The Mis-education of Comparison Anchor Charts -

In my quest for sample Comparison Anchor Charts to add to our curriculum documents (you know that phrase, "work smarter, not harder"), I must admit I was overcome with sadness when I spent minutes scrolling Google images (and Pinterest) only to find alligator-themed posters over and over again. So I decided it would be my joy to support educators in embracing more sound practices that support the MATH for students.

Although I've never been a fan of the "alligator" comparison symbols (I was a connect-the dots girl- *see below); it doesn't matter...a trick is a trick. A trick is NOT math. So, over the years, I've invested in studying the mathematical research behind the concept of comparing numbers.

Why is this concept - COMPARISON- so difficult for students?

A 5th grader (quite proficient in math) approached me for help because she failed her STAAR test by 1 question. When I poured over her data/answer choices released by the state, I realized she only had minor mechanical/computational errors (chosen answer choices meant to highlight those misconceptions). But then I asked her to tell me the story that data DOESN'T often tell. Her response was,

"I just don't get which way the comparison symbols go when trying to indicate which decimal number is larger."

Stunned that a student who has been comparing since Kindergarten was about to enter Intermediate school with this misconception, I was compelled to be the one who changed "the game" for her! Quickly, I sketched an open number line on a scratch sheet of paper much like the one below- end arrows and all.

        < (less than)                                                     (greater than) >

                                     
I gave her two whole number and asked her to point out where they would go. Then I increased the value of those numbers by 100 and again by 10,000 until I finally asked her what did she notice about the smaller numbers and the larger numbers. To which she responded that numbers with smaller value lean more towards "this end" of the number line and larger numbers towards the other end. I shared with her that the arrows on those respective ends indicate the appropriate comparison symbols.

So 3 < 4 because 4 contains the value of 3 (note to teachers: also called magnitude of numbers and hierarchical inclusion). But more cool than that is that the 3 is close to the "<" arrow. And likewise, 4 > 3 with 4 being closer to the ">" arrow; so essentially I can record both number sentences and be accurate in my statement. Her mind was BLOWN!
You see, the number line isn't a trick, it's a strategy that helps students understand the magnitude of a number in relation to another number.



So if a 5th grader who excelled in math, still misunderstood the comparison symbols is it safe to say that its possible somewhere along her math exploration journey, she learned the symbols (through some cute alligator-like, kid friendly method) or prematurely? Perhaps her exposure to the symbols was justified by the notion that she was "capable" of understanding or "ready" for it.

Can I submit to you, however, that a change is badly needed? 

With that being said, I'd like to share grade appropriate anchor charts with each approach that include research based strategies and TEKS-aligned pedagogy. *Keep in mind anchor charts are built WITH your students, so these are pictorial BIG ideas to help guide your lesson approach so when its closure time and your students are brain dumping from their lessons and station exploration, the results should look similar to these pictures.

Kinder and 1st grade teachers: (Concrete) Provide your kiddos with a colossal number of opportunities to build two numbers (use (FUN) counters, linking cubes, ten frames, rekenreks, etc) and sentence stems to orally express which is greater and why. Don't forget to model sharing "which is fewer/less" as well.

"..the word less proves to be more difficult for children than the word more. To help children with the concept of less, frequently pair it with more and make a conscious effort to as "Which is less?" as well as "Which is more" questions."
~Teaching Student Centered Math K-2
(John Van DeWalle)

BOY Kinder (to 10) Day 1 example

BOY Kinder (to 10) Day 2 example

BOY Kinder (to 10) Day 3 example

                               
 

BOY Kinder (to 10) Day 4 example

BOY Kinder (to 10) Day 5 example with both comparison statements paired

So you'll see how each anchor chart "pairs" comparison sentences as an example of a conscious effort on a teachers' part to pull this out of students through exploration.


1st and 2nd grade teachers: (Concrete/Pictorial) Provide your kiddos with an appropriate number of opportunities to build two numbers (use ten frames, beaded number lines, linking cubes, base ten blocks, etc) and sentence frames coupled with symbols to express which is greater and justify why using number lines.

*BOY 1st Grade Whole Numbers (to 20)

2nd and 3rd grade teachers: (Pictorial/Abstract) Provide your kiddos with a model of how to use number lines (annotated and open) as a pictorial model followed by expanded form (and notation) to compare numbers as a more abstract scaffold.

