## Sometimes a Visual Model is all you Need!

After I read the task prompt I had a feeling that it had to do with the dimensions of the new cube but as soon as I sat at my table and drew some visual models I was like bingo! I took it back to the my colleague and showed her the picture and she was like yes, that makes sense. Here is a pic of the model I drew, with some added notes for this blog to explain what I was thinking.

The model on the top was the first one I drew  to represent what the students had did by visually showing the 240 gram cube times by 3. Immediately you can see that only one dimension of the new shape is 6 cm and it is also not a cube. The model on the bottom I drew next to help visualize what the 6x6x6 cube would look like with the smaller 2x2x2 cube inside it.

How can this help students?

I think using this misconception could be very powerful for students to see if they take the time and use a visual model to represent their thinking it can help them see that their current thinking on how to solve the problem is not on the right track. They could also use the model to check their solution for reasonableness when they are finished. Maybe we wouldn’t have seen as many solutions with this misconception if it hadn’t been in the relay part of the day but I have often seen students make this type of mistake. Drawing a visual model is a great problem solving strategy and is sometimes overlooked similarly to using a concrete model with manipulatives as being for younger students. It is not and is always an effective way to make sense of the math!

## 7 Key pages from the Parrish Number Talks Resource

I have recently been modelling a lot of number talks for new teachers in my schools. We often debrief on the strategies, key ideas and models that were used. Still there are seven key pages in the resource that I always direct teachers too that I think sometimes get overlooked. These pages can be very helpful to get teachers familiar with the structure of running a number talk and ways to get started.

These seven pages start on page 25 and end on page 31 and are part of Chapter 2: How to Prepare for Number Talks. The pages focus on how to keep students accountable and how to start with 5 small steps. See the pic below:

I am not going to describe each of the six ways to keep kids accountable or the five small steps for starting as I am sure you can read them. I will though list them and highlight a few that I think really stand out. Here we go!

Six Ways to Develop Accountability with Students

1. Ask students to use finger signals to indicate the most efficient strategy.
2. Keep records of problems posed and the corresponding student strategies.
3. Hold small-group number talks throughout each week.
4. Create and post strategy charts.
5. Require students to solve an exit problem using the discussed strategies.
6. Give a weekly computation assessment.

In her book each of these six ways has a description detailing what each one means, some are more self evident than others. I  personally think number 2, 3 and 1 are super effective.

Number 1 allows you to get formative assessment on the spot about whom has clear idea about which strategies are more efficient. For example a problem like 28 + 39 has been done and the strategies on the board are: counting on, counting on from the larger or splitting/place value. You then ask the students to show 1 finger for counting on, 2 fingers for counting on from the larger number and 3 fingers for splitting. If the majority of your students hold up 1 finger for counting on as being the most efficient strategy then you quickly see you have some challenges!

Number 2 is often one that is overlooked in many classrooms. It is also the reason why I don’t like doing the recording and modelling of the number talk on the smartboard. Often when it is used the previous thinking and prompt are not visible once the new prompt in the string is put up. It takes away from the idea that the previous thinking can be used to help solve the new prompt and  connect ideas. Leaving the student thinking up during the talk and then creating the strategy chart is key. I like using old school chart paper or VNPS whiteboards.

Number 3 is crucial to help get to those students that may be struggling with moving through the strategy development. Having a chance to do a number talk in a small group with them may be all they need to really move forward. It also gives students who may not share as much with the whole group a chance to share their thinking and have a math discussion with other students.

I believe all six are important but those three really stand out in my opinion.

Five Small Steps to getting Started

1. Start with smaller problems to elicit thinking from multiple perspectives.
2. Be prepared to offer a strategy from a previous student.
3. It is alright to put another student’s strategy on the back burner.
4. As a rule, limit your number talks to five to fifteen minutes.
5. Be patient with yourself and your students as you incorporate number talks into your regular math time.

As with the last list I believe all these are important but I want to focus on 1, 2  and 3.

