If you work in one of the 45 states that adopted the Common Core standards to start this year (and chances are you do), then you are probably aware of one major glaring issue regarding the implementation of the math standards.
That is: how can we teach this new, deeper style of math when it supposes that students have had previous years of training and learning in a similar style?
Common Core supporters will say that’s exactly the reason we needed these standards, but that misses the point. The point is that until a class of students has received Common Core instruction from Kindergarten to the end of their careers, it’s not going to make sense. So, my students, who are brand new to the Common Core way of math, are starting out way behind where Common Core supposes they should be.
This should have been a discussion before Common Core was green lighted and rushed into our classrooms. It might have gone something like this:
Logical Thinker: “Here’s the thing. If we start Common Core in every grade this year, it can’t possibly work for all students.”
Skeptic: “Of course it can.”
Logical Thinker: “No, it can’t. Think of it this way. We’re going to ask third graders, for instance, to do math based on what they would have learned in second grade. Only thing is, they didn’t learn Common Core math in second grade, so they don’t have the prerequisite skills to do the Common Core math in third grade.”
Skeptic: “You know what? That’s a good point. And that’s going to be true for all grades, except…”
Logical Thinker: “Kindergarten. If we start Common Core with them this year, then they can do Common Core in first grade because they’ll have done it for a year. It all builds on the previous year so kindergarten is the only grade we can really start with without totally demoralizing kids, teachers, parents, and administrators.”
Skeptic: “This actually makes sense.”
Logical Thinker: “Of course it does. So we start with kindergarten in 2012. Then do kindergarten and first in 2013. Then kindergarten through second in 2014, and continue implementing it in waves until the 2012 kindergarteners are at the end of their career.”
Skeptic: “Okay. I get that. But what about all the kids who don’t get Common Core? We’d just be failing them.”
Logical Thinker: “No, we wouldn’t. Implementing something like this requires logical thought, and that’s why I’m here. We will fail them only if we cause them frustration by expecting them to do math they can’t be expected to do. You can’t build a building without a foundation. Where is their foundation?”
One of the unexpected pleasures of emerging from testing season with two months of time left is the fact that I’ve been encouraged to keep kids motivated through project-based learning.
What a breath of intoxicatingly fresh air. We know that creativity has less and less of a place in our elementary schools. The kids wear this knowledge on their sad little faces as they flop test prep packets onto the desk and fall asleep over highlighted pages of nothingness.
Our current literacy unit involves research. There are four groups in my class and each is responsible for a New York City landmark of their choice. I have three goals for the unit: 1) have students direct their own learning about the landmarks; 2) give them transferable skills; 3) keep them engaged and having fun.
So, while kids are doing research, they are also creating a mural. First, everyone in the class sketched their landmark, paying close attention to details. On a tri-fold board, I sketched general parameters for each image. Then, they each drew their picture on the board, creating a bit of a mosaic of New York City. Students used rulers and measurements to maintain neatness. They are also picking up some art skills as they mix the paints to create desired colors, learn effective ways to use a brush and paint small areas, and visualize how items must overlap in order to look the right way. Maybe most importantly, the mural involves a good amount of group work and cooperation that, for the moment, is more effective in art than in research.
Once the mural is finished, I will use it to extend our math unit, which is focused on multiplication and division. As an example, students will be asked to compute the number of windows in the Empire State Building on our mural (the windows are arrays, which is a current focus). They’ll be able to measure different elements on the mural and compute areas and perimeters. I’ll figure out a way to have them review fractions through the mural, too.
Until the mural is complete and ready for us to use it for math, we are working on multiplication and division in context, tying them to, what else, New York City? Word problems don’t say “Sally shared 21 cookies with 7 friends. How many did each friend get?” but they do say, “21 tourists got into 7 taxis. How many tourists got into each taxi?” They are motivated by the New York-centric theme and, if I do say so myself, I am seeing a nice output on their parts.
Given the license to go with projects, you better believe I’m going to drive with it. Students are getting their kicks and their concepts, and it’s phenomenal.
