## Posts tagged ‘math’

### Engaging math for all learners

Buy this Eratosthenes Net on a PiFactory tee-shirt

The UK extends its revised secondary (high school) national curriculum to 15-year-olds from September. The focus is on engagement, and in particular engaging all learners — regardless of ability — with rich, varied and compelling math activities.

And to reflect the new priorities, the UK’s public examinations — the General Certificate of Secondary Education GCSE — will boost assessment of applications and problem solving from 20 per cent to 50 per cent.

“This does not mean that technical competence is no longer important, rather that just being able ‘to do’ mathematical techniques will not be sufficient,” writes Sue Pope of the UK Qualifications and Curriculum Development Agency^{*} in Mathematics in School.

“Students will need to be able to think for themselves and decide when and how to use their mathematics to tackle problems within mathematics and in other contexts.”

The new curriculum is ambitious and defines “an entitlement of experience for all learners.

“Rather than labelling learners and restricting access, the richness of the entire progamme of study needs to be made available to all,” says Sue Pope. “Whilst this may seem daunting, particularly if you are used to teaching level by level… it can also be liberating.”

Mick Waters, director of Curriculum at te QCDA: “If we want young people to do well in mathematics, it helps if they enjoy the subject… to see that the subject is fascinating and exhilarating, to see the way it affects everyday life and helps to change the world in which we live.

“We have to strike a balance between the challenge of incremental steps in understanding, knowledge and skills, and the joy, wonder and curiosity of learning.

“It is not about ‘basics’ and ‘enrichment, all children should have a rich experience.”

For students to develop problem-solving and mathematical thinking schools “their classroom experiences need to be rich and varied”:

A rich mathematical task…

❏ Engages everyone’s interest from the start,

❏ Allows further challenges and is extendable,

❏ Invites learners to make decisions about how to tackle the activity and what mathematics to use,

❏ Involves learners in speculating, hypothesis making and testing, proving or explaining, reflecting, interpreting,

❏ Promotes discussion and communication,

❏ Encourages originality and invention,

❏ May contain and element of surprise,

❏ Is enjoyable,

❏ Allows learners to develop new mathematical understandings.

The QCDA worked with some 30 UK schools to develop programs of rich tasks aimed at “combining understanding, experiences, imagination and reasoning to construct new knwledge”.

Tasks and case studies are spelled out in the downloadable Engaging Mathematics for all Learners.

^{*}Shortly after the election of a new Conservative government in May, the UK Department of Education announced legislation will be introduced in the autumn to close QCDA.

### Tilting towards Nrichment

Buy this Eratosthenes Net on a PiFactory tee-shirt

TILTED SQUARES from the inspirational UK site nrich.maths.org is a great example of an open-ended math problem that leads to some deep math thinking… even among students who don’t normally show much interest in their math lesson.

The task at first sight appears to be quite simple. A square drawn on square-dotted paper is tilted by raising the lower right-hand corner vertically by one dot, and a new square is then drawn on this tilted base. The question is, what is the area of the new square? And then, is there a pattern to the areas with continuing tilts, as the lower right-hand corner is raised by a dot at a time? The immediate response of almost all the students in my classes was, the first square and the second square were exactly the same. This, even from my most accomplished mathematical thinkers. The Nrich site has an interactive program, so the square can be repeatedly tilted and projected. Even on tilt three or four students were still insistent the areas remained the same. Only when the square had tilted to become what students call a diamond, was there questioning.

My two most inquisitive students held out until after they’d carefully drawn and cut out the first two squares and placed one on top of the other before they would countenance any change.

Problems from Nrich seem to self-differentiate: two students who rarely are able to engage in more traditional exercises, quickly constructed on dotted paper a pattern of 18 tilted squares. They were excited. They were animated. And they wanted to talk about, and show what they had done. Others insisted on constructing the pattern on the interactive whiteboard. They did this while some of my more “analytical” students struggled to visualize and draw the squares, unable to identify the corners of a newly-tilting square.