Number 1 is important to get people involved and to build confidence in sharing their strategies with the class. It’s also a way to differentiate the number talks, I sometimes add a smaller numbered problem into the string to give more students an opportunity to have a strategy even if their development may be closer to the beginning of the continuum of operational development. There is almost always a way to connect it to the rest of the string. Even if you are focusing the rest of the string on more challenging problems this a good way to start. For example if you are working on developing the key idea of the distributive property and you are ready to move on to help students leverage it when working with 2 digit numbers by 1 digit you could create a string like this:

2 x 5

5 x 5

10 x 5

12 x 5

Most students can enter in on the first 3 problems with strategies that are inefficient but it still allows them to share. They may make the connection from the earlier problems to 12 x 5 and begin to use the partial product strategy with 2 digit by 1 digit or they may need more time but they at least have a way to enter in to the number talk.

Number 2 I think is sometimes done but with the teacher jumping in to say how they may solve a problem. As she warns in the book, that is sometimes taken by the students to mean that is the way you want it done! Just shifting this slightly by saying you are modelling another students strategy from another class or year can shift the students mindset and start opening up the sharing of their own strategies.

Number 3 is one I often see teachers struggle with and one I still struggle with myself. When students start explaining their strategies sometimes it is very hard to decipher what they are saying and even harder to model it on the board. Don’t be afraid to put a strategy on the back burner or as Parrish says in her book politely tell the student you will meet them later in class to work through their strategy. This still honors their work but doesn’t slow the class conversation down.

I love revisiting resources and mining nuggets of information out that I may have not looked at as in depth the first few times through. There is so much information in the Number Talks book that it takes many visits to get to all of it. I hope these little nuggets may be useful to you!

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## “Mr. Stamp the deck you asked us to construct is actually 6 squared km!”

I have been working in Mr. Sontag’s grade 7/8 room for the last few days wrapping up a performance task we had the students complete that had them choose an authentic house floor plan and then create a quote for the cost to floor the house using materials that they had to research online. It covered a variety of concepts:

• calculating the area of a rectangle, triangle, trapezoid and circle by choosing and applying the correct formula
• make metric conversions (and metric to imperial conversions)
• decompose irregular shapes into simpler shapes so the are can be calculated easier
• Estimate costs of materials based on prices you see at different online stores to make comparisons without having to calculate each example you find
• Use mental math calculation strategies when estimating material costs.
• multiply and divide with whole numbers and decimals
• recognize unit rate prices and solve unit rates to calculate cost of materials
• reason and prove that your quote is reasonable, a good product and decent price etc.

The focus of the assessment of learning was on their understanding of area, applying and calculating using the the correct formula, making metric conversions and the math process of reasoning and proving. Mr. Sontag was using the information gathered on the other concepts as diagnostic assessment for further learning, while also seeing assets students may have in those areas or possible gaps in learning.

When I cam into class yesterday one of the students said, “Mr.Stamp why did you make the deck we had to construct onto our house plan be 6 squared kilometres?” Mr. Sontag and I had already went over the student work and had noticed this misconception so it didn’t surprise me that this question came up! Below is picture of  the solution from the student who asked the initial question.

The prompt for this part of the task was this: Construct a deck to add onto your floor plan that has an area of about 600 000 squared cm. Then calculate the cost for the floor of the deck into your quote.

You can see in the picture above that this group divided by 100 to get 6000 squared metres. This was a common misconception. Below you can see in this picture of another groups work they did something similar but divided by 1000 and got 600 square metres.

I took this picture after they had used their feedback to figure out their misconception as you can see they did. You can still see the erased part of their original answer which was 600 squared metres. Only two groups didn’t make this conversion error so we knew we needed to maybe do a mini lesson/guided group. Two groups were able to figure out why they had made this error but the rest needed to see the mini lesson.

What is the misconception?

The students who are dividing by 100 are viewing the conversion as a linear measurement conversion instead of an area measurement conversion which is squared. They knew that 1 m is equal to 100 cm so they divided 600 000 squared cm by 100. This was the most common error in the class regarding this conversion.

What did we do about it?

We started the mini lesson with this think-pair-share prompt:

After a solid discussion most still felt that there wasn’t a misconception and that the first option was correct. They couldn’t see why it wouldn’t be divided by 100 since 1m was equal to 100 cm. Then we moved on to this:

We had the students look at the model we built in class and asked these questions: What do you think the big square and the little square represent? (You can see the little 1×1 cm tile above the big square in the pic on the right.) After some think time they were able to say the big one is a square metre and the little one is square centimetre. Then I moved the square centimetre tile into the bottom corner of the square metre and asked: How many of these squared centimetre tiles what it take to fill up the square metre? I must say we had some wild estimates but it didn’t take long for some hands to go up and say 10 000. I asked, “How did you get that?” and one student explained that they used the rectangle area formula and multiplied 100 x 100 because the base and the height would each be 100 cm because 1m is equal to 100 cm.

