Teaching Numeracy, Critical Habits 3 & 4

If you’re just joining us, we’re reading and discussing Teaching Numeracy, 9 Critical Habits to Ignite Mathematical Thinking, by Margie Pearse and K. M. Walton.

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Use the links below to read previous posts.  Last week’s conversations were lively and thought-provoking!

Reading Schedule

Habit 3: Identify Similarities and Differences, Recognize Patterns, Organize and Categorize Ideas, Investigate Analogies and Metaphors

“Math is a science of patterns; it is much more than arithmetic”

I have a little secret for you…I never set out to be a math teacher. When I left the business world to pursue teaching, I thought for sure that I would be a reading teacher, because I loved reading (still do…). I interviewed at a well-respected elementary school in Houston ISD that was also very close to my home, and I really wanted a job there! The principal and I hit it off, and before I knew it, she was offering me a job…as a 3rd-grade math/science teacher. Wait, what? I smiled, graciously accepted the job, and the rest is history. Why do I tell that story? Because I think it is directly connected to the quote I used to introduce this section. When I was in school, I was taught arithmetic, and I did not find it at all interesting. It was only when I began teaching math, and studying how to teach math, that I saw the beauty in the patterns and became hopelessly hooked! For example, follow the evolution of a simple number bond and see how the concept of decomposing numbers connects so many different skills–skills we typically teach in isolation.

Referring to work by Marzano, Pickering, and Pollock (2001), Pearse lists four highly effective forms of identifying similarities and differences: comparing, classifying, creating metaphors, and creating analogies. I’d love to hear comments about how each of those looks in your classroom. I think I regularly incorporate comparing and classifying activities, but I’m not so sure about metaphors and analogies.

I had the pleasure of hearing Jana Hazecamp present a session on building mathematical vocabulary last week, and she shared an activity that is both simple and powerful.  Perfect for this chapter!

Habit 4: Represent Mathematics Nonlinguistically

“Nonlinguistic representations are necessary to students’ understanding of mathematical concepts and relationships.  Representations allow students to convey mathematical approaches, arguments, and conceptual understandings.”

With my love of the concrete, representational, abstract (CRA) sequence of instruction, this chapter was music to my ears! I heard a lot about CRA this past week at the National Conference on Singapore Math Strategies. They call it CPA, with the P standing for pictorial, and it’s at the heart of how math is taught in Singapore.

The research on nonlinguistic representations is compelling, and I think we’ve all experienced what happens when students are rushed to the abstract stage of learning. My experience with struggling students is that they have no pictures in their heads for the math they are trying to do, which fits perfectly with the authors’ statement that “For every symbol that students write, there must be a concrete referent in their heads of what that symbol refers to.” (p 38) I think this gets back to the problem of focusing on quantity over quality in our classrooms. I’d rather have students work on one great problem–build it, draw it, talk about it–than ten routine problems.

So many great, practical ideas in this chapter! I particularly like the list of Good Questions to Ask When Incorporating Visual Representations (p 42). How about taping a copy of that in your lesson plan book as a reference for planning?

Finally, what a great list of math read alouds (p 49-52). I’ve always used literature to engage students, but I didn’t think about how powerful read alouds are for helping students visualize. You might want to check out this blog post for additional online resources for literature for teaching math.

I’m looking forward to another lively discussion about this week’s reading!

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  1. The quote at the beginning of the blog made a connection with me too. I’m always telling students that our brains crave patterns. It’s always on the look out for a pattern. And first graders find patterns everywhere they can. They see patterns in everything! I think expanding simple patterns from earlier grades to more mathematical patterns like skip counting, repeated addition, and telling time helps numbers make sense to kids.

    I love the references to Marzano throughout the book. Our district has been big on having us use the different Marzano strategies. Often I have to really think about how to apply the strategies to first grade. For me, math can be more difficult than reading. This book is helping me see ways I can apply them. One example that came to mind was how the relationship between addition and subtraction works for creating analogies and fits perfect with common core. The relationship can be overlooked by teachers who are more concerned with rote math like hopping on the number line.

    One question that I would love to get feedback on has to do with visualization. As I was reading, I thought my strength is having kids work with concrete models. I have a more difficult time differentiating for those who already have a good grasp of visualization. Those kids who can look at a word problem and know the answer because they see the picture in their head. How can I help them to keep growing and thinking deeper while I pull out the counters, place value mats, linkers, and all the other things that the non-visual ones still need?

