angles.

10:20 TAG boys firing salvos from a deck full of cannons, with many seemingly many direct hits… “we’re trying to fix them, but now we’re lost” they say laughing!

TAG girls… “we’ve got it animated, but the squid is shooting back! It’s going the wrong way”. Encourage and help to examine the functions… they’ve added the distance 1000m inside the x-function

TAG(2) girls are working on the math… trying to get the cannon ball to come out of the cannon which is on the poop deck along with an animated Jack Sparrow! Discuss how they could do this.

TAG boy (loner) struggling to understand how cos and sin work and why we use them. Brief lesson on trig with some scribbled cannons and right triangles.

09:38 (lesson ends 09:40) some students still working to complete and email to me.

I am just flabbergasted! Unbelievable! I’m bouncing around the ceiling!

At the next lesson (p4): “Class keen to see its work projected from last lesson. Lots of pride. Show a range of the work.”

We move onto plotting moving cars and finding whether or not they will crash.

09:50 TAG boys solve, they don’t collide. So… how do you make theme collide? Change speed car A. What about car B? What if a car starts earlier or later? What if a car takes a different route?

TAG girl takes time to remember the variable that will change is t… we have 2-minute dialogue where discuss everything about a car journey with kids in the back seat moaning till she hit on time is what makes them impatient. She gets the coord change +45*t immediately. Changes car A coord v quick too. She checks in she has it correct… she needs lots of validation, even tho v smart.

TAG who struggled last class, gets it quick this time.

09:57 TAG boys “they crash” Now what do you want us to do? I want you to decide how to change the problem… change time, direction.

10:05 couple of pairs need support to change the coordinate. Need to emphasise idea of time driving x and y. Up till now x has always driven y, now t for time drives x and y

TAG boy (loner) completes last lesson exercise showing the two cannon balls. The target (the eye) on the squid is clearly marked in red, points labelled, explanation included. Meticulous completion of the task, with reluctance to move on till completion. Moves on to new problem. (So, why are tests timed? What are we testing? People work at different speeds to their own different standards)

Pix of cars off internet being put on screen.

10:23 TAG girl creates a car out of points and drives the car.

TAG boys make four cars travel diagonally till all hit pedestrian in the center of the screen.

TAG boy draws on-screen map, has a miss and a collision, carefully color-coded and labelled. Other students build the Starwars battle.

10:28 TAG girl now has two cars built of points that collide… suggest she changes the nature of the crash… full on, glance, 

10:32 TAG girl 2 crashes five cars… boys sneer, they’re going too slow. I can make them go faster responds girl.

Lower ability girl, who is rarely engaged, continues to work with furrowed brow trying to change the coord to get cars to meet. Ask if she’s ok and get told she’s fine.

New boy knows exact speed change. Can run the actual collision in slow-mo back and forth. Suggest he changes time of start of one of the cars. He realizes straight away that he needs to change t in some way… he multiplies it by ¹/₂… hence making the car travel at half the speed… suggest he needs to think about just making the car start one hour later… he gets (t-1)

10:40 TAG boys getting 8 points to crash into one point… no longer talking cars talking “points” More serious in tone and attitude.

All students achieve basic aims of the investigation with many going much further. Solid engagement in a fun atmosphere for 70 minutes.

TAG girl reports the exercise on her Facebook page… suggest she posts the parametric coordinates. “I will,” she says.

10:47 TAG boys now have 12 cars crashing.

The other classes seem to pick up too. The log for the p2 lesson on cars crashing records:

The concept of miles per hour times t gives a distance not much of a problem, but need a little pushing to go there. Class not prepared to ask itself questions, but will respond to questions asked.

08:45 Model on board to make sure everyone roughly at the same spot. Class engaged + feedback is flowing freely. Working in pairs v cooperatively. Brief discussion about nature of time, no wrist watches any more (kids all show their left wrist), clocks used to tick, the only sound when no cars, etc… Someone asks how did old fashioned clocks work…

Suggest consider changing the problem. How do we ensure they meet? What if one car starts at a different time? What if a car decides to drive a different direction. Talk briefly about how math developed… by changing the problem and asking what if?

09:01 class silent and completely engaged.

J (struggles with math, but tries really hard, concepts v slow to hit home) Shows me how he has got the cars to collide, what he has changed, how he has changed the coordinates + how he has adjusted the speed of the animation so that he can see the collision clearly plus he has put a point on the screen to show the point of collision.

