**What’s a decimal place?**

It is the amount of digits after a decimal point, so for example if a question asks you for a number too two decimal places it could be 1.23 to one decimal place could be 1.2 etc.

]]>Exposition by the teacher has always been a fundamental ingredient of work in the classroom and we believe that this continues to be the case. We wish, though, to stress one aspect of it which seems often to be insufficiently appreciated. Questions and answers should constitute a dialogue. There is a need to take account of, and to respond to, the answers which pupils give to questions asked by the teacher as the exposition develops. Even if an answer is incorrect, or is not the one which the teacher was expecting or hoping to receive, it should not be ignored; exploration of a pupil’s incorrect or unexpected response can lead to worthwhile discussion and increased awareness for both teacher and pupil of specific misunderstandings or misinterpretations.

It’s interesting that back in 1982 Cockcroft felt the need to highlight the importance of dialogue. Exposition makes us think of *teacher talk* where the ‘more knowledgeable’ teacher stands at the front and instructs or explains but this is not how it is defined here. I am fortunate to observe many lessons, trainees, NQTs, appraisal observations, research observations and I often see an interesting mix of *instruction* (ie the teacher telling pupils some information) and *questioning* (ie the teacher eliciting information or uncovering knowledge from pupils). Often, I believe a teacher asks a question when telling the pupils would be the better choice and I worry that this has been inculcated into many of us in the profession through a misunderstanding of statements like this one from the National Numeracy Strategy (2001).

High-quality direct teaching is oral, interactive and lively. It is not achieved by adopting a simplistic formula of ‘drill and practice’ and lecturing the class… It is a two-way process in which pupils are expected to play an active part by answering questions, contributing points to discussions, and explaining and demonstrating their methods to the class.

In other words, “if my lesson is not interactive and lively I’m doing something wrong.” Have we lost, or did we never have the confidence to know when the classroom should be interactive and when we should have the kind of silence that Tom Sherrington writes about here (point 9) or do I continue to see teachers show me what they think I want to see?

I would like to hope that my classroom is ‘interactive and lively.’ Discussion and questioning is common and pupils feel comfortable stopping the lesson for clarification or asides. But I would also like to hope that I understand when and how to use this approach. Exposition, in my view, means knowing when to talk and when to listen. It means knowing how to create dialogue in lessons, and for the right reasons.

The NNS suggests elements of “good, direct teaching” including directing (“ensuring that pupils know what to do”) and instructing (“giving information and structuring it well”). Mercer & Dawes (2008) note that, “in many classrooms, most of the talk is not only asymmetrical, it has a particular structure: teachers ask closed questions and children provide brief answers on which the teacher then makes evaluative comments.” I would contend that this comes from not knowing when to instruct and when to question. Closed questions have great value when quickly checking knowledge or stimulating recall (Mercer calls this *elicitation*) but they limit opportunities for pupils to express or evaluate ideas of their own (Tanner et al 2005). Here is an interesting quote from Westgate & Hughes (1997) observing a discussion in a nursery class

[The children’s replies are] evaluated for their closeness of fit with ideas she wishes them to take account of. This procedure appears to lead to some confusion which might have been avoided by the teacher simply sharing at once what it was she wanted the pupils to notice.

This is backed up by Mercer & Hawes when they write, “A teacher wants children to become actively involved, which is a good idea, but this may not be the best way to achieve it.”

Going back to Cockcroft, over 30 years ago he stressed the importance of how “exploration of a pupil’s incorrect or unexpected response can lead to worthwhile discussion and increased awareness”. Now this can happen with closed, elicitation questions but is much more likely with well designed dialogue. Dialogue goes beyond the Initiation-Response-Feedback (IRF) triad described by Mehan (1979).

- Initiation: The teacher prompts, or more commonly asks a question
- Response: A pupil respond, usually by answering the question
- Feedback: A follow-up comment by the teacher, usually evaluative.

I’m much more interested in exposition going beyond listening to and evaluating pupils’ answers. I’m not even fully sure that IRF results in really assessing what pupils know since one of the characteristics of linguistic communication is that it is almost always potentially ambiguous (Wells & Arauz 2006). I think this is as true in the maths classroom as it is in any other.