         

2nd Grade Whole Numbers (to 1,200)

3rd Grade Fractions (with like numerators)

4th and 5th grade teachers: (Pictorial/Abstract) Your approach shouldn't differ much from 2nd and 3rd, but starting over with a concrete approach as you bring in decimal models (via money models, base ten blocks, grids, and number lines) as well as fraction models (towers and tiles) would be ideal.

4th (and part of 5th) grade decimals

As one of my favorite bloggers and practitioners (Math Coach's Corner- Donna Boucher) would say, "There are no swamps (and thus no alligators) in Math!"

Let's ensure our practices reflect the MATH rather than the TRICKS!

Nix the Tricks Free eBook: Download the book here

Will or Skill?

For years, this whole notion of "Will over Skill" has been a controversial topic of discussion in the education sector. I remember listening in on my very first conversation around this matter- those locked into the discussion were passionately sharing their justifications behind their points of view.

One argued for the issue of will over skill, by stating that hiring teachers who have the will to learn is much more profitable in the long run than holding out to wait for a teacher who has the skill and may not be available for hire at the time of need. After all, due to the will of the teacher readily available, he or she might end up being more loyal (less likely to leave) and mold-able to the point of having the same level of skill as the skillful teacher, in years to come!

The other argued for the issue of skill over will, by refuting those statements with their own ideals. Their vantage point was predicated on the premise that a skillful teacher will not only bring their experience and knowledge to the table, but also the same (if not more) "will" of the will-only teacher. Not only that, the skill-grounded teacher would produce greater results quicker than the will-based teacher.

While I can see the validity in both arguments; and while it's quite possible that I, at one point, may have benefited from being hired because of my will more so than my skill (right out of college, and when transitioning from the classroom to an instructional leadership position), I have a hard time saying that one approach is the only way to go.

For example, every teacher who has been given leadership experience, and received their Admin certification deserves a chance to at some point be chosen as an AP. Not every interviewed AP needs to have had previous experience as such. But even in a "will over skill" situation like this, it's poignant to understand that such a candidate should be acknowledged for the volume of leadership experience (thus skill) they are bringing to the table. This same ideal could be applied to teachers leaving a certification program (having had time with a mentor teacher in a student-teacher experience); as well as a to an educator moving into some other type of leadership position. Some consideration of experience (skill) should be taken into account and thoroughly scrutinized prior to hiring. I spoke with a friend recently, and she told me about a rigorous interviewing 3-step process she endured and how mentally exhausting it was as she wondered whether or not she would be chosen despite her extensive experience and skill.

The first step was an in-depth interview with content experts, asking deep questions around her knowledge of the TEKS, her abilities to traverse the ins and outs of curriculum research. She said, as a curriculum minded person, it impressed her that the department cared so much about her background knowledge as to screen her in this way.

The second step included her sitting in front of a board of leaders of various kinds (Execs, Admin, Superintendents, etc) inquiring about leadership scenarios and support. The final step in the process was a performance task, where she was ushered into a room that held the current department plan on the wall. She was asked to critique it and provide solutions to any holes or gaps she found. As I sat and listened, I was highly impressed. This is the type of process that weed outs those with limited skill and an overabundance of will, but allows for those with an equal measure of skill and will through.

On the other side of the coin, there are certain situations & positions, (I believe) that require an equal amount of skill and will or possibly a bit more will than skill but nonetheless a considerable amount of skill. Certain leadership positions that require one to serve as a leader over others in that area, leading the charge with various tasks related directly to the knowledge and skills needed in that area; well I have a hard time seeing a person with more will than skill leading such a charge. It can be very stressful for those who serve under such a leader, to not only respect said leader but also to endure working on projects, getting critical decisions made, etc when that leader is operating predominantly off of will. When that leader has no sound knowledge or concrete experience (good and bad) to build off of, it could be detrimental to those who are directly (and indirectly) impacted by their leadership. Take for example an educator who gets chosen to serve in an Executive position leading other Administrators, but has had no administrative experience of their own (sound absurd right?) yet has had various other leadership positions and perhaps has even worked alongside other Administrators might exemplify the will to serve in this position; to learn from various professional learning situations and perhaps even befriends several administrators to learn from them (on the job). How might this new Exec provide the mentor-like encouragement, the sound advice that has worked in their favor (or even has failed them) when a new Administrator seeks out advice?