TAG girl raises the fact the car is going in the wrong direction… girl who has not attended class for three months(!) suggests need a negative.

On to the “final”. Ditch the common assessment and go for a Gx-based investigation: a sister/brother is using a hose to water a basket of flowers hanging above brother/sister who is sleeping in the sun underneath the hanging basket. What parametric coordinates get the water into the basket, without drenching the dozing brother/sister… or, drench the sibling and miss the basket? I add: How can you then change the problem?

p2 the first to go. We hit difficulties because I’ve misjudged some of the numbers. But now I’m not the teacher just someone in the classroom so I don’t get phased. Fixing the numbers just becomes part of the problem. The log records:

Despite the difficulties all class sticks with the problem for allotted time of 65 minutes.

Interesting… recall by students is a lot less than teachers would like to imagine. Even TAG students need a lot of reminders even though we’ve been doing this sort of problem for two weeks. Each time they catch on quicker and some can recall bits well, but few can recall all aspects. Taken back to traditional teaching this means that few students probably remembering v little in a sense that includes understanding.

The log for p5 records:

This class not able to complete this investigation without considerable amount of support, however all pairs remained on task for allotted time of 70 minutes with most completing the basic investigation. Two pairs (working together) managed to work out how to generate a stream of drops of water by adjusting the variable t… (t + 0.1).

Although this class has been working on this type of problem for several weeks using Gx, retention is an issue. The big success with this class is that students who adamantly refuse to engage with the subject, (now) do so with enthusiasm and determination when using Gx.

The log for p4:

Students willing to go further. All round good engagement. Even those who struggle, ask questions and move forward slowly.

Pair work seems to be accepted as beneficial, even where several students sit together each with a laptop!

This class v keen to draw a pic of the problem inside Gx, including placing artwork culled from the internet. All pairs manage to get to a solution where the water does not hit the  basket, but drops from above into the basket. Many plot a stream directly at the person asleep under the basket.

Explain that the parametric point represents the path taken by just one drop of water. Recount the example of the park in France where water is fired in short bursts over the pathways into barrels on either side of the pathways… bizaar effect of short isolated strings of water flying along a perfect parabola through the air. Suggest as inspiration of how to take the problem forward… how to fire a stream of drops one after the other… need to adjust t… is it t + 0.1… t – 0.1?

One student picks this up and works hard to get it to work. She does so.

One student asks why we would want to expand the problems once got an answer… explain idea that math developed through this process (heuristics).

Good end to the year.

¹ The development of Geometry Expressions is funded in part by grants from the National Science Foundation. The author sits on the NSF committee monitoring the project. The logs are being used as part of an NSF-funded research project un by Oregon State University looking into issues raised by using technology in math classroom.

We should be teaching mathematical thinking

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The use of IT in the math classroom raises implications for pedagogy in the math classroom.

This brief discusses the purpose of math education and tests the arguments for thinking more in terms of teaching creative math thinking skills instead of the current practice of teaching a series of mostly algorithmic skills. In particular the brief argues for a new form of assessment reflecting students attempts at math thinking as opposed to an ability to demonstrate facility with applying algorithms.

The brief concludes that such changes could mean more students feel success in math, grow to enjoy their math and attain more useful math skills — math thinking skills.

+++

WHAT’S the point of teaching and learning math?

For some time now, I have increasingly felt that a major issue in math education is that math teachers — plus those that create and administer the structures within which math teachers work — are not clear in giving good reasons to students about why it is so essential to study mathematics, exactly what is it we are all trying to achieve.

Current course structures, strict curriculum, standards, the emphasis on testing… all do not help teachers to reflect on or explain the purpose of what they do. Probably a majority of students, but certainly significant numbers, remain confused&hellp; and uninspired.

Part of the problem arises from the fact that just about all involved — except the students and the parents — are at least reasonably good at math, they get it, they enjoy it, they value it for its own sake, they see its value in the wider world. Few will have experienced the debilitating confusion, demoralization, despair that is the lot of substantial numbers of their students.

They see math education as turning out kids like themselves, mathematicians or one sort or another. This is fine when they encounter kids who get it, enjoy it, value it, etc.

