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How Do I Teach Conceptually AND Cover the Curriculum?


I first heard the phrase ‘slow down to speed up’ in my Master’s program- they were talking about the importance of taking the time to create common values, vision and missions within organizations.

 

 

These processes are expensive and time-consuming so many organizations don’t do it….but the culture and therefore efficacy of the organization often suffers as a result. We learned, by using many real-life examples, of how slowing down to do this work led to much more productive and therefore profitable organizations. In fact, in a Harvard Business Review study, “the companies that embraced initiatives and chose to ‘go, go, go’ to try to gain an edge ended up with lower sales and operating profits than those that paused at key moments to make sure they were on the right track. What’s more, the firms that ‘slowed down to speed up’ improved their top and bottom lines, averaging 40% higher sales and 52% higher operating profits over a three-year period.”

So what has all this got to do with math education? Let’s Explore!

‘I have such a huge curriculum to ‘cover’, how will I have time to do it all?’

The question I hear most often from teachers at professional development sessions that I lead is: “This all sounds great, but it takes so long to use manipulatives, have students explain their thinking and dig deeper into a concept; I have such a huge curriculum to ‘cover’, how will I have time to do it all?”

This is a really good question and a totally valid concern. I have often felt that same panic; it’s February and I’ve ‘covered’ less than half (or even a quarter) of the curriculum – it’s a terrible feeling! So, my answer to this BIG question is multi-faceted. I’ll unpack it in this blog.

Slow Down to Speed Up

Just like what I learned about organizations, I’ve found the exact same phenomenon with students. When teachers slow down and teach concepts, especially foundation concepts conceptually, visually and in contexts, students develop a much deeper, long-lasting understanding. This ends up saving time later on in the year when they are learning new concepts that are connected to existing concepts (because all of math is a web of connections and relationships).

I’ll give you a couple of examples. The first was when I was teaching grade 8 math. I taught integers in October and usually (when I used to teach in procedures) I would have to re-teach (not even review, actually do it all again) this concept again in May before I could start my Algebra unit. BUT, when I taught the students conceptually, I didn’t even have to review it in May and because the Algebra Tiles works so similarly to the integer chips, they figured out how to use them and to connect to integers immediately, thus saving me a lot of time. I also noticed this effect on every concept I taught.

A more recent example is when I was visiting Terrace last May and had the opportunity to teach in Lisa’s class (see my blog post on this) for 3 lessons (over 3 days). Her grade 4 students had been taught conceptually all year and had just finished fractions. When I came in and introduced decimals, the whole class understood them within those three days! They were able to compare, build with blocks, add and some were even subtracting with them already! Because they saw all the connections, they immediately understood that hundredths are smaller than tenths because the denominator is so much larger, therefore the whole is split into many small pieces. I saw more understanding in 3 days with these students than I often see in grade 6’s!

I’m guessing that regardless of the grade you teach, you are going back and re-teaching many of the concepts from the previous several grades. That was always my experience…until we started working as a whole school and changing our practice to become more conceptual. Then, the students were retaining their previous learning much, much more.
I truly believe that once we are all teaching conceptually we will see a large shift in these time pressures (side note: I see this shift is happening now as we are being offered the required training and resources to make these changes in our practice). Imagine if you didn’t have to re-teach all those concepts you do, but instead only had to review quickly and build on them? You would have the time to teach all of the concepts in your curriculum. In fact, I would predict that once students have a firm conceptual understanding they will be able to build on this at a much quicker rate than teaching ever could.

Covering’ ≠ Learning

You’ll notice that throughout this I am using quotes for ‘covering’ the curriculum. I do that to highlight that covering procedurally does not equal learning nor understanding conceptually. There are several decades of research that have shown us that teaching procedurally doesn’t lead to understanding the math nor the ability to problem solve.

I’ve seen teachers blast through the curriculum quickly when they are teaching procedurally. The problem is that the students don’t understand the vast majority of it conceptually and therefore don’t retain it and so the next year, it’s as though they had only covered 10% of the curriculum.