Again, I understand the notion of both view points, but I am more inclined to take skill over will than will over skill; or at least if I did have to hire someone with will over skill, I'd be looking for someone with a considerable balance of the two with the ability to pass through a rigorous screening process. I've seen a lot of nepotism and favoritism in education in the area of hiring practices. I've tried to determine which is more detrimental to a system centered around students and I've come to find them both somewhat equally deplorable and disrespectful to the skilled staff that serve under such hired leaders as well as to the students we (in education) claim to "do it" for! At the end of the day, such hiring practices are much more self-serving than others focused! With the fact that education receives the catfish of salaries, the short end of the stick from government in terms of support and funding and yet demands the most hours and effort - it saddens me how political and not-about-kids it can become when leaders are put in a position to hire. We should be more cognizant of these practices by tightening up the hiring process and doing everything we can to develop and build proper culture to retain our skilled teachers, administrator and leadership staff, to reduce the possibility of being placed in such predicaments. Let's get behind the notion of skill over will so we can show students we value their education and those in the corporate world that we take our profession seriously.

Composing and Decomposing IS NOT adding and subtracting

I remember when the NEW TEKS first released, I was super confused about the composing and decomposing language. It sounded like "putting together" to me, so I naturally associated it with addition. And decomposing means to break apart numbers, so I naturally associated it with subtraction.

But after surveying many of the the state's supporting documents and reading one of my now favorite books, Teaching Student Centered Mathematics (by John Van deWalle) and Developing Number Concepts by Kathy Richardson; I began to get a deeper understanding of the concept of composing/decomposing.

"To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers." (Van deWalle 2006, pg 26)

This was such a great quote that helped me make the connection that composing and decomposing is the conceptualized foundation that helps support students understanding for addition and subtraction.

So bottom line is composing and decomposing is NOT addition and subtract, but it lays the foundation for students being able to fluently add and subtract, mentally!




Watch this video:

Does McDonald's Sell Cheeseburgers? Well,...does it?

Of course McDonald's serve cheeseburgers- what kind of question is that right?

I mean isn't this the thief we are referring to?

Or perhaps THIS is the thief that we should really be addressing. I must raise my hand and be the first one to admit that I was the preacher for this riddle/rhyme when teaching my students long division. I cared not about students' frustration with learning acronyms, giving them steps that had no real mathematical meaning and thus robbing them of whole number- number sense and more importantly robbing them of the true meaning of division! For me, the vocabulary for this unit was:

• Outside number
• Inside Number
• Number on top
• Divide
• Multiply
• Subtract
• Bring Down
• Check
• Answer

Did you count the true academic vocabulary in that list? I did not hold my students accountable for academic vocabulary, mathematical thinking nor articulation of repeated subtraction with efficient methods.

Research says that division isn't only similar to subtraction in that it involves the separating mechanics, but that it is (just like subtraction) a difficult concept to grasp for students. And difficult concepts should NOT be approached, tackled nor attempted with abstract methods. If not all concepts, those which are most complex should be gradually approached with concrete manipulatives and connection to pictorial representations. Consistency in practice should take place at these levels before we make the connection to a more abstract algorithmic method.

So I share this video (based on approaches that I've read are more developmentally appropriate for students) and I wish I would have used when I was a classroom teacher. Now, I get chances to work with students in small groups and I claw at every chance I get to expose them to these strategies and raise the level (bar) of mental math for them. After all, students can learn at high levels if we simply embrace those methods and take the risk of exposing them to those strategies! Will you try it?

To Coach...or not to Coach?

Serving as an Instructional Coach can be a very daunting and overwhelming task. It can also be varied and uncertain in terms of tasks and expectations. Have you ever heard that joke about the "fine print at the bottom of a contract"? If any position in Education relates to this joke, it's that of an Instructional Coach (or Specialist).

A colleague came to me not long ago and expressed their interest in returning to the classroom from their time as an Instructional Coach. I must say, having spent some time investing in this colleague as a mentor, I was a bit devastated.

*Side Bar* I take pride in being able to spot a servant-hearted person; one who takes ownership of their professional learning by spending their time studying research-based practices (rather than searching TPT and Pinterest for activities) and seeks chances to turn such learning around immediately. That's worth admiring and when I observe such qualities in someone I'm inclined to move on such a hunch, by investing time and energy modeling, mentoring and maturing said person into their potential. But hey, it's not about me, right? *Okay I digress*

So I asked this colleague why they were considering returning to the classroom (which there is NOT a law against, by the way). Their response to me was that their true passion lies in working directly with kids and that they miss the classroom. Now, at face value, I took the response to mean that they had this aching desire to be in the classroom where they felt they could have the biggest impact! Few words/ideals are more true than that and I wouldn't dare fight such a theory and noble desire.