This approach is compounded by political forces, which can include some mathematicians and their associations such as the NCTM, which view math education as little more than a utilitarian function at the service of corporate America.

For this latter group in particular, the maximum of math teaching is the delivery of a numerate workforce. And, it should be acknowledged, that is an aim of math teaching.

However…

There are swathes of young people, children, who do not get it, do not enjoy it and do not see any value in their math education. Yet many of these young people may, indeed, have math thinking abilities. They will certainly need math thinking skills, math understanding, in the fast-changing world in which they will live.

Crudely put, these children are not always well served by mathematician math teachers. Or, are not well served by mathematician math teachers who do not reflect wider on issues of pedagogy or wider (social, political and philosophical) concerns about education in its widest sense.

One dire result is that math education in the public schools is often confined and restricted to training children to be numerate, with little more.

The emphasis on standards, state testing and curricula driven by textbook adoptions, militate against wider reflection. Math teachers simply have too much on their plates to reflect… or they do reflect, and then knuckle down.

Students themselves give the clue to solving the dilemma when they ask — and they always do — “when am I going to use this in real life?”

The frank answer is most are not going to use much of it at all, and certainly not the math that gets tested in multiple-choice computerized testing. One realistic answer to the question is that math education is attempting to teach abstract thinking skills, or problem-solving skills.

But that doesn’t go far enough. It’s not just about what and why students are learning&heliip; but also how they are learning. Indeed, the how can really be the embodiment of the what and the why.

Focussing on how children learn math may be the answer to the student-question. If the experience in the classroom is totally focussed on math thinking — with the student feeling in meaningful control of the progress, mentored Vygotsky-style within the zone of proximal development by adult guidance and peer collaboration — then the teaching pedagogy itself may give real meaning, be itself the explanation of what math education is all about.

And if the assessment supports this approach, helps guide it forward, focussing on helping the student to find their own thinking skills — rather than seeking to reward or punish — the student will not so much be learning math thinking skills but experiencing using math thinking skills.

The how becomes the what and the why.

+++

The how does not include didactics, the pressure of tests and quizzes, points, grades.

The how does include peer collaboration, teacher as mentor, student control… and time for the child to play, think and work out their own solutions.

That can be done without technology. But there is no doubt that technology can really aid the approach, providing the classroom atmosphere and activities allow the child to get lost in their thought, explorations and discussions. Indeed, the creative use of technology almost demands a new approach to pedagogy in the math classroom.

It also demands a new approach to assessment…. still working on this!

Avis foregoes the restroom and expresses some geometry

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AVIS was in my Geometry class this year. Or, occasionally he was in my Geometry class. For a few minutes at a time.

Avis is not disruptive in class, as much as he simply leaves class. He’s frequently found just wandering the halls.

He has numerous accommodations to account for his behavior. Some teachers respond with referrals, but I’ve decided little is achieved by repeated referrals when the aim is learning.

In my class Avis asks — insistently — to go to the bathroom. Again and again. On his return it can be minutes before the demand is repeated. If I explain he has just been, he disappears from the room the moment my attention is elsewhere.

And when he is in class, it is hard for him to sit in one place. Avis doesn’t do much geometry, though he will turn in some work if he can sit next to an accommodating pretty girl willing to help.

Earlier in the year the class had used laptops to do various constructions and investigations using Geometer’s Sketchpad. The exercises had not been as successful as hoped, more likely because of the didactic approach I had adopted, making the investigations quite formal.

But Avis’s attention had been grabbed, and the frequency of his bathroom visits slowed. Avis had spent much of his time playing, using the program to draw pictures. As play is a positive start to learning, this seemed a welcome step.

At the end of the year I decided try again with the laptops, using a different program, Geometry Expressions, and adopting a looser teaching style.

Geometry Expressions ¹ is a geometry-algebra system. Algebraic notation is used to “constrain”, or fix, the drawings. The outcomes can also be expressed as algebra. The focus of the mathematical thinking involved is different to that used in attempting to do traditional straight-edge and compass geometric constructions.

If you constrain a triangle side-angle-side, and then try to add a second triangle using the third, unconstrained, side Gx forces some interesting thinking if you try to impose constraints on the second triangle.

Teachers and academics who have worked with Gx readily acknowledge its power for older or higher-level students — my own AP Calc class produced some lovely work based on animating hypocycloids after their exam.