‘I have yet to find an administrator who doesn’t support this approach to teaching students well instead of quickly.‘

I’ve taught numerous students who can memorize and reproduce any procedure you give them and therefore scored well on procedural tests, yet when I sit with them and do an assessment, they don’t even have basic number sense, nor much understanding of place value, fractions, decimals or anything more than basic counting principles. These are the unfortunate many who end up hitting the “math wall” at some point, usually in grade 10 or 11.

My philosophy here is that it’s better to teach conceptually half the curriculum and have students retain 80% than to ‘cover’ all of it and have them retain 10-20% of it. In fact, if you do the math it’s about twice as effective

Covering the curriculum for the sake of covering it doesn’t do our students any favours. Furthermore, I have yet to find an administrator who doesn’t support this approach to teaching students well instead of quickly.

The last part of my response is to take another close look at the curriculum and decide where you can combine concepts and where you can teach the concepts in other subject areas, thus freeing up some more time in math class to really dig deeper into number concepts an patterns. Numeracy is present and relevant in all subject areas, just as literacy is.

Unfortunately, due to our culture, we often don’t see it or know it is present. Furthermore, few parents do daily numeracy with their kids in the same way they do literacy, so our students are a lot less experienced with seeing math everywhere. The great news is that we can support parents in this by sending home ‘homework’ that helps to build number sense and recognize the math all around us (that’s going to be my next blog post!).

Here are some examples to help you get started:

Teach data analysis in social studies and science or during a morning routine where you ‘check in’ with your students. You could pose a question such as: “If your mood could be described as weather, what would the weather report be?” and, as a class, you could generate 3-5 responses such as ‘sunny, cloudy, rainy, sunny with some clouds, blizzard’ etc. Then you write these on the board and the students put their name under the heading that best describes them. Once this is done, you have a set of data to analyze! You could draw bar graphs or pictograms to show your results (or line graphs or circle graphs). You could talk about the most common and least and how many times more than/less than they are. You could do fractions, decimals, percentages! AND you are allowing them to express how they are feeling and communicating this with their class. Win-win.

 

You could also insert geometry, symmetry, translations into art (see my last blog on art in math).

 

Probability can be taught with games and even when you’re reading with your class. We ask students to make predictions about what will happen in their story (or piece of informational text), you could go a step further and ask “is that very likely, somewhat likely or not likely? Why do you think that?”

You can combine teaching multiplication with area by using area models and arrays. You can use Cuisenaire rods and a ruler (see our lesson for this) to teach multiplication and division to combine with measurement. You’d be astounded at how many students don’t know how to use a ruler properly!

You can also combine teaching about perimeter with adding and subtracting (including decimals for older students) Insert your financial literacy into your decimal lessons regularly. For younger students, insert your financial literacy into adding and subtracting (maybe even multiplication and division). Use financial problems as your contexts for practicing adding and subtracting.

Lastly, talk with your colleagues and let them know what you spent a lot of time on and what you didn’t get to as you transition your students to the next year. At one school all the grade 4 teachers didn’t get to the same concepts and they let their grade 5 teachers know this so those teachers were better prepared for the students coming in. As a teacher who has worked mostly in middle schools, I can tell you that we have students coming from several feeder schools and have a huge range of skills/knowledge but what makes our job so much easier is if they have basic number sense, which means they understand:

1. How numbers can be broken up and put back together

2. Relationships between operations and numbers

3. How to use flexible strategies

I also love it when they can model math and use manipulatives as tools for problem-solving. When I taught high school, I wanted the same thing

My Journey with Aboriginal Education – Including Math!


My Journey with Aboriginal Education

 

I recently spent a couple of weeks teaching several classes and groups of teachers the First Nations game Lahal (Slahal). I’d like to share it with you as it was well received in all of the classrooms (grades 6-8) that I visited and it is a great way to integrate First People’s perspectives and history into math class!