But after some thoughtful consideration, I mauled over their statement and proceeded to dig a bit deeper. Could this person mean by "they miss the classroom", that they feel as if they don't get enough time in the classroom? So after a series of probing questions, I hit gold! They were trying to articulate that the position of a Specialist/Coach was super overwhelming and inundated with so many other tasks that being where teachers needed them (modeling and supporting kids) was near impossible.

This is true in some settings. I remember my first year as a Specialist even though it was a blur. It was filled with trivial tasks that had nothing to do with Math, busy tasks that helped teachers but may or may not have been content related, included pulling data, being in meetings and helping teachers plan. I rarely sat with a kid or put into practice what I preached (so a teacher could see my value). The following year, my job title was protected by leadership as boundaries and expectations were clearly set and consistently held to. But this didn't take away the urge to sit behind my desk and complete tasks for teachers. I had to plan for our PLCs, plan to be prepared for our planning sessions, amongst the normal data pulling, analyzing, program coordinating (GT, testing, RtI, etc). So when was I going to do all of this AND be in a classroom for 70-90 min with a teacher and students? If perhaps a teacher did nail me to model or co-teach, I had to spend time preparing...more time away from classrooms.

But I learned that some things needed to happen "off the clock" and I needed to MAKE TIME for teachers. So I organized my schedule by placing priority on a specific few grade levels (supporting K-6 meant I needed to pick both a testing and primary grade to focus on).

1. After my PLC and planning times were set, I put my office hours during the lunch block (11am-1pm) and filled everything else in with classroom time!

2. When I sat in daytime (45 min) planning with teams, I looked for opportunities to come into their classes to support and model. If I heard them talk about not knowing how to approach something, after I gave an example, I followed my spill with "I can come in and model or co-teach"! This statement probably won me the MOST time invited into classrooms.

3. Sitting in PLCs discussing data were prime opportunities to invite myself into classrooms! Once a teacher inevitably asked, "how can I reteach this?..." I found my window of opportunity and put them on my schedule!

4. While observing in classrooms, rather than standing at the back taking notes, I sat with kids and helped them. I engaged in the classroom learning as if I was a student. This way when the teacher stopped me later and asked what I thought...I found a way to ask if they needed me to support them with anything- another window to slide into classrooms!

5. After each co-teach and modeled lessons, I made it a priority to follow up with the teacher and consistently ask, "Let me know if you need me to come in and help!"

6. Offering to stay after school with them to help them look up resources and flush out their lesson plans!

These practices, along with building relationships with teachers: eating lunch with them (although I'm an introvert and prefer time to myself or was busy and had other things to do), and chatting with them after school while they planned all conspired to build trust and before long I was invited to classes (rather than inviting myself).

So yes, there are 99 problems in Instructional Coaching, but getting into classrooms and working WITH teachers, should NOT be one of them! It should be a front burner priority. It is how we impact learning and make footprints in best practices. The desire to "BE IN THE CLASSROOM" should never die out. It should be the motivating factor for Instructional Coaches and Administrators alike. If we ever lose the hunch and drive to be in a classroom (after we've taken a different role than teacher) then we've forgotten why education is important. Now, while I personally don't believe climbing the "corporate" ladder should be a mindset for Educators, I do understand (and don't "shade" people for trying to financially improve - we all know Educators are the catfish of salaries). But there's so much more reward in the touching of lives, that I feel our focus should be pursuing what we are passionate about and gifted in. It should be about impacting more teachers and students with best pedagogical practices with the intent to stay as close to the classroom as possible.

As a District Math Program Coordinator, not much has changed. Going from serving 20-24 teachers to now 32 campuses (roughly 800 teachers), the mindset has to be the same. My plate is not only full, it looks like a buffet overflowing with food. But I make it my purpose to set aside a day a week to be on campuses, showing my face in classes, highlighting the wonderful things our dedicated teachers do everyday and looking for open doors to be in PLCs, planning and/or classrooms by invitation. I still get that tingly feeling when a teacher sees me in their class, and invites me to join in the lesson. No greater joy than working with kids and gathering authentic data to not only continue my growth as a lead learner but to make ripples in the lives of Educators and students alike! 

