What worked in my maths classroom this year? What engaged my students, improved their knowledge, and developed their mathematical skills?
Twenty years after I left primary teaching, I returned – and improving my knowledge and skills in maths teaching became (and remains) my highest priority.
I study, read and view maths books and webinars, listen to maths podcasts, talk and plan with my maths mentor best friend, Margarita Breed, and learn from maths teachers on Twitter.
So what teaching practices have worked or seen the biggest shifts in student attitudes, knowledge and/or skills?
Unpacking Marking Guides
Ensuring that students are clear about the terminology of maths marking guides has been important in my maths classes (See Marking Guide: Evaluating our designs and answers below).
As we unpack each term or phrase, I write in a way that students will understand. As we work, I regularly draw them back to these dot-points, to monitor if they’re on-track. Students also submit work that we review according to the marking guide dot-points.
I ask students for specific evidence that they do meet specific aspects of an assessment standard. Key here is going beyond talk. So they highlight where it meets the standard. If it doesn’t, I’ll ask them to examine and unpack examples that do meet the standard, and go from there.
At times, we’ll develop review checklists (See Open Number Line Review below). Students then mark where their work meets the requirements, and if it doesn’t – note what is missing and fix it.
Throughout these processes, students become much clearer about what they’re working towards. They have ready access to examples and non-examples. They also recognise that ultimately it’s about approximating closer and closer to their end goals – in respect to achieving the achievement standards (or aspects of such).
Copyright Greg Curran 2021
At the senior level, there is much focus on naming and explaining the mathematical strategies we use, along with providing clear representations of our thinking.
I teach students how to write mathematically with clear scaffolds and exemplars. As they become familiar with the structures, they’re able to make them more their own.
Here’s an example from our Number Properties (Prime Numbers, Square Numbers, Triangular Numbers) Unit:
- Name them and define them
- These numbers are ______ numbers which means they….
- Provide an example and explain it (include a diagram)
- For example, …
- (Optional) Provide a non-example and explain it
- By contrast, ____ is not a _____ number because …
In another example, students had to name and explain their mental strategy, with examples.
- Name your mental strategy
- I chunked (partitioned) each number into tens and ones.
- I also used my place value knowledge of multiplying by tens.
- Give examples and explain them
- For example, I chunked (partitioned) 55 into 5 tens and 5 ones and 36 into 3 tens and 6 ones
- For 50 x 30, I thought of ten 30s which is 300
- Then I scaled up by five (to get to 50 x 30), 5 x 300 = 1500
I also teach students to use key phrases in their explanations. Here’s some examples:
- We know this because…
Using prior knowledge
- We know that …. so…
- Since we know that…. (prior knowledge), then…
Sequencing our steps
- First we…
- Then we…
- Next we..
One of the most significant impacts on learning in maths – in my class – is the use of stand-up whiteboards.
Working in mixed ability groups – of no more than 3 students – has had most impact in my class. The following rules are also important:
- The student who speaks doesn’t write. This reduces the opportunity for the speaker to solve the task without discussing it with their team-mates.
- The speaker must clearly communicate their ideas so that the recorder (of ideas) understands and is able to reproduce their thinking on the whiteboard.
- Students must all be standing and their boards must be visible to everyone in the class.
Other key strategies I use – in connection with stand-up whiteboards are:
- Highlighting: Using a different marker colour to the students, I draw attention to interesting ideas or strategies, clear ways of laying out maths thinking, key mistakes or queries (where students don’t know what to do next). These highlights then become the focus of an explicit teaching session or a student lead think aloud or presentation.
- Collect an idea or strategy: After working for a while, I ask students to check out the boards of other groups with the goal of collecting something useful for their team. As they do so, I’ll ask some teams to share what they collected, and why they thought it was beneficial to their team.
- Notice and wonder: There are different possibilities here. I might ask students to wander and notice similarities or differences between the groups’ approaches to a maths task. I might ask students to highlight something that makes them wonder and write a question for the group (like, ‘why did you …?’)
- Team-work strategies: At times I will draw students’ attention to what is working well in their teams, and what it looks like and sounds like. At other times, we’ll focus on what isn’t working well, what that looks and sounds like, and what we can do to improve in this area. Having students write their focus and particular strategies to implement at the top of their whiteboard can be a useful reminder here.
Rich Maths Tasks
Regardless of the achievement standards I am teaching, I seek out rich tasks to teach skills and build knowledge relating to such. The tasks I favour are often low floor, high ceiling meaning that students can access them at their knowledge and skill level, and then challenge themselves in terms of where they take the task.
I regularly use the subscription site, Maths300 since it is linked to the Australian Achievement Standards in Maths, has detailed teaching plans, with all the materials needed to teach such. The tasks are fun, hands-on, and encourage high level thinking. Students also learn about planning, team-work, strategising, how to be efficient, and the importance of making mathematical connections. There’s so many possibilities with Maths300 tasks.
Here’s a recent example (adapted from Maths300).
Prior to completing this fraction task, students had extensive experience estimating fractional parts of a whole (a short piece of rope). For example, I asked them to place a marker ¼ along the length of the rope. Students then checked their estimates without using a measuring tape or ruler. One method we devised was to fold the rope. So for quarters, we folded the rope into four parts to check our estimates.
Then we began the rich task, focused on identifying fractional parts of a whole (our school building).
