Sequencing

Sequencing comes into play in multiple areas of math, so it’s important to make sure which aspect of sequencing you are thinking about or learning about.

• Counting to 100 and back to 1 by ones
• Naming Days of the Week, Months, Continents, etc.
• Skip Counting and Coin Sequences
• Multi-Step Procedures (like long division)
• Following Multi-step Directions
• Solving Word Problems

To keep it really interesting, the reasons why sequencing is difficult can vary! I’ll address the reasons with each aspect of sequencing.

COUNTING UP

Let’s start with counting to 100 by 1’s. This is a great example of a math task that is conceptually fairly straightforward, as long as students have 1-to-1 correspondence, but can be very difficult to execute despite that understanding. This is one of those skills that can have a deep impact on learning. If you have trouble counting up, learning addition can be very challenging.

A student who is struggling with counting might sound something like this:

“One, two, three, um, one, two, three, four, five, um, six, seven, um” Starts building up some momentum, “One, two, three, four, five, six, seven, eight, nine, ten, umm…”

Expressive Language: A student who is having difficulty counting because of expressive language issues will benefit from hearing someone else count with them while using objects to count. For students  who are having a particularly difficult time with producing the names of numbers, I will write the names on base-ten blocks to help support the process of learning to count.

Word Retrieval: A student might know the counting sequence well and all the names for the numbers, but when they go to grab the word, it’s not there. You know that feeling when a word is on the tip of your tongue, but you can’t quite find it? Imagine that happening every day, throughout the day. Assuming the student can read, writing out the words for the numbers can help support this student. You could write them out on a list off to the side, or list them in order. Practicing naming the numbers out of sequence will also help.

Auditory Memory: Remembering the days of the week, counting, multiplication, and the seasons can be difficult due to weak auditory memory. This also shows up in writing, when a student can only remember part of a sentence that they are trying to write. Auditory memory can be improved simply by practicing sequences, recalling sentences from memory (start at a manageable level and then make it longer over time), and by listening to simple tunes and reproducing them on a piano. If this seems to be exceptionally hard, I usually recommend that parents consult with an audiologist.

Attention: If a student is losing their place in the sequence because they start talking about something else, or pause and lose track, give them something physical to count. The physical stimulation is usually enough to keep their attention on the task. Adding in a metronome can also help. You set the speed to match the rate at which the student is counting.

Processing Speed: For student who can count to 100 but slowly and laboriously, I use a number sheet in rainbow colors and have them alternate from naming the colors to naming the number, and then combine that with a metronome at a pace that matches their counting. I’ll sometimes add in a wiggle board as well, so they have to keep track of the rhythm while balancing. This engages many senses at once and makes it so challenging that once you take away the wiggle board and the metronome, counting seems easier!

Working Memory: If a student has trouble switching between the tens place, “18, 19, . . . .20” and then zips through the 20’s until “29……30!” then I encourage them to practice counting by tens, and then focusing in on “19, 20. 29, 30. 39, 40…” until it becomes more automatic. This is not something I’ll spend a lot of time on unless there is such a significant delay that they start to derail completely.

COUNTING DOWN

Once a student can easily count up to 100 and understands how the numbers repeat at each new set of ten, I have them practice counting down from 100 to 1. That’s a significantly harder task!

I scaffold this process by making sure they can easily count down from 10 to 1, then 20 to one. Then 100 to 0 by tens. And then we combine it all. Some students need extra support at first. Sometimes I hold my fingers up to keep track of the 10’s, or I give them a quiet verbal cue for the next lower ten. I have worked with some students who consistently need to figure out the lower ten before  starting with the nine. For instance, they might say “…94, 93, 92, 91, 90, 80, 89, 88, 87…” – as long as I know they are just cueing themselves, I chalk that up to an effective strategy.

Number Sense

The biggest challenge to understand why number sense matters, and isn’t just something that kids have or don’t, is that if your focus on math instruction is to simply get to the correct answer, then it makes no sense to teach with a focus on number sense. It takes time, there’s a lot of exploring, and it’s not about finding efficient ways to solve problems but rather creative ways to solve problems. That being said, I think it makes no sense to teach math strictly from a procedural perspective. But I didn’t always think that way!