    1. Sue, I had the amazing opportunity last week of spending a day and a half in sessions with Yeap Ban Har, a master Singapore math teacher and professional developer. Phenomenal! I’d like to share a couple of thoughts that might answer your question.

      First, he stressed that students should always be recording their work, he called it documenting their thinking, in their math journals. It’s a part of pretty much every math lesson for them. So their anchor task uses concrete materials, but they also represent it by drawing and writing.

      One of the techniques he modeled that really stuck with me was how he differentiates instruction for his students. It goes something like this: “Students, we have solved this problem 5 different ways. Record two ways that make the most sense to you in your math journal. If you’re fast, record more ways. If you’re really fast, come up with another way.” Loved it!

      All teachers face the issue of students who don’t think they need to show their work. Last year, we really stressed the Standards for Mathematical Practices as the habits that mathematicians use when doing their work. We referred to students as mathematicians whenever possible, and reminded them that mathematicians communicate their thinking in words and pictures. It’s a more purposeful way to say “show your work.”

      So your kiddos will be growing as mathematical communicators when they document their work in their math journals!

    2. I often have kids record their two or more ways to solve the problem in their math journals. I started doing it a few years ago and it makes a huge difference in how well they listen as other kids are explaining their thinking. My students are now very invested and interested in hearing other kids’ strategies.

      I would love to go to a Singapore math conference.

      The Math Maniac

    3. Donna I would love your thoughts on math journals. Do you start these with kids in kindergarten? What does a math journal look like in different grades? Habit 4 definitely had me envisioning using the journal to have kids draw a math story that they are visualizing as I read it. With the kids at this age just learning to write, I ma not sure if I would get much from them in terms of writing about their strategies, but maybe this is where the drawing and nonlinguistic representations take shape. Thoughts? Advice?

    4. There is a great blog kindergartenkindergarten.com that does a wonderful job of showing math journals at work in Kindergarten. It hasn’t been updated in nearly a year but it really got me going on using math journals with younger kids.

      The Math Maniac

    5. Karen, yes, math journals should start in Kindergarten. Of course it would be lots of pictures at that point. But, for example, students could draw a picture of two of something and write the numeral 2. As your working on composing and decomposing numbers, students could draw all of the combinations for different numbers. Lots of uses!

      Tara, thanks for sharing that website! Always nice to hear ideas from a classroom teacher who is living it!

  2. I enjoyed these two chapters as they reinforce the style of teaching I enjoy. A few ways that I promote patterns in the classroom is with number lines. I like to use Marilyn Burns “Do the Math” strategies. She uses number lines to show students the patterns for multiplication. The repeated groupings shown in arrays are easy to tie to number lines. For example, 2 x 3 can be shown as jumping from 0 to 3 and then from 3 to 6. Addition and subtraction can also be shown on a number line. Even division makes sense on a number line.

    We also have a bulletin board that displays Singapore Bar Model problem solving. The consistent format for the students creates a picture in their heads to allow them to understand what is happening in the various situations. We have found that about 80% of problems can fit into the bar model format. Using this format has also helped students break down a problem. The bad habit of pulling the numbers and applying an algorithm is decreasing and accurate answers are increasing. I see this as a sign that number sense is increasing.


    1. Ellen, I couldn’t agree more about bar model drawing! It’s such a powerful strategy for helping students visualize their thinking, and it really helps them attack unfamiliar problem types. It definitely takes practice, and there’s a pretty steep learning curve for teachers. We implemented model drawing in grades 1-5 last year, and I’m excited to see this group of kiddos coming up. I have always used model drawing in my classroom, but not bar model drawing. There’s something to be said for kids coming up from one grade to the next and already having that tool in their tool box.

    2. I am completely unfamiliar with bar model drawing. Can you give me some insight? Should I google Singapore math, or is there another site you recommend?

    3. Karen, I would google “bar model drawing” and you’ll find lots of information. You might want to check out Thinking Blocks. It’s an online site for bar model drawing. One thing that is very cool is that it walks the students (and teachers!) through the process, so you can see what it looks like. If you are in Kinder, thought, bar model drawing doesn’t start until late 1st grade. As with everything else, it starts very concrete–lining up counters in a line that looks like a bar.