I’d like to share a small piece of my learning journey, with respect to Aboriginal education before I jump into this fun and engaging game. As a non-indigenous teacher, I have been feeling very torn for the past few years because I really want to work towards learning and teaching the truth of Canada’s history and also be a part of reconciliation. I am torn because I’ve never been very sure about what my specific role should and shouldn’t be. I was afraid of appropriating, misinforming and to be honest didn’t feel like I knew enough to authentically teach about First People’s cultures and world views.

I’ve been extremely fortunate to have some great mentors in the past few years that have helped me tremendously, not only in gaining more insight and understanding of our shared history but also how I can be an ally and educator of these important concepts.

I’d like to acknowledge those who have helped me along the way; Nella Nelson, Paola Bell, Craig Shellenberg, and Sarah Rhude, all of whom are part of the ANED team at school district 61 (we are very lucky!). I’d also like to raise my hands to Eddy Charlie and Kristin Spray who have inspired me and shared so much of their experiences and teachings with me. Eddy and Kristin started Orange Shirt Day Victoria and have presented at Cedar Hill school twice. They have introduced me to other inspirational people such as Phyllis Jack-Webstad and Bear Horne. It is because of these people that I have become passionate about working towards a much needed change in education. I strongly believe it is all of our responsibility to be properly educated about the Indian Act, residential schools, the 60’s scoop and the long lasting effects of these atrocities on First Nations cultures, communities, families and individuals. I know very little at present but the more I learn, the more passionate I become.

For the past year and a half I’ve been trying to authentically integrate more First People’s content and perspectives of learning into math lessons. I’ve been stymied for much of that time. However, I’ve learned a few things: most importantly, I’ve learned that teaching math in the way that I do (and how my lesson plans are structured on Educating Now) is aligned with First People’s principles of learning. When we allow for students to make meaning, draw, talk about their ideas, construct their own ways of understanding and doing math, we are honouring them and their learning process. If you look at the First People’s principles of learning you will see that most of them promote teaching practices that are best for all learners (in other words, they are simply ‘good teaching’). Furthermore, when we change the WAY we teach we will make far more of an impact on improving student achievement than focusing on content.

Lahal Game

So, if you are in a similar position, where you
are wondering how on earth to integrate aboriginal education into your
math class as it is prescribed by our curriculum, I encourage you to
start by engaging in the math curricular competencies (the way in which
we are teaching math).

As for integrating content authentically,
I’ve done a few things thus far. I partnered with a colleague who was
doing a native plants unit with his students and we looked at some of
the math involved such as: area and perimeter of the park, measuring and
charting growth of Camus, finding approximate ratios of native to
invasive species in the park. Next, I took a great resource: Star
Navigation, which is based in Alaska, and created a local version that
incorporated: angles, ratios, navigation, and the solar system aimed at
grade 6 (their Science curriculum involves the solar system so there was
some nice cross curricular connections). My most recent lesson sequence
was Lahal. I feel like the Lahal lesson is easily taught and can be
used (with adaptations) at grades 4-8. It is also one of the suggestions
in our curriculum (in the elaborations).

If you are a teacher in the Greater Victoria School District, there are
3 kits that can be borrowed from the ANED resource center (they include
tapes with the music so you’ll need a tape player!). Because this game
was also played by semi-nomadic tribes, pre-made kits are not necessary.
You will need 5 sticks per team (tally or score keeping sticks) and
then two sets of ‘bones’. Popsicle sticks could be used for this. We are
going to make some Lahal kits at our school and will likely use some
small dowel pieces for the bones and then we will sand and paint our
tally sticks out of scrap cedar. Students could easily go outside to
gather their sticks but you’ll want to plan ahead for the bones or you
could use rocks and just put tape around two of them (ensure they are
small enough to fit into students’ hands).

 

How Do I Assess This New Math Curriculum?


This is probably the most frequent question I get about mid-way through a professional development session on teaching math conceptually.