When I first started tutoring math in college, I was a firm believer that kids were just missing some content and if they just had the missing skills filled in, they’d be set. I was very successful in traditional mathematics education despite having weak number sense. I could memorize procedures easily, was always one of the first people to finish the timed drills, and even passed AP Calculus in high school by mimicking answers – with absolutely no clue how it all fit together or what it meant. In college, I saw some of the teacher education courses where they were working with manipulatives and I remember thinking I would never have studied math if I’d had to use those. Once my classes got rid of the numbers and turned to pure mathematics I was thrilled! But, despite my success in math class, I couldn’t even think about calculations without a pencil in my hand, I learned math by studying the answer keys, and I hadn’t understood a word problem since third grade!

It’s hard for me to believe how much my views on mathematics have changed. Once I started working with kids who struggled in math, I noticed that the issues they were having in “basic” math were similar to the issues I had in “higher-level” math with one major distinction – they thrive on visualizing and building mathematics – while I have to fight with my brain to get it to engage. It took me several years to figure out how to design a lesson to physically build 35. I was floored by the richness of the conversations that the students were having while they worked on building a representation of the problem! The hardest part for them was they kept trying to build the answer.

So what is number sense? Most people agree it’s hard to define and that it involves mental math, flexible thinking with numbers, and estimation. It underlies learning the basic facts, place value, and comparing numbers.

Here’s a great example of thinking flexibly with numbers from youcubed.org. This approach still hurts my brain a little, but that just means I still have room to grow!

18 x 5 from YouCubed on Vimeo.

I am a fan of including numbers sense activities across the grades, and not limit it to K-2 instruction. A popular approach to including numbers sense instruction is using number talks.

Just search online for “number talks” and you’ll find a wealth of information. The basic concept is that students find an answer to a math problem using mental math, and think about other strategies as they wait for their classmates. And then solutions are discussed. If you’d like to dig deeper, here are some great options for books:

Another activity for building number sense is to work with number systems from different cultures. I love to work embed the history of math into class because it helps students see that there are different representations and how much it can change. It’s hard to find history of math books that don’t get bogged down into the math and the history, but here are some fun books on number systems:

It’s important to check a student’s understanding of place value because it is one of the best examples of how number sense can impact understanding. For instance, I will ask a student to show me 534 blocks with base ten blocks. Usually, they can mimic the answer of what it should look like with the blocks. Then I rearrange the order of the blocks to 3 tens sticks, 4 cubes, and 5 five hundreds sheets. The ones who struggle with number sense will simply answer that there are 345 blocks. There’s no real connection between the actual number of blocks, the representation of place value using the blocks, or preservation of number. I set those blocks aside, have them show me 345 blocks and then compare them to 534 blocks. Most students immediately catch what they have done, but I have worked with several students who have to go through the process a few times.

Another example is when students are regrouping numbers for addition and subraction, I will ask them what the 1’s digit is worth wherever it appears in their problem. Invariably, student who struggle with number sense, despite using base ten blocks to learn regrouping, will answer that it’s worth 1 regardless of its place value.

One of the things that becomes obvious about teaching number sense, is that you can’t rely on written answers to check understanding. You have to hear what students are thinking. And, bonus – talking about math with peers and teachers helps to clarify understanding and increase retention of the material.

Estimation is another example of a number sense activity. Estimation180 is full of great resource to integrate activities into your daily routine. Estimation is a skill that is used outside of the classroom and often comes more naturally to students if they think about what they would do in the real world instead of in math class. And it encourages critical thinking, helping students learn to think about get the gist of an answer rather than relying on exact information. It leads you into questioning if an answer makes sense, rather than just blindly following a procedure. And we certainly need more of that type of thinking!

Math and Movement

There is an unfortunate tendency to think of math as something that happens in the brain, but it is deeply embedded in the body. I’m not recommending movement just because it gets the blood pumping or is fun for kids who struggle in math, but because the body has ways of knowing that are not tapped into in traditional math classrooms. And, don’t even get me started on the importance of unstructured play to develop critical thinking skills. That will have to wait!

In the meantime, I will rant about the importance of movement. I recently discovered Dr. John Ratey’s work, who “has embarked on a world-wide mission to re-engineer schools, corporations, and individual lifestyle practices by incorporating exercise to achieve peak performance and optimum mental health.”

I’ve also had a chance to talk with several OT’s and learned how many issues with learning are rooted in the body.  Did you know we have more than five senses?! The vestibular system helps with balance and movement. Proprioception, unlike the other six senses, doesn’t take in any information from the outside world, but relies on information from the joints, muscles, and other parts of the body to figure out things like how hard to grip your pencil. And they both explain why I keep walking into walls!

