Cling to the main vine, not the loose one.
Kei hopu tōu ringa ki te aka tāepa, engari kia mau ki te aka matua

Thoughts on Teaching and Learning of Mathematics

Lesson #4 • Revised 9/1/20

Number and Algebra

- to be clear about the purpose of learning Number and Algebra
- to be aware that what is taught is not necessarily what is being learned 

Number is everything. In the beginning there was one! This is my sense of humour acting but Anthony Harradine challenged me one day by asking what is the most important mathematical concept and I answered "The number 1; without one there is nothing. From "1" we create mathematics." I reminded him of this conversation when we met again at NZAMT #13 CONF in Lower Hutt 2019. It was good to catch up - relationships!
 
There is a curious article on Reality in New Scientist (2016) which suggests that at the root of everything, on the quantum scale of size, is a binary number system which might underwrite the universe. Quarks are Up and Down, Left and Right, Spin or No-spin so all in pairs or in binary. We could also solve every problem using counting if we had enough time. Luckily there are some curious properties of number that make being numerate a much more rewarding enterprise than simply counting. We all count things however and often do not notice when we do so. An example is counting loose change (coins). Some golfers have trouble counting their stokes but that might be for another reason.

Algebra allows us to generalise patterns and across all mathematics to solve and to prove. Algebra is a language that transcends geographical borders, age and time, gender and race and mathematics itself. The language of algebra is adapted and invented as new ideas are generated. New technology demands new language and algebra provides the opportunity. Algebra as we know it today had its roots in ancient Babylonia. The Arabian influences gave us zero and along with Indian help the numerals. Great mathematicians seem to pop up in all races and places and times.

Here is an interesting resource by Anthony Harradine that puts a different spin on learning algebra. Algebra Families - His copyright message covers this use.

Algebra has traditionally been a "sorting gate" for students progressing to senior school courses in mathematics and some schools persist in this silly requirement as though mastering elementary algebra is some indicator of future success. I feel this ludicrous requirement is more about teachers protecting the false sanctity of their own knowledge and power. The "I know something you do not" syndrome. This false power is sourced in a lack of big picture knowledge and personal capability and should be noticed and exposed. A better "sorting gate" would be "high perseverence".

Deep understanding of a concept is about identifying key understandings that form the "arch of sense and knowing.  I like to explore deep understanding with teachers and students and ask them to try and write a statement that captures the concept being discussed. This helps to develop big picture mental models and position the teacher in a journey rather than at a destination. 

Anyone can blindly stumble through a forest but to comprehend the way plants interact to form a forest and create complexity requires a different approach. Mathematics is like that as well. Deep understanding is about knowing the subject but also knowing about learning. This reminds me of a book written by Liping Mah in which she says "Know how but also know why." I would recommend this book to help understand the need for understanding! 

Here is a useful resource on counting and what happens next. Worth reading.

We learn number by counting objects in the first instance and progress through operations using an increasingly complex description and understanding of number and numbers. But at any stage of development the same deep understanding unfolds...

The Deep Understanding of Number
Being enabled to take numbers apart, operate on them, and reassemble them to solve problems.

With a little pondering it should be easy to see how we use strategies like tidy numbers in addition, factors in multiplication, parts in proportion, place value in all of these to make a problem more accessible, to operate and to then reconstruct the numbers to solve a problem.

A simple example is 18 x 5. Here we could see the  18 as 10 + 8 and then operate 10x5 and add 8x5 to get 50 + 40 which is 90. This is using the Distributive Law and is a multiplicative strategy or an indicator that a student is operating at early NZC L4 (being multiplicative). We could also see the 18 as 2 x 9 (or as 9 x 2) and then write 18x5 = 9 x 2 x 5 and see the answer as 9 x 10 which is 90. This uses the factors of 18 and is a more complex multiplicative thinking. My goal for all students in early years is to get them into this multiplicative thinking ASAP!

Likewise in Algebra it is possible to make a deep understanding statement.

I notice patterning is a very early event for all living creatures. For example, it takes a few minutes for a new born  human baby to get a reaction to a cry and a few hours to generalise that this is a good way to control need of warmth, hunger and comfort. Who is controlling who? Who has become a generalist  in this case? Another example is when my dog knows when it is walk time, food time, and time to play. The magician (brain) is at work! Noticing patterns and making generalisations from these is a very "life engaging and aware" thing to do. Patterning is nature.