Students lined up at one end of a building and estimated the half way mark (to the other end of the building). We began walking together, stopping when we thought we had reached the half way mark. Later we did one quarter, and 3 quarters and so on. Then we had to check our estimates, identifying how far over or under we were, and seeking to get more accurate in our estimates over time.
The joy of rich tasks besides learning maths skills in a real-world, applied context is that you often learn more than you expected And often the learning, in this case relating to measuring accurately, comes from mistakes or unexpected occurrences.
It all started so well with some distance between groups’ lengths of strings.
And then we began to get tangled and tempers flared.
The utter chaos of a mass of entangled strings was the result of everyone starting their measuring at the same point and location, and not being mindful of giving space to each other. This lead to a ‘stop everything and review’ moment where we came up with ways to measure without becoming entangled.
Students also suggested other ways of measuring such as with our feet. There were a number of measurement problems here too, in respect to maintaining accuracy. So I observed each group then demonstrated to them what I saw. For example, many weren’t placing their feet heel to toe, and were instead inserting a space between their feet. This lead to in-the-moment discussions about how to improve their accuracy.
Students also kept getting distracted with their foot count due to others talking loudly or other student activity in the immediate area. This meant they had to keep starting over. This lead to a a whole class discussion about how to keep track of our count. Students suggested using tally marks or marking milestones like 20 or 50 or 100 feet on the ground.
Given we were making comparisons across the groups, students recognised the importance of measuring our foot size and indicating that in our results. When it came to converting feet to metres, the question then arose what was the standard in terms of a foot, and how did that compare to our foot size.
Rich tasks often extend over a couple of sessions so making connections with prior learning is essential. Before we started estimating each day, I asked students what they knew already that could help them. For example, we already knew half-way was X feet so one quarter was half of that.
Good team-work is critical in rich tasks. I set time-limits and we discussed how to be more efficient and complete the task in the time frame. Afterwards, we reviewed whether we had achieved our efficiency goals, and what had helped or hindered us.
I’m always on the lookout for maths in my everyday life that I can bring into the classroom. When we were studying angles, the following image prompted much discussion.
Being able to interpret data is becoming an increasingly important real-world skill. Students became familiar with interpreting weather graphs with the following prompts:
- What do you notice?
- What do you wonder?
- Write 3 to 5 sentences about the weather in April 2021, as recorded at our local airport.
Over time, students become more sophisticated in the statements they’re able to make as they become aware of the layout, what each set of numbers and lines referred to, and what the terminology meant (such as Max Temp, Min Temp).
The following photo from the local outdoor store provided many mathematical opportunities to compare prices – in respect to best value. As can be seen from the student responses below, it allowed for different entry points in terms of how they approached the task, and the level of detail or depth they provided.
Sometimes students take you to an area you hadn’t thought of. For example, in the first piece of student work below, they ended up calculating prices per gram which I was able to connect to the unit pricing used on product labels in supermarkets.
Number Boxes and Open Middle Problems
I was first introduced to Number Boxes by Jenna Laib (who was a previous guest on my education podcast) and Mark Chubb. They quickly became a winning warm-up (as in often requested) in my class. All you need is the visual and a dice.
Here’s a couple of examples I’ve used from Mark Chubb’s website.
There’s many add-on possibilities with this activity. For example:
- what strategies did you use to ensure the largest number?
- does this strategy always work? Why or why not?
- what’s the value of specific numbers (For example, a 6 in the tens column = 6 x 10 = 60)
- put yourselves in order from smallest to largest number
- what’s the difference between the smallest and largest number?
- what number is closest to the middle (between the smallest and largest number)?
- use two strategies to prove your answer (as in number boxes involving operations)
Mark Chubb and Jenna Laib also provide many add-on possibilities or extensions, in the links provided above. I also discussed Number Boxes in a previous blogpost.
After discovering Number Boxes, I made connections with Robert Kaplinsky’s Open Middle Problems. They have the empty boxes (as in the Number Boxes) but have an additional challenge. For example, using the numbers 0 to 9 once only, to make a true equation.
Students can get frustrated without the immediate or readily worked out answers but with the necessary support, encouragement and modelling they’ll become an important part of our maths routines over the year.
Listen to My Maths-Related Podcast Episodes
Favourite Math-Reads and Listens
Maths Is Figure-Out-Able Podcast with Pam W Harris and Kim Montague
- Short and straight to point. Always with takeaways that I readily implement into my maths classroom. Also check out their website, and click on the Learn Now tab.
Teaching Mathematics: Foundations to Middle Years – Dianne Siemon, Kim Beswick, Kathy Brady, Julie Clark, Rhonda Faragher, Elizabeth Warren
- My must have maths reference text book. It’s my go to when I start planning a unit and whenever I have a query. It provides clear explanations of the foundational maths skills.
Becoming The Maths Teacher You Wish You’d Had by Tracy Johnston Zager
- Another must-have maths reference text. Filled with ready to implement maths activities. Challenges the business as usual, traditional approaches. Especially useful around the value of productive struggle which I discuss with Margarita Breed in Episode 55 of Pushing The Edge with Greg Curran.
- A regular in my maths classroom. Michael has built an extensive collection of maths games arranged in year levels or via math topics. He clearly demonstrates each game in a video with his child.
- Jenna’s blog is another regular read of mine. She captures in detail her maths teaching along with student responses, and extensive reflections about such.