The question has been could the progam help younger or lower-ability children with their mathematical thinking? Avis was not the student I had most in mind when we tried our first exercises using Geometry Expressions.

Students were invited to draw some triangles and then constrain — fix — side lengths, angles, to create right triangles.

The class then had a variety of problems involving Pythagoras. They were asked to solve the problems using Gx to recreate the problem. One particularly interesting task was to find all the possible sides of a right-triangle if two of the sides are lengths 10 and 15.

Below are extracts from the lesson log written as the the lesson progressed (TAG = “talented and gifted”, RR = “rest room”, AA = Avis):

30 mins in, most students able to change sides a and b for numbers + observe how the formula calculates the answer.

Some students use the program to draw pictures not related to the exercises. (This also happened when students introduced to GSP).

40 minutes in… higher level of engagement with activity/focus better than normal for this class. Students interested in confirming answers in discussion.

TAG-type student complains the exercises can be done more quickly without the computer… explain am training him to use the program with aim of using it as a thinking tool + more difficult problems later. Agrees to give it a second go on a more complex problem.

45 mins in… one of most difficult students (usually can’t focus at all, frequent trips to RR, etc) still engaged and asking questions… and has not visited RR! Call this student AA. AA working with girl BB who does not find subect easy, but who tries.

50 mins in… some signs of disengagement among some students.

60 mins in… TAG student happier… still feels he’s working slower, but is seeing the program can aid his thinking.

Two students raise calculation that gives pi/2. Leads to brief discussion that this is 90 degrees (this class not done radians).

65 mins… engagement now down to about one in three students. This is an improvement for this class. Class also much more quiet than normal.

Allow internet access for last 20 minutes… some students still continue with Gx.

Girl BB + boy AA still v engaged and 70 mins in are now trying to work out how to constrain a side using a radical. Still no visit to the RR by AA!

TAG student comments that Gx good “verification tool”, useful for checking answers you’re not sure of.

Pack away 80 mins in.

AA cleans board, still quiet (!!!) and no visit to RR. Wow! AA smiles genuinely brightly when I congratulate him on not going to the RR.

Two days later the same class works on using Gx to solve problems involving the equations of circles. This time Avis is very much the focus of my attentions.

Below are extracts from the log written as the class progressed:

blitz start to lesson… doors locked, etc. Computers eventually arrive from another classroom.

Students specifically told to work in pairs.

13:30 all students working. Enough computers.

AA + BB work together and ask frequent questions (on task).

Questions from students make it v easy to answer with a question focusing on why?

14:48 Class engaged. One TAG student (girl) asked for help, but figured answer while I worked my way round. Questions focus on basic use of Gx… partic how the animation works. These are easily solved questions and students pick up quick.

Talk in room is almost exclusively on the task.

pair work v successful.

13:54 Deep discussion between two TAG students on equations of circles.

Respond to earlier request from BR for help… “I got it! it’s ok…” Doesn’t look up, stays completely focussed.

LI, BB, AA have discussion about where the “variable”s go… these students never talk about “variables”!

14:03 Five students pulled out of class for OAKS state testing… they’re really upset! Includes LI who is working really hard on task.

14:12 AA + BB want to discuss what are the two things on which they must agree for them both to draw the same circle. BB using words like “segment”. AA has not been to the RR!

Two students working in Gx, but drawing pictures. Only occasionally are students caught on the internet.

Easy to get students moving.

Three TAG students now working on the extension activity.

14:12 BB and AA in deep discussion looking at the calculated equation of a circle.

14:22 AA + BB still on task. HT ok writing out definition of circle with help. BB points to an equation and asks “does this make sense to you?” We spot that she had not replaced r for radius with a specific value.

14:24 Tell students to start to shut down… BB shouts out to AA “quickly, let’s do this one.” And they do.

Sadly, departmental decisions about the need to deliver aspects of the curriculum before the end-of-semester test meant an end to the experiments with Geometry Expressions and a return to textbook-based didactics for Avis’s class. At the following lessons Avis mostly left the classroom, though he did frequently ask whether or not the class would be working on the computers.

But the brief experiment did demonstrate that Geometry Expressions can motivate and aid the mathematical thinking of younger and lower-ability children. This was confirmed by similar reactions by lower-ability students in Algebra 2 classes where it was possible to use the technology over a much longer period.