On a practical level, Mercer & Dawes suggest the following strategies

- make it clear that some parts of lessons are expressly intended to be discussion sessions, in which questions and diverse views on a topic can be expressed;
- during whole-class discussions, allow a series of responses to be made without making any immediate evaluations;
- if some different views have been expressed, ask pupils for reasons and justifications for their views before proceeding;
- precede whole-class discussion of particular questions or issues with a short group-based session, in which pupils can prepare joint responses for sharing with the class. It may help to offer pupils a set of alternative explanations, contentious statements or ideas on a topic, and ask them to decide which are true/false, and why;
- before providing a definitive account or explanation (of, for example, a scientific phenomenon), elicit several children’s current ideas on the topic. Then link your explanation to these ideas.
- use whole-class sessions to gather feedback from children about how they have worked together in groups. Are the ground rules working? Do the rules need to be revised? Do they feel their discussions have been constructive? If not, why not? And what could be done about it?
- ask pupils to nominate other pupils in whole-class discussions, so that the teacher does not always choose who should speak.

Now, as with anything, I must highlight the importance of context, of appropriateness. I hope that teachers have the confidence to know when to use an approach like that above. It takes time to turn an oil tanker, and we are all oil tankers in our beliefs and values. It’s hard to change a ‘style’ that is hard wired. What I tell the trainees I work with is, above all, plan your questions, think about what you are going to ask and why you are going to ask it. Sometimes they do. More often they don’t. More often none of us do. So perhaps we’d be better off just telling the pupils what we want them to know.

References:

Cockcroft, W. (1982) Mathematics counts, Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr WH Cockcroft, London: HMSO

Department for Education and Employment. (2001) (DfEE) Key Stage 3 National Strategy: Framework for teaching mathematics. London: DfEE

Mercer, N & Dawes, L (2008) The value of exploratory talk. In Mercer, N. & Hodgkinson, S., (eds). Exploring Talk in School. London: Sage.

Mehan, H. (1979) Learning Lessons: Social Organization in the Classroom. Cambridge, Mass: Harvard University Press

Tanner, H., Jones, S., Kennewell, S. & Beauchamp, G. (2005) Interactive Whole Class Teaching and Interactive White Boards. Mathematics Education Research Group of Australasia Conference (MERGA 28)

Wells,G & Arauz (2006) Dialogue in the classroom, The Journal of the Learning Science, V15(3) pp 379-428

Westgate, D. & Hughes, M. (1997) Identifying ‘ Quality ’ in Classroom Talk : An Enduring Research Task. Language and Education

]]>I’ve been teaching Mathematics for nearly 20 years. When I trained, at Newcastle University, the six strands of mathematics teaching recommended by Cockcroft were drilled into us:

- exposition by the teacher;
- discussion between teacher and pupils and between pupils themselves;
- appropriate practical work;
- consolidation and practice of fundamental skills and routines;
- problem solving, including the application of mathematics to everyday situations;
- investigational work.

We learnt this through the statement, “Your teaching should be *SPICED.*” Solving problems, Practical, Investigations, Consolidation, Exposition, Discussion. It’s always stuck in my mind and I’ve always used it when planing lessons. If we take as read that the learning done by the pupils is the most important aspect of any lesson, then the task follows to decide which of these methods best supports that learning; or if a mix of approaches will work.

However. even though Cockcroft writes, “we are not saying anything which has not already been said many times and over many years,” (pg 72) I think it is important we analyse each suggestion in turn.

The first item on Cockcroft’s list is exposition by the teacher. I write about what we mean by classroom talk here.

Over the next few weeks I hope to write about the other items on Cockcroft’s list.

]]>

Bill Amend’s **FoxTrot** (January 25, and not a rerun) puts out a little word problem, about what grade one needs to get a B in this class, in the sort of passive-aggressive sniping teachers long to get away with. As Paige notes, it really isn’t a geometry problem, although I wonder if there’s a sensible way to represent it as a geometry problem.

