WELCOME TO MULTIPLICATIVE THINKING
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Perhaps the earliest systematic use of a symbol for zero in a place-value system is found in the mathematics of the Mayas of Central and South America. The Mayan zero symbol was used to indicate the absence of any units of the various orders of the modified base-twenty system. This system was probably used much more for recording calendar times than for computational purposes. The Maya counted essentially on a scale of 20, using for their basal numerals two elements, a dot representing one and a horizontal dash representing five. The most important feature of their system was their zero, this character as illustrated, which also had numerous variants
McNeill, S.A. (2001). The Mayan zeros. Mathematics Teacher 94,7, 590-592.
The Mayan symbol for zero is of interest to us as zero plays a significant role in multiplicative thinking.
For a brief introduction to multiplicative thinking, access the PowerPoint through the 'Getting Into Multiplicative Thinking' link
For some children's work samples to start you thinking, access the PowerPoint through the 'Do children in your class do or say these things?'
BRIEF HISTORY OF OUR WORK
We began our work on multiplicative thinking in late 2013. Since then we have learned more than we thought possible about this important idea. Initially, we worked with small cohorts of primary school children in Perth, Western Australia, and used a one-on-one semi-structured interview to gather data. While this provided us with very rich information, it was time-consuming and by the end of 2014, we had developed our written instrument, the Multiplicative Thinking Quiz, which enabled us to gather data from class groups. The quiz has been modified and improved to its current form.
During the project, we worked with over 120 teachers and 2000 students from primary schools in Western Australia, Victoria, Dunedin (New Zealand) and Plymouth (United Kingdom). To enable us to provide assistance to teachers, we developed a collection of over 200 teaching tasks to follow up the analysis of data from the Multiplicative Thinking Quiz. The assessment instruments and teaching tasks are all available on this website.
WHO WE ARE
DR CHRIS HURST
Adjunct Research Fellow
Perth, Western Australia
I trained as a primary teacher and taught in various jurisdictions for 37 years before completing my PhD in 2007. From 2006 to 2020, I was an academic in the School of Education at Curtin University in Perth, Western Australia. I am currently an Adjunct Research Fellow at Curtin and am an active researcher. My main interests lie in multiplicative thinking, children’s capacity to make appropriate computational choices, and mathematical content knowledge of primary teachers.
WHAT WE DO
We offer a research-based consultancy service on multiplicative thinking for primary/elementary and middle years educators.
We will work with teachers, education assistants, support staff, and parents, as needed.
We will assist teachers to assess children's multiplicative thinking through the use of diagnostic interviews and/or written quizzes.
We also offer consultant advice in all other areas of primary and middle years mathematics.
We provide professional learning for whole school staffs, small groups, and individual educators.
A professional learning day for education assistants.
A child was asked to explain with the use of bundling sticks what 7 x 15 meant. She demonstrated it and then regrouped the seven groups of five into three ten bundles and five singles.
LET US START TO THINK ABOUT MULTIPLICATIVE THINKING
An interview with Karl (Year 6)
Karl was asked for the answer to 7 × 15 and immediately said ‘105’.
INT: How do you know it’s 105?
KARL: First I timesed 7 by 5, which is 35, and I leave that 5 in its place – the ones place – then I times 7 times 1 equals 7. Then 3 plus 7 is 10.
INT: Do you normally write them down like that?
KARL: I normally write them down but sometimes it pops up in my head.
INT: Can you show me how you would do it if you wrote it down? [Karl wrote down the vertical algorithm, but did not show the carried tens].
INT: Can you show me what seven lots of 15 equals 105 looks like with the bundling sticks?
KARL: [after a pause] I don’t know. I’ve never worked with sticks before.
INT: OK, there’s 15 [bundle of 10 and five singles] What would you need to do to show me that there? [indicated the written algorithm].
KEC: Can I work in my own way? [He did show seven ten bundles and grouped them to make 70 and then seven groups of five which he grouped together as 35. However, he did not regroup them to make ten tens and five singles].
WHAT DOES THIS TELL US?
MORE FROM KARL
Karl was shown the example 29 x 37 and was asked how he would do it.