  3. Wow! Habit 3 has got me all fired up! I love the quote “If identifying similarities and differences is basic to human thought and boosts student achievement, why, then, are we still content to settle for number crunching and formula writing as the dominant form of instruction?” This quote along with your story about being taught arithmetic and not being all that interested in it got me thinking about my own journey in learning math. My first attempt at leaving a comment got me way off on a tangent and I ended up writing an entire blog post about why math should be about more than number crunching and formula writing. Here goes attempt #2 and hopefully I can focus on the book this time!

    My big take away from habit 3 is the idea of creating metaphors to deepen thinking. I have never done anything like this with students but see the potential it would have. It made me think of my sixth grade class from this past school year and how much fun they would have had with things like this. I love the idea of the teacher starting this and having the kids create their own as the school year progresses. I need to start thinking of metaphors and writing them down because this is so new to me!

    I love the quotes from habit 4 about making sure kids do not have only procedural knowledge and superficial understanding. “It is not okay to simply teach procedures!” That is one I want to scream from the rooftops! I am so glad you are doing this book study Donna! I think it should be required reading for teachers of math! I love that it really applies to all grade levels. I am definitely adding the authors to my list of great math teachers that I want to meet someday!

    The Math Maniac

    1. You make a great point, Tara. Research clearly indicates best practices for teaching math, yet the required paradigm shift is huge. So thankful that the Internet gives us the opportunity to “gather” and share ideas online!

  4. Gosh, these conversations are so inspiring! I love to read your thoughtful responses and find myself visiting the blog for a daily pick-me-up. How sweet is that!

    As a math professor, I teach students from different K-12 backgrounds and abilities. From this experience, I have learned more about what is most important in cultivating deeper mathematical thinkers. Now, if I could go back and change one thing about my middle school teaching days, it would be to really emphasize the relationships, comparisons and patterns between mathematical concepts.

    My college students who can recognize similarities, differences, patterns, and trends are at a great advantage because they are able to transfer the relationships they recognize and use them more practically across the board. For example, those who can understand the patterns in comparing 1/7, 1/2, 1/4, 1/9, and 1/12 are better able to transfer that knowledge when analyzing annuities. Unfortunately, if I had a dollar for every sophomore who told me that “1/12 is bigger than 1/2 because 12 is bigger than 2”, I’d be able to treat my family to Disney World.

    I have found that using questions to promote comparison thinking is critical to transferring what is known about one mathematical concept and seeing how it applies to something else.
    I find myself constantly asking students questions like,

    “What do you notice about these numbers, strategies, concepts, etc.? What is similar about them? What is different? What patterns do you recognize? How can you show those patterns using pictures? How do those patterns work in a different setting? How does identifying these patterns help make math easier for you?”

    1. Hi Margie! I am so glad you are participating in this book study! I really love your last question, “How does identifying these patterns help make math easier for you?” I think that is a super motivating question for students. Recognizing patterns and making connections makes math much easier and much more fun! Getting students to see this is a big part of our job!

      The Math Maniac

    2. Agreed, Tara! I remember one time talking to a class about reading a thermometer. I turned it sideways, and when they realized it was just like reading a number line you’d have thought I’d done some magic! Ha ha. Those connections are priceless and make math much more enjoyable. Who doesn’t like working puzzles?

  5. A big piece for me from the reading this week was all about mental images. I have been studying this a lot recently, and these mental images are critical to all forms of comprehension. The key time in life to be building mental images is in the early years, based on lots and lots of world experiences. Then as kids come to encounter symbols, they have the image to call back on.

    This is easy for me to understand in terms of vocabulary and language, but thinking about it in terms of math is really stretching me.

    So, let me know if I am thinking about this in the way you understood it. Here is one example: a child is given a story problem, and they need to be able to visualize the components in the story and how they relate (adding, subtracting, etc). Another example: we are discussing sizes of different animals, and the child needs to have a mental image of the difference between an ounce and a lb, or a cm and a meter, to mentally compare the animals. Third example, I ask a child if they would prefer 1/2 a cookie or 1/8 of a cookie, and they need to have a mental image of the size of the fraction?

    Do these examples convey what we are talking about? And if so, then the best way to build the mental images is through lots and lots of exploration of concrete manipulatives. Am I understanding it correctly?