Ruben Bolling’s **Super-Fun-Pax Comix** superstar Chaos Butterfly appears not just in the January 25th installment but also gets a passing mention in Mark Heath’s**Spot the Frog** (January 29, rerun). Chaos Butterfly in all its forms seems to be popping up a lot lately; I wonder if it’s something in the air.

Hector D Cantu and Carlos Castellanos’s **Baldo** (January 27) shows Baldo enthusiastic about a particular chain of multiplications, and I can’t say I care for Gracie’s harshing on him. A problem you’re interested in is a good way to learn things, regardless of why you’re interested.

Nate Frakes’s **Break of Day** (January 27) is an anthropomorphic-numbers joke that Frakes is probably slapping himself for not saving until March 14th.

Bill Schorr’s **The Grizzwells** (January 27) is another bit of subverting a word problem, though I have to admit it’s a variant on the plane-leaving-New-York problem I don’t remember reading in a long while.

Francesco Marciuliano’s **Medium Large** (January 28) is Marciuliano’s typical sort of pop culture-referential subversion of number-trivia panels. I think there’s something to be learned from it all.

Chris Browne’s **Hagar the Horrible** (January 29) is strikingly close to **Baldo** in its premise, really, and it caused me to wonder just how far back blackjack goes. If Wikipedia’s to be believed the game goes back at least to the time of Cervantes, who was the first to describe it, although the name “blackjack” didn’t become synonymous with “twenty-one” until the 1930s, when Nevada casinos briefly offered a ten-to-one payout for the player drawing an ace of spades and one of the two black jacks (clubs or spades). Blackjack is famous among card games for being kind of winnable, in the long term, if the player is able to keep some track of how many of each value card are still in the deck, so that one can bet more or less depending on how likely the player is to beat the dealer. So it makes a nice example of probability — for each hand — and statistics — for playing the whole game — as well as, yeah, addition.

Last summer my love and I were at the Waldameer amusement park in Erie, Pennsylvania, and they had a neat redemption game similar to Fascination: you rolled balls down a wooden track, scoring points based on which hole the ball falls into, with the objective being to get that glorious 21, or at least some of the interesting possible combinations. It was a fun game, and we played enough rounds to take home a park triangular penant that, yes, we could have got more cheaply from the souvenir shop, but where’s the chance to roll balls down wooden tracks in *that*?

Richard Thompson’s **Richard’s Poor Almanac** (January 29) sees the return of the typing monkey, although in this case the monkey isn’t *specifically* being tasked with the problem of producing a copy of Shakespeare.

Of this set of comics, I have to credit **Hagar the Horrible** for teaching me the most. But **Medium Large** gave me the funniest moment since, like most people, I don’t actually know whether Dick and Jane books even actually exist or if we just tell jokes about what we imagine them to be.

If you can figure out the simple pattern that these odd Pythagorean triples make, you can predict the next one in the sequence FOREVER by squaring only one number, and without ever taking a single square root!

Every odd number is the short leg of at least one primitive Pythagorean triple! Here’s how I came to realize this amazing fact:

Last week I was thinking about the difference of two squares applied to integers in general

as I looked at this multiplication table:

Everyone knows that the numbers in the boxes outlined in red are perfect squares, but most people do not realize that the numbers inside EVERY other colored square on this multiplication table can be expressed as the difference of two squares. The larger of those two squares will be the perfect square that is the same color.

For example, look at 55. It is three squares away on a light blue diagonal from the 64 that is outlined in red. That means that 55 = 64 – 3^2.

As I looked at the multiplication table I realized that some even numbered squares are colored and some are not, but EVERY odd number square is in color! That means that EVERY odd number on the table can be expressed as the difference of two squares in at least one way.

The light blue 9 from 1 x 9 is four squares away from the 25 that is outlined in red. 9 = 25 – 4^2. Since all those numbers are perfect squares, we can write 3^2 = 5^2 – 4^2 or the equivalent in Pythagorean Theorem form 3^2 + 4^2 = 5^2.

When an odd number is squared, the resulting perfect square is ALWAYS an odd number. Since all odd numbers can be written as the difference of two squares, The square of all odd numbers can be written as the difference of two squares. In other words, every odd number is a leg in a Pythagorean triple!