Not only are we tasked with teaching math differently, but also with assessing math differently. Did I mention we first have to actually learn the math differently? Yikes! Don’t panic. We are here to support you – Educating Now’s videos help you, the educator, to first understand math conceptually. I do want to point out that you were likely never taught math this way, nor were you instructed on how to teach math conceptually, so be patient with yourself as you learn math anew. Our videos also provide you with lesson plans that incorporate many of the curricular competencies so that you are teaching the new curriculum.

Last year I wrote a detailed blog on formative assessment, which you may want to read because this is the most VALUABLE type of assessment in terms of how it affects achievement and learning.

And I am going to delve a bit deeper into Assessment of learning in the next blog. So this post is really dedicated to summative assessment. We will answer the questions:

1. How do I know what to assess?

2. How often do I assess?

In the next post we will explore:

1. How are my students involved in the process (besides being test takers)?

2. How do I assign a letter grade to the assessment?

Let’s start with: How do I know what to assess?

This used to be the easy part right? If the chapter was on square numbers and square roots then we used to give something that looked like this:

The problem with this test is that it is mostly procedural and if I hadn’t taught this concept visually, the word problems would be very difficult for students to solve. The other problem is that even if a student gets all the questions correct, I don’t know what they actually understand, but rather I only know what they can do. These are not the same thing. Students can perform operations without any understanding at all. This happens all too often in fact, and then they forget the procedure shortly after the test is done as they’ve moved on to memorizing a new procedure. Also, with respect to the curriculum, all that we are assessing here is the content area of square and square roots (knowing) and a little of the doing (with the word problems), and again we are not testing for real understanding.

If we want a chance of having this concept retained then we need to teach the concept (not just the procedure), use visuals, use contexts, and use words to make sure they fully understand the concept and then practice all of these, including the procedures.

 

 

Here is an example of an assessment that would encapsulate all of the above suggestions:

 

 

Can you see the significant difference in the information about your students’ understanding you will gather between these two assessments? Can you also see all of the competencies we are assessing here (visualizing, explaining, justifying, demonstrating understanding pictorially, solving problems, etc.)?

How Often do I Assess?

The short answer is: all the time! Most of my assessments are formative, as this drives my instruction. Assessment should be so embedded into your teaching that the two are almost indistinguishable.

Below is an example of formative assessment: if I were teaching squares and square roots, I would ask students to use individual whiteboards to write on as I gave them pieces of the puzzle, they would give me the other pieces. This is how it happened in a grade 8 class recently. I wrote either the words, symbols or pictures on the board and they wrote the

other 2 pieces on their whiteboards. So, when I wrote: three squared or the square of three, they showed me: 32 =9 &

This formative assessment in my lessons tells me where my students are at with their understanding. This informs what I will do in the next lesson for example:

1. Do I need to use another model, like a number line or multiplication table to help them to understand this further?

2. Do I need to allow for some students to continue developing understanding while challenging some others to solve problems involving these concepts because they have demonstrated a firm grasp of the concept so far?

Formative assessment is NOT GRADED nor is it “for marks”. It is for you and for the students as a way of identifying how close they are to the learning targets or goals.

In terms of how often do I do a summative assessment, that totally depends on the formative assessment, the concept and the students in front of me. Although formative assessment happens every day throughout the entire lesson, a summative assessment ONLY happens when the learning is mastered and the students are now demonstrating their understanding.

I allow re-tests for this reason. I believe it is unrealistic to expect all students to have fully mastered a concept by 10:00 am on Thursday (or whenever your test is scheduled for). Our goal is not to compare them with each other at any given time, it is to help them fully understand the concepts and this will not happen at the same time for all students. Thus re-tests are there to allow for students to continue persevering and working towards full mastery. So, in terms of how often do you assess – there is no structured answer like ‘every week’. Unit tests or concept tests will happen after lots of formative assessments and opportunities for learning and will depend on the concept. I do them before we move onto a new concept.