Some students explored finding the areas of the square by measuring, and others tried Pythagoras.

The first found the results frustrating as they estimated fractions of a millimeter change in length, squaring the answers and ending up with lots of decimals and no clear pattern. The Pythagoreans built up tables and areas and tried to spot a pattern in the growth. They quickly spotted a pattern which they could verbally explain and extend, but could not or would not commit their analysis to paper, other than to list the answers.

Students were encouraged to try with a differently-sized first square. With a smaller starting square, the growth in area is more quickly apparent. The Pythagoreans could verbally list the areas with ease. But they refused to countenance any search for a method that could be committed to paper.

The majority drew several squares, and tried to estimate the area by counting squares formed by the dots — or just counting dots as an estimate. They could not extend to the results of the Pythagoreans, but were much more open to watching some hints on the interactive whiteboard: Why not a square with a horizontal base around the outside of any tilted square?

The area of the outside square, subtract four triangles and you have the area of the tilted square. The visual approach appealed to our early pattern artists, and provided the clue for the majority who were searching for a calculation.

Quickly more areas were forthcoming, while the Pythagoreans reluctantly drew some titled squares with exterior square… only to dismiss the approach with open scorn.

What if there are 99-tilts? The Pythagoreans raced for the answer, but could not agree.

Meanwhile another hint: Build up a table breaking each area calculation down into all of its detailed parts, including some reference to the tilt number. Use different colors for the numbers from different parts of the calculation, red for the tilt number, green for the side length of the outside square and so on. Is there a pattern?

What if the tilt number is n, representing the n^{th} tilt? Can n be identified in the pattern?

The early doubters confidently give the answer for the area of the 99^{th} tilt. The Pythagoreans are still arguing about the mental math, though there are signs on paper of the formula that was, indeed, in their heads.

And the artists had now produced colored titled squares and were demanding their works were now put on display.

Now that’s enriched mathematics. And nothing feels so Good!

❏ For more discussion about using open-ended questions to promote mathematical thinking… plus thoughts on what exactly constitutes math thinking see About Nrich: research plus articles

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

Buy this Eratosthenes Net on a PiFactory tee-shirt

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.

### Win, win not fail, fail

Buy this design on a PiFactory tee-shirt

GRADING day, end of the first quarter. After a long day writing individual assessments, I have some envy for my colleague teachers who have been punching in percentages into their computers for the past few weeks.

As the email reminded us at the beginning of the day: hit the yellow button and the magic software will turn all the numbers into a grade and get the report cards ready for dispatch. Bingo! Easier than punching in 180+ grades and descriptive assessments by one-by-one.

I note one of my colleagues’ students has a percentage of 92.827. Well, that’s an A, then. To three decimal places!

Clearly the assessments that built this grade had not just 100 criteria, not 1,000, not 10,000… but 100,000. Some grading!

But what is this percentage of..? What exactly has our student achieved 92.827% of..? What, exactly, is being measured?

Percentages are a mathematical nonsense, unless they are of something. Just what did our student fail to do to miss out on the last 7.133%? Nothing on the report card gives an explanation.

Even being more sensible about the three decimal places (the very expensive software used across school district spewed those out, not I), just what would, say, a rounded 90% actually mean?

More important what about the student who got a 65% and got an F? What practical advice does the 65% contain to tell the poor unfortunate who has been branded a failure need to do to become a success?

The A, B, C, D or dreaded F may contain no help in specifically describing what a student has or has not learned… and certainly contains no help in telling a student what they need to do to improve. But it does label the student.

This might be ok for the (albeit stressed-out) student labelled an A or B… but it’s not so hot for the student labelled C, D or F. Labelled a failure… but given no clue as to what to do.

And, believe me, where these percentage-based letter grading systems are used with enthusiasm, then these numbers have been pinned up fresh every week in classrooms… raising stress levels in all the students weekly and forcing them to focus and re-focus… not on the joys of learning, but on the terrors of the grade.