Generalisations of patterning logically follow and this leads to proof and bold statements of truth in mathematics. Finance and politics have generalisations as do food and sport. If you could not generalise about your surroundings you would wake to a new world everyday and that would be an amazingly scary place and a reality for some old and all young people. Generalising is about being alive, engaged and aware. Imagine the issues of not being able to generalise about rooms in a house where you sleep and having to explore the place to find the toilet in the middle of the night. Generalising that the room you seek is in the same place as it was before you went to sleep is a very useful thinking ability.

The Deep Understanding of Algebra
 Being enabled to notice, find and record patterns and relationships, use visual interpretations, to make generalisations and make sense of the world in solving problems.

Visual solutions have always been been a  favoured way to communicate for a very long time.  It is natural to draw sketches in the sand just as did the original designers of the Landrover (http://www.landroverusa.com/our-story/index.html)  in the 1940's and making their visuals become over 3,000,000 vehicles called Series 1, Series 2, Series 3 and Defender. The Australian Aborigine used visual pictures and recorded 50,000+ years of habitation (First Footprints also a TV documentary). The NZ Maori had a visual history through carving.

How many other projects and plans began the same way? A square number is represented by a square and this also exposes other properties all connected to geometry. Adding two equal sides and a one makes a new square. Visualisation is a powerful method and many famous mathematicians used geometry to help develop and communicate their understanding.

This square of happy faces shows how adding two equal sides and one makes a new square.

An odd number is made up from 2 equal parts and one as well, in this example 3+3+1. We generalise that all square numbers are an odd number away from the next.  We could generalise that every odd number sits between two square numbers.

By squaring all the odd numbers we discover an infinite series of Pythagorian Triples. EG 9^2 = 81 so (81-1)/2 = 40, 9^2 + 40^2 = 41^2. 9, 40, 41 are a Pythagorean Triple. There are an infinite number of odd numbers so there are an infinite number of such triples and we have just made another generalisation and actually a proof.

Different shading shows the difference between consecutive squares are the odd numbers. The sum of n odd numbers forms the square number n^2. Here we see 1 + 3 + 5 + 9 = 16 or 4^2. This is another pattern and a generalisation. More patterns happen in one and three dimensions.

                
 
The triangle numbers form a staircase and making these important numbers with interlink blocks shows how two consecutive triangle numbers always form a square. This is also a generalisation and an important one. I claimed to a group of teachers at a workshop, in 2003 "if you cannot draw a picture of what you are teaching do not teach it." Since that day I have been trying to prove this statement wrong. Visual mathematics and models are very very powerful.

    

Here are two representations of triangle numbers. I strive to have students generalise the sum of these numbers to n as n(n=1)/2. This can be done by putting two of the same Triangle numbers together and looking at the rectangle that is made. EG 2xT3 numbers form a 3x4 rectangle or 3(3+1) rectangle. Pulling the two parts apart represent division by 2 and so we get the formula. Proof.

More about Visual Models

Self Test your Visual Skills and Mathematical Understanding with Fractions
(a) Draw a model explaining 3/4 (Answer will show the 3, the 4 and the relationship to the whole 1)
(b) Draw a model explaining 3/4 divided by 2 (Answer will show dividing and the answer 3/8, showing the new 3 and the new 8]
(c) Draw a model explaining 3/4 divided by 1/2. (Answer will show all numbers, the answer 1 and 1/2, and divided). (Warning...the "one" changes!)

I have used this self test with teachers to help explain why students find "derision" so hard. Lewis Carroll was absolutely correct having the Mock Turtle describe "the four branches of arithmetic as ambition, distraction, uglification and derision". Always have a "growth mindset" about drawing and encourage students to draw ideas. In the self test above, first two (a) and (b) are relatively easy but the third (c) requires the re-unitisation of what the whole or the "one" is. Think of (c) as "How many 1/2's are there in 3/4?" or "How many 0.5m lengths are there in 0.75m of cord?" and you may well see sense. Lipping Ma explained this to me in her book (ref above).