At this point I think we could see some light bulbs starting to turn on. We now asked these two questions: Now that we know that 1 square metre is equal to 10 000 squared centimetres. How does that change what we are looking for when we asked you to convert 100 000 square cm into square metres? Why can’t we divide 600 000 by 100?

We got some quality responses:

• One student said it changes what we looking for in my mind because I see now we are looking for how many 10 000’s are in 600 000.
• Another student then added in that they think we can’t divide by 100 because we are not looking for a length measurement but an area measurement. So it is squared.
• Just like we saw how many 1 cm by 1 cm tiles fit into the bigger square metre model this would be looking at how many of those big square metres would be in 600 000 square centimetres. And to do that it makes sense to think about how many square centimeters is equal to one square metre and that is 10 000.

Do we feel all the students understand this now, no we don’t but more certainly do. There is more work to be done here with some of these students to help them consolidate this idea. I suggested some more small guided groups or possibly some one on one time.

If you really dig in here this really comes under the umbrella of the big idea of unitizing. Seeing a group of, as a unit. Making any conversions really relies on having built a solid understanding of unitizing. Whether it is seeing 1 m as 100 cm (linear) or in this case 1 square metre as 10 000 square centimeters (area/squared measurement) which is more challenging for students, it comes down to being able to unitize and see the unit you are using and then multiplying/dividing by 10, 100, 1000 etc which is also built on unitizing!

Hope you enjoyed this one!

## Something I never would have done before the Fraction Pathway Action Research Project!

Today I was working with my principal on the next assignment for the RMS Virtual Learning session we are currently doing as a school. Part of the session is  solving some problems individually and then sharing with our colleagues how we solved the task. It is in preparation for the next session. The tasks have been focusing on how students develop towards multiplicative thinking and helping us to look at it from an LD learners perspective. This was the task for this part of the session:

Bike-A-Thon Problem:

In a bike-a-thon, cyclists will find

-water stations every two-thirds of a kilometre

-medical stations every three-halves of a kilometre

-bike repair stations every three-fourths of a kilometre

John has reached the first medical station, Alexa is at the second water station, and Sima is at the first bike repair station.

Who is further along the course? By How much?

Here is how I, my principal and one other colleague solved this task:

Mine is the top solution. The reason I wanted to share this blog is because recently on Twitter some educators had been sharing some fraction tasks with student work. People were discussing that they didn’t know what they didn’t know and the fact fractions have so many key ideas and levels of understanding that they were beginning to have their eyes opened to the content knowledge you need to have and understand to teach fractions is effectively.

Well I certainly would have fell in that boat, my schooling left me with a severe lack of understanding of fractions, a hate of fractions may better describe it. You can imagine how confident I felt then when it came to teaching it, which I feel many teachers are in that boat. When I had my math awakening and started researching and learning myself I began to feel more confident with fractions, then I moved into a Instructional leader position for my board. In my second year I had the opportunity to be part of the Fraction Pathway Action Research project that was lead by Dr. Cathy Bruce, Tara Flynn and Shelley Yearly and came to realize that I still didn’t truly understand fractions until I met and learned with this crew! Wow to say I made some leaps in my fraction sense is an understatement. If you haven’t seen their research check it out on edugains.ca under the Fraction Pathway.

This leads me back to the point of this post, when I looked at this question the fraction parts just made sense to me. I didn’t need to convert them to decimals as I was taught to do in school. Even the part that said the medical stations were every three-halves made total sense. I also kept everything on my number line mostly in proper and improper fractions to keep the unit fraction visible. I find it easier now see 4/3 and 6/3 then to convert to mixed numbers. I went straight to a number line which was never in my fraction vocab until I was part of the pathway research. I visualized my partitions and even for the last part subtracted fractions to find how much further John was along the course. It just makes me giddy to see how comfortable I am using fractions now when even 6 years ago I would have said it was a weak spot for me. And if you went back to when I was in school I would have been the student so frustrated with fractions that I wanted to cry and run for the hills.