Here are a few programs if you are interested in going further down the rabbit hole:

• Rhythm of Math
• Bal-A-Vis-X
• Interactive Metronome
• Brain Gym

Rhythm of Math

I haven’t had a chance to try Rhythm of Math out yet, but I am so thrilled to discover it exists. I’ll be sure to report back when we have a chance to pilot this program. I focus a lot on tactile learning (fine motor/sensations on skin) but not as much on kinesthetic (moving large muscle groups) so I’ve been looking for a way to add in more movement.

From the website, “The Rhythm of Math engages students in learning and applying essential mathematical concepts, while performing, studying, and composing rhythms. Rhythm Blocks are a technique that is easy to learn, even for teachers and students with little or no music experience. They have certain mathematical qualities that make them ideal for learning properties of natural numbers, flexible ways of conceiving of multiplication and fractions, division, ratios, proportions, and measurement.”

Bal-A-Vis-X

I was completely floored by the impact of Bal-A-Vis-X on our classroom culture at My City School. It requires students to attune to each other and pay attention to visual, auditory, and movement cues. While I have no idea if it has any direct impact on learning content, my impression is that it has increased our students’ ability to actively participate in group lessons. As far as I can tell, there is no well established research on this program, but there is a case study.

Interactive Metronome

I have been through the first level of training in Interactive Metronome, and it is tough!  I have had several students go through the program with an OT and have seen significant improvements. There is a lot of research out there on IM. For me the question to ask is not “Does it work?” but rather “Who benefits the most?”  I try to be efficient both with time and money when I make recommendations. It’s unclear to me if it is a good choice for a general ed classroom teacher to use or if a smaller SPED class would be a good choice or if it’s best delivered in an OT setting.

Brain Gym

Brain Gym is a fun and easy way to introduce movement into a classroom for brain breaks. So far, research on Brain Gym points to an overstatement of claims and an over-reliance on “anecdotal evidence”. I’ll eventually explain my thoughts on research in general, but for now, here’s a case study. I don’t recommend using Brain Gym as a stand-alone intervention but I do find it useful as a way to get kids moving and to note who has issues with movement. Yoga might serve a similar purpose! Mindful movement, especially set sequences, help with motor planning and brings awareness of your body moving through space.

As always, I’d love to hear about any other movement programs – especially if they have a direct connection to learning mathematics!

Build*Draw*Talk*Explore

Welcome to Transforming Math Class!

Here you will discover a ton of resources, gain a deeper understanding of how students learn mathematics, and get inspired to take a risk or two in your classroom!

“In mathematics, the art of proposing a question must be held of higher value than solving it.” Georg Cantor

Some of the major themes we’ll be exploring:

• Shifting from teaching math to students to exploring math with students
• Creating time in the classroom
• Identifying and supporting a variety of challenges for all of your students
• The cognitive underpinnings of “simple” math tasks
• Identifying your teaching bias and curriculum mishaps
• Cultivating a classroom culture that encourages failure and deep thinking
• Discover why the history of math is more than a side-page in a text book

This will be a non-linear journey, but eventually, there will be a framework available to make it easier to navigate!

Foundational Skills

Here you will find a collection of resources I have on some foundational skills:

• 1-to-1 Correspondence
• Subitizing Patterns and Quantity Sense
• Visual-Spatial Relationships
• Rapid Automatic Naming

You’ll notice these areas may need to be supported by an Occupational Therapist (OT) or Speech and Language Therapist (SLP). For the most part, I don’t reference age ranges because by about 7 or 8 years old, these skills are assumed to be fine in the classroom and aren’t looked at. I have met adults who struggle with these skills. But, if you have a student who is struggling to learn math, it’s worth checking these out. From 4-7 years old, it should be fine to work on developing these skills as long as you aren’t pushing into a place of frustration and anxiety.

I would love to hear any recommendations you have in these areas. Please try to categorize them based on the four areas (or add a category of your own!)

1-to-1 Correspondence

When you count objects one-by-one, in theory, you say a number each time you touch an object. For students who struggle with counting, organization, impulsiveness or tracking, this can present a challenge.

Activities for 1-to-1 Correspondence

For younger kids, I start with larger objects. For seven years and up, I use paper clips because they are a little harder to work with. I put out a pile of objects and watch how they count.