Division, or the word I prefer "derision" has two meanings. We can look at 6 divided by 2 as "How many groups of 2 are there in 6" or "if I split 6 into 2 groups, how many would there be in each group?". The context of the problem determines which meaning applies. The answer remains the same! Students will pick the interpretation needed once they understand the problem so my advice is to allow this to happen naturally. Derision is hard! Like fractions.

One of the issues to successful learning of mathematics is "experience". Students with a broad experience of many different life adventures have little trouble constructing meaning. In too many classes in 2018 I asked students to list all the life experiences such as catching a fish, visiting the city, traveling to a nearby town, riding on a train, flying in an aircraft, hitting a golf ball that they have experienced. Alas I think I stumbled upon a major barrier to learning. Most students do not have broad life experiences and so are not able to make connections to problems such as "I am riding on a 1km long train travelling at 6o km/hr. It enters a 2km long tunnel. How long is the train in the tunnel? " The student questions begin..."Can a train be that long? How long is 1km? Where do you sit on a train? Is the tunnel bent? What is a tunnel? Is it dark in the tunnel? How do you make a tunnel? What if the train hits the tunnel?". Without contextual knowledge students struggle to make sense of the context and actually the problem presented is not the most important problem. Remedy - take students on context broadening experiences and advise parents to do the same. More on this in a later chapter.

Why do number and algebra often become a barrier for people learning mathematics? This is a wicked problem actually. See NZCER http://www.nzcer.org.nz/nzcerpress/key-competencies-future for an explanation of "wicked problem". Mathematics is a complex and highly connected multitude of ideas and representations. It is a language and is continually evolving for new situations, or being discovered as Paul Erdōs would claim (https://en.wikipedia.org/wiki/Paul_Erdős#cite_note-23) from "The Book". Everyone should read about Paul. He said "Coffee turned thoughts into mathematics." he also traveled from place to place overstaying his welcome for his entire life.

Number Stages
Number begins with thinking, continues with thinking, and does not end without thinking. Young children develop a COUNT thinking which parallels their view of the world of "Me", "I", "touch", "in the mouth" and all this within a few centimeters of the little face. Eye contact is the single most important development tool a mother can give her child. It is all about "Me". All information travels through this visual and causes brain development. Add the strong negative feedback loop and when "touch" causes "pain" and the brain very rapidly learns a lesson. Neglect visuals in early life and the brain becomes sluggish. Why would it need to develop? Stimulate and grow. Gain experiences. Play safe but explore!

Counting is what I call a zero dimensional concept and has no direction. It does not connect or link. It happens. Learning to count is important but be aware that a 2 year old reciting numbers to 100 is not evidence of counting. Counting is being aware of the quantity involved and matching objects to words.

I asked my 7 year old grand-daughter "How many blocks in the staircase (see below) and she immediately asked for a calculator and proceeded to add 1 + 2 + 3 + 4 + ...+20. Her strategy was sound and her method modern and persevering, after a while, she confidently announced "289". We asked Siri...who said "210".

Repeating the quest on a calculator rather than the iPhone app she confirmed 210. She suggested the iPhone was not a good calculator!

I rearranged the 1 to the 19, the 2 to 18 and quite quickly she finished the pattern. "Ahh" she said "it is 10 lots of 20 and 10 more. 10x20 +10 which she did with a calculator and of course confirmed the 210 answer. I rearranged them to 21 groups of 10 for her. So here we see in my 7 year old a counted solution and not a lot of developed base ten ideas yet. That will happen however and very soon. We had fun making the staircase, photographed it on her suggestion and sent this to Mummy and her teacher.

The barrier to progression from counting is the brilliant "place value" concept with it's canon "ten" rule. The Romans missed this one. This causes students to ask "what comes after 9, 99, 999?" Learning to control the digits and see the multiplicative pattern of re-use, each time with a more important (place) value takes time and make no assumption that anyone understands this system until they can explain it to you. Take a look at the Power of Ten to convince yourself of the Power of Ten. Only 43 such powers contain all we currently measure in the Universe. My 17 Year Old Physics students would ponder and walk the "power of ten wall" in amazement.

CHECK POINT CHARLIE TIME
Do you  understand and appreciate place value?

