Eddie Woo has been annoying for a long time. Eddie knows much less than he realises and his smiling inanities, which are invariably swallowed whole, are a continual distraction from real issues and real solutions. But he’s gotten worse. Eddie Professor of Practice Woo has graduated from being a distraction and an annoyance to being an active menace. Continue reading “Eddie Woo’s Mental Connections”
Tag: multiplication tables
Maths Anxiety Is Not a Thing, But Let’s Talk About It Anyway
A couple days ago there was an article in the SMH, titled,
Bad with numbers? You might have maths anxiety
Yeah, maybe. Or maybe you just suck at maths. It’s a conundrum.
Continue reading “Maths Anxiety Is Not a Thing, But Let’s Talk About It Anyway”
Maths Anxiety is Not a Thing
I am, of course, wrong. Prove it. Continue reading “Maths Anxiety is Not a Thing”
ACARA Crash 12: Let X = X
(With apologies to the brilliant Laurie Anderson. Sane people should skip straight to today’s fish, below.)
I met this guy – and he looked like he might have been a math trick jerk at the hell brink.
Which, in fact, he turned out to be.
And I said: Oh boy.
Right again.
Let X=X.
You know, that it’s for you.
It’s a blue sky curriculum.
Parasites are out tonight.
Let X=X.
You know, I could write a book.
And this book would be thick enough to stun an ox.
Cause I can see the future and it’s a place – about a thousand miles from here.
Where it’s brighter.
Linger on over here.
Got the time?
Let X=X.
I got this postcard.
And it read, it said: Dear Amigo – Dear Partner.
Listen, uh – I just want to say thanks.
So…thanks.
Thanks for all your patience.
Thanks for introducing me to the chaff.
Thanks for showing me the feedbag.
Thanks for going all out.
Thanks for showing me your amiss, barmy life and uh
Thanks for letting me be part of your caste.
Hug and kisses.
XXXXOOOO.
Oh yeah, P.S. I – feel – feel like – I am – in a burning building – and I gotta go.
Cause I – I feel – feel like – I am – in a burning building – and I gotta go.
OK, yes, we’re a little punch drunk. And drunk drunk. Deal with it.
Today’s fish is Year 7 Algebra. We have restricted ourselves to the content-elaboration combo dealing with abstract algebraic expressions. We have also included an omission from the current curriculum, together with the offical justification for that omission.
LEVEL DESCRIPTION
As students engage in learning mathematics in Year 7 they … explore the use of algebraic expressions and formulas using conventions, notations, symbols and pronumerals as well as natural language.
CONTENT
create algebraic expressions using constants, variables, operations and brackets. Interpret and factorise these expressions, applying the associative, commutative, identity and distributive laws as applicable
ELABORATIONS
generalising arithmetic expressions to algebraic expressions involving constants, variables, operations and brackets, for example, 7 + 7+ 7 = 3 × 7 and 𝑥 + 𝑥 + 𝑥 = 3 × 𝑥 and this is also written concisely as 3𝑥 with implied multiplication
applying the associative, commutative and distributive laws to algebraic expressions involving positive and negative constants, variables, operations and brackets to solve equations from situations involving linear relationships
exploring how cultural expressions of Aboriginal and Torres Strait Islander Peoples such as storytelling communicate mathematical relationships which can be represented as mathematical expressions
exploring the concept of variable as something that can change in value the relationships between variables, and investigating its application to processes on-Country/Place including changes in the seasons
OMISSION
Solving simple linear equations
JUSTIFICATION
Focus in Year 7 is familiarity with variables and relationships. Solving linear equations is covered in Year 8 when students are better prepared to deal with the connections between numerical, graphical and symbolic forms of relationships.
I – feel – feel like – I am – in a burning building
ACARA Crash 11: Pulped Fractions
We’re still crazy-nuts with work, so, for today, it’s just another fish. This one is from Year 7 Number. and appears to be the sum of fraction arithmetic in Year 7.
LEVEL DESCRIPTION
As students engage in learning mathematics in Year 7 they … develop their understanding of integer and rational number systems and their fluency with mental calculation, written algorithms, and digital tools and routinely consider the reasonableness of results in context
ACHIEVEMENT STANDARD
By the end of Year 7, students use all four operations in calculations involving positive fractions and decimals, using the properties of number systems and choosing the computational approach. … They determine equivalent representations of rational numbers and choose from fraction, decimal and percentage forms to assist in computations. They solve problems involving rational numbers, percentages and ratios and explain their choice of representation of rational numbers and results when they model situations, including those in financial contexts.
