Whole Numbers to decimals - an easy transition?

If you have been following my work over the past few years, you will know that my PhD research focused on the assessment and teaching of whole number place value.
Part of this work has been, and continues to be, helping teachers to develop a deep understanding of their student’s knowledge of whole numbers. This means teachers are better able to teach whole number place value, and make data-driven decisions around when their students are developmentally ready to be introduced to decimals. 

If we look at the Australian Curriculum Version 9.0, the NSW syllabus document or the Victorian Curriculum Version 2.0, decimals are first introduced in Year 4.

I have found over and over in government, private and catholic schools across Australia, that many (not all) Year 4 students lack a sufficient depth of knowledge in whole numbers, and thus find decimals very difficult to comprehend. I have written about my concerns introducing decimals to these students in this blog. 

For this reason, I always encourage teachers to complete my PVAT assessment to accurately determine each student's level of whole number knowledge, before they embark on teaching the decimals.

Teachers who complete the PVAT often ask how do they know when their students are ready to explore decimals?

My answer has always been, once students reach the final stage, Stage 4, in my whole number place value developmental progression (PVDP) (as determined by the PVAT assessment), they are ready to explore decimals.

Lately I have been looking more closely at PVAT Stage 4 student data with several schools who are part of my Numeracy Teachers Academy.

Once students reach PVAT Stage 4, I encourage teachers to administer the Stage 5 decimal place value assessment I developed (this assessment is available in my PVAT mini course).

Having looked at both Stage 4 and Stage 5 data (I love unpacking PVAT data!) I can see that there are a group of PVAT Stage 4 students who struggle to transfer their whole number knowledge to decimals.

While I am working with schools to unpack the 'why' behind these observations-I have a few theories that I thought I might share.

I believe there are some students who we can describe as ‘apparent experts’ in whole numbers. They can correctly answer some PVAT questions despite having a superficial understanding of important place value ideas.

For example, students may use an algorithm when asked to multiply by ten.

Or they may use an algorithm in a context where renaming or counting may be more appropriate.

For example,

Whilst these strategies allow students to correctly answer the items, to me, the methods they used are not indicative of a deep and flexible understanding of the place value system. Having this superficial, procedural-based understanding makes transferring their knowledge to the decimal context very challenging.

It means their whole number foundations are shaky.

This brings me to an important point related to assessment. In an assessment we are often looking for simply a 'correct' or 'incorrect' response.

While accuracy is important, it doesn't always provide us with a 'full picture'.

In both samples shown above, the students have arrived at the 'correct' answer.

But... we can see the methods they used may indicate a superficial, procedure-based understanding of the content (this may not be the case, perhaps the student thought they 'needed' to answer in this way). For this reason, it is important we spend some time engaging with the actual student responses (and not just looking at spreadsheets or data summaries) and discuss responses with students. Recognising these strategies as possibly indicative of a lack of understanding is very important knowledge for us, as teachers, to possess.

One of my concerns is the pressure teachers feel to move students through the curriculum. This can lead to us being forced to 'help' students by introducing 'tricks' or 'short cuts' that allow them to find the correct answers, but do not necessarily lead them to develop a deep conceptual understanding of the content.

This ‘rules without reason’ instruction can lead to 'short term gain', but 'long-term pain'.

For me, this is yet another reason why we must continually strive to develop on our PCK. Having knowledge and skills to help students develop a deep understanding of maths concepts is critical to high-quality instruction.

Over the past few weeks, I have been reading Sarah Cottingham's book "Ausubel's Meaningful Learning in Action". This book provides a great summary of David Ausubel’s research into learning. David was an American psychologist and psychiatrist and his research centres around deep learning.

Among other things David's research talks about the importance of avoiding compartmentalizing knowledge, instead working on ensuring learning is connected. In the context of place value, this means helping students to see how whole number place value is connected to decimal place value. Often we teach these as two 'separate' ideas.

Ausubel talks about the importance of a student's prior knowledge in a particular area, and using this to create ‘anchors’ for new knowledge.

When I read about the importance of these 'anchors', I thought of our Stage 4 'apparent experts'.  I suspect their superficial learning of ‘rules’ and procedures means they have poor structures in place, and thus they find it difficult to connect their decimal learning to their, already shaky, whole number knowledge.

This means that the transition to decimals is not as seamless as it could/should be.
My suggestion is that we slow down our teaching of place value and continually check that students can explain the 'why' behind rules and short-cuts.

I am all for efficiency (in fact I pretty much live my life looking to find more efficient ways to do everything- including wearing my running gear to bed so that I don't waste time getting dressed when my 5am alarm sounds!😂). But the 'flip-side' of efficiency is understanding- and this is critical for our students to develop.

So, this week I encourage you to think about your students. Have you noticed some struggle to apply their whole numbers knowledge in the decimal context, or if you are in the junior grades, have you noticed they struggle to apply their knowledge of doubles, when halving? It may be that they have compartmentalized their learning and need your assistance to make and see connections between the knowledge!

Have a great week!

Ange🎲🎓

P.S. You can download the PDF version of this blog to print or share with colleagues here.