"I got 99 problems (things to do) ...

...but being in classrooms AIN'T one of them!"

Hyped over Hyperdocs!

So one of our District Goals this year was to provide Review Resources to our teachers so they could focus more on having collaborative discussions around their data and creating their instructional lessons to align to the standards. So as I fumbled over the first few modalities I would use to send this MASS REVIEW resource out to teachers, I stumbled across this amazing website!

It gives samples of Hyperdocs (a google doc on steroids with hyperlinks to various resources) created by teachers for various subjects. I grew particularly fond of this one Hyperdoc Game board template and began to play around with it.

6 months later, I have a compilation of various Benchmark & STAAR review Hyperdocs created for teacher clickable ease!

Check them out!

Kindergarten Benchmark Review

1st Grade Benchmark Review

2nd Grade Benchmark Review

3rd Grade Benchmark Review

4th Grade Benchmark Review

5th Grade Benchmark Review

3rd Grade STAAR Review

4th Grade STAAR Review 

5th Grade STAAR Review

Facts or Fractions?

                VS                 
Lately I've had this re-occurring question pop up and not far behind has been it's coupled demand...

Question:
"Shouldn't students have a good grasp on multiplication facts before they manipulate with fractions?"

Demand:
"We need to put Multiplication/Division unit before the Fraction unit in the Scope and Sequence. It makes sense that students need to master their facts before they work with fractions!"

I beg to differ and here's why:

3.3F (Equivalent Fractions) TEXAS standard says students are to...

"Represent equivalent fractions with denominators of 2, 3, 4, 6 and 8 using a variety of objects and pictorial models, including number lines."

If we isolate the content (equivalent fractions) from the WHAT or skills students are to do (represent) we get a very concrete and representational approach to instruction.

The CRA (concrete-representational-abstract) process is evident in the progression from 3rd to 4th grade.

Check out the verb difference...

3.3F - Represent...

4.3C - Determine...if two given fractions are equivalent using a variety of methods.

Do you see the progression? Third grade is the place to develop "Fraction Sense" (conceptual and visual foundation of breaking fractions in half) through the lens of the concrete and representational lens. Fourth grade is the place to make connections and build on their fraction sense by using abstract methods to find equivalent fractions. 

So perhaps we use 3rd grade as a training ground for exploring with manipulatives and creating visual mental pictures of how to break fractions in half; leaving students with an ability to find methods that make "fraction sense" when they get to 4th!

I do resolve that Multiplication should precede Fractions but so that students have a longer "school year" to master their facts, NOT for the purpose of using the knowledge to master fraction equivalencies.

Here's a quick video to clarify!

Instructional Nourishment

What was the last meal you had? What all did it consist of?
Better yet, can you give an account for the last three meals you've had (breakfast, lunch and dinner)?

I'm hoping your breakfast was full of protein, oats and fruit! Typically, lunch might not be as wholesome. Perhaps some seafood, salad, maybe a sandwich and chips with some veggies on the side. And for dinner...lets say you make a nice thick piece of beef or chicken with a starch like scalloped potatoes, a green leaf like a salad or spinach and some bread. Okay so you thoughtfully attempted to balance your meal with doses of the proper nutrients and minerals to give your body the nourishment it requires to help you function. Let's even go so far to say you're a health nut and you enjoy preparing very balanced, high in (insert nutrient here) with just the right number of calories to help you with your personal weight goals.

What you don't do is fill your refrigerator and pantry with vitamins and mineral supplements with the intention of tossing a Vitamin C pill into a hot skillet and cooking it sunny side up. Nor do you empty a half bottle of Fish oil capsules in between two slices of bread to get your dosage of meat for lunch. And you certainly don't place an arrangement of various sizes and colors of supplemental vitamin pills into the sectionals of your dinner plate, pull out a fork and knife and dig in!

How silly, right!? Because supplements are meant to take up for where the planned, well-balanced meals might fall short in terms of nutrients. They're the back up plan when your Plan A happens to be missing some critical components. And even in their use, they're not intended to be consumed continuously in place of food, unless you're physically unable to intake the foods that contain those essential nutrients.