One thing I see A LOT, especially in high school math classes, are quizzes each day or once a week. These are awesome if you are using them for FORMATIVE assessment but if students are getting marked on these and this is becoming part of their grade I would ask: “where is the opportunity to think, learn and really understand the concepts?”. Too often math becomes a performance subject where students are always being graded and so don’t engage in the thinking we want because they will be penalized for mistakes and students respond by just memorizing so that they can perform well on their quizzes. The same applies for marking homework – this sends a very clear fixed mindset message that mistakes will be penalized and there is no room or time for actually developing understanding (which happens through making mistakes).

Dylan William gives a nice example of how we often assess (especially in math when it comes to unit or chapter tests) in this 2-minute video:

https://www.youtube.com/watch?v=sYdVe5O7KBE

 

When rethinking your assessments, I encourage you to do the following:

1. Start by looking at the curriculum and decide upon the content, competencies and big ideas you are focusing on for the assessment and create questions that incorporate those competencies and big ideas (where applicable)

2. Ask yourself: is this going to show me what they know, understand and can do with this concept?

3. Include visuals and written explanations

4. Ask yourself if I am promoting a message that I have high expectations for all students and that the goal of the assessment is to determine the depth and breadth of their understanding?

5. Ensure you have provided plenty of opportunities for students to develop a deep understanding of this concept.

6. Is this formative or summative?

 

Next post I will include some rubrics and suggestions for engaging students in the process of assessment (including some assessment as learning) as well as discuss how to put a grade on assessments using these new-fangled assessments.

Check Out What Lisa is Doing in Her Classroom!


I can’t believe it’s June!

 

This year has just flown by. I’m sure you’re all feeling just as ready as I am for summer break! I wanted to share this amazing experience I had in a classroom in Terrace BC earlier this month.

I was invited to Lisa Pushong’s class to team teach. 2 years ago Lisa was teaching math in a very common way, here is how Lisa describes it:

“I taught it the way I was taught in school. I stood up in front of the class and told students how to solve problems and then gave them worksheets so they could do exactly as I had shown them. My students had no conceptual understanding of what they were doing or why they were doing it. I truly had no connection, emotion or feeling when teaching math, and of course, this rubbed off on my students. They followed my rules and procedures and we all just went through the motions completing worksheet after worksheet.”

 

Lisa came to a workshop I presented in Terrace 2 years ago and that workshop started her on a journey to transform her math class. She dove right in and began to immediately see changes in her students and so kept trying new things.

“While listening to Nikki describe how we are doing an injustice to our students by only teaching them rules and procedures I literally wanted the floor to open up and take me away. Everything she was describing was how I taught math. At that moment I knew that there had to be big changes in the way I was teaching math. When I left the workshop that day I really didn’t realize the impact it had had on me. That Monday I started on my journey to becoming a better math teacher, and haven’t looked back.”

Lisa says that now her math classes couldn’t be more different from how they were 2 years ago:

“It’s hard to describe my feeling towards teaching math now. Its excitement, curiosity, the desire to always learn more and challenge myself, and the best part is that my students all feel the same way. You can feel it during our math classes, math is no longer a boring subject where we all sit at our desks silently completing worksheets with all the rules and procedures we have been taught. Math class is dynamic and multifaceted, it’s challenging and rewarding, everything it never used to be.”

I experienced Lisa’s classroom first hand and it was amazing!

She learned how to use manipulatives to explore each concept by using Educating Now videos and reading books. She focuses on the big ideas and concepts rather than on how to do procedures and the magic happens – Lisa has transformed her math class and I was blown away by what I witnessed. She has also been surprised by what happened as she discovered that teaching math has become “a passion that I have not felt in my teaching career”, which is also what happened to me when I started teaching this way (and the passion grows!).

Here is what I experienced in her classroom:

1.) Can do attitude towards math – These kids just kept on going no matter how many times they made mistakes. They puzzled and questioned and begged me for more challenging work. Because they weren’t plagued by self-doubt or frustration, they were so much better able to learn.

2.) They listened to each other and responded to each other as a true learning community. They said things like “I agree with Kate because….”, or “I saw it differently than Cole because….”. Active listening was a part of how they operated. It felt respectful and just plain awesome.

