This morning Gail is talking about thinking about fractions as unit fractions. The information here is from the work done by H. Wu out of Berkley. Fraction instruction starts on a number line – not with a pie diagram or other physical model. She is showing us an Nspire document that is counting by 1 fourths. The idea is multiples of unit fractions. So in the app when you are at 2 on the number line, the app shows “10/5″ as well as “ten 1/5″. We have ten copies of 1/5.
Equality is defined as having the same place on the number line. When we start with pie diagrams we are creating misconceptions. For one – we wind up not being able to be greater than one. Also we need a different circle for each fraction type.
Gail takes a minute to tell us that words in the common core matter much more than they ever did before. When the core talks about “unit” it really means it.
Next we talk about unit squares. How could we prove 15/6 = 5/2. A number line model or a unit square model will work. Now think about a unit square with 3/4 of it filled in. That is your unit. Then a half filled in unit square would be 2/3. A unit square split into 8 rectangles with 3 filled in has a value of 1/2. One and one half filled in unit squares has a value of 2. You can see these examples here.
On to addition: – we think about additional as copies of unit fractions. Multiplication is sort of a traditional area model.
Lots of states have adopted the Common Core Standards – but that doesn’t mean much. At this point all they have agreed on are a list of standards. How that plays out is going to be a states rights issue.
One thing we don’t do a good job of is tracking student growth. One of the critical promises of the assessment systems being considered is that we will have a record of how students are progressing. This is based on the assumption that all testing will be computer based so the data is easily accumulated and archived.
(The idea of college and career readiness is an awkward thing now. There is a preponderance of soft skills that companies want (team work, communication, etc). What they can’t come up with are the hard skills that kids need coming out of high school.)
One issue coming out of the testing that is meant to tell how students are doing, is that they are talked about being used to measure teacher effectiveness. The danger is in using a test that is made for one thing (assessing students) to try and draw conclusions about something else (teacher effectiveness).
As Dr. Kepner shows us some test items, what we need to realize is that there are not any items ready for release – all that we can see are current items that may be something like what we might maybe see. As a side note - in order to take these computerized tests, students will need to have practiced with these mathematical tools (sketchpad, calculator type apps) well prior to the actual test dates. The consortium(s) will be preparing reports for each state (which will be handed to the governor) outlining the cost that each state will have to pony up to get ready. Are the governors ready for their 5 billion dollar invoices?
Each consortium is also building intermediate assessments (benchmark assessments) that can be used any time during the year to measure where students are at. These optional intermediate assessments will be built out of the same (similar) items that appear on the official end of the year test. So the goal should be to take advantage of whatever is done “to us” in order to use the tools we are given to benefit the students and the instruction that we create. Dr. Kepner recommends that as much as we can, we are raising our hands and volunteering to look at items and use our kids to pilot tests.
www.smarterbalanced.org is a place that we should be looking at periodically for information on what is going on. Their document on Content Specifications is an important one because it was given to each state that joined and then agreed on in order to move forward. There are 4 claims made in this document: (Sample problems themselves can be found on the Smarter Balanced web site.)
Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency. These are the types of test items that we are most familiar with. The cluster heading in the standard is what this is aimed at.
Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. These are contextual problems which may lead to claim 1 problems. This is the math wars – we will see nice neat clear math problems but also contextualized.
Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.
Students can analyze complex, real world scenarios and can construct and use mathematical models to interpret and solve problems. These are long maybe 45 minute type items.
The current state of Smarter Balanced is looking like 40% traditional multiple choice (claim 1) and then 20% each of the other claims.
Wow. That is quite a title. He starts with the problem: If you continue to do what you have always done, you will continue to get what you have always gotten.
Steve believes, as I do, that Algebra II is an impossible course to teach and it should be. What the hell is it good for? (Please don’t tell me that they need it for Calculus.) The point is that we have been party to the process of sorting kids out for a world that no longer exists.
Steve then showed us some data and asked what we saw? Two lists: {40, 10, 20} , {4,2,4}. His point is that we can get a lot out of just discussing. For example by asking what is next, we can start hearing what others are thinking and critique. We are making conjectures and engaging in deductive reasoning and inferential reasoning. Steve completes the table and asks us where we are. (What?) Then he adds a first column which shows things like Roller Coster, Ferris Wheel, Fun House. Now that we know where we are we try to give meaning to the original two columns. They could mean anything that we can justify. Like the number of people on the ride versus the number of riders that puke on average per ride. Imagine the discussion. The thing is, if you tell them from the outset what the columns are, it takes so much away from the activity.
Steve started with this (we are now 20 minutes in) to illustrate that it isn’t the standards that matter – it is the translation of those standards, it is processes that are as important a content, and we need to give kids a reason to care. The days of being a Middle school cowboy (screw them – we’ll do what we want) are over – we have to work together to get this done.
Steve has 5 perspectives that we need to consider:
It is a great time to be convening as teachers of mathematics. Steve tells a great story about THe Bill Gates Foundation and the research that SRI and ARI did which showed he was wrong. Check this out. Check out Harvard student questionnaire for 8th graders which turns out to be a great predictor for future success.
Kids have nothing going for them if we don’t do what we need to do as mathematics educators. Why are state test scores so high in Massachusetts? (Yes, high standards, but what more so?
We need to teach in a way different that we have been taught.
We have to collaborate if the Common Core is going to work. (Steve admits he drinks the CCSS Kool-Aid – or is at least very optimistic) See page 5 of CCSSM for a great quote on how the standards are not new names for old ways of doing things. Steve told the story of how he and others drew a line in the sand and went to Achieve threatening to use his big mouth to show how the “New” standards were nothing more that a rehash of old crap. The final draft of the standards shows fairness, clarity and spiraling.