CONTENT
determine equivalent fraction, decimal and percentage representations of rational numbers. Locate and represent positive and negative fractions, decimals and mixed numbers on a number line
ELABORATIONS
investigating equivalence of fractions using common multiples and a fraction wall, diagrams or a number line to show that a fraction such as 
is equivalent to 
and 
and therefore 
expressing a fraction in simplest form using common divisors
applying and explaining the equivalence between fraction, decimal and percentage representations of rational numbers, for example, 
and 
, using manipulatives, number lines or diagrams
representing positive and negative fractions and mixed numbers on various intervals of the real number line, for example, from -1 to 1, -10 to 10 and number lines that are not symmetrical about zero or without graduations marked
investigating equivalence in fractions, decimals and percentage forms in the patterns used in the weaving designs of Aboriginal and Torres Strait Islander Peoples
CONTENT
carry out the four operations with fractions and decimals and solve problems involving rational numbers and percentages, choosing representations that are suited to the context and enable efficient computational strategies
ELABORATIONS
exploring addition and subtraction problems involving fractions and decimals, for example, using rectangular arrays with dimensions equal to the denominators, algebra tiles, digital tools or informal jottings
choosing an appropriate numerical representation for a problem so that efficient computations can be made, such as 
or 
developing efficient strategies with appropriate use of the commutative and associative properties, place value, patterning, multiplication facts to solve multiplication and division problems involving fractions and decimals, for example, using the commutative property to calculate of 
giving of 
exploring multiplicative (multiplication and division) problems involving fractions and decimals such as fraction walls, rectangular arrays, algebra tiles, calculators or informal jottings
developing efficient strategies with appropriate use of the commutative and associative properties, regrouping or partitioning to solve additive (addition and subtraction) problems involving fractions and decimals
calculating solutions to problems using the representation that makes computations efficient such as 12.5% of 96 is more efficiently calculated as 
of 96, including contexts such as, comparing land-use by calculating the total local municipal area set aside for parkland or manufacturing and retail, the amount of protein in daily food intake across several days, or increases/decreases in energy accounts each account cycle
using the digits 0 to 9 as many times as you want to find a value that is 50% of one number and 75% of another using two-digit numbers
CONTENT
model situations (including financial contexts) and solve problems using rational numbers and percentages and digital tools as appropriate. Interpret results in terms of the situation
ELABORATIONS
calculating mentally or with calculator using rational numbers and percentages to find a proportion of a given quantity, for example, 0.2 of total pocket money is spent on bus fares, 55% of Year 7 students attended the end of term function, 23% of the school population voted yes to a change of school uniform
calculating mentally or with calculator using rational numbers and percentages to find a proportion of a given quantity, for example, 0.2 of total pocket money is spent on bus fares, of Year 7 students attended the end of term function, of the school population voted yes to a change of school uniform
interpreting tax tables to determine income tax at various levels of income, including overall percentage of income allocated to tax
using modelling contexts to investigate proportion such as proportion of canteen total sales happening on Monday and Friday, proportion of bottle cost to recycling refund, proportion of school site that is green space; interpreting and communicating answers in terms of the context of the situation
expressing profit and loss as a percentage of cost or selling price, comparing the difference
investigating the methods used in retail stores to express discounts, for example, investigating advertising brochures to explore the ways discounts are expressed
investigating the proportion of land mass/area of Aboriginal Peoples’ traditional grain belt compared with Australia’s current grain belt
investigating the nutritional value of grains traditionally cultivated by Aboriginal Peoples in proportion to the grains currently cultivated by Australia’s farmers
ACARA Crash 10: Dividing is Conquered
This Crash is a companion to, and overlaps with, the previous Crash, on multiplication. It is from Year 5 and Year 6 Number. and is, as near as we can tell, the sum of the instruction on techniques of division for F-6.
ACHIEVEMENT STANDARD (YEAR 5)
They apply knowledge of multiplication facts and efficient strategies to … divide by single-digit numbers, interpreting any remainder in the context of the problem.