3.) They wanted to figure it out because they knew they could. This is a growth mindset. They didn’t give up and say (as I often hear) “can you just tell us how to do it”. I was introducing them to decimals for the first time and they wanted to figure them out all on their own…and they did!

4.) Because they had such strong conceptual understanding of fractions, they picked up the concept of decimals very quickly by relating decimals to fractions. Learning the big ideas and really understanding them visually (and being able to explain them) helps students to learn new concepts.

5.) So, how does Lisa do it? How did she create such a wonderful class? I wish I knew all the details because I really think she should teach a Master Class on it! What I do know is this:

6.) She teaches and models a ‘can do/growth mindset each day. This means she doesn’t allow students to take the easy road by giving them answers that they can figure out. She gives them deep thinking tasks, rather than procedure focused worksheets. She provides feedback and nurtures their growth mindsets.

7.) She uses manipulatives and structured talk. A lot. This is how she teaches math. Students know the WHY as much as they know HOW and WHEN to apply it to solve problems. They can talk about their understanding, show it and teach it to their peers.

8.) She models for them a genuine wonder and love of math and that creates an entire class filled with kids who LOVE math. Just as students pick up on and, sadly, mimic a teacher’s dislike, discomfort or anxiety for math (there’s some interesting research on this here), they also mimic when they see and feel the passion of an educator who loves math. Here’s the really cool thing….this love of math education is quite new for See below some examples of building 2.37 in many ways and naming them properly.

Lisa shared some insight that many of you might feel as well:

 

“Of course there have been struggles along the way. When I first started on this journey, I really had no clue what I was doing. Using manipulatives was intimidating in the beginning, they were loud, kids touched them all the time and I wasn’t really sure what to do with them. But it got easier and now I can’t picture math without them. I think that once you can overcome the feeling of not knowing all the answers you will truly see the beauty that teaching math this way has to offer.”

I also want to note that in the 3 days that I was teaching in this class, students only wrote down a few things. Most of the lessons were introducing the concept of decimals and so we focused on language and visuals (see the short video). Then we paired the language and visuals with symbols. I mention this only because one of the things I see often is a rush to procedures and symbols in math. Understanding decimals doesn’t mean knowing how to write them, but rather knowing what they represent. Take a look at some of the pictures and listen to the video to see what we were doing (we used the flats as 1 whole).

 

Students found many ways to show 1.37. On day 3 we used grids instead of blocks to represent decimal numbers.

 

 

 

You can transform your classroom just like Lisa has – Educating Now makes it easy for you! All you need to do is:

  1. Watch the video for your next lesson
  2. Print the lesson plan and try it in your classroom.

Even experienced teachers find this useful as it reminds them of the important language and activities they can do to help their students really learn. Sign up now for lifetime access to Educating Now. The current price for Educating Now is a great deal and will be rising sometime in the next 2 months – so sign up today and transform your class starting in September!

Finally, thanks to Lisa for opening her classroom to me and for inspiring me with her teaching practice and passion!

Basic Number Sense Part 3: One and Two More/Less and Rounding


This is the third blog post in a series about building number sense foundation.

 

I want to ensure that you have read the previous two as this post assumes you understand why you need to go ALL the way back to where students currently are to build that solid foundation as well it assumes that you have worked on subitizing, benchmarking to 5 & 10 and partitioning.

As with the other number sense basics, this skill may seem really obvious to you, but as with the other basics, I see a whole lot of students who don’t automatically know the relationships between numbers in this way. For students who already have this skill, you can extend this to finding 10 or 100 more/less (looking for similarities and differences between finding one/two more and less and finding 10 or 20 more/less). For example, if a student is working on explaining that 10 less than 54 is 44 but says that they take 1 away from the 5, then they are not using proper place value language and so are likely not thinking in place value but rather in digits. To build number sense (not digit sense) we need students to be very aware of place value.