    I also really appreciated the list of read alouds, and especially that they were aligned with certain math topics. One of my favorite math read alouds of all time is “Snakes, Long, Longer, and Longest.” It has really brought measurement to life for my students, and I am yet to have a group of students that is not absolutely enthralled with the book!

    1. Yes, Karen, you’ve got the right idea about visualization and how to move through the concrete, representational, abstract sequence of instruction. A key point is that this is not really tied to age–it’s essential to go through this process whenever a new concept is being taught. Another important point is that the stages should overlap. So students are working with concrete materials first. Then they would work with concrete materials and draw representations, so the concrete and pictorial are overlapping. Then add in the abstract (symbols). Thinking in terms of fractions–first lots of concrete practice, then they’re using manipulatives and drawing pictures, finally add in the symbolic notation along with either pictorial or concrete. Finally, and I think where the process often goes awry, you can’t rush through the stages. It can’t be, “Oh, we’ve used manipulatives for 2 days, now it’s time for abstract practice.”

  6. First off let me just say that as I keep reading these chapters I keep thinking to myself, how am I going to do this with my kindergarteners, then boom, there are the early elementary ideas. Thank you so much for including those in the book! All of those ideas are so simple and so doable!

    I really am looking for a way to make math more meaningful at a kindergarten level. I am not a fan of the kindergarten workbook that comes with the math series we use, but since that is what our district wants us to use, I do. Most of the time I tell the students that even though they have already done the problem for us, we are going to double check by doing it ourselves to make sure that they got the correct answer. Then we use the manipulatives and do the problem.

    I loved the list of books to use as read alouds for math. I can say that is one of the good points of the math program we use, it usually lists a book to read that goes with that lesson or unit.
    Oh man do I relate with this statement, “It is not okay to simply teach procedures and hope students will ‘pick up on it later,’ that ‘it will suddenly make sense later.” First off as soon as I read this statement it took me back to my junior year in HS when I was enrolled in Algebra II/Trig. This was most likely the worst math experience I ever had. The teacher of the class basically taught to the smart kids, the sophomores in the class, when you’d ask a question he would look at you like why don’t you get this. He never could explain it so that the rest of us could understand.

    I never have understood why after a certain grade level (maybe that has changed now) the some teachers stop using manipulatives. As I read through these math books I always think if I had been allowed to use manipulatives through my years in school I would have a better understanding of certain math concepts.

    1. Interesting you mention the textbook. When I was attending Year Ban Har’s session at the conference last week, he kept referring to the fact that “this is a lesson from the textbook.” The lessons were so deep and rich, that I was having a hard time picturing how they would be presented in a textbook. I raised my hand and asked how the textbook is used in Singapore, and he said they only open the book after the anchor lesson. So the textbook reinforces what they’ve just done in the hands-on lesson. It’s used for guided practice and independent practice.

      Maybe that’s something you could do. Look at the lesson in the textbook and ask yourself how you would teach it with the book closed.

    2. I like this idea. Teach it with the book closed. Definitely going to try this and encourage others to give it a try, too!!

  7. I have enjoyed reading everyone’s responses. I learn so much and think differently about what I read and have found myself saying, “Why didn’t I think of that?” In Chapter 3, I think comparison thinking really hits at a higher level of understanding and sometimes a way of thinking that makes students feel uncomfortable. Yet, it is so important. I have used analogies with my students in math. Sometimes as a bell ringer or as a launch for the day. At first, it takes time for them to “get” the format. I really like how it forces students to look at things differently sometimes. Once students become more proficient at analogies, I have them start to create their own…or should I say attempt to create their own :)! I would love to hear some examples of a few other math metaphors if anyone has an idea. I thought of…how is area like a puzzle? But was hoping to branch out from the concept of area. The idea of a Working Word Wall (35) was music to my ears. Too often word walls get put up and forgotten about. I like to view a word wall as a living math word wall that is updated, active, and revisited. In this way, students can manipulate the words, see relationships between the words, and own them!

    In Chapter 4, a powerful statement for me was, “Representations provide visual confirmation of the learning (37).” I tell my students the only way I know what they are thinking is by what they tell me and show me…I can’t get into their brains. In the past, my students have looked at architectural pictures whether from a book, online, or around school and identified geometric features. On page 44, I really like the idea of asking students to defend why each polygon is placed where it appears; is it for aesthetics or for structure. Hm…definitely something to think about! There are so MANY wonderful ideas here. Can’t wait to use some of them when school begins in fall.