It’s 2012, so of course that means that one of the most ubiquitous tools at our students’ disposal is also one of the most reviled in NYC. Cell phones are simply not allowed in schools. There are too many people in positions of power who see them as texting, calling, and gaming devices as opposed to cameras, computers, and encyclopedias (ie. something that could enhance one’s education rather than take away from it like, I don’t know, test prep).
My kids are too young to have smartphones, but I’m a big boy so I get to have one. Today, it came in handy.
This year’s class got their first experiences using my set of digital cameras today. I thought I had one per customer, but as it turns out, I was two short. Nearly everyone was armed and ready to go on a scavenger hunt collecting pictures of arrays, but I had to improvise for the two who got shut out. So, one got my iPad and the other got my, you guessed it, phone.
There they were, traipsing about the halls, looking for arrays. Flashbulbs popped here, flashbulbs popped there. A girl held an iPad up and snapped away. And there was my cell phone user, happily capturing arrays all over the building.
Without a cell phone, she would have been excluded from the activity. That’s the way some would prefer us to have it, but it’s not the way I prefer to operate.
Without a cell phone, at least one of my students would not have been able to participate in our array scavenger hunt today. Instead, she was able to complete the same task as her peers.
I introduced one of my favorite math topics, fractions, to my third graders last week. To do so, I drew on the experience of an assistant principal and our math coach. The pivotal point is that students understand that fractions are equal parts of one whole, so I really tried to stress that during the lesson. Maybe you’ll try these with your own class!
Pizza Pie Fractions
Using magnetic fraction circles, I told my class a story about dinner last week when I was so hungry that I decided to order a whole pizza for myself. When my sister arrived to bring me something, she saw the pizza, and being hungry herself, asked to have some. I gave her one slice, but she said, “That’s not fair! I want the same amount that you’re eating!” So we decided to split the pizza in half and we both got two equal pieces. Right as we were about to take the first bite, her husband called her, saying he was starving. She told him we were having pizza and he should come join us. I tried to give him one slice, too, but he also complained that everyone should get the same amount, so we cut the pie into three equal pieces.
You get the picture? This continued to fourths, fifths, sixths, and tenths. I made it funny by saying things like, “Just as I was pouring the garlic on my pizza, the bell rang again!” as well as, “At this point, I was wondering if I should just order another pizza,” and “I only had six seats at my table!” They loved the story and the visuals helped support their understanding that a fraction is an equal part of a whole.
Pizza pie fractions - hold the anchovies, double the fun and understanding.
Homemade Fraction Bars
The next part of the lesson involved students creating their own fraction bars. Each student received eight strips of brightly colored paper and followed my directions on how to create fraction bars.
The folds for thirds, fifths, sixths, and tenths were extremely difficult for the students, so I would advise having extra paper on hand!
At any rate, each time they opened their freshly folded paper, students were able to count the number of sections and therefore easily identify the fraction they created. These strips will be invaluable when we begin to study equivalent fractions!
Fraction bars - made for the students by the students.
On a side note, during the lesson, in order to help students realize that fractions didn’t just come in circles or rectangles, I encouraged students to discuss other things that were or could be made into fractions. One girl noticed her glasses could be divided into thirds (the lenses being one-third, and each arm being a third.) One student bent his arm to indicate a half (roughly). And one student mentioned that an orange could be divided into fractions. Conveniently enough, I had one in my bag, so we fractionalized and everyone enjoyed one-sixteenth.
This week, I invited the math coach in to teach a couple of lessons. Watching a master teacher work is an opportunity for me to be reminded just how far I have to go.
(Since someone on Twitter asked me to clarify what a math coach is in this context, I will do the same here. The math coach’s roles are many and are not limited to: modeling lessons, observing lessons and making suggestions, joining teachers on intervisitations, providing ideas and resources, and serving as a planning partner).
Here are some takeaways from the lessons modeled in my classroom this week:
Math needs to be made as concrete and relevant as possible. In teaching perimeter and scratching the surface of area, the coach began with a story about her backyard fence being broken and her dog escaping. The story set up a problem she asked the students to help her solve: Based on the size of her backyard, how much fencing would she need to buy at the store?