• Write the number "eleven thousand, eleven hundred and eleven".

• If I rearrange the digits of a number and subtract the smaller from the larger, why is the result always divisible by 9?

• http://www.flashlightcreative.net/swf/mindreader/

Share your answers and see what understandings you expose.

Adding or Combining in a Linear way.
Following the development of counting strategies such as counting in twos and the ideas of place value we soon see that adding the place values gives the correct answer and we can now know the sum of every combination of every number. We can develop a strategy or an algebraic pattern to do that for us. Hence basic facts and adding strategies like "tidy numbers", the common algorithm and a calculator. There are some fascinating insights that can be discovered in adding.

Additive thinking is however one dimensional or linear thinking. More complex than counting and more connected but quite straight forward!

2 + 3 to a "counter" is different problem than 2 + 4 but to an "adder" is "just one less" or "just one more" depending on how you ask the question. The two problems are actually connected.

It is this connected thinking and its development that separates "a counter" from an "an adder". The connection shows a more complex form of thinking and this shows development from NZC Level 1 to NZC Level 2. With place value ideas becoming established the progression is assured.

We learn to read big numbers and solve problems like "How many legs on 3 dogs" by adding 4 + 4 + 4. We see addition as the way to solve most problems and for many people this additive way becomes a final platform. Most people in any civilisation are predominantly "additive". Sad but true. Sad because we allow this happen and true because I notice it all the time in every place I venture. Why is this the case?

My view is that too many teachers are only additive thinkers in mathematics. I say "in mathematics" because that is where I am noticing the sad fact. It is highly likely that this form of thinking is across all learning areas (and life) for those teachers and I am trying to get evidence to confirm this prediction. The brain goes with the person so why would thinking in other contexts be more, or less, complex?

Dilly dally and continue to revise addition at your student's peril. My advice:- just like counting, once you have learned to add get out of there and and explore multiplication. No one needs 50 ways to add. You can learn more about adding when the numbers become fractional and negative.

CHECK POINT CHARLIE TIME
What strategies do you use to solve these problems?

Add 997 to 5.
What is 299 + 37?
Add up 1 + 2 + 3 + 4 + 5 + 6 + ... 998 + 999.

More on Number!
The NZ Numeracy Project made an error here, in my view, and allowed primary teachers who are predominantly additive thinkers (my observational data and there is a research paper on the nzmaths site) in number to learn, promote and develop many different and horribly inefficient ways to add. This hampers mathematical development by pausing too long and causing boredom and reducing need. There are additive curiosities that can be explored but depart and leave the "Fifty Ways to Add Your Answer" to Simon and Garfunkel  ["50 Ways to leave your Lover", a great song]. When you discover a new set of numbers like integers, decimals or fractions,  re-explore addition as need arises.

My First Target of Learning Maths, Multiplicative Thinking.

Multiplication is, in my head,  two dimensional thinking. The person with this ability holds the idea of 3 groups and the idea of 4 in each group "at the same time" and this makes 12. It is "How many legs on three dogs?" or "I have 4 bags each with three lemons, how many all together?" Either way the answer is 12 and with the context supplying the meaning of the 12.

An array of 3 rows and 4 columns is the visual model of 3 x 4 = 12 and explains every part of that equation. We see the "3", the "4", the "12", the "=" and that the "x" multiplication is rectangular. Multiplication is explained by the shape, a rectangle, that the array forms. Every multiplication problem between two numbers is rectangular. Getting students into multiplication ASAP should be every teacher's goal. Once there we can do a lot more, move on, explore. Become multiplicative!

"Every student is to become multiplicative by the end of Year 10" is a worthy "First Target" for all junior students and please state this clearly and boldly in every classroom. I was criticised for this comment but after explanation to my critics that it is not a final goal and that for new to school Year 9 students only about 15% are in fact multiplicative, I am usually approved. Typically, and by the end of Year 10 this data point has become about 75%. It is indeed a worthy initial goal. In one school last year (Feb 2016) the Y9 cohort began at 11% multiplicative and one year later were measured to be 60% (Feb 2017). That is acceleration and we deliberately caused it to happen. In 2018 I helped improve a whole cohort of 220 students by twice expectation (100% acceleration) and  nearly 60% became genuine multiplicative thinkers. These students opened many doors by doing so. [More on data - Measuring and Monitoring in a later Chapter.]