As you can see both my colleagues went straight to converting the fractions to decimals and they both shared that they were confused with the wording of “every three-halves of a kilometre” in the beginning. They both shared they felt uncomfortable working with the fractions and it was because they felt they struggled with fraction sense in school and feel like they shy away from fractions. Many of our students are also in this boat but I feel many that are being taught using ideas from the Fraction Pathway, for example, focus on unit fractions, use visual models and build their fraction sense are going start to be more confident and hopefully not have to wait until they are in their late thirties to have a fraction awakening like I myself did!

Like so many concepts I have re-learned since I have become a math  geek and started down this math journey to show students that math is awesome and that there are better ways to learn math then the rote procedures and memorization tasks I was forced to do and failed miserably at during my schooling, fractions can me learned in a way that makes total sense!

## The Art of not Stealing the Learning!

There is a very fine line when working with a student or group of students between prompting them to move their learning forward or stealing the learning from them by giving to much info away or doing it for them. I tell teachers I work with that it is an art form and takes a lot of practice. I am by no means a master of doing this or even close to where I want to be in my ability to move students learning forward when prompting. Still I do feel I am getting better at it and am in a position to help people learn to do it better. One of my principals I work with is moving along the journey too, and she would be the first to tell you she use to shut the learning down or steal it from students when she sat and worked with them. She is so proud of herself now when she sits with a group and is able to push thinking forward with timely prompts. She often takes notes and has questions for me when I return to her school about interactions she has has with students. We discuss the moves and prompts she gave and how it could be improved or not. More teachers in my buildings are doing this too which has been awesome.

Here are few points I think are super important for providing prompts to students to push thinking forward and not to steal the learning from them. 1) Know your content well, not just well super well! If you don’t know your math content its hard to give timely feedback. 2) Use the Five Practices to plan your lessons. The anticipation part has you do the math first in as many ways as you can. Plus it has you think of any questions they may have or hints that you could provide to move them forward on the task. 3) Ask questions don’t provide answers. 4) Create hint cards before the task that you could provide students with before they access you for feedback 5) Take the time to sit and listen to groups or individual students while they work. Observe and have conversations. If you don’t do this you will never be able to give timely feedback to push their thinking. I know there are more that could be on that list but its a start.

I have uploaded a video below with a quick example of how easy it can be with a simple prompt to push students thinking to other levels.

In the video I all asked these two students were, “Is there another way to arrange your groups of two in way that is easier to see and count?” I didn’t tell them or show them how to put them into lines of ten or eventually into an array like they did when they had finished. I asked a question and then watched as they did the thinking and came up with the idea to arrange them in lines. They didn’t know they were creating an array, they said lines but still they thought about it and came to see that its easier to see and count them when they form them into lines of ten. My prompt was about organizing the counting piles to make it more efficient to count but in doing so it moved them from using equal group piles of 2 to forming an array but also to realize its more efficient to count my tens not two’s! Here is the video of the final count.

## Do Students really Understand the Operation they are Performing?

I recently read this great article from NCTM called “Capturing Children’s Multiplication and Division Stories: Learn the value of having students create their own stories and pictures to represent number sentences as classroom assessments.” Here is link to the article:

It is definitely worth a read and really inspired me to go out and try it in a class. I asked one of my grade 3 teachers I work with if we could try it in her class and give them a diagnostic assessment to see how well they understood multiplication. In the article the study that was done had them do a story and a drawing for a multiplication prompt and a division prompt. We only had the students complete a multiplication prompt. This class has done some work with multiplying but not much this year so far. I have added the curriculum expectations for grade 2 in Ontario below:

This what students in grade 3 have learned previously in grade 2 about multiplication and division. We also know students develop along a continuum for operations so the students in the class have a wide range of where they could be in their understanding of early multiplication.

Here is the prompt we gave them:

Make up a story and a picture about marbles for this number sentence: 4 x 3=

(We said they could change the topic from marbles if they wanted to.)

Here are some of the student work samples:

As you can see all three of these students knew the answer but were unable to make up a story that describes a multiplication situation. The two that were able to make a story up described an addition situation to arrive at 12.