Let’s look at the **factoring information for 371** and use it to find some Pythagorean triples.

- 371 is a composite number.
- Prime factorization: 371 = 7 x 53
- The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 371 has exactly 4 factors.
- Factors of 371: 1, 7, 53, 371
- Factor pairs: 371 = 1 x 371 or 7 x 53
- 371 has no square factors that allow its square root to be simplified. √371 ≈ 19.261

371 is odd and has two odd prime factors. Here are some Pythagorean triples involving 371 as the short leg:

**371**, 68820, 68821 (This triple is a primitive from the odd primitive sequence.)**371**, 1380, 1429 (A primitive, not from the odd primitive sequence. I’m also making a note that √(1380 + 1429) = 53 and √(1429 – 1380) = 7.)- 7, 24, 25 multiplied by 53 becomes
**371**, 1272, 1325 - 53, 1404, 1405 multiplied by 7 becomes
**371**, 9828, 9835

371 is also the hypotenuse of a Pythagorean triple: 196, 315, **371** which is primitive 28, 45, 53 multiplied by 7.

Than a number, because it isn’t made;

And though I resist

The thought, I can’t shake it:

Something can exist

Without a maker to make it.

xoxo

]]>**Debal Sommer – Mathematics**

Style: Tech House, Deep House

Record Label: Tonkind

Catalogue Number: TOK040

Release Date: 20-01-2015

Total Tracks: 4 Tracks

Source: WEB

Tracks:

1. Debal Sommer – Mathematics (Original Mix) (7:33)

2. Debal Sommer – Crazy People (Original Mix) (8:44)

3. Debal Sommer – Fugibeat (Original Mix) (8:12)

4. Debal Sommer – C’est La Vie (Original Mix) (11:16)

Server | Quality | Size | Format | Link | Report Dead |
---|---|---|---|---|---|

Uploaded.net |
320kbps | 82MB | RAR | Download |
Report |

http://minimalfreaks.pw/2015/01/debal-sommer-mathematics-tonkind/

Deep House, Tech-House

Debal Sommer, Mathematics, Tonkind

Here’s the first lecture :

]]>

If you are using a personal tutor to support your child, you should be making sure there is plenty of time left to look at the wide variety of questions contained in the test.

I would recommend at least 20 sessions with a tutor to ensure your child has experienced every type of question which the test will throw at them. Of course, I *would* say that as I am a tutor myself!

But I really think that even high ability children, who are consistently hitting Level 5 in mathematics and English, need time to rehearse strategies for the 11-plus.

The test is a very specific way of assessing your child, and students are unlikely to have come across its type of question before.

I know – way back – when I took my 11-plus (or 12-plus as it was in those days), I did plenty of practice beforehand, using what were known as General Progress Papers! When it came to the 11-plus itself, I felt confident and nothing in the test surprised me. I felt I was in a position to do justice to myself.

That is all we can ask young people to do – to feel confident and perform to the best of their abilities. So if you think that weekly or twice-weekly lessons with a private tutor can help, then please contact me and I will be pleased to advise you.

]]>If I was at home, I could sleep = Unreal Present

- If I was at home, I could sleep = un (real [meaning] + present [time])
- If I was at home, I could sleep = unreal + unpresent
**not**.(If I was at home, I could sleep) =**not**.(unreal + unpresent)

*Multiplication Sub-step 1:*

.(**not**** If **I was(

I was not at home, so I could not sleep = real + (**not**.unpresent)

*Multiplication Sub-step 2:*

.(I **not** not at home, so I **was**** could** not sleep) = real + (

**I AM NOT AT HOME, SO I CAN NOT SLEEP = REAL PRESENT**

- I
not at home = Real~~am~~~~Present~~

~~ PRESENT ~~~~ ~~~~PRESENT ~~

- I be not at home ≈ Real

- I (pronoun) ≈ Real (adjective)

- Subject (noun) ≈ Real (adjective)

- Subject + ive (adjective) ≈ Real (adjective)

- Subjective (adjective) ≊ Real (adjective)

- Subjectiv(e) + ity (noun) ≊ Real + ity (noun)

**SUBJECTIVITY (noun) = REALITY (noun)**

The Maine DOE will begin visits next month to provide targeted technical assistance to schools preparing to award proficiency-based diplomas.