Mooovin On, as the cow said!
But what about fractions and decimals? These ideas need some knowledge and here there are a couple of axioms in mathematics that are almost never exposed but are vitally important. [Referred to in an earlier chapter]. We need also to count, add, subtract, divide and multiply this fractional and decimal world. Decimals are sometimes called decimals fractions.

But first the axioms.

Axiom #0
One can be anything I choose one to be.

Axiom #1
When we combine or compare anything in mathematics we do so using the same size bits.

An axiom is a self evident truth. That "one can be anything I choose one to be" is not at all obvious. In the task above of dividing 3/4 by 1/2 the "one" being referred to at first to establish the fractions changed mysteriously to another "one" (the half) and never once did "your hands leave your arms!". We should identify "the one" in all problems and normalise this approach. 

The naming of the axioms above, by myself,  happened because like the Laws of Thermodynamics the most important and more fundamental Zeroth Law was discovered after the others! I think that "one can be anything I choose one to be" is most important. Anthony Harradine of ACEMS asked the question "What is the most important concept in Mathematics?" I said "one". You can explore his fascinating website to find other answers and resources.

The combining and comparing axiom likewise is not obvious and is exposed when we ask the sum of 1/2 and 1/3. Most young students first say 2/5 adding everything in sight and most older students, in the western world, pull out an algorithm and say 5/6ths but have no idea why. These two methods are the same boat from my point of view, both in equal peril!

By understanding that the bits being added need to be the same size and using number knowledge of common multiples we see 3/6 and 2/6 which can now make 5/6. There is numerator (the count) and denominator(the part) knowledge needed here as well.

The visual shows the 1/2 being renamed into 6ths and likewise for the 1/3. Now that the "ones" are the same size and we are talking about "the same size bits" we can combine them and see 5/6ths as the answer. Do not underestimate the complexity of the knowledge that underpins the learning to "add fractions".

CHECK POINT CHARLIE TIME
How well do you understand these two key theorems?

• Draw two different models of 1.
• What does half a small pie and half a big pie add up to?
• When can 1/2 + 1/3 be 2/5?
• What is 2/3 of 3/4? (See visual, and explain the twelfths!)

The NZ Numeracy Project (http://www.nzmaths.co.nz) labelled operating with fractions and decimals, using rates and ratios as proportional thinking. You can not be a proportional thinker operating on factions without being a multiplicative thinker either so this earlier goal is even more vital. One researcher in the Rational Number Project reported only 15% of the planet's human population are capable of this complex form of thinking. I am a bit skeptical of that figure but certainly it is not as high as 50%.

This researcher was Sue Lamon and at a Numeracy Conference in Auckland said "You do not become a proportional thinker by growing older". I love that blunt message.

She was adamant that, as the poster above says, deliberate acts of teaching are needed to have people become proportional thinkers. She challenged us with the problem  "There is a tank of 200 fish. 99% of the fish are blue and the rest red. What has to happen to make the tank have 98% blue fish?" It is not that hard but just deceptive. How can a small %change cause such a large number change or vice versa?

To me proportional thinking is many dimensional. I wrote a .ppt for a conference workshop on this topic to help explain my thinking. It is fun.

I was driving to a workshop one morning and the radio announcer on ZM came up with math problem competition called Farmer Brown. This became another power point and I recorded many solutions here as well.

Farmer Brown.

The proportional problem about hot and cold water filling a bath is worthy of attention as well. The hot tap fills a bath in 60 minutes, the cold tap in 30 minutes, if I turn both taps on full how long to get half a bath of warm water?

(a) 60 minutes    (b) 30 minutes    (c) 90 minutes    (d) 45 minutes    (e) 24 minutes    (f) 12 minutes    (g) something else    (h) who cares (i) all the above1
I usually add (g), (h) and (i) to multi-choice.

Math Mooovies
There is a classic proportional problem on Harvard's collection of math in movies. http://www.math.harvard.edu/~knill/mathmovies/swf/league.html. The problem is explained in the movie with a variety of solutions. If Joe can paint a house in 3 hours and Sam can paint the same house in 5 hours, how long would it take if they work together? The Math Movie website, above, is full of curiosities, impossibilities, great theorems and ideas all in a mathematics context. Some right and some wrong!