Just like with all of the other concepts visuals are key here. It is really valuable for a student to literally see on a 10 frame that 7 is two greater than 5 or that 5 is 2 less than 7 (students need more practice with LESS than because it is used less often in daily life). When they are just figuring these relationships out the visuals along with the numerals will help them to make the needed connections. Number paths or number lines can also be used but eventually, we want them to know these facts and so we need to give them plenty of time to practice.

 

 

To give students a chance to practice this skill you can give them dice (I like to use different sided dice to differentiate – see the picture below for examples of different sided dice). Then students roll their die and say what 1 more and 1 less of the number is, before passing it to their partner. The partner ensures that it is correct before they have their turn. You can do the same activity for 2 more and less once they’ve mastered 1 more and less. If you spent some time with the 10 frames when working with subitizing and again when working with partitioning then they should be pretty familiar with how numbers relate to 5 and 10 so now we are just going to be more explicit about it. You can ask questions for 8 like: “What is 1 more? What is 2 more? What is 1 less? What is 2 less? How do you know? Is it closer to 5 or 10, how do you know?” For those students who are still counting up or back by 1 or 2 when asked to do this, they should have a 10 frame so that they can start seeing the numbers and use more of their subitizing strategies rather than relying on counting. Number lines are a great visual too.

 

 

For those students that need more of a challenge, you can give the larger sided dice and ask them to discuss the place value connections they are making. For example: if a student knows without counting that 8 is 2 less than 10 but then when they roll 30 and they don’t know that 28 is 2 less without counting you can use 10 frames to show them the connection between 8 and 10 as well as 28 and 30 (you can print filled in and partially filled in 10 frames that are great to use with larger numbers here from Carole Fullerton – she also has Canadian money that can be used as an another visual to work on the same skill:

https://mindfull.files.wordpress.com/2006/11/part-ten-frames-pink.pdf

https://mindfull.files.wordpress.com/2006/11/full-ten-frames-blue.pdf

 

 

This concept is the same when we add or subtract 10 or 100 to a number. We want to be explicit about the place value rather than using tricks like add one to the tens place. We want to actually ensure they understand that they are adding 10 or 100, not 1.

Rounding is one of those concepts that we often teach by telling students “look to the immediate right of the place value you are rounding and if it’s 5 or greater roundup and if it’s less than 5 round down”. Unfortunately, this doesn’t help them to understand what it means to round AND this rule only works in base 10. Considering that in our Canadian money, we now round to the nearest nickel (base 5), this method does not work at all.

Instead of this rule, use manipulatives or number lines to show that what we are really doing when rounding is figuring out what number we are closest to. It is just that simple. For example:

Round 437 to the nearest 10. Let’s look at this on a number line:

 

 

It is obvious when we see it that 437 is much closer to 440. So, what happens with 435? Because it is the exact middle it is an important number to explore and discuss. We can let our students know that by convention we round UP when we find ourselves at this midpoint but, in reality, it isn’t close to either number because it is in middle. For all other numbers, it is better to think about it as ‘what is it closer to’.

This same concept applies for decimal and fractional numbers too:

 

 

We can also use base 10 blocks to help us round in terms of ‘what is it closest to?’

You can give your students some tasks such as:

Round 1 497 to:

 

a) the nearest thousand _____________

b) the nearest hundred ______________

c) the nearest ten ____________

d) the nearest five ___________

 

I hope you found these 3 blog posts useful. Although it is challenging to go so far back when we have such a range of abilities and skill levels in our classes, the students who are missing these skills need them to build a strong foundation. The students who already have these skills can be engaged in other forms of problem-solving, logic puzzles (solving puzzles and playing games are curricular competencies). There are great problems at this site: www.nrich.maths.org that will challenge your ‘at grade level’ learners while you spend some time on these basics. Remember that these activities can also be sent home as parents can help to support these basics too.

Here are some freebies for you to help you along with working on these skills. I want to write blogs and vlogs that will help you, please send me topics you would like to see posts about or questions you have or add them to the comments below.