Learnzillion – check it out as an alternative to Kahn.
Steve talked about the amazing assessment items that are being produced. Also about the adaptive part of it. http://hdd.apec.org Example 4 is what will be on an 11th grade smarter balance test. 8th grade algebra is going away. You cannot condense 6,7,8 into two years and send one-thinrd of the kids to Algebra 1 in 8th grade. This is the 1st time in our lifetime that we have the possibility of a coherent, integrated, balanced system that we can run with.
Karim is rewriting curriculum (aligned with the common core) around real-world topics at Mathalicious. He is going to start today by showing that a skills first approach is not as effective as a real world exploration approach.
So, skills first… traditionally we might take a fraction like six-tenths and have students divide 6 by 10 and then move the decimal place over twice. Imagine explaining baseball to a foreigner that has never seen a game. (I have, his name was Alexi- he was from Russia and we went to a Brewer game together.) Isn’t the fraction example a lot like the baseball example?
So, how do we contextualize this? Karim looked at a video clip of two guys shooting free throws. (btw, this is from his activity “From the Line”) The question is who did better? Skip forward to looking at a table and wondering who might score more when each takes 100 shots. All of this makes sense. We aren’t making any moves that don’t make intuitive sense to the students. We want to minimize those times when students are wondering why we are doing a certain thing. (Like moving the decimal place over – or even multiplying by 100).
The next activity is Wheel of Fortune – the next question is “How can we determine if Wheel of Fortune is rigged?”
We talked a little bit about what percentage of the time does bankrupt come up that would make us suspicious. (We should expect 11% by looking at the wheel) We watched an episode next. The percentage of Bankrupts was 15% of the spins. Would a 6th grader be suspicious of this? Probably – and this leads into an interesting discussion.
Now we are talking 7th grade solving proportions. Obviously cross-multiplying (puke) is the preferred traditional method. To contextualize this – if Walmart needs to produce a banner 24 inches wide that will show an iPod on it – how tall does it have to be? We look at an animation of growing the iPod. One way by adding 2.2 inches and one way by multiplying the width by 1.96. After seeing that multiplication is the way to go we now have a method for solving proportions: What do we multiply one dimension by to get the other?
Now on to Big Foot Conspiracy. The question: What if Nike charged for shoes based on the size (essentially weight)? So we have kind of an incomplete table with sizes 9, 9.5, 10, 11, 11.5 and associated weights. The cost of the shoe is $160 per pair. So what if we use size 10 as a base and then adjust the other models accordingly?
Now a quick look at 8th grade. Calculate the equation of the line going through (16,499) (32,599). Traditionally we look rise over run, divide. Plug slope in then tell the kids to pick either point (what?), plug it in and solve. Ouch.
The activity here is iCost. The first question is what is the y per x? How much does Apple charge for the iPad given the cost of two different models? The idea is that we can have a really smooth conversation about cost per GB and have it lead intuitively into the equation of a line. Then look at a 64 GB and you see that the pricing is actually not linear. Karim makes the claim that this activity covers 28% of the 8th grade common core (Exclam!)
Next for high school we are looking at exponential growth. The activity is called XBOX Xponential. The discussion we eventually have is this one: Is there a point when video games are so realistic that we will eventually plug in like the Matrix? Do we want that? The point is that the math is what gets us to this point.
Next is the 51 foot ladder. This looks at right triangle trig. Turns out you can climb a 50 foot wall with a 51 foot ladder. What is the shortest ladder you would be willing to climb?
Gail started by playing a little game with us. On her TI-Nspire software she drew a triangle and measured one of the sides . Then she locked the “area” and then moved one of the points. She asked “What do you think is going to happen?” Ultimately: What does a trace of all the places you can move one of the points look like? (Try it!)
Next game is what happens if we instead lock the perimeter and trace? What do you think will happen? (Try it – your hint is that it has something to do with the sum of two of the sides of the triangle.)
Gail’s point is that the only reason we ask questions is to find out what students are thinking. To probe or to push their thinking. If we aren’t doing that with our questioning, then our questions aren’t worth it. Tasks that push and probe should be our focus. A possible task – Which glider goes farther: one that starts at 25 meters high and goes 185 meters horizontally, or one that starts at 25 meters high and goes 155 meters horizontally. (One audience member asked which distance) This is an example of different context for the same concept. In terms of the Common Core this type of task hits a lot a the Math Practices.
Next: Draw triangle ABC. Construct the perpendicular bisector of sides AB and BC. Make a conjecture about the perpendicular bisector of AC. Move point A. What do you observe? Cool – but how can we make this problem more interesting? October recipes from No More Cookbook Math (Harper and Edwards 2011) has a rubric for thinking about Teacher Centered vs Student Centered.
Stein 2000 talk about choosing solutions, sequencing solutions, managing solution strategies, ask questions, consolidate the math using student work. Gail would call this “purposeful walking”. The work on formative assessment says that hands should never be up unless the student is asking you a question. You do the planning and choosing.
Say you poll the class on a particular problem (maybe TI Navigator style) what do you do with disparate results? You could have a representative from each solution justify their responses…or what about just asking the whole class what the people who answered A. were thinking, and what were the people thinking that chose whatever. Or what about pairing kids up with the goal of convincing each other that they are right. Then re-poll the class. Now you can deal with the outlying kids one on one.
Next we looked at the Nspire activity What_is_a_Solution. A simple and brilliant activity.
Lastly we did a sequence activity with the Fibonacci sequence and I’ll try and recreate it here if I can find time. In the mean time – here are Gail’s slides from the presentation:
