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.

Don't forget to get your set of Deca-Deck Cards

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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Conceptual Fluency: 3rd (& 4th) grade look at Multiplication Facts

I must say I have a tough time explaining why conceptual fluency is so critical in my eyes.

But I have to defer to the fact that research says that neither conceptual nor procedural fluency is better than the other. That they both are a dynamic duo that serve to make a student flexible with numbers and able to tackle abstract concepts with more than just speed and accuracy. They help as student be flexible in their approach to numbers and concepts/skills.

Let's say I wanted to multiply 7 x 7 but UGH, I don't know my "sevens". But I do know my "fives". So I could decompose 7 into 5 and 2 (as addends) and multiply (or skip count by 5 until I get to 7) seven times 5 to get a partial product of 35. Ooooh and I also know my twos!! So I could skip count by 2 until I get to 14 as another partial product). Well when I combine 35 and 14, I get the same product of 49. Although that abstract understanding will follow; this way I have built a better foundation of understanding behind why I even multiplied 7x7 to find that product rather than a rote "because my teacher said to memorize my fact" or listing out all of my sevens by counting up seven on my fingers and listing each product on the side of my paper! Both take some time, but one approach helps a student in a more scaffolded approach to better grasp their understanding of what 7x7 (or any fact for that matter) is as a final product!

So I stick to my argument that conceptual focus BEFORE procedural focus, builds a better equipped WHOLE student.

Think about these two concepts approached in 4th and 5th grade respectively. What happens when students begin to multiply larger whole numbers and eventually decimal numbers? They already have an understanding of decomposing numbers as well as using partial products to support this approach. This is all about decomposition...which builds from a conceptual understanding of numbers!

So, for my 3rd grade teachers who serve our babies on the front lines of fact fluency.
Here's a video series for you, that encompasses teacher and student stations for your classroom. All around facts (multiplication) from a conceptual standpoint! Before a baby ever gets on a computer to play a fact game, this helps build their foundation of numbers so their fluency truly is procedural in that they can manipulate numbers rather than "rotely" spew out facts with speed!

I would encourage pre-assessing your student with a running record to see what strategies they might already know. Click here

Learning from a 5th grader

I was graced with the opportunity to learn a few things from a 5th grader this weekend.

1. When your family knows you are an educator, that innate desire to tell every kid in school (regardless of age or grade) to ask “Auntie for help” seems to follow. So my 3^rd cousin inevitably had her “math lesson” that cousin Kim was purposefully placed there to help her with.

2. Homework truly does have a ZERO effect size (Hattie). This girl was given 4 pages of worksheets for her 3 day weekend. One of which had over 20 three-digit by two-digit multiplication problems for standard algorithm only (no word problems, no area models). Two pages of multiplication of whole numbers and fractions). When we did begin what the older generation calls “her lesson”, I found that she had been poorly taught (read: not taught at all) how to multiply fractions and whole numbers as well as division. This teacher who was not only seeking to suck the life of this student, was also failing to facilitate the learning that was truly capable, from this student. She had been given 2 other pages of second and third grade level work in this daunting packet. I was as exhausted and unmotivated as she was. Again, help me understand how will 5 pages of rows and columns of work I either know how to do or do NOT know how to do, going to better me? *shoulder shrug*

So we began working the multiplication of whole numbers and fractions pages first and I showed her a picture to help her develop a strategy to solve the problem. *watch this video

We went through talking about the meaning of a denominator and how to apply that meaning to the picture; then we repeated the same steps for the numerator and began drawing on the picture to help solve the problem. Honestly, it got a bit easier for her until she reached a problem she didn’t want to represent. So we looked at another way to draw a picture that might better help. *again watch above video

This time by problem ten, she had developed her own strategy. She figured, instead of drawing the picture, she might divide the whole number by the denominator and then multiply her quotient by the numerator. Without articulating this as I just did, I watched her talk to herself out loud, “4, 8, 12, 16…so 4 dots in each circle. Then three circles of 4 is 12”

This was it! This was the learning! John Van de Walle talks about invented strategies in his Teaching Student Centered Mathematics. How students can take from their conceptual understanding and create their own strategies without us as teachers needing to “model” it to them! I hadn’t even taught her this abstract process that she’ll learn in intermediate school…I had only showed her how to use concrete and representational models to get her solution and she had devised the abstract method on her own! Invented! I praised her and watched what was once a very distracted and apathetic 5^th grader who took any and every distraction as a break move to a focused and determined astute student complete the remainder of her page with pride! I was amazed! 

Perhaps because I’ve been out of the classroom for so long, or because I saw something I read often materialize before my very eyes! I must admit, I was the type of teacher who frequently modeled for my students and slowly morphed into a teacher who facilitated a healthy struggle. By the time I began to embrace that method of approach, I was swept away into a Specialist role. So this phenomenon was life changing for me. 

Conceptual Fluency or Procedural Fluency: A look at conceptual stations for 1st and 2nd grade Teachers

I must say I have a tough time explaining why conceptual fluency is so critical in my eyes.

But I have to defer to the fact that research says that neither conceptual nor procedural fluency is better than the other. That they both are a dynamic duo that serve to make a student flexible with numbers and able to tackle abstract concepts with more than just speed and accuracy. They help as student be flexible in their approach to numbers and concepts/skills.

Let's say I wanted to decompose five-fourths to express why it's equal to one whole and a fourth.
Knowing all of the different ways to decompose the number 5 (1+4, 2+3, 4+1, 5+0, 0+5 and 3+2) all serve to help me understand that I can decompose five-fourths using the facts (1+4 and 4+1) into four-fourths and one-fourth. That coupled with the understanding that four-fourths is related to one whole, conceptually helps be better prove why 5/4 = 1 and 1/4. I don't have to worry about dividing 4 into 5 and finding the remainder. Although that abstract understanding will follow; this way I have built a better foundation of understanding behind why I even divided to find that mixed number rather than a rote "because my teacher said to do it this way".

So I stick to my argument that conceptual focus BEFORE procedural focus, builds a better equipped WHOLE student.

Think about this. What happens when I need to subtract 300 - 234 and I don't know how to subtract across zeros? Yes, I could learn how to regroup in 1st and 2nd grade, but what is regrouping anyways? It's simply decomposing a number different ways. So 300 can really be two hundreds, and one hundred ones (or ten tens) which might look like this 299. NOW, I can easily take away 34 ones from one hundred ones or three tens from 9 tens and so on! This is all about decomposition...which builds from a conceptual understanding of numbers!

So, for my 1st and 2nd grade teachers who serve our babies on the front lines of fact fluency.
Here's a video series for you, that encompasses teacher and student stations for your classroom. All around facts (addition/subtraction) from a conceptual standpoint! Before a baby ever gets on a computer to play a fact game, this helps build their foundation of numbers so their fluency truly is procedural in that they can manipulate numbers rather than "rotely" spew out facts with speed!

*********************ADDITION STATIONS*************************

FILL A FRAME (Station 1)

DECOMPOSE THEN MAKE 10 (Station 2)

DOUBLE AND NEAR DOUBLES (Station 3)

*******************SUBTRACTION FACTS*********************

DOWN THROUGH 10 (Station 4)

TAKE FROM 10 (Station 5)

Have Fun!