Manipulatives are great for manipulating, and there needs to be time to allow exploratory and free use of them. The coach used geoboards and rubber bands and allowed the students three minutes to use them in any safe way they desired before moving on to a more structured use. She also used the geoboards as response cards of a sort, checking for understanding of the properties of a rectangle by asking students to make one on the geoboard and show her.
It is important to slow down – as calmly as possible – when the students show resistance. My students all have disabilities and are English Language Learners, too, so it is important to really think about the most basic knowledge that they need to have before progressing to the more complex. In this case, it became clear to me and the coach that we had to really break down the use of the geoboard and how to properly count (rather than begin at 1, begin at 0) for accuracy’s sake.
Love the Earth, save the paper! The students used geoboards and the SMARTBoard for about 55 minutes before transferring their knowledge to paper. By then, they were solid and able to make the transition from concrete to abstract.
It is always beneficial to see others teach, especially those with all the years of experience. If you don’t have a coach, try to set up an intervisitation with a colleague! You won’t regret it.
An interesting conversation came up yesterday regarding the proper way to grade students’ understanding of multiplication and division. Part of our most recently completed math unit was the Common Core task, which required students to use a table of information to compute one and two-step word problems using multiplication and division concepts.
Throughout the unit, we made a point of teaching various methods of multiplying and dividing. Students learned how to use arrays, pictures, tables, bar models, area models, repeated addition, repeated subtraction and more. However, based on the rubric provided in the task, if students used those methods and not explicitly written multiplication or division expressions, they could not receive credit for the work.
The discussion was spirited. Many people felt that since we taught so many different ways, it was just fine (and in fact, desirable) for students to choose the one with which they were most comfortable as long as they applied it properly. The other side of the discussion held that the task was designed for the purpose of measuring students abilities to recognize those methods as multiplication and division, and indeed, to convert their understanding to writing the sentences.
Since this task does not count for a grade and because it is our first foray into such a thing, we arrived at a consensus that a note should be made that in many cases, points awarded were not indicative of the learning that occurred or the work students did. The plan is to consider this for next year when we come up to the task again and place a greater emphasis on teaching students how to relate the chosen methods back to multiplication and division.
Three lingering thoughts I have from my students’ work:
If the work of the problem should have been (4×4)+ (4×5) and a student simply wrote 16+20 (meaning they multiplied in their head), is that insufficient?
If a student can multiply 3×2 mentally (or add 3+3 mentally), do they need to write it in order to receive credit when they give the correct answer of 6?
If a student accurately computes division using pictures, is it division by another name?
I would love to read your thoughts on this, especially if you are familiar with your state’s Common Core task.
It would be disingenuous to ignore the range of Khan’s knowledge or his capacity to produce so many videos. However, to claim that he and his style are the answer to the ills of education, I think, is a bit much. In my eyes, like pretty much every other reform idea, Khan’s videos may work for some, but they won’t work for many.
It is clear Sal Khan never taught special ed.
(Or ELLs, for that matter).
Recently, I was looking for some video options to reinforce multiplication concepts, and I watched Khan’s “Basic Multiplication” video. I wanted to incorporate some visuals and videos to help engage some of my more reluctant learners.
Whenever I watch videos or consider content, I have to do so from my students’ perspectives. So, while something may make perfect sense to me as is, I know that, usually, my students will not accept it in the same way.
I thought I’d give Khan a try. Watching the video from my students’ perspective, though, it was obvious that there was no way it was going to work in my classroom (a self-contained special education class of 100% ELLs at intermediate or beginner levels).
For starters, the amount of text in the video would be overwhelming. I am guilty of sometimes having too much going on at once in my class, but at least I’m there to help filter out the extraneous information (or erase it!) and help students refocus. In this multiplication video, Khan writes the word “Multiply” and puts “2 x 3” on the left, but then reviews addition (2 + 3) for about a minute on the right.
I'm concerned with the amount of text on this screen, as well as the lack of visual delineation between mathematical concepts.
There is no clear designation about what concept is what. The potential for confusion is too great, in my opinion, for this to be effective for many students.