Teachers, a guaranteed viable curriculum is the critical component in Tier 1 or Initial instruction for students. A guaranteed viable curriculum is comprised of a healthy knowledge (depth and breadth) of the standards and content you are tasked with facilitating learning around. It involves a healthy balance of backwards design in planning out lesson approaches (rigorous formative assessments), low floor, high ceiling tasks that engage and hook students and peak their interests, hands-on math tasks that encourage exploration and evoke questions while promoting visual stimulation. It's a class full of peer to peer discourse and movement, students testing their theories and learning from their mistakes. It's students receiving small group support or one-on-one conferencing to adjust their goals. It's students eagerly anticipating the stations their teacher has so diligently prepared so they can independently or collectively explore their own ideals about math. When a teacher takes this type of approach to their initial instruction, they are providing a well-balanced instructional meal to their students.

The textbooks are simply a supplement. They serve the purpose of supporting the well-balanced healthy instruction that occurs initially, in the areas where the student didn't get enough of the nutrients from the initial lesson. They are created to take up the slack, just like a supplemental vitamin. Now, if you're the novice teacher and more specifically, you're dealing with little to no supports from a district or infrastructure, the textbook might account for a larger percentage of your initial instruction meal for a while. But nothing a few trips to a local professional development won't cure. Investing in PD can knock the crutch of depending on the textbook "to teach" from under you and get you to walking the right path soon enough. But the general idea here is that as we (teachers) move from the keeper of the knowledge to the facilitator of the knowledge/guide in helping students find the information; our job is to prepare the well balanced meal in such a way that we can guide students to the correct information and guide their uncovering of the appropriate learning. We do this rather than shove textbooks in front of them (like pills down someone's throat). After all we can't call ourselves educators when textbooks and worksheets eat up our instructional minutes anymore than we can fill our dinner plates with Omega-3 and potassium pills and say we had a hot meal for dinner.

Let's nourish our students by showing them we are willing to plan out instructional meals that will make them want to feast at the table of learning and leave the supplements on the shelves.

25 simple steps to solving Word Problems

I'm surprised you even clicked on this blog, after reading a title like that. Seriously though, who wants to learn (or has time, let alone mental capacity to absorb) 25 whole steps (which is NOT simple, by the way) to solve word problems?

But this is the type of absurd pressure we place on our students in Elementary schools in efforts to cram comprehension down their throats, get them to pass their annual accountability assessments, or convince them that they have some false level of mastery. You know what i'm talking about. All of the colorful, cute posters that we spend endless minutes dressing up and hanging confidently on our walls for students to reference when we leave them to solve word problems in isolation! Let me jog your memory...

Any of these look familiar to you!? I can honestly say I used that last one (RUBIES) one year, after seeing it at a local conference. I was duped into believing that my students who had 1-2 years of gaps, were struggling in ELA and read on levels lower than their current grade would be able to take a few doses of this poster medicine and magically make growth or passing grades in math.

Think about the title of this blog and how silly that sounded...that's just how silly that RUBIES acronym proved to be to my students. Some of them came from other districts where they had been exposed to (through drill and kill) various other Acronym posters forced upon them and here I was saying, "Forget that smut and learn this NEW-TRIED-AND-TRUE method!" At that rate and under those conditions, by the time a 3rd grader exited elementary, they could have easily been exposed to 25 "simple" steps to solving word problems!

I tutor this 5th grader (weekly) who attends school in a different district than the one I currently work in. This student has been the sad victim of several teachers who were inexperienced and/or had crumbled under the pressures of being a teacher, leading to their quitting in the middle of the school year. After weeks of re-teaching concepts from a conceptual and representational standpoint, I tested the waters with introducing him to abstract learning through the lens of word problems. I had him read (and re-read) the word problem and tell me what he thought was going on in the problem.

I had him talk me through the meaning of each sentence and then I asked the question that caused me to fall out of my chair:

"What do you think the problem is asking you to do? How do you know?"

To which he replied, "I think we are dividing because it says 'which'." 

*crickets...blank stare*

I could have screamed! Why did he think "which" meant to divide? Perhaps it had something to do with those posters? After some brief investigating, I uncovered that his teacher had taught him the following:

"Which" means divide

"How much" means to add

"How many" means to subtract

...and so on. 

I stopped him and mentally scolded that teacher. I was up against a much bigger giant than just closing his years of math concept gaps, and firming up his ability to comprehend word problems...now I also had to break his bad habits and train him in new ones. 

This is an example of the gaps we can create and the ultimate fate of giving students cute-sy short cuts and fast food feaux learning tricks rather than helping them understand how to think through a problem. 