• Is the counting synchronized to the movement? Do they double count because they are counting faster than they can move? Or are they under-counting because they are moving faster than they can count?
• Do they keep the objects organized? Is the miscount due to losing track of what they have and haven’t counted yet?
• Are they randomly counting and moving objects?
• Once they can count smaller and smaller objects, can they count dots? Do they lose track visually? Do they need to tap or make marks with their pencil?
• Do they have the physical and organizational requirements for 1-to-1, but are struggling with the language of counting?

If they are having issues with 1-to-1 correspondence, I have them recount a pile of of objects until they get the same number twice in a row. I usually don’t show them how to improve, they have to work it out for themselves.

One exception to that rule is if their frustration level is getting too high because they are doing their best and just can’t control their motor movements well enough. An OT referral might be the next step.  Or, if the language ability is the issue, then I focus on supporting language development. More on that later!

Subitizing Patterns and Quantity Sense

At the most basic level, subitizing is hardwired. We can recognize from 1 – 4 objects without counting. That’s why we have special words for first, second, and third – but switch to fourth, fifth, sixth, etc. And why early counting has the words “one, two, many” as the names for numbers. From there, you can leverage subitizing and pattern recognition to develop quantity sense. Students who have trouble memorizing addition and subtraction facts likely struggle with subitizing patterns and quantity sense. In a nutshell, they struggle with taking parts into a whole. They may also struggle with memorizing sight words. Visual memory plays a role in all of this as well, but we’ll get to that down the road.

Back to quantity sense, I have worked with kids who can’t reliably count the fingers on one hand at the age of seven.

Me: Count your fingers on one hand.

Student: “One, two, three, (skip a finger), four”. I have four fingers on one hand.

Me: Awesome. Let’s count again.

Student: “One, two, three (double count a finger), four-five, six”.

Me: How many fingers do you have on one hand?

Student: Six!

It does not phase them in the slightest that they have different numbers of fingers on the same hand. This is my favorite example of issues with quantity sense. A student who has issues counting, but has strong quantity sense, will recognize that they counted wrong. A student with quantity sense issues may have memorized that they have five fingers on their hand, but not care that they counted four and six.

Activities for Subitizing Patterns and Quantity Sense:

There is a great app for the ipad to practice this skill: Subitize Tree

And, I have been working with a lovely developer to design a game that develops quantity sense using Mayan Numbers. It’s in very early stages, but feel free to take it for a spin: Mayan Numbers Drill or Mayan Numbers Matching

Visual-Spatial Relationships

This is another area that might lead to an OT referral. I am not an OT so I still need to sort out the difference and relationships between visual-perceptual, visual-motor, and visual-spatial, but what I do know is that one of the most easily overlooked fundamental areas of learning math is this visual realm.

Some of the more obvious ways it shows up is in handwriting and organization on the page, Cartesian graphing, and copying shapes. If you want to check this one, have a student copy a dot design and see what they see. And ask them to draw a square on grid paper. This might not happen to everyone, but most of my students will draw something like a 5 x 6 “square”. I ask them how they know it’s a square and they tell me the sides are the same length. I ask them to count the the length of their sides, and they discover the error. Obviously, any of us could misdraw a square. I am sharing this to illustrate how someone can understand a definition but not be able to demonstrate the meaning.

Activities for Visual-Spatial Relationships

This pinterest has some great examples of what your student might find challenging. I use this types of images as an assessment and as an activity. I don’t just look to see if a student can do it, but what strategies they use, how much erasing is needed, and how frustrating the activity is.

Here is a massive collection of activities from Your Therapy Source. I have not used this particular resource, but it is an OT’s collection. You can make example activities if you are trying to figure out which one will benefit your student before purchasing. The overall look is pretty young, so if you are working with an older student, it might not be the best choice.

I also love to use Critical Thinking Company’s Building Thinking Skills. I don’t follow the grade-level recommendations. I use the first few pages of each section as an assessment. And, for a language boost, where it says to write information, you can have students give answers verbally instead. Also, I do have to provide more foundational work for students who can’t complete the activities as written in the book.

Rapid Automatic Naming

Rapid Naming is the ability to look at an object or number and call it by name without hesitation. Difficulty with rapid naming contributes to issues with learning to count and vocabulary in math, and contributes to issues with reading.

If you like to read research, here’s a study that showed “… counting and RAN were powerful predictors of arithmetic and reading fluency.”

Activities for Rapid Automatic Naming

I love BrainFlex 4 -8 years old and 9-15 years old . It is written by an SLP and has a variety of activities for reading and math.

I have developed a series of activities to work on RAN and processing speed but need to put together the directions before making them available. Let me know if you’re interested and I’ll speed things up!