He wrote a vertical algorithm and worked it out very quickly [See work sample – note how he did not put a zero in the second line, but left a gap].
INT: What’s this little gap here?
KARL: Nothing . . . I’ll say zero.
INT: Why did you put the gap there?
KARL: [hesitated] I don’t know really . . . it’s my Maths teacher tells me that’s how it works.
INT: But you wouldn’t put that 87 under there [indicated the 10 and 1 column – see sample]. Is there any reason why?
KARL: No, I don’t know a reason why.
INT: Do you think it might be something to do with that you’re multiplying that by 30 so it is ten times bigger?
KARL: [No response].
INT: That was really quick how you did that one.
KARL: I do fractions too.
WHAT IS THE MESSAGE HERE?
CONSIDERING ALL THAT KARL CAN DO, WHY IS HE DEEMED TO BE A STUDENT AT RISK?
In the vertical algorithm for 7 × 15 and 29 x 37, he calculated correctly, but did not show the carried tens in either example.
The comment that he’d ‘never worked with sticks before’ is telling. It begs the question as to how he learned the algorithms.
While he wrote a vertical algorithm and worked it out very quickly, he could not explain why he left a gap in the second line, stating that his teacher tells him that ‘that’s how it works’. When probed as to whether it was ten times as many, he did not respond.
Karl has a very quick response and can do an algorithm correctly, but he does not know why he does what he does.
In another question, he could not use bundling sticks to show 90 ÷ 7, yet he mentally gave the answer, again very quickly.
Karl could do each example using a procedure but really struggled to use the materials. This suggests that procedural knowledge has been developed at the expense of conceptual understanding.
Finally, Karl mentally solved 23 x 4 very quickly. When asked about the answer for 400 x 23, he said he ‘added two zeros to the 92' from the first example – another use of a procedure which is not helpful.
FEEDBACK THAT WE HAVE RECEIVED FROM TEACHERS OF LOWER SECONDARY STUDENTS SUGGESTS THAT STUDENTS LIKE KARL ARE COMMON AND THAT THEIR FUTURE LEARNING IS BEING CONSTRAINED DUE TO A LACK OF CONCEPTUAL UNDERSTANDING.
A MULTIPLICATIVE SAMPLER
We have developed a comprehensive assessment tool in the form of a Multiplicative Thinking Quiz (MTQ). It is in four sections, each of which matches one set of multiplicative connections in our array-based Multiplicative Thinking Model. In this sampler, we provide a copy of the model, one of the MTQ test items, and a Key Task designed to remediate children's understanding as identified by the test item.
We strongly recommend that teachers engage with our professional learning to learn more about multiplicative thinking before embarking on a teaching program.
MULTIPLICATIVE THINKING MODEL
We developed this model for multiplicative thinking over four years. We believe that multiplicative thinking has four elements, each of which is based on the array. These elements are
Connections 1 - Equal groups, factors and multiples
Connections 2 - Times as many relationship
Connections 3 - Development of the multiplication algorithm
Connections 4 - Fraction, ratio, and proportional reasoning
SAMPLE ASSESSMENT ITEM
This sample assessment item is based on Connections 1 and is designed to assess children's capacity to generate an appropriate word story to match a given multiplication and division fact. It requires children to understand how the array shows a number of equal groups and that the first number in the number fact tells us the number of groups (rows in the array) and the second number tells how many are in each group (or how many in each row of the array). In our research we found that many children struggled to provide an appropriate story, possibly indicating that they had not had much experience of doing so
SAMPLE KEY TEACHING TASK
This sample task is titled Tell Me A Story. Children are required to make an array and then write a story based on the 'number of groups x the number in each group' convention. Following that, they are asked to re-write their story as a division story. The task is structured the same as all of our teaching tasks. It contains links to key mathematics, links to the proficiencies, links to the assessment item with preferred responses, details of how to conduct the task, and notes for teachers.
ARE YOU INTERESTED IN LEARNING MORE ABOUT MULTIPLICATIVE THINKING AND GAINING ACCESS TO OUR FULL RANGE OF TEACHING TASKS AND RESOURCES?