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I would like to add a thought about “once they understood it.” Until recently, I would explain briefly and then show them examples. I didn’t realize I had trained them to zone out until I showed the examples. Sometimes I would question them to make sure someone understood it, but the main take-away was procedure. Then they would ask, “Do I multiply or add? Where do I put the zero?…” Now when they ask questions like that, I help them reconstruct the conceptual explanation. Now the main take-away is “what would be a sensible step?”
I must disagree that having to consult the internet means there is a problem. There is a huge problem with math education in this country, but it is because too often we math teachers have presented math as a series of compartmentalized, memorized steps. That’s what most adults remember, but I wonder if they are aware of how few consider themselves to be good at math. There are some instructional techniques that research shows develop solid number sense that goes with the math procedures, connecting them and making them meaningful, easier to remember. Those techniques need names. I have had to look up a few myself. Better yet, I have watched a few demonstrated on teachingchannel.org. It’s an exciting time to be involved with math instruction because, for the first time, I believe we are making a huge step toward making mathematics within easy reach of all students.
I don’t know enough to define “place value” definitively here, but I have learned to see the importance in the standards. Place value connects with “like terms” in all of math. In two-digit arithmetic, place value makes the difference between 20+5=25 OR 20+5=70, because we add like terms: tens to tens and ones to ones. Otherwise students memorize “line up on the left or line up on the right” and get those rules confused later. That’s why working with tens boxes or something similar is crucial in the lower grades. With decimal numbers, students often believe 0.25 > 0.8 because they don’t understand place value very well. With fractions, students confuse “multiply straight across” with “add straight across” if they do not understand we can only add like terms. In algebra, 2x + 3y = 5xy if a student doesn’t understand we can only add like terms. Pedagogically, I favor using interactive presentation software like Notebook. I type 536 on top of a white rectangle behind which I hide 500, 300 and 6. I drag the hidden numbers out from behind the white rectangle so students can all clearly see the value of each digit. I have to do this for some kids who make it all the way to high school, still fuzzy about place value.
I think you are correct that the standard is focused on the structure where a single digit is multiplied by a multiple of ten. As a high school teacher I use the K-8 standards to remediate, and many of my students are surprised when I point out (as they are reaching for a calculator) that 2 x 80 is simply doubling eight tens. Worse, most of them think they need to put a one under the 2 to multiply 2 x 1/8. So it makes sense to have a time of focus on the particular structure where a single digit comes first. However, that doesn’t negate 3.OA.B.5 where students apply the commutative property.
Ha! I just graded a unit test and I saw a lot of: sq root 11 < 3 2/3 > 3.51. Because I never know exactly what vocabulary my students will encounter, I read 0 < x < 10 as “x is between” and sometimes say, “Zero is less than x which is less than 10, but isn’t that a mouthful?” Between notation is succinct for describing domains and ranges. Is there a way to connect the terminology “order symbols” to a solution set like {x | 0 < x < 10}?
You raise a very important issue. We definitely need to get rid of mouths because x<6 is interpreted as x eats 6. Wide-part-as-large, in my opinion, seems mathematically sound and works very well with my Algebra students, particularly when analyzing word problems: “She needs to make at least $50, so what needs to be large?” After seeing <> for many years, many of my 14-year-olds still get them mixed up. I write <ess frequently, but still… So after my initial surprise to see wide-is-large in the progressions, I felt affirmed in my practice. It makes sense to me to have mental images like 8<10, 0.05 < 0.5, 1/2 > 1/3, driving the recall. We all have “hooks” that are necessary for retention. So if they can recall an example like 8<10, then they can recall “less than” if they pause a second to think 😉
Oh, of course. I was looking at pages 78-85 from EngageNY and the treatment of the commutative and associative properties. It looked like overkill to me but maybe I just don’t have the right thinking on that.
How important is the word partitioning over dividing in 1.G.3 “Partition circles and rectangles into two or four equal shares…” I see the difference but noticed that a 2nd grade textbook I am evaluating consistently uses “dividing.”
Thank you for directing me to EngageNY. I also realized MA has some good stuff http://www.doe.mass.edu/candi/model/download_form.aspx
I have some questions after looking over EngageNY for Algebra I at:
http://www.engageny.org/sites/default/files/resource/attachments/algebra-i-m1-teacher-materials.pdf
but I’m not sure this the best place to open such a discussion. What do you think, Dr. McCallum?There are lots of great task examples here: http://www.illustrativemathematics.org/HSF-BF.A.1
Explanations for F.BF standards begin on page 11 of the High School Functions progressions:
http://commoncoretools.me/wp-content/uploads/2012/12/ccss_progression_functions_2012_12_04.pdfOh, I should have also mentioned Illustrative Mathematics has a couple other great tasks you might like to see. Illustrative Mathematics is linked to this site, but here’s the particular tasks:
I’ve been watching teachingchannel.org (and Phil Daro videos on vimeo.com) these days and I envision this excellent number story playing out like this: Kids work in pairs or small groups to figure out the problem. Some kids would draw pictures of 7 bags, each with 3 apples and figure it out that way. More advanced students would, as you say, write two equations. Even more advanced students write one equation. At each level, the teacher serves as coach, drawing the best out of the students as they think and talk with each other.
Next, a whole class discussion begins with the picture-students who explain their thinking. The discussion then morphs to students who used two equations and capped with the single equation. Lastly, the students are given time to think about how subtracting before multiplying would mess up the answer.
3.0A.8 says, “Represent these problems using equations with a letter standing for the unknown quantity.” In your task, the variable is already isolated. At some point we need the students to be able to solve this: Julie has some bags of apples, each with three apples. If she has 21 apples total, how many bags does she have? 21 = 3b. The next level of complexity would be something like, “Julie empties the bags onto the table and Amanda takes one of them away. Julie sees there are 20 apples left. How many bags did she have?” 20 = 3b -1
This might seem “over the top” with a class of third graders, but working in groups to solve problems like this tends to pull their thinking skills impressively upward! What we don’t want is for students to see there are two numbers in the problem and automatically use their favorite operation to “get the answer.”I think the answer you are looking for is on Page 8 of the progression, “Ratios and Proportional Relationships,”
Where it says, “Recognizing proportional relationships. Students examine situations carefully to determine if they describe a proportional relationship…Students recognize that graphs that are not lines through the origin and tables in which there is not a constant ratio in the entries do not represent proportional relationships.”I totally get this. One would have to be some sort of omniscient prophet to anticipate exactly how the standards will play out in all respects: vertically, horizontally with science classes, etc., in a society with evolving technology beyond “Google Glass.” One of the reasons I have grown to trust your reflections is your tenacious humility. In a Country where democracy has been quickly fading, you have refused to become a dictator, and I really appreciate that.
I lead my students in a discussion of factoring sums and differences of cubes by comparing similarities with factored differences of squares. For example, it is easy to see why $2x-3y$ might be a factor of $8x^3 – 27y^3$. Rather than memorize the pattern for the other factor, my students divide out $2x-3y$, generating the other factor.
[2013-12-06: Typo corrected]