It’s not only the text in the video that concerns me. It’s Khan’s delivery. Clearly, he is a well-spoken man with great depth of knowledge. However, delivery of that knowledge in a way that is too dense for students to understand means he might as well be speaking a different language. And for many ELLs, I imagine when they hear sentences such as the following, English all of a sudden does sound like a different language:
And this is probably the first time in mathematics that you’ll encounter something very neat: that sometimes, regardless of the path you take, as long as you take a correct path, you get the same answer.
Say I’m eight years old. I’m a beginner or intermediate ELL, or I’m fairly new to the country. I just heard all these crazy words: encounter, neat, regardless, path, and as long as. I’m totally lost. I need someone to help me understand the context and meaning of those words. I need someone with a little more sensitivity to my needs than Sal Khan.
Khan, shortly after that long-winded statement, says that, in considering other representations of multiplication, he will continue by drawing rows of lemons so he can continue, “our fruit analogy,” (he referred to raspberries and blueberries previously). Then:
An analogy is just when you kind of use something, as, as an – well, I won’t go too much into it.
After a while, it becomes uncomfortable – and inefficient – to listen to Khan’s colloquial manner of speech and his many verbal pauses. His video is neither concise nor succinct, and therefore it enables the mind to wander, rather than be inspired.
More verbal garbage from Khan can be found. He draws an array of lemons to talk about why multiplication is useful as an expedited form of counting. In my class (as in any class of ELLs), the critical point of arrays when they are introduced is learning what a row is and what a column is.
Khan begins to introduce what a row is:
A row is kind of a, the side-to-side lemons. I think you know what a row is. I don’t want to talk down to you.
Yet, unfortunately, with a statement like that, Khan is talking down. Because he assumes that everyone knows what a row is, he cuts off populations with his pomposity and makes it difficult to access the information.
I think if I made a list of all the words and phrases Khan uses in the video that would be stumbling blocks for ELLs and/or students with disabilities, I would come off as a whiner. However, in my estimation, it’s a fairly long list.
Look, there is some value to what Khan is doing. Just watching the video gave me some ideas of ways I could approach multiplication with my students. However, the mission statement of the Khan academy is not to help teachers teach. On the web site’s about page, it says:
We’re a not-for-profit with the goal of changing education for the better by providing a free world-class education to anyone anywhere.
Hmm. Well, if “anyone anywhere” means kids who are fluent in English and have the ability to follow dense text peppered by colloquial speech, then these types of videos will be fine. However, if “anyone anywhere” means, truly, anyone anywhere, then Khan has quite a long way to go.
I am sharing the video I analyzed so that you may do the same, if you choose. Would this video work for ELLs? Do you know students with disabilities who would be overwhelmed by it? Does it serve the needs of all students? See for yourself and determine your own answer!
I think a lot of people who don’t fully understand differentiated instruction mistakenly believe that a good differentiated lesson has to involve menus, six different groups doing six different things, integration of every one of Gardner’s multiple intelligences, and 12 different ways to demonstrate understanding. Or they believe that differentiation a lesson means teaching every student the same lesson and then flying around the classroom and reteaching or enriching for each group. Because differentiation is misunderstood, it gets a bad rap.
In truth, differentiation often is many of those things all in one lesson. It can be overwhelming to think about planning lessons like that for every subject every day. Yes, it is a lot of work, but, the more I hone my ability to differentiate, the more I see my students investing in the work and the less I see them becoming frustrated.
So today, I will share a very successful, highly differentiated math lesson I taught this week that had every student engaged at their own level and also incorporated a variety of concepts and modalities. I do this in the hope that it helps others better understand differentiation and see the benefits of it and perhaps provides a springboard for others’ pedagogical growth.
We are working on multiplication in my class. We kicked off the lesson by reviewing the facts for 0 and 1. We started without referring to the charts on which we had written them the day before, but of course, some kids felt more comfortable reading the facts when I called on them, so I allowed them that option. Also, even though one of my students knows many of his multiplication facts (and certainly 0 and 1), I included him in this part of the lesson because it gave him an opportunity to work on the commutative property, which is a new concept to him.