¹ The development of Geometry Expressions is funded in part by grants from the National Science Foundation. The author sits on the NSF committee monitoring the project.

‘Just weighing a pig doesn’t fatten it’ — Obama hint on testing

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PRESIDENT Obama spoke, briefly, at his June 11 Green Bay town hall meeting on education as well as health… and gave a hint of some hope and change that may not be as bad as many educators feared when he announced the appointment of the militaristic, test tsar Arne Duncan as his education secretary.

Local teacher Matt Stein of 20 years challenged the president: “One of the things that I’ve learned in education in the last 20 years is that the system is not broken. And it bothers me when I hear politicians, and even my President, say that our educational system is broken.

“This system works in cases. There are great things happening in Green Bay and Appleton and all over the UP. And there are things that can be reproduced. My question is: When will the focus be on reproducing those things — smaller classrooms, creating communities in your classrooms — and moving the focus away from single-day testing and test-driven outcomes?”

The president responded:

“So here’s the bottom line. We’ve got to improve, we’ve got to step up our game — which brings me to the next point in your question, which is, how do we do that? I agree with you that if all we’re doing is spreading around a lot of standardized tests and teaching to the test, that’s not improving our education system.

“There’s a saying in Illinois I learned when I was down in a lot of rural communities. They said, ‘Just weighing a pig doesn’t fatten it.’ You can weigh it all the time, but it’s not making the hog fatter.

“So the point being, if all we’re doing is testing and then teaching to the test, that doesn’t assure that we’re actually improving educational outcomes.

“We do need to have accountability, however. We do need to measure progress with our kids.

“Maybe it’s just one standardized test, plus portfolios of work that kids are doing, plus observing the classroom.

“There can be a whole range of assessments, but we do have to have some kind of accountability, number one.”

Obama’s remarks heavily focused on the need for a change in health care in the US. They were reported as such.

But as the town hall was held in a school — SouthWest High School — a likely question on education was predictable. Obama’s remarks may have been less off the cuff than appears. Buried and almost lost in the extensive arguments on health care, the remarks on education could be seen as an early precursor of things to come.

Arne Duncan is at present on a “listening tour”.

When Arne Duncan was first appointed as education secretary there were fears across the education community, based on his draconian approach to bringing change to education in inner-Chicago — see Hope and change in my classroom.

But balancing Duncan, Obama also appointed the much more radical Linda Darling-Hammond, professor of education at Stanford, as head of his education policy transition team, a champion of constructivism and reduced testing. She was the choice of those who understand how children learn as Alfie Kohn put it.

Could it be that we’re going to see Duncan as the hard cop in the public eye, while some of the soft cop’s more thoughtful and intelligent policies make it into the classroom?

Stanford’s teacher training program now includes on its reading lists for aspiring teachers the radical (for the US) pamphlets and research of Kings College London’s Prof Paul Black and friends.

The record of the president’s remarks can be read at: www.whitehouse.gov

Below are the president’s remarks in full on education:

Well, let me — first of all, thank you for teaching. My sister is a teacher, and I think there is no more noble a profession than helping to train the next generation of Americans. (Applause.)

I completely agree with you that there is a lot of good stuff going on in American education. The problem is, is that it’s uneven. (Applause.) Well, let me put it this way. There are actually two problems. In some places it is completely broken. In some urban communities where you’ve got 50 percent of the kids dropping out, you only have one out of every 10 children who are graduating at grade level — this system is broken for them.

Q Crime — (inaudible).

THE PRESIDENT: Well, I’m going to get to that. We can’t have too big of a debate here. You got your question. (Laughter.) Don’t worry, though, I’m going to answer your question.

So there are some places where it really is completely broken. And there, yes, a lot of it has to do with poverty and families that are in bad shape. There are all kinds of reasons. And yet, even there, there are schools that work. So the question is, why is it that some schools are working and some schools aren’t, and even in the worst circumstances, and why don’t we duplicate what works in those schools so that all kids have a chance?

Now, in other places, Green Bay and Appleton and many communities throughout Wisconsin and Michigan, the average public school is actually doing a reasonably good job — but can I still say that even if you factor out the urban schools, we are falling behind when it comes to math; our kids are falling behind when it comes to science. We have kind of settled into mediocrity when we compare ourselves to other advanced countries and wealthy countries. That’s a problem because the reason that America over the last hundred years has consistently been the wealthiest nation is because we’ve also been the most educated nation.