Last May, the Department released six options by which districts could apply for extensions to fully implementing systems that support the awarding of proficiency-based diplomas starting in 2018 as a 2012 Maine law requires. Extension options five and six were designed for districts early in their transition and thus in need of the most structured support from the Maine DOE. It is the 50 or so districts who applied under one of those options, which extend full implementation to 2021, that will be visited by teams of up to four Department staff.

The half-day visits are intended to positively support schools in progress toward their benchmarks; identify and validate progress since the application; inform the Maine DOE of district needs; and as needed, to clarify Maine DOE understanding of district work and benchmarks.

- Annual Certification of Superintendent, due Jan. 31 (Reminder)

More Notices | Administrative Letters | Reporting Calendar

Maine schools wanting to give the upcoming Maine Educational Assessment for Mathematics and English Language Arts/Literacy developed by Smarter Balanced as a paper-pencil test rather than online must let the Department know by Wednesday, Feb. 4. So far, only one school has indicated they will do so. | More

In preparation for Maine’s application for a renewal of its existing ESEA waiver, the Department is seeking feedback on how the State’s current accountability and improvement system is serving Maine’s students and schools. | More

Maine educators who will be helping to administer the upcoming online State assessment but couldn’t attend one of the Department’s recent regional workshops can now see the entire training online. The Maine DOE has posted a four-hour video on its YouTube channel filmed at the southern Maine workshop session on Friday, Jan. 16. | More

An iOS 8 compatible secure browser application that will be used for the upcoming administration of the Maine Educational Assessment for Mathematics and English Language Arts/Literacy is now available. | More

The Maine DOE is seeking recommendations through May 1 for the 2015 Maine Educator Talent Pool. | More

The Maine DOE’s State Personnel Development Grant and the Maine Department of Labor’s Division of Vocational Rehabilitation (DVR) have partnered to offer schools a free e-workshop that helps students discover employment possibilities. | More

The ACCESS for ELLs® testing window is set to close Friday, Jan. 30 and districts are required to pack and return all testing material to MetriTech by Feb. 6. | More

The Maine DOE and the State Board of Education have reopened for further public comments the rules regarding the qualifying examinations for credentialing purposes for teachers, educational specialists and administrators. | More

A NEO facilities training has been rescheduled to Feb. 12 due a snowstorm this week. | More

The Maine DOE has updated its physical form for bus drivers, which can be found here on the Department’s website along with instructions. | More

Maine schools are invited to enter a free digital storytelling competition that asks students to share their visions for the future. | More

More Dispatches | Press Releases | From the Commissioner

]]>

Are there more orange or blue?

Are there more orange or blue?

You will be able to tell how your child’s concept of quantity is developing in their answer. The child in early stages of developing will commonly answer that there are more orange in the second picture based on area (the orange bobbins are spread over a larger area). Use a larger quantity for older children to see how they are developing.

As usual I brought this into the kitchen but trying the same theory with capacity. I had My Toddler pour equal amounts of water (1 cup) into two different shaped glasses – one tall slender and one wider glass.

I then asked which glass had more water in it to which she answered the tall slender glass as the water came up higher than on the wider glass. When I tried to explain that we were using the same amount of water as we had used the 1 cup measure each time she was blown away.

I just love and truly appreciate child development.

By doing these fun exercises, you will help your child grasp this important concept which is a common weakness found among school children.

Most importantly as always make it fun! Try using smarties on cupcakes to practice next time you bake. Spread the same amount of smarties in different patterns.