John Van de Walle (as if I could write a blog without quoting my Mathematical boyfriend) says its profitable for students to draw pictures, act out and model (with objects) word problems as early as Kindergarten so they (in true slow cooker form) grow up into a deeper, more solid foundation of how to think through and solve word problems. The TEKS even make provisions in their explicit wording that yearly building on students' developmental abilities should suffice in regard to word problems. 

Model the action of joining and separating... 

(K.3A)

Solve Word problems using objects and drawings...

(K.3B)

Explain the strategies used to solve problems...using spoken words, concrete and pictorial models... 

(K.3C)

Use objects and pictorial models to solve word problems involving joining, separating and comparing... 

(1.3B)

Represent word problems involving addition and subtraction...using concrete and pictorial models... 

(1.5D)

Explain strategies used to solve problems...using spoken words, objects, pictorial models... 

(1.3E)

Represent and solve addition and subtraction word problems... 

(2.7C)

Not to mention the explicit use of strip diagrams in 3rd-5th for multi-step problem modeling. 

Let's commit (for the NEW YEAR) to cut the tricks out of our lesson plans and get down to supporting student thinking. This means that among the other numerous hats we wear, we must also be reading teachers! Guided Math and Math Workshop are great vehicles through which we can foster such conversations with students. We have that small teacher to student ratio (uninterrupted time) to have a two-way conversation with students about thinking through each sentence of a word problem, providing them manipulatives and scratch paper to model and draw out their thinking. I can guarantee this method is NOT simple. But for my class of 22 (full of students with severe gaps and low reading levels) this proved successful as I watched my students gain confidence and make significant steps in growth. 

If you have ideas of innovative and non-cliche methods that have worked in YOUR classroom, drop your ideas, videos and/or pictures in the comments. Or follow me on Twitter @kloneal2 and leave me a comment! 

Kindergarten Numeracy Activities

        In Kindergarten, when considering the notion of comparing numbers (and even if we back up to the ideal of magnitude of numbers), we tend to jump ahead of the development process for students. Donna Boucher talks about how “Alligators are for swamps, not for comparing numbers” and this is completely accurate. However, we lean towards this animation of a numeracy concept simply because we don’t know how to help students retain the understanding of comparing two numbers.

Our state standards address, very directly, the method with which we can support students development in comparing numbers. John Van de Walle encourages this approach in his book, Teaching Student Centered Mathematics. Kinder students should focus heavily on not only building numbers (to support their one-to-one correspondence) but also have exposure to multiple representations of numbers (to support their perceptual and conceptual subitizing skills) while comparing such models with words. I would even highly suggest that students use two comparison statements so they are acclimated to the notion of “fewer” and “less” as critical comparisons.

A member of my PLN (Professional Learning Network) on Twitter shared these cards and I found them quite helpful in supporting the idea of students using them to compare numbers. They are multiple representations of numbers and can be used for various reasons. 

Here are a few:

            
          Comparing numbers. Have students shuffle the deck and place them face down. Then with two players, each player turn over one card and build (with linking cubes) their number and then express the comparison(s) using words.

          More Less or Equal. Have students shuffle the deck and place them face down. Have them spin a spinner that has options for “More, Less and Equal”. Once pulling a card and spinning the spinner, have students build (with linking cubes) a number more, less or equal (based on the results from the spinner) to the number they pulled from the deck.

         Build the number multiple ways. Have students shuffle the deck and place them face down. After pulling a card, have students build that number using double sided counters on a five or ten frame. Then have that student build or represent that number as many different ways as possible. (example: if they pull the number eight with fingers five and three, have them build eight as five and three on the ten frame and again as four and four to represent the same number). They might even record all of the ways they built the number.

         Decompose it. Have students shuffle the deck and place them face down. Draw a line down the middle of a plastic bag with a permanent marker. Turn over a card and place that number of beans inside the plastic bag and seal it. Have students slide beans around on both sides of the line to represent different ways to decompose the number. Once the number has been decomposed different ways, pull another card and replace the beans in the bag with the new amount.

Just a few numeracy activities that would support students development if they remained in centers as TEKS based station activities for Kinder! The great thing about these activities are they can be extended and used with numbers 11-20 and reused at the beginning of first grade as scaffolded concepts.

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Example of Stations listed above that I created for my District Kinder Teachers from ideas I gathered from reading John Van de Walle and viewing other resources.
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