CONTENT (YEAR 5)
choose efficient strategies to represent and solve division problems, using basic facts, place value, the inverse relationship between multiplication and division and digital tools where appropriate. Interpret any remainder according to the context and express results as a mixed fraction or decimal
ELABORATIONS
developing and choosing efficient strategies and using appropriate digital technologies to solve multiplicative problems involving multiplication of large numbers by one- and two-digit numbers
solving multiplication problems such as 253 x 4 using a doubling strategy, for example, 253 + 253 = 506, 506 + 506 = 1012
solving multiplication problems like 15 x 16 by thinking of factors of both numbers, 15 = 3 x 5, 16 = 2 x 8; rearranging the factors to make the calculation easier, 5 x 2 = 10, 3 x 8 = 24, 10 x 24 = 240
using an array model to show place value partitioning to solve multiplication, such as 324 x 8, thinking 300 x 8 = 2400, 20 x 8 = 160, 4 x 8 = 32 then adding the parts, 2400 + 160 + 32 = 2592; connecting the parts of the array to a standard written algorithm
investigating the use of digital tools to solve multiplicative situations managed by First Nations Ranger Groups and other groups to care for Country/Place including population growth of native and feral animals such as comparing rabbits or cane toads with platypus or koalas, or the monitoring of water volume usage in communities
LEVEL DESCRIPTION (YEAR 6)
use all four arithmetic operations with natural numbers of any size
ACHIEVEMENT STANDARD (YEAR 6)
Students apply knowledge of place value, multiplication and addition facts to operate with decimals.
CONTENT (YEAR 6)
apply knowledge of place value and multiplication facts to multiply and divide decimals by natural numbers using efficient strategies and appropriate digital tools. Use estimation and rounding to check the reasonableness of answers
ELABORATIONS
applying place value knowledge such as the value of numbers is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a natural number by a decimal of at least tenths
applying and explaining estimation strategies to multiplicative (multiplication and division) situations involving a natural number that is multiplied or divided by a decimal to at least tenths before calculating answers or when the situation requires just an estimation
deciding to use a calculator in situations that explore multiplication and division of natural numbers being multiplied or divided by a decimal including beyond hundredths
explaining the effect of multiplying or dividing a decimal by 10, 100, 1000… in terms of place value and not the decimal point shifting
ACARA Crash 9: Their Sorrows Shall Be Multiplied
We still have no time for the deep analysis of this shallow nonsense. So, we’ll just continue with the fish.
Below are two content-elaborations combos, from Year 5 and Year 6 Number. As near as we can tell, that’s about the sum of the instruction on techniques of multiplication for F-6.
ACHIEVEMENT STANDARD (YEAR 5)
They apply knowledge of multiplication facts and efficient strategies to multiply large numbers by one-digit and two-digit numbers
CONTENT (YEAR 5)
choose efficient strategies to represent and solve problems involving multiplication of large numbers by one-digit or two-digit numbers using basic facts, place value, properties of operations and digital tools where appropriate, explaining the reasonableness of the answer
ELABORATIONS
interpreting and solving everyday division problems such as, ‘How many buses are needed if there are 436 passengers, and each bus carries 50 people?’, deciding whether to round up or down in order to accommodate the remainder
solving division problems mentally like 72 divided by 9, 72 ÷ 9, by thinking, ‘how many 9 makes 72’, ? x 9 = 72 or ‘share 72 equally 9 ways’
investigating the use of digital technologies to solve multiplicative situations managed by First Nations Ranger Groups and other groups to care for Country/Place including population growth of native and feral animals such as comparing rabbits or cane toads with platypus or koalas, or the monitoring of water volume usage in communities
LEVEL DESCRIPTION (YEAR 6)
use all four arithmetic operations with natural numbers of any size
ACHIEVEMENT STANDARD (YEAR 6)
Students apply knowledge of place value, multiplication and addition facts to operate with decimals.
