## Posts tagged ‘classroom practice’

### Engaging math for all learners

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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

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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

### Working inside the black box

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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

### Testing… a teachable moment

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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.

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