From reviewing those facts (an assessment to see who understands the concepts, by the way), we moved into arrays. It was not the first time we ever talked about arrays, but since students had never really made their own, I knew I wanted to spend the majority of our math block focusing on them and gaining a better understanding.
The essence of arrays is that they are organized sets of equal rows and equal columns. Being that my students are all ELLs, over 90 % were unfamiliar with the words “row” and “column”. So, to introduce those crucial concepts, I first drew an array. Then I boxed out a row and introduced the word. I did the same with a column. To further reinforce the concepts, every time I said the word “row,” students put their arms out horizontally. Every time I said the word “column,” they put their hands up in the sky.
With their new knowledge of these words, we watched a BrainPop Jr. video about arrays. This kept the students’ attention because it was visual, amusing, and all around entertaining. I liked the video because it really emphasized rows and columns at every turn. As we listened and watched, I paused at different spots. Often, I reinforced the physical representations of rows and columns with arms. Total physical response is huge for ELLs and students with disabilities, and because we had established a physical anchor for an important concept, my paras and I were able to consistently and constantly revisit it once we moved on to the next phase of the lesson.
After the video, each student was challenged to create an array using counters. This was the most obviously differentiated part of the lesson. Each student worked on making arrays for a different number. The number was based on their current math abilities. (Note: I use the word “current” because the expectation is that all will improve. Not using the word “current” implies students are where they are and will stay there. Back off the soapbox).
So my student who struggles the most in math worked on an array for 6. The student who knows multiplication better than others and generally is one of the better math students worked on 45. Everyone else was somewhere in between. Their work during the next 30 minutes was some of the most exciting work I’ve seen in my career. Every student was engaged and, so importantly, challenged. The student making an array for 6 took quite some time to figure it out, but she got it. The student making an array for 45 took about 5 or 6 tries until he figured it out, but he got it. It was really exciting to witness, and the kids were truly invested.
The majority of the student work this period involved students figuring out arrays for numbers based on their math abilities.
Throughout their work, I circulated to check in and monitor for understanding of those crucial row/column concepts. For some students I simply asked “How many are in the row?” and they answered easily. For others, I needed to say “Show me a row with your arms” in order for them to remember that rows run across. For others, it went one step further, and after they showed a row with their arms, we touched the counters together to show horizontal (or in one student’s case, laid a pencil atop the row to see a linear representation). This is all differentiated formative assessment.
As students figured out their arrays, they were required to write the corresponding multiplication sentence. The students were quite proud as they finished, but I kept the rigor of the lesson going and challenged them by saying, “Make another one.” (Side note: My only assistance for most students was a reminder that the rows and columns must be equal. They were on their own more than they have been all year). They generally looked at me with mock shock and then got busy. A student who I had last year as well, and who has struggled mightily with math, actually managed to figure out four arrays for the number 10 – with minimal adult assistance. A multiplication master in the making!
Once students figured out the different possible arrays for their number, they took a piece of construction paper and drew one of their arrays using dots. After they adjusted for neatness, they took stickers and covered the dots to make the arrays look a little prettier. Then they wrote the multiplication sentence for their array and the corresponding fact based on the commutative property (ie. 4 x 7 = 28 and 7 x 4 = 28).
Each student finished and no one judged or questioned the different numbers used. It was truly wonderful.
I wrapped the lesson by having students cut out flash cards for the 2s times tables. We also sat with a 12×12 times table grid and talked about how to use it. Differentiation here came in the samples we worked on. Everyone did the same work, but there were different points of access. I had the students figure out the answers to problems like 3 x 5, but also ones like 12 x 8.
Why was this lesson a success?
Students were invested and engaged from the start.
They were working on grade level concepts at an appropriately challenging level based on their current abilities.
They had opportunities to talk, listen, watch, touch, move, write, and do art.
They experienced multiplication in several different forms (visually, orally, aurally, in arrays, on tables, and on flash cards).
Everyone felt comfortable and everyone had a point of access to difficult material.
This is what differentiation can look like. It does, indeed, take some extra thought during planning time, but it is an easily managed lesson and it is just so effective for the students.
Please share with us if you try something like this. Or share your own differentiation success stories!