The first pic shows the students understood it can be repeated addition but was unable to draw a picture that models 4 x 3 or make a story up about the situation. The second pic down knew the answer was 12 but drew an equal group picture that represents 6 x 2 not 4 x 3 and also described a addition situation in their story. The bottom pic was interesting in that this student knew the answer immediately. This student said work is done on times tables at home. This is a great example of students memorizing answers to times tables questions without conceptual understanding. The student may know answers but will struggle to be multiplicative if this is not addressed. The pic shows an addition situation as the story and an addition picture not even an equal groups for the picture.

I wanted to also show this piece of student work because I thought it was neat they tried to model it with a number line as their picture of the situation.

There is a few simple errors but they understood its three equal groups of four. The number line doesn’t start at zero and there are two groups that are 3 not four but they tried! I like that they tried to use the number line.

Overall  this assessment showed a clear lack of understanding of multiplication but was very informative for the teacher, she now knows she has a lot of work to do! I am going to show this too my grade 1 and 2 teachers and continue the conversation that multiplicative thinking needs to be developed simultaneously with addition and subtraction. I feel sometimes there is a thought that it needs to start after addition and subtraction and that it is something looked at more in grade 3. The foundations have to start in FDK to grade 2 with learning to unitize (seeing groups of as one unit) and starting explore what multiplication and division mean. Start counting equal groups, looking at equal group situations and solving basic multiplying/division prompts by direct modelling and fair sharing. Students in these grades don’t need to hear the word “times” or multiplication but that doesn’t mean they can’t start learning it by exploring what I listed above. I like the language Kathy Richardson uses in her book “How Children Learn Number Concepts”. She suggests using:  groups of, piles of,  rows of, stacks of, cups of instead of times.

I plan on trying this in some junior classes too, I suspect we will find some students with very similar misconceptions to this grade 3 class. Try it out in your classes, I think you will be very pleased with the information you find out about your students.

## The Importance of Making 10 and Connecting across Strategies

As I dig deeper in this journey of operational development and students movement towards number sense fluency (proficiency) I wanted to write a post about how important it is for students to develop the Make Ten Strategy. Some people also call it Break Apart and Make Ten or Using Up/Down over Ten, whatever name you choose it is all the same strategy and the power of this strategy for students developing number sense fluency can not be overstated. It is often said Make Ten is the most powerful strategy in addition/subtraction development because of its longevity or how far it carries over into more complex strategies that are used when the problems become multi-digit. This leads me to the second part of the title of this post and that is highlighting that connection across strategy development.

I was asked to come into a grade 6 class recently and model a number talks for the teacher that would help make these connections for her students. The teacher recorded the number talk so now I am going to share it with you! Below is the number talk broke apart into four videos. Videos 1-3 cover the the 3 problems in the string. The fourth video is the quick consolidation we did to connect the Make Ten strategy across the more complex strategies. The string of problems I chose for this number talk  were these:

8 + 7=

28 + 7=

28 + 17=

Before you see the videos here are some pics from Doug Clements book “Learning and Teaching Early Math: The Learning Trajectories Approach” where he speaks to the importance of Make Ten and highlights how it is used in Japan which I found very interesting. The pages are 96 to 99 in his book if you are wanting to read the whole section.

I also like that he emphasises that students need to also see and learn other strategies too, to build fluency but by focusing on Make Ten it will help with later multidigit computations.

Here are the four videos that help show how Make Ten is connected across multiple strategies. I hope it is helpful and like I say about all my posts its not perfect and there may be some mistakes and I am always open for comments and feedback. Thanks!

Video 1- (8+7=)

Video 2- (28+7=)

Video 3- (28+17=)

Video 4- (Consolidation)

## Breaking Down an EQAO Prompt through the Lense of Multiplicative Thinking

Today as my principal and I were having a meeting looking at our EQAO results we came across a great example of a prompt where the importance of being a multiplicative thinker really becomes evident. As we chatted we discussed how it tied in with our focus at the two schools which is really digging in deep and knowing our learners in their operational development from counting to multiplicative thinking. (Note: There is a stage higher than multiplicative thinking and that is exponential thinking) We choose this prompt to look at because it was the question this group of grade 6’s scored the lowest on. Only 18 percent of the students answered it correctly. Here is pic of the prompt:

The prompt is actually coded as a thinking prompt under the strand measurement. Take a minute and break it down into what the students need to know and do in order to solve this task. Then look at the list I made in the meeting with my principal below. Maybe I have missed some and if I have leave me comment. I would love to see other people’s thoughts.

(Note: I only looked at the concepts needed not the problem solving strategies etc.)