You might get an A one week on your assignment. The next, you miss it. That means you’re at best 50% and failing wildly. There’s plenty of teachers who practice, and defend this as a perfect reflection of their students’ learning. You get a perfect 100% A the following week… that still does not lift you back up to passing!

You might be well on top of the learning… but you can’t meet deadlines = F! What’s important here?

So, what’s on your mind? The beauties of that Shakespearean sonnet or doing something desperate to scrabble together a few more percentage points? Or, just call it quits? You can’t win.

What if you are the kid who doesn’t get As? And you miss an assignment? And you work nights?

Is this education… or just a confusing nightmare? Life was much more fair in Catch-22. Welcome to school.

❏ So, what is to be done?

Drop the As, Bs down to Fs. Learn the lessons of pre-algebra and accept that percentages without a definable “of” are a mathematical nonsense.

Instead, describe in student-friendly phrases just what it is they need to learn, what they have learned, and what is the next achievable step they need to take to improve.

And give them as many chances as they need to do it, to learn. It’s the learning that’s important, whenever and wherever it finally happens. Not the grade.

That’s a win, win. Not fail, fail.

### Working inside the black box

Click here to buy this Eratosthenes Net PiFactory tee-shirt

YET MORE research is now available to confirm that alpha-numeric grading and traditional testing not only do nothing to promote learning… but actually lower achievement.

Since the late ’90s the team assembled around Prof Paul Black at King’s College, London, has steadily chipped away at the traditionalist approaches to assessment with their Inside the Black Box series.

“In terms of systems engineering, present policies in the US and in many other countries seem to treat the classroom as a black box,” Black and Dylan Wiliam argued a decade ago. “Certain inputs from the outside — pupils, teachers, other resources, management rules and requirements, parental anxieties, standards, tests with high stakes, and so on — are fed into the box.

“Some outputs are supposed to follow: pupils who are more knowledgeable and competent, better test results, teachers who are reasonably satisfied, and so on.

“But what is happening inside the box?” they asked.(October 1998 issue of Phi Delta Kappan, Vol. 80, No. 2.)

Their books and articles have since influenced tens of thousands of teachers and educational professionals around the world. But while their reasoning has changed both thinking and practice in the UK and Europe, it has yet to take a hold in US public schools which remain fettered by traditionalist, formulaic practice and teaching children to collect points.

Now Black, Wiliam and colleagues have produced a new series of slim pamphlets taking the argument further and giving specific advice and a wealth of teaching points tailored to key subject areas, with an overall summary Working Inside the Black Box.

The pamphlets, published by GL Assessment, focus on maths, english and science.

Working inside the black box means focussing on assessment inside the classroom with the priority of serving student learning: “It thus differs from assessment designed primarily to serve the purposes of accountability, or of ranking, or of certifying competence.

“An assessment activity can help learning if it provides information to be used as feedback, by teachers, and by their pupils, in assessing themselves and each other, to modify the teaching and learning activities in which they are engaged.

“Such assessment becomes ‘formative assessment’ when the evidence is actually used to adapt the teaching work to meet learning needs.”

Research experiments have established that, whilst pupils’ learning can be advanced by feedback through comments, the giving of marks — or grades — has a negative effect in that pupils ignore comments when marks are also given (Butler, 1988).

“These results often surprise teachers, but those who have abandoned the giving of marks find that their experience confirms the findings: pupils do engage more productively in improving their work,” says the Black team.

“The central point here is that, to be effective, feedback should cause thinking to take place… the assessment of pupils’ work will be seen less as a competitive and summative judgement and more as a distinctive step in the process of learning.

“The key to effective learning is to then find ways to help pupils restructure their knowledge to build in new and more powerful ideas… that learning was not a process of passive reception of knowledge, but one in which the learners were active in creating their own understandings. Put simply, it became clear that, no matter what the pressure to achieve good test and examination scores, learning cannot be done for the pupil; it has to be done by the pupil.”