CONTENT (YEAR 6)
apply knowledge of place value and multiplication facts to multiply and divide decimals by natural numbers using efficient strategies and appropriate digital tools. Use estimation and rounding to check the reasonableness of answers
ELABORATIONS
applying place value knowledge such as the value of numbers is 10 times smaller each time a place is moved to the right, and known multiplication facts, to multiply and divide a natural number by a decimal of at least tenths
applying and explaining estimation strategies to multiplicative (multiplication and division) situations involving a natural number that is multiplied or divided by a decimal to at least tenths before calculating answers or when the situation requires just an estimation
deciding to use a calculator in situations that explore multiplication and division of natural numbers being multiplied or divided by a decimal including beyond hundredths
explaining the effect of multiplying or dividing a decimal by 10, 100, 1000… in terms of place value and not the decimal point shifting
UPDATE (29/50/21)
we’ve just discovered some multiplication techniques tucked inside some division elaborations, as indicated in this companion Crash. The two Crashes should be considered together (and should have been just one Crash, dammit.)
You Got a Problem With That?
We’ll take a day off from bashing the draft curriculum, in order to bash the draft curriculum. This one’s not a Crash post, but it gets to the disfigured heart of the draft.
Yesterday, a good friend and colleague, let’s call him Mr. Big, threw a book at us. By Alexandre Borovik and Tony Gardiner, the book is called The Essence of Mathematics Through Elementary Problems. The book is free to download, and it is beautiful.
There is much to say about this book. It is, unsurprisingly, a collection of problems and solutions. By “elementary”, the authors mean, in the main, in the domain of secondary school mathematics. Note that “elementary” does not equate to “easy”, although there are easy problems as well.
The problems have been chosen with great care. As the authors write, the problems are included for two reasons:
-
- they constitute good mathematics
- they embody in a distilled form the quintessential spirit of elementary mathematics
As indicated by the the Table of Contents, the problems in The Essence of Mathematics are also arranged very carefully, by topic and in a roughly increasing level of conceptual depth, and the book includes interesting and insightful commentary. Their twenty problems and solutions embodying 3 – 1 = 2 is a beautiful illustration.
The Essence of Mathematics also contains an incredibly important message. Here is the very first problem in the book:
1(a) Compute for yourself, and learn by heart, the times tables up to 9 × 9.
Regular readers will know exactly where we’re going with this. Chapter 1 of Essential Mathematics is titled Mental Skills, it includes simple written skills as well, and the message is obvious. As the authors write,
The chapter is largely devoted to underlining the need for mastery of a repertoire of instantly available techniques, that can be used mentally, quickly, and flexibly to analyse less familiar problems at sight.
In particular, on their first problem,
Multiplication tables are important for many reasons. They allow us to appreciate directly, at first hand, the efficiency of our miraculous place value system – in which representing any number, and implementing any operation, are reduced to a combined mastery of
(i) the arithmetical behaviour of the ten digits 0–9, and
(ii) the index laws for powers of 10.
Fluency in mental and written arithmetic then leaves the mind free to notice, and to appreciate, the deeper patterns and structures which may be lurking just beneath the surface.
What does all this have to do with ACARA’s draft curriculum? Alas, nothing whatsoever.
The draft curriculum is the antithesis of Essence. The “problems” and “investigations” and “models” in the draft curriculum are anything but well-chosen, being typically sloppy and ill-defined, with no clear direction or purpose. The draft curriculum also displays nothing but contempt for the prior mastery of basic facts and skills required for problem-solving, or anything.
Essence is not a textbook, but the authors clearly see a large role for problem-solving in mathematics education, and, with genuine modesty, they can imagine their book as a natural supplement to a good curriculum. Such a role can mean slow and open-ended learning, or at least open-ended teaching:
Learning mathematics is a long game; and teachers and students need the freedom to digress, to look ahead, and to build slowly over time.
The value of such digressions and explorations, however, does not negate the primary goal of mathematics education:
Teachers at each stage must be free to recognise that their primary responsibility is not just to improve their students’ performance on the next test, but to establish a firm platform on which subsequent stages can build. …
The effect [of political pressures] has been to downgrade the more important challenges which every student should face: namely
-
- of developing a robust mastery of new, forward-looking techniques (such as fractions, proportion, and algebra), and
- of integrating the single steps students have at their disposal into larger, systematic schemes, so that they can begin to tackle and solve simple multi-step problems.
Building systematic schemes out of the mastery of techniques. Or, there’s the alternative:
A didactical and pedagogical framework that is consistent with the essence, and the educational value of elementary mathematics cannot be rooted in false alternatives to mathematics (such as numeracy, or mathematical literacy).
There is problem-solving, and there is “problem-solving”. ACARA is shovelling the latter.