The fact that this measurement prompt sits on top of multiple number sense concepts is worthy of another blog post entirely!

Even if I missed some other pieces in my break down of the prompt I wanted to focus on what I have highlighted in yellow. The idea that students need to move to multiplicative thinking in order understand scaling, unit rates and proportions becomes fairly clear in this prompt. If they can’t compare the quantities in a multiplicative way they are going to struggle with this task. I like Kathy Richardson’s explanations, here are some pics from one of her books:

She clearly states that “Children need more than how to get answers to multiplication and division problems. They also have to develop multiplicative thinking, because multiplicative relationships underpin many number-related concepts, such as fractions, percentages, ratio and proportion, similarity, functions and graphs, rates of change and algebra.’  As you can see by the list of concepts in the excerpt from Kathy’s book, multiplicative thinking becomes one of the most important milestones in students math learning. It can stall much of a student’s learning from junior level on if they don’t successfully make the transition.

## Subitizing and Quantity- Important Pieces to Operational Development

I have done a couple video blogs before and they are challenging for me. I am not very comfortable filming myself! This is a short video blog on subitizing and how it fits in with operational development. I try my best to make them easy to understand as sometimes math edu-babble can overwhelm people when they are first learning. Its another reason why I thought seeing it as a video may help. There may be mistakes, still I think I have caught the essence of why subitizing is so important. There are four parts to the videos. I also try to make the connection to purposeful practice. Hope you enjoy!

Part 1

Part 2

Part 3

Part 4

Video | Posted on by | 5 Comments

## Learning Basic Facts in Classrooms Today!

I blogged about this last year with some posts about using guided math groups where the students were learning key math ideas that were helping them to develop basic fact fluency. With so many articles in the news lately about “back to basics” etc. because of the newly released EQAO scores for grade 6 in our province I inspired me to make this my first post of the new year. Most of the articles from the media want to create drama that gets people to click on their article of buy the paper by using terms like “discovery math” and not enough “basics” in the article title to try and highlight that as the problem with our math classrooms. That is another post completely but I want to quickly point out that there has never been a label called “discovery math” in any document that I can find  released for math education in the province of Onatrio. They all promote a balanced math program.

Here are  two pics are from our Effective Guides to Instruction; Volume #1-Foundations of Math Instruction. I wanted to share these two pics because it again shows our Ministry documents promoting a balanced math program with Guided( which includes some direct instruction), Shared and Independent math.

I wanted the focus of this post to highlight that teachers are still teaching basic facts or fact fluency in their classrooms. The difference is many are now doing it in a way that is foreign to most of the classrooms our students parents and myself would have attended. I want to share personal story before I detail some of the differences we now see. I hated math in school, despised it with all my heart, now it is passion and I see it in such a different light. One of many areas I struggled in was memorizing my times tables and simple addition and subtraction facts. I couldn’t learn from flash cards or endless games of “travel around the classroom” that would have seen me shaking and sweating with nervousness as the person who was flying around the class beating everyone moved closer to my desk. First of all my mind doesn’t think fast like that, sometimes I need process time and even if I did know it I would cease up with anxiety. This was majority of my math experiences, memorize this fact, this formula and then watch the teacher do it and then try and repeat it many times even though I knew I didn’t understand it in the first 10 questions! I was never shown other ways to do problems or math concepts and I was certainly never shown that there are strategies, patterns and games that can help you develop auto-recall of your facts. Surprise! Writing math facts out hundreds of times didn’t work for me. Maybe it did work for you and that is great but for many students it didn’t work and even for some that it did work for they maybe didn’t understand what it meant. Here lies the focus of this post, and that is to highlight how we are getting students to develop auto-recall of their basic facts and have a strong number sense at the same time. Let’s make it clear nowhere in our curriculum or in our ministry documents has it said students don’t need to have automaticity of their facts. If there is I have never seen it.

Here are some pics of some of the Ontario Ministry documents that shine a light on the focus of learning basic math facts and computational fluency :

Below is a pic of one of the Effective Guides to Instruction: Volume 5- Teaching Basic Facts and Computations. This whole guide is dedicated to teaching students how to become proficient at basic facts.

