Formative assessment, developmental stages and starting the year well

The goal of formative assessment should always be to find out what each student NEEDS next, rather than focusing on identifying what they already know. When we truly know what students need, we can plan for their progress, however it’s more powerful than that…

When we know what each student needs, we automatically take that into account throughout the course of the day. We ask students the right questions to make them think. We look for evidence of their progress just in passing. When we are checking in with students, we know what to focus on. It helps us be far more effective all the time, not just in our lesson planning.

Formative assessment at the start of the year is a powerful strategy to identify the learning needs of a child. However, asking a challenging problem as regular formative assessment each week is arguably of even more benefit as it keeps us up-to-date on our kids. There is no reason not to do both. Information is included below on how to know what a student needs, how to find out, diagnostic testing for different purposes and incorporating regular formative assessment within your practice.

Identifying what a student needs:

I mostly want to assess a child’s understanding of number. I want to understand which stage a child is working towards, so that I know what is likely to be appropriate and I can plan for effective learning. The following image shows our current correlation of stages of student learning. We made this sequence by correlating several thousand student samples, however is it not yet available in formal, peer-reviewed research.

The image above identifies 10 stages of understanding of number that we have found tend to occur across primary schooling. The stages are a guide only, as they try to put together key concepts in understanding of number. While any one child might be working across stages, typically, a student who could collect and draw 10-12 objects might only be able to partition 5 or 6 objects (stage C).  The aim of term 1 testing should be to identify what a student needs, or more simply, what stage a student is working towards. That way we can plan for progress.

Typical understanding at each stage:

Stage A1:

• Collects 2 or 3 objects but has trouble collecting 4 objects consistently
• When asked to draw their hand or foot, it doesn’t have the right number of fingers
• When a teacher moves 4 objects around on a desk, the child needs to keep counting them to make sure that there are still 4. Note: sometimes children count because they think we want them to. Make it clear that they can just tell you how many if they know.

Stage A2:

• Successful with most Stage A1 tasks
• Has trouble drawing 4-5 objects when not in a straight line (e.g. teacher makes a square from 4 cubes for student to examine, covers it, student has trouble drawing the square from memory)
• If the teacher places 5 or 6 blocks in a coffee mug then shakes it, the student thinks the number of blocks still in the cup might have changed.

Stage B1:

• Successful with most Stage A tasks. Foot or hand drawing looks pretty much like a foot.
• When student closes their eyes and tries to show 6 fingers, they have some trouble. E.g. might show 5 and 1, but has trouble working out 3 and 3, and 4 and 2.
• Has trouble drawing structures made of 6 or more blocks (e.g. drawing 6 arranged in a circle, or a 2×3 rectangle constructed of cubes)

Stage B2:

• Successful with most Stage B1 tasks.
• Missing part partitioning task: Show 8 blocks. Cover 5, leaving 3 visible. Do they know that 5 are hidden? Increase number of blocks up to 12. Students in this Stage tend to find this task hard up to 12 blocks.
• Working towards drawing arrays (or rectangles cut into squares) for up to 12 blocks. Finding multiple arrays for the same number can be tricky. Some kids find it hard to make arrays of odd numbers that are composite (e.g. 9), because they just make 2s. Drawing a tens frame is a typical task at this level, but other arrays should be included as well.
• Working towards understanding teen numbers as 10 and some more.

Stage C:

• Successful with most Stage B tasks.
• Working hard to add and subtract to make teen numbers (e.g. 7+8, 13-6…)
• Working towards consistently making two-digit numbers as tens and ones. This includes the idea of adding and subtracting ten, placing numbers on hundreds charts, identifying how many tens and ones in each number, knowing that 23 is not 32.)
• Working towards drawing multiple arrays for teen numbers

Stage D:

• Successful with most Stage C tasks.
• Working towards addition and subtraction of two-digit numbers (including regrouping or using negative numbers)
• Working towards making, naming and regrouping three-digit numbers
• Changing from a array model to an area model for drawing multiplication (e.g. a rectangle cut into 3 rows and 5 columns rather than 3 rows of 5 circles)
• Has some trouble positioning numbers 2, 3 and 8 on an open number line between 1 and 10. Numbers will be in the correct order, but the positioning of the numbers will not allow the right spacing to add on the missing numbers.
• Thinks folding a rectangle in half diagonally will not make the same size “half” as folding the same rectangle in half vertically or horizontally.

Stage E:

• Successful with most Stage D tasks.
• Consistently understands hundreds, tens and ones, but is working towards adding and subtracting three digit numbers.
• Working towards understanding that halves, quarters and eighths all have to be “fair”. This means “the same size”, rather than “the same shape”. This includes fractions of groups.
• Working towards multiplying/dividing where one of the two factors is 1-5 and the other is 6-10.
• Working towards placing tens on a number line between 0 and 100.

