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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 mission is for teachers to feel great about their impact on student learning. I make it easy for teachers to prepare and deliver lessons that will change lives.

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