    1. Another neat way to have students interact with the word wall is to play a mystery word game. Give students clues about a word on the wall and let them try to figure out which is your mystery word. Such a quick little activity–great for those spare minutes that need filling.

    2. I love your thinking! In fact, I am teaching a new class this semester on math methods in PreK-2. That’s a little new for me, so I find myself printing out all of your wonderful responses and I put this blog first in my new syllabus as the go-to place for fabulous ideas! Thank you!

      I can’t wait to try all the worthwhile ideas around a working word wall. I also love to use Janet Allen’s Concept Circles (a circle split into fourths) as an exit ticket. I just pick three of the vocab words and leave the fourth sector blank and ask teams to brainstorm another word that would connect with the three and explain why. Sometimes I include all four words and ask students to explain what is the connection and then challenge them to create a heading that would make sense and why. Then, other times, I leave the circle (with 4 sectors) blank and challenge students to choose four concepts from the word wall that would help summarize the day’s lesson and have them use these words in a short GIST statement or summary. I like to use formative assessments that are somewhat more open-ended. I think it pushes the thinking the bit. Would love your thoughts!

    3. I love that idea, Margie! I’m so glad we’ve got all these great ideas captured and can go back to them in the fall.

  8. My comments for this week include pictures to help make my point so I can’t post it all in here. But if you want to read it, I posted it to my blog here: http://www.therecoveringtraditionalist.com/stop-using-base-10-blocks-teach-algorithms/

    It is title “Stop Using Base 10 Blocks to ‘Teach’ the Algorithms.” I discuss how we need to let students make sense of the manipulatives for themselves and let them decide how to use them to solve problems.

    1. Awesome blog post! In addition to the great math you shared, I love the fact that you cited research and also featured technology that teachers can use to support their lessons.

    2. This is fantastic! My blood literally boils when teachers uses base 10 blocks to teach the algorthim and then call it hands on math and good teaching practices!

      The Math Maniac

  9. I love your blog post too! I am trying to find the video clip I’d like to show it to my student teachers. Is there an easy way to find it?


  10. I finally got my copy of the book, Teaching Numeracy…. and have begun my reading. I think the authors did a great job under Habit 3 (pg 31) Key Ideas where they talk about identifying patterns and parallel relationships. Teachers need to highlight to their students that mathematics is a science of patterns. Creating learning experiences where students are allowed to work thru recognizing and discussing patterns as they work thru solving real life word problems is essential.
    Early on in the school year, an activity I do with my students ( I am a math specialist who works with students in Grades K-4), are lessons on identifying patterns . I give each pair of students a baggie of color tiles and ask them to create a pattern .Then, they are to use color pencils and recreate the pattern on graph paper (cm paper for Grades 3-4 and 1″ graph paper for Grades K, 1 & 2).
    Then, I allow time for each pair to share their patterns, while I pay close attention and write on the board key vocabulary used during the each pair of students’ discussions.
    This activity allows students the opportunity to explore creating patterns using color tiles. After much discussion, I then have them write their own concluding statements about their patterns in their math journals. We then have each pair share as we discuss their journal entries. An example maybe…”There was 1 blue square, then 1 red square, then 1 blue and then 1 red.” “I can also write the pattern like this…”
    1 2 3 4 5 6 7 8 9 10
    B R B R B R B R B R.
    “All the odd color tiles are Blue and the even color tiles are Red”. The second graders may then say, “I think the 12th tile will be Red and the 19th tile will be Blue”.
    Using color tiles with this activity, although seems like a simple activity that most elementary teachers have used, I shifted the focus of the lesson first by allowing the students to create their own patterns, then having the students recreate their color tile patterns on graph paper, then share with the class with the teacher facilitating the discussions on patterns while highlighting key vocabulary on the board and then finally having each student write their own concluding statements provides a rich learning experience. Jeanine Brizendine, Math Specialist

    1. Great lesson, Jeanine! You definitely elevated this activity by having the students describe their patterns in different ways and make generalizations.

  11. Proof of habit 4: A teacher was in my classroom standing in front of my desk talking to me about one of our students. As we were talking she was looking above me at a math poster that had multiplication representations on it. Her eyes got bigger and her face showed amazement. “I get it, I get it!” she exclaimed. She told me thank you and that she had never understood multiplication, but now she did. A simple picture was all she needed.

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