Stage F:

• Successful with most Stage E tasks.
• Has trouble working out that fractions such as 1/3 need to be fair, just like fractions with even denominators (e.g. will call a circle broken into one half and 2 quarters “thirds” or “uneven thirds”).
• Working towards placing hundreds and tens on an open number line between 0 and 1000.
• Working towards multiplying and dividing all single digit terms (including division with remainders).

Stage G:

• Successful with most Stage F tasks.
• Working towards connecting decimals and fractions, including seeing fractions AS a number rather than as cutting whole objects into pieces.
• Can place simple fractions on a number line from 0-1, but is working towards placing improper fractions on a line from 0-2.
• Working towards connecting division and fractions (e.g. when this is achieved a child dividing 12 by 5 can picture this as a rectangle with a length of 5 and a width of 2.4 or 2 and 2/5ths, rather than answering as 2 remainder 2).
• Working towards multiplying multi-digit numbers by single digit numbers or two-digit numbers using a distributive model.

Stage H:

• Successful with most Stage G tasks.
• Working towards operations with fractions.
• Working towards connecting fractions, percentages, decimals and chance.
• Working towards generalising arithmetic properties to algebra (e.g. ab = ba as 3×4=4×3), including order of operations.
• Working towards understanding of using unknowns and formulae.

Diagnostic testing and tracking of growth:

All of our diagnostic testing is made freely available to help teachers identify what their students need. The testing comes in two types, which you choose between depending on your purpose.

Type 1 tests:

The first type of tests are easier to administer and understand than the second type. They are also more useful for checking growth (e.g. effect size) across a whole cohort, so they are often recommended as a way of getting started. However, as the first type of tests have set questions rather than adaptable questions, they do not give as good information for knowing exactly what each student needs.  They are great for identifying patterns in the understanding of students across a whole class and therefore helping teachers know where to start. Typically, we recommend the following tests for cohorts of students:

• Foundation and struggling Year 1 students: Oral Addition and Subtraction test (Tests number to 10)
• Year 1 and Year 2 students: Written addition and subtraction test (Tests number 10-100. Feel free to read every question to students.)
• Year 3-4 students: Place Value test (Tests 2 and 3 digit number) + Multiplication and Division test
• Year 5-6 students: Fractions test + Decimals and Percent test, OR Upper Primary and Junior Secondary Diagnostic Test (checks all concepts, but in less detail). Note: make sure you also download the Interpretation with Student Samples.
• Years 7-9 students: Upper Primary and Junior Secondary Diagnostic Test (checks all concepts, but in less detail). Note: make sure you also download the Interpretation with Student Samples.

Type 2 Tests:

The second type of tests are designed to find students’ stage of learning. They contain a mix of concepts that is shown in the picture of Developmental Stages above. These tests are designed to use two phases. Phase 1: a handful of oral questions to roughly group students. Phase 2: more detailed oral or written testing based on the initial grouping, but then adapted as needed. Using this format, teachers can more easily focus questions to identify exactly what each student needs and track their growth over time. These tests are not great for large-scale data analysis (e.g. effect size of intervention), as they are individualised to each child. They give far better information at an individual level. Typically, we recommend that these tests are used by teachers who have been implementing Back to Front Maths for a while, or for profiling individual students.

• Foundation to Year 2 testing sequence – Tests stages A-D
• Years 3-5 testing sequence – Tests stages D-F
• Years 6-8 testing sequence – Tests stages F-H
• Number Tracker for individual students – very useful for for junior primary teachers and students with IEPs.

Regular Formative Assessment:

While initial testing is useful, ongoing formative assessment that tells you what students need to know is typically more helpful for ensuring growth. One very effective way of incorporating this is through using complex and challenging tasks. These tasks should be hard enough for students to be “a bit wrong”, so that you can figure out what students need next. The last few questions on NAPLAN tests provide some good examples of questions that require hard thinking, but have relatively simple content.

Back-to-Front Maths includes Journal Problems or Novel Problems for this reason in the Lessons Bank section. Each problem is designed to be tricky, so that it identifies underlying student misconceptions. The lesson plans help you identify student responses and also contain questioning to address any misconceptions found. For the F-2 lessons, the first lesson in each sequence for a topic is always a Novel Problem. For the Years 3-7 lessons, simply look for brackets with the words (Journal Problem), or abbreviation (JP), following the name of the lesson. If you are interested in finding out more about How Back-to-Front Maths works, or having a look at our peer-reviewed research and data please follow the link.

Whatever decision you make regarding formative assessment, be sure to focus your assessment on identifying what a student NEEDS rather than just what they know, so that teaching is as effective as it can be.

Interested in fixing the misconceptions you have just found? Check out our acclaimed Interventions series.