Teachers in the UK involved in a two-year experiment demanded to learn more about the psychology of learning after working on such formative assessment practice in the classroom:”Learning is not just a cognitive exercise: it involves the whole person. The need to motivate pupils is evident, but it is often assumed that this is best done by offering such extrinsic rewards s merits, grades, gold stars and prizes. There is ample evidence that challenges this assumption.

“Pupils will only invest effort in a task if they believe that they can achieve something. If a learning exercise is seen as a competition, then everyone is aware that there will be losers as well as winners: those who have a track record as losers will see little point in trying.

“Thus, the problem is to motivate everyone, even though some are bound to achieve less than others. In tackling this problem, the type of feedback given is very important.

“Feedback given as rewards or grades enhances ego — rather than task — involvement. It can focus pupils’ attention on their ‘ability’ rather than on the importance of effort, damaging the self-esteem of low attainers and leading to problems of ‘learned helplessness’ (Dweck 1986).

“Feedback that focusses on the needs to be done can encourage all to believe that they can improve. Such feedback can enhance learning, both directly through the effort that can ensue, and indirectly by supporting the motivation to invest such effort.

“When feedback focusses on the student as a good or bad achiever, emphasising overall judgement by marks, grades or rank order lists, it focusses attention on the self (what researchers have called ego-involvement).

“A synthesis of 131 rigorous scientific studies showed this kind of feedback actually lowered performance (Kluger + DeNisi 1996). In other words, performance would have been higher if no feedback had been given.

“This is because such feedback discourages the low attainers, but also makes high attainers avoid tasks if they cannot see their way to success, for failure would be seen as bad news about themselves rather than an opportunity to learn.

“In contrast, when feedback focuses not on the person but on the strengths and weaknesses of the particular piece or work (task-involving feedback) and what needs to be done to improve, performance is enhanced, especially when feedback focuses not only what is to be done but also on how to go about it.

“Such feedback encourages all students, whatever their past achievements, that they can do better by trying, and that they can learn from mistakes and failures (see Dweck 1999).”

Quotes taken from:

Working Inside the Black Box, Dept of Education + Professional Studies, King’s College London by Paul Black, Christine Harrison, Clare Lee, Bethan Marshall and Dylan Wiliam

Mathematics Inside the Black Box, Dept of Education + Professional Studies, King’s College London, by Jeremy Hodgen and Dylan Wiliam.

❏ If this article was of interest, try… No prizes in points

❏ PiFactory’s descriptive grading rubrics can be found at: pifactory.net/catalog/assess_page_one.html

### What really was the point?

Buy this design on a PiFactory tee-shirt

AT THIS year’s Commencement ceremony I idled the time reading the list of graduates. Names from the past triggered memories.

I watched the recalled names walk across the stage in cap, gown and garlands, looking so proud and so much taller than my memory. Was that really mischievous D’Arcy? He seemed so much smaller then, now he could cradle the principal under his chin.

Jeff, two years of struggle, of repeated assignments undone and promises of better to come and broken promises… who came alive when asked if he could help make a film about the math in Tom Stoppard’s Arcadia.

Ashley who tried so hard, turned everything in, pages and pages of detail and colorful annotations… and almost every answer not even within the realm of what we were doing. Lindsay, who would not move to the next question till the current question had been confirmed, pristine and correct in every way.

Kyle… Kyle… ah, yes, Kyle. Richard who was so correct, now sporting a giant mohican.

I wondered… do they remember anything whatsoever we struggled over? And, does it really matter?

The oh-so-important assignment, grade, topic, algorithm, nuance that seemed so damnedly frustrating and vital two, three years ago seemed so irrelevant now. Our year (or two) together through Indian fall, and the long, grey, wet months, praying for a snow day before seizing on an early glimmer of sun would, at best, be the briefest of memory. You remember that math teacher…?

Each semester I do a student feedback survey. It prompts some reflection and student self-assessment and seeks some feedback on the lessons, what worked, what didn’t. The memories are always a surprise… certainly not the meticulously-planned and brilliantly-executed three-part lesson.

It’s usually the spontaneous moment. The something that popped out of no-where, or, the go-with-the-flow moment or experiment. Lesson-of-the-day: Go with the flow.