]]>Here a sample series is given with detail procedure:

1+2+4+7+11+16+22+…(nth term)

First find the successive sums of the terms given:

1 = 1 = S(1)

1 + 2 = 3 = S(2)

1 + 2 + 4 = 7 = S(3)

1 + 2 + 4 + 7 = 14 = S(4)

1 + 2 + 4 + 7 + 11 = 25 = S(5)

1 + 2 + 4 + 7 + 11 + 16 = 41 = S(6)

1 + 2 + 4 + 7 + 11 + 16 + 22 = 63 = S(7)

Write the sums(S) in a list corresponding the given series term

integers:

n S(n)

1 -> 1

2 -> 3

3 -> 7

4 ->14

5 ->25

6 ->41

7 ->63

Form the column of 1st differences by

writing beside each number the difference

between it and the number just below it

n S(n) diff(1):

1- 1 – 2

2- 3 – 4

3- 7 – 7

4- 14 – 11

5- 25 – 16

6- 41 – 22

7- 63

Form the column of 2nd differences by writing beside each number the difference between it and the number just below it

n S(n) diff(1) diff(2)

1 – 1 – 2 – 2

2 – 3 – 4 – 3

3 – 7 – 7 – 4

4 – 14 – 11 – 5

5 – 25 – 16 – 6

6 – 41 – 22

7 – 63

Form the column of 3rd differences the

same way.

n S(n) diff(1) diff(2) diff(3)

1 – 1 – 2 – 2 – 1

2 – 3 – 4 – 3 – 1

3 – 7 – 7 – 4 – 1

4 – 14 – 11 – 5 – 1

5 – 25 – 16 – 6

6 – 41 – 22

7 – 63

Our goal is to find a column of difference as constant. Here the column of 3rd differences is

constant, so we assume a 3rd degree equation

for S(n):

S(n) = an^3 + bn^2 + cn + d

Since we have 4 unknowns, we only need

the first four sums:

S(1) = 1 = a*1^3 + b*1^2 + c*1 + d

S(2) = 3 = a*2^3 + b*2^2 + c*2 + d

S(3) = 7 = a*3^3 + b*3^2 + c*3 + d

S(4) = 14 = a*4^3 + b*4^2 + c*4 + d

Simplifying, we have this system of

4 equations with 4 unknowns:

a + b + c + d = 1

8a + 4b + 2c + d = 3

27a + 9b + 3c + d = 7

64a + 16b + 4c + d = 14

Solving those 4 equation we get,

A = 1/6, B = 0, C = 5/6, D = 0

So the equation

S(n) = an^3 + bn^2 + cn + d

becomes

S(n) = (1/6)n^3 + (5/6)n

Or, S(n) = n(n^2+5)/6

Now just set value of n to get the sum of nth terms of the given series.

Using the sum we can find any term of any position of the series.

#the method is valid for any integer series. ]]>

In fact, a comprehensive reading of Glanville’s papers places him firmly in an epistemological tradition that radicalizes the concept of the observer, never flinches from attempting to learn to perceive, closely watches negations inherent in selections of explanatory constraints, and considers objects to be just another example of eigen-values stemming from recursive operations. Indeed, nothing is more important than paying relentless attention to the distinction between ideas, notions, and notations or between world, cognition and description.

Yet Glanville’s writing and lectures may also be seen to follow up ideas pursued by older philosophical traditions. His own reflections on philosophy lead him back to Ludwig Wittgenstein, who in *Tractatus Logico Philosophicus* recommends treating one’s own observations as traces on a screen leaving everything behind a mystery. Light can also be thrown on Glanville’s thinking by reference to Plato’s *Sophist,* who deep mistrusts yet nevertheless relies on negations, contradictions, distinctions, and other kinds of polemic not only to verify but also to produce plausible explanations. I also wonder how Johann Gottlieb Fichte would have welcomed Glanville’s modelling facility with regard to both observers and objects when, in his various takes on a *Wissenschaftslehre*, a “doctrine of scientific knowledge,” he seeks to comprehend any object *X* and any identity *A = A* as dependent on, and thus as elusive as an *I* positing *itself*.

I mention these grand themes only to prepare the ground for an even grander one. I suggest it might be fruitful to read some of Martin Heidegger’s works within the perspective of both first-order and second-order cybernetics in general and Glanville’s take on them in particular…

Read more: The Be-ing of Objects at ssrn…

]]>