UPDATE (28/05/21)
Mrs. Big, AKA Mrs. Uncle Jezza, has given the draft curriculum a very good whack in the comments, below. As part of that, she has noted an excellent quotation that begins the Preface of Essential. The quotation is by Richard Courant and Herbert Robbins, and is from the Preface of their classic What is Mathematics?
“Understanding mathematics cannot be transmitted by painless entertainment … actual contact with the content of living mathematics is necessary. The present book … is not a concession to the dangerous tendency toward dodging all exertion.”
While we’re here, we’ll include another great quote, from the About section of Essential, by John von Neumann:
“Young man, in mathematics you don’t understand things. You just get used to them.”
Understanding is a fine goal, but it can also be a dangerously distracting goal. ACARA’s “deep understanding” is an absurdity.
ACARA Crash 8: Multiple Contusions
OK, roll out the barrel, grab the gun: it’s time for the fish. Somehow we thought this one would take work but, really, there’s nothing to say.
It has obviously occurred to ACARA that the benefits of their Glorious Revolution may not be readily apparent to us mathematical peasants. And, one of the things we peasants tend to worry about are the multiplication tables. It is therefore no great surprise that ACARA has addressed this issue in their FAQ:
When and where are the single-digit multiplication facts (timetables) covered in the proposed F–10 Australian Curriculum: Mathematics?
These are explicitly covered at Year 4 in both the achievement standard and content descriptions for the number strand. Work on developing knowledge of addition and multiplication facts and related subtraction and division facts, and fluency with these, takes place throughout the primary years through explicit reference to using number facts when operating, modelling and solving related problems.
Nothing spells sincerity like getting the name wrong.* It’s also very reassuring to hear the kids will be “developing knowledge of … multiplication facts”. It’d of course be plain foolish to grab something huge like 6 x 3 all at once. In Year 4. And, how again will the kids “develop” this knowledge? Oh yeah, “when operating, modelling and solving related problems”. It should work a treat.
That’s the sales pitch. That’s ACARA’s conscious attempt to reassure us peasants that everything’s fine with the “timetables”. How’s it working? Feeling good? Wanna feel worse?
What follows is the relevant part of the Year Achievement Standards, and the Content-Elaboration for “multiplication facts” in Year 4 Algebra.
ACHIEVEMENT STANDARD
By the end of Year 4, students … model situations, including financial contexts, and use … multiplication facts to … multiply and divide numbers efficiently. … They identify patterns in the multiplication facts and use their knowledge of these patterns in efficient strategies for mental calculations.
CONTENT
recognise, recall and explain patterns in basic multiplication facts up to 10 x 10 and related division facts. Extend and apply these patterns to develop increasingly efficient mental strategies for computation with larger numbers
ELABORATIONS
using arrays on grid paper or created with blocks/counters to develop and explain patterns in the basic multiplication facts; using the arrays to explain the related division facts
using materials or diagrams to develop and record multiplication strategies such as skip counting, doubling, commutativity, and adding one more group to a known fact
using known multiplication facts for 2, 3, 5 and 10 to establish multiplication facts for 4, 6 ,7 ,8 and 9 in different ways, for example, using multiples of ten to establish the multiples 9 as ‘to multiply a number by 9 you multiply by 10 then take the number away’; 9 x 4 = 10 x 4 – 4 , 40 – 4 = 36 or using multiple of three as ‘to multiply a number by 9 you multiply by 3, and then multiply the result by 3 again’
using the materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication
using known multiplication facts up to 10 x 10 to establish related division facts
Alternatively, the kids could just learn the damn things. Starting in, oh, maybe Year 1? But what would we peasants know.
*) It has since been semi-corrected to “times-tables”.
A Simple Message to Primary Schools About Multiplication Tables
Dear Primary Schools,
If your students are not learning their multiplication tables, up to 12, by heart, then you are fucking up.
If you think giving your students a grab-bag of tricks replaces multiplication tables, then you are fucking up.
If you think orchestrating play-based, student-centred theatricalities replaces multiplication tables, then you are fucking up.
If you think quoting the Australian Curriculum gives you license to not teach multiplication tables, then you are fucking up.
If you think quoting some education twat gives you license to not teach multiplication tables, then you are fucking up.
Thank you for your attention.