SOLO TAXOMONY
The counting, adding, multiplying and proportional stages of number thinking development can be paralleled to Bloom's and others Taxonomy's. From recall to application. The SOLO taxonomy is another and is explained here. What is happening is an increasing complex way of connecting ideas and processes in thinking. It happens in all learning areas of course however I think we can measure this quite nicely using number. More later on this when you discover the LOMAS test ( another Chapter or page in this book).

The brain, or the magician as I sometimes refer to it, is a complex entity, a wicked problem! See http://www.brainwave.org.nz to read about some of the developments in what we know about the brain in the last few years. For good health ...exercise your brain...learn to do Cryptic Crosswords! Here is a good link start...Cryptic Starter

MORE ABOUT ALGEBRA
Algebra at this level is the generalisation of number and number patterns. We are all very good at generalising and do not realise this. We learn from a few minutes of age that if crying makes food. It works and seems to go on working. When I wake every morning I already know what the room looks like and how to find essential rooms. If I could not generalise the world would be a very very scary place. It would be a new maze every step and turn and highly confusing.

Learning to generalise and making this explicit is important to learning mathematics. Learning how to record these generalizations and manipulate them to make new understandings or generalizations is called algebra manipulation. Often algebra is confused with new procedures for a new problem and tedious examples where I make the same mistakes over and over again. Learning early on that algebra is the generalisation of number is a magical illustrator of algebraic manipulation.

An example is the Distributive Rule where 2x(3+4) = (2x3)+(2x4). This is generalised as a.(b+c)=(a.b)+(a.c) I have used "." to represent "x" or the binary operation we call multiplication. The number version illustrates that the left side is equal to the right side and mathematical truth is preserved. It is a leap of faith to extend this to all numbers but we prove this works with Real Numbers and then check every time we find a new application.

Thoughts on HOMEWORK
Do we need homework? What do I learn by doing heaps of problems? Am I not better off by pondering what I learned today? What do I learn if I keep making the same mistake. Like reading books, homework should be easily accessed, 95% of words recognisable in reading and likewise in maths. So why do it?

I think it better to have students leave the classroom pondering mathematics. What is infinity? What is past that place? What is 0/0? [Ask Siri!] Sometimes students need to practice or memorize methods and facts and then there is a case for homework. I want to see students playing games and interacting, arguing and explaining, planning and building, learning to participate, collaborate and contribute. That is the real and better learning.

There is an abundance of resources for learning number and algebra in the form of text books, worksheets and on the WWW web.  The NZ Numeracy Project is the largest mathematics site in NZ and is located at http://www.nzmaths.co.nz. We do not have a resource issue but we do have a use of resource issue. Investigate NRICH, youcubed.org, Otago Maths, Starter of the Day, Cut the Knot, NLVM, MAV, AMC...this list is long and growing daily.

Noticing what students are doing when solving problems, noticing their use of mathematical language and curiosity, involving them in the daily discourse, having them reflect on their learning, using number and algebra to develop those more important traits of perseverance, collaboration, creative thinking, being critical and reflective, developing and using the language of mathematics and connecting to other learning areas is what learning is all about. See Team and Problem Based Learning in Chapter 11.

Thinking like a mathematician and that statement draws this chapter to a close and links to that main vine again.

Lesson #4
Key features of any year level is monitoring the thinking and skills of each student. Knowing the broad stages of number development and monitoring this progression is a vital action for any teacher of mathematical thinking. Being multiplicative predicts future success in mathematics and a broad range of occupations. Being multiplicative opens the door of proportional thinking. The more mathematics you know the more mathematics you will use. Place value knowledge underpins all number and this is also a strongly multiplicative idea. Be visual.

Teacher TASK
• How would you describe the Deep Understanding of Number and the Deep Understanding of Algebra
• Describe how you develop number and algebra thinking for a typical Year 7, 8, 9 or 10 student.
• Find out how you rate on the "Being numerate" scale.
• How do you develop visual representations? Do you draw pictures of problems?
• What are your favorite on line resources?

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