It used to be by a pretty sizable factor we had the highest high school graduation rates, we had the highest college graduation rates, we had the highest number of Ph.D.s, the highest number of engineers and scientists. We used to be head and shoulders above other countries when it came to education. We aren’t anymore. We’re sort of in the middle of the pack now among wealthy, advanced, industrialized countries.

So even with the good schools, we’ve got to pick up the pace, because the world has gotten competitive. The Chinese, the Indians, they’re coming at us and they’re coming at us hard, and they’re hungry, and they’re really buckling down. And they watch — their kids watch a lot less TV than our kids do, play a lot fewer video games, they’re in the classroom a lot longer. (Applause.)

So here’s the bottom line. We’ve got to improve, we’ve got to step up our game — which brings me to the next point in your question, which is, how do we do that? I agree with you that if all we’re doing is spreading around a lot of standardized tests and teaching to the test, that’s not improving our education system. (Applause.)

There’s a saying in Illinois I learned when I was down in a lot of rural communities. They said, “Just weighing a pig doesn’t fatten it.” (Applause.) You can weigh it all the time, but it’s not making the hog fatter. So the point being, if we’re all we’re doing is testing and then teaching to the test, that doesn’t assure that we’re actually improving educational outcomes.

We do need to have accountability, however. We do need to measure progress with our kids. Maybe it’s just one standardized test, plus portfolios of work that kids are doing, plus observing the classroom. There can be a whole range of assessments, but we do have to have some kind of accountability, number one.

Number two, we do have to upgrade the professional development for our teachers. (Applause.) I mean, we still have a lot of teachers who are — we’ve got a lot of teachers who are well-meaning, but they’re teaching science and they didn’t major in science and they don’t necessarily know science that well. And they certainly don’t know how to make science interesting. So we’ve got to give them the chance to train and become better teachers. We’ve got to recruit more teachers, train them better, retain them better, match them up with master teachers who are doing excellent work so that they are upgrading their skills.

If after all that training, the teacher is still not very good, we’ve got to ask that teacher, probably, there are a lot of other professions out there; you should try one. (Applause.) I mean, I’m just being blunt, but we’re going to have to pick up the pace.

Now, the key point I want to make is this: We should focus on what works, based on good data. And Arne Duncan, my Secretary of Education, this guy is just obsessed with improving our education system. He is focused a hundred percent on it, and he is completely committed to teachers. We think that teachers are the most important ingredient in good schools. We’re going to do whatever works to help teachers do a better job — (applause) — we’re going to eliminate those thing that don’t help teachers do a good job. Some of it is going to require more money, so in our Recovery Act, we have more money for improving curriculums, teacher training, recruitment, a lot of these things. But you can’t just put more money without reform, and so some of it is demanding more accountability and more reform.

There’s one other ingredient, though, and that is parents. (Applause.) We’ve got to have parents putting more emphasis on education with our kids. That’s how we’re all going to be able to pick up our game. (Applause.)

UK moves closer to ending damaging tests in favor of teacher judgement

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UK SCHOOLS are further ahead than most of their competitors in developing and using formative assessment, assessment for learning, in classrooms. But the pressures of traditional testing still appear to be holding back further progress.

In 2005 the UK’s chief of Qualifications and Curriculum Authority, Ken Boston, predicted that external summative tests for 11-year olds and 14-year olds will eventually be replaced by moderated teacher assessment, possibly by 2015. He said that teachers in England will one day be allowed to select tests for their pupils from a bank of assessment tasks and tests and choose when the tests should be taken.

Research focussing on a cohort of pupils over eight years, extending from before the introduction of national testing for seven-year-olds in the UK, has revealed that after the introduction of the external tests, teachers’ own classroom assessment became more summative.

Before the introduction of tests, pupils felt that teachers’ assessments helped their learning but they later noticed that their teachers increasingly focused on performance outcomes rather than learning processes. Pupils themselves began to adopt summative criteria in commenting on their own work.

Reporting on the findings, the pamphlet The Role of Teachers in the Assessment of Learning concludes that opportunities to help learning — and reduce the gap between higher and lower achieving children — are being missed.

The pamphlet has been produced by members of the influential UK Assessment Reform Group as part of the Assessment Systems for the Future Project funded by the Nuffield Foundation.