This next picture is from our curriculum where it clearly states multiply to 7×7 and divide  up to 49 by 7 by the end of grade 3. Some would argue it doesn’t clearly state memorize but after my post I hope it will be clearer that it means the same. We want students to be able to do this by the end of grade 3. Some teachers think that the second part, up to 9×9 which is for the end of grade 4 probably should be included with the grade 3 expectation and I would agree with that. Why stop at 7×7 when kids are moving to proficiency!

What is the difference between how we learned facts (in most cases, I have heard of some teachers in the past teaching strategies) in the past compared to now?

The simple answer is students are allowed to develop strategies that help them access facts and answer problems until, with enough practise they become automatic. Alex Lawson’s book “What to Look For” has been a valuable asset to help guide teachers to give students the best learning opportunities to move from direct modelling to proficiency. Lawson has an article on edugains where she explains this very clearly and compares it to how most of us were taught. Here is pic of her developmental continuum for moving from direct modelling to proficiency:

Dr. Lawson explains it like this, imagine if all the strategies on the above continuum between counting three times (direct modelling) and proficiency were covered up. That is what most of our past teaching and learning of basic facts would have looked like. We would start in Kindergarten/Grade 1 by answering simple addition and subtraction problems by direct modelling the problem on counters or our fingers and then counting to get the answer. Then we would have quickly moved to trying to memorize the facts through activities like flash cards, worksheets or writing them out multiple times until we hopefully became proficient. We would therefore not learn or be introduced to all these strategies that research now shows kids actual develop through on their way to proficiency. You often talk to many adults who developed many of these strategies on their own because it makes the math easier to do! The same principle applies to multiplication and division. I chose Lawson’s continuum because our board is using it a lot currently. There are other resources that promote this same instructional approach. (Clements, Richardson, Small, Van De Wall are some other good examples). Here is a pic of Lawson’s continuum for multiplication and division:

How do we develop students through these strategies until it becomes automatic for them?

Teachers are using many different instructional strategies for this but two that stand out are number talks and math games. Number talks allow students to share their thinking and strategies with their fellow classmates while seeing different strategies modelled on the board with an appropriate model to support conceptual understanding. Math games which is just as important as the number talks allows the students to practice their strategies and computations in fun way. There  are others but I wanted to highlight these two especially games because in essence it is drill (practice) but in much more  effective setting then say flashcards or drill worksheets.

The last piece of this post is I wanted to share three videos that show one student’s journey along the continuum from counting on to proficiency. The journey goes from beginning of grade 1 to near the end of grade 1. He now knows all his addition/subtraction facts to 20 automatically  and is now applying the strategies to larger number combinations while also beginning to work on the multiplication continuum. Mrs. Kennedy had her students playing games, doing number talks and also working in small guided groups with her on key ideas that help students develop these strategies. Enjoy his journey!

Video #1- October 2016 Beginning of Grade 1

Video #2- January 2017 Middle of Grade 1

Video #3- April 2017 End of Grade 1

Here is pic of a continuum that highlights his journey with each strategy he used along the way.

Hope this helps show that students are most certainly still learning their facts and becoming proficient to the point of auto-retrieval. Maybe this isn’t happening in all classrooms but it is in the schools I work at and I certainly see a lot of like minded educators on twitter teaching this way. It matches what our Ministry of Education in Ontario  and our board promotes. It also works! The journey is different now which opens it up for all kids to learn their facts!

I am not afraid to admit that I did not know all my facts when I became a teacher, I do now! It comes from being introduced to these strategies and just playing with the math! I still have to think about some, for instance 8×7 and 9×7 confuse me sometimes but I now can quickly do 8×5=40 and 8×2=16, put them together and you get 56. I just used the partial product strategy (distributive property) in under 5 seconds to access my fact I was stalled on. Think of the power this gives a student who use to think if they didn’t know the answer to the flash card/worksheet they just moved on and didn’t get it! That was me! Not anymore and I work now to give all students this opportunity. If you were a memorizer and you got it, congratulations but in my experience that just isn’t the case with most students. Teaching facts may have prevented a lot of math phobia that many parents show today. My 75 year old mom shared a story last week with me that highlighted the anxiety and sick feeling she would get when her teacher use to draw a clock on the board, then point to two numbers and pick someone randomly in their seat to answer. If you didn’t get it then you got centered out in front of the whole class. She still remembers it to this day 60 plus years later! That is just another reason why the process we use now makes me so happy. Thanks for reading.