Nik complained all year, when are you going to do those puzzles again? Next week. At last he spelled it out insistently… the three baby elephants, the telescope, tweezers and the jamjar. How do you catch three baby elephants and put them in a jamjar armed only with a pair of tweezers and a telescope before mummy and daddy elephant spot what you’re doing? Those thinking puzzles, the ones where the answer wasn’t the point. And remember, we needed to punch some holes in the jamjar lid. They really made him think, Nik explained.

What they all asked — Jeff, Kyle, Lindsay, Ashley, Nik, D’Arcy — is when are we going to use this in real life? And that is the question all math teachers need to ask… and be able to answer. That’s really the point. And depending on the answer, depends what students remember and take away from that walk across the Commencement platform.

If all we did was merely deliver the curriculum, they probably take away little if anything. If the answer is the purpose was to learn to think and be creative in problem solving and that all students will need such skills in real life, students may, it is to be hoped, take away something.

But, as Marie, our school psychologist, commented as students filed out: “They got here. We did our job.”

### Testing… a teachable moment

Buy this design on a PiFactory tee-shirt

TESTING can provide some teachable moments.

Imagine. The desks are in rows. One child per desk or sitting alternately on opposite sides. Different colored tests. “No talking,” written on the board. Adults think kids cheat is the message.

And then one student, child, asks something along the lines of I-don’t-get-this-can-you-help-me? The answer? Well, in the traditional classroom it’ll be something along the lines of No-this-is-a-test.

Let’s look at this.

If the purpose of a test is to find out if a student, child, understands an idea… then the question I-don’t-get-this-can-you-help-me? seems to provide the answer with little doubt. Incontrovertible. So, why the No? In any other lesson the same question from the same student would be seized upon, or, one would hope.

Imagine you’re the hapless student. You don’t understand, you ask for help and the teacher says No. You then have to sit there, in silence unable to do the natural thing — ask your neighbor if they can help. Children are hard-wired to talk, ask questions, communicate. That’s how it works in the real adult world too. But not in a test. Ask your neighbor, and you’re a cheat. So much for teaching co-operation, social skills and collaboration.

As the student sits there, confused as well as stuck, what does that do for their self-esteem?

Personally, I agree with the research that is conclusive — testing reveals little about a child’s knowledge and mostly does harm to the learning process. But the US education system seems wedded to testing and not to the conclusions of contemporary research data. And in my school it’s on the up-and-up. Testing is becoming an obsesssion.

Testing is what teachers talk about daily — planning the test, reviewing for the test, putting off the test and then, heads wagging in disbelief, incomprehension as to the results… they-just-don’t-get-it.

Apart from abandoning testing, is there an alternative that meets the needs of those who believe in testing and, more importantly, the confused child who needs help?

This is what I tried this week.

I took the departmentally-agreed questions and buried them inside three half-page assignments each on different colored paper. The green sheet with the word question had two other word questions — not identical questions or the same question with the numbers changed, but questions around the same idea.

The other calculation test questions on the pink and blue sheets were buried in groups of similar questions exploring much the same ideas.

Students were told to bring their working and their answers to me as they completed each question or group of questions. OK, a bit of queue formed, but I was able quickly to spot what was going on with each student and give instant feedback accordingly.

I was also able to build up a list of common mistakes, misconceptions and approaches. I was able to mark some answers to share with colleageues later. I was also able to note the inadequacies of our commonly-decided questions, the ambiguous wording, how students interpreted our questions.

The only questions on which students received no immediate feedback were the magic test questions. Those I just noted, right or wrong. No student noticed as we discussed the surrounding questions.

I didn’t need to shift the desks into rows. There was no big sign saying No Talking. The students go to socialize. Each got individual and immediate feedback without having to ask for it. I got a detailed formative assessment as to the thinking, approach and understanding of each student to guide the next lesson. No one risked being accused of cheating.

And whoever is interested in the test statistics got what they need too.

Testing with teachable moments. Everybody happy.

Recent Comments