“Worse perhaps is the distorting effect on assessment for learning… Many (UK) schools give the impression of having implemented AfL when in reality the change in pedagogy that it requires has not taken place. This may happen,for example, when teachers feel constrained by external tests over which they have no control. As a result they are unlikely to give pupils a greater role in directing their learning, as is required in AfL, in order to develop the capacity to continue learning throughout life.

“The nature of classroom assessment is dictated by the tests.”

The report acknowledges that the use of teacher assessments for summative purposes is not without its problems. But, it adds, these problems need to be judged against “the ample evidence that a system based on tests is flawed”.

“Systems relying heavily on tests results are found wanting in several respects, particularly in their ability to give a dependable, that is, both valid and reliable, account of pupils’ learning… the negative consequences of summative assessment for learning and teaching can be minimised by more appropriate use of teachers’ judgements” it concludes. It adds:

❏ Testing-based assessment fails to provide information about the full range of educational outcomes that are needed in a world of rapid social and technological change and therefore does not encourage the development of these skills.These outcomes include higher-order thinking skills, the ability to adapt to changing circumstances, the understanding of how to learn, and the ability to work and learn collaboratively in groups as well as independently.

❏ It inhibits the development of formative assessment (or assessment for learning) which is proven to raise achievement levels and reduce the gap between higher and lower achieving pupils.

❏ The data it provides are less reliable than they are generally thought to be. For example it has been estimated that the key stage (KS) tests in England result in the wrong levels for at least a third of pupils at the end of KS2 and up to 40 per cent at the end of KS3.

❏ The weak reliability of tests means that unfair and incorrect decisions will be made about some pupils, affecting their progress both within and between schools and beyond school.

❏ There is no firm evidence to support the claims that testing boosts standards of achievement.

❏ It reduces some pupils’motivation for learning.

❏ It imposes stressful conditions that prevent some children from performing as well as they can.

❏ It encourages methods of teaching that promote shallow and superficial learning rather than deep conceptual understanding.

The authors also add the use of test results, from league tables to target setting, is “too simplistic”.

Much summative assessment restricts the range of learning outcomes that can be assessed and excludes many of the higher-level cognitive and communication skills and the ability to learn both independently and collaboratively. The high stakes attached to the results encourage teaching to the test and excessive practising of test-taking.

This can result in pupils being taught to pass tests even when they do not have the skills that are supposedly being tested.

A study commissioned by the Department for Education and Skills (DfES) concluded that while drilling 11-year-olds to pass national tests is likely to boost results it may not help pupils’longer-term learning. The narrow range of learning outcomes assessed by tests contrasts with the broad view of learning goals reflected in the DfES Every Child Matters policy document.

The authors argue, it is crucial that assessment covers the learning that will be essential for young people who will live and work in a rapidly shrinking world and changing society. Two key sets of goals in any subject are:

❏ learning with understanding;
and

❏ understanding learning.

The first refers to the development of “big ideas” — concepts that can be applied in different contexts, enabling learners to understand a wide range of phenomena by identifying the essential links between different situations. Merely memorising facts or a fixed set of procedures does not help or a fixed set of procedures does not help young people to apply learning to a range of contexts.

The second set of goals relates to the development of awareness of the process of learning. It is widely recognised that ‘students cannot learn in school everything they will need to know in adult life’. School must therefore provide the skills, understanding and desire needed for lifelong learning. “Since what is assessed has a strong influence on what is taught and how it is taught, we must look critically at what is assessed. If the required outcomes are not included, then alternative methods of assessment are needed.”

❏ Testing can reduce the self-esteem of lower-achieving pupils and can make it harder to convince them that they can succeed in other tasks;

❏ Constant failure in practice tests demoralises some pupils and increases the gap between higher and lower achieving pupils;

❏ Test anxiety affects girls more than boys;

❏ Teaching methods may be restricted to what is necessary for passing tests (eg neglect of practical work).

Instead “the negative consequences of summative assessments may be minimized by giving teachers a greater role in assessing individual pupils” concludes the report.

Reports from the Assessment System for the Future project seminars can be found at: www.assessment-reform-group.org/ASF.html

A PiFactory review of the ARG’s earlier review of research on testing, Testing, Motivation and Learning is at The research gives testing an F.