Category Archives: Middle School Math

To de-track or not to de-track; that is NOT the question*

Although the debate over ability tracking (placing students in classes by mathematical level) in the mathematics classroom has been alive and well for some time, the heat has exponentially increased recently after California looks to adopt a new framework where students of all levels of mathematics will be mixed in together. This framework aims to address academic, social, and racial disparities in the mathematics classroom. The new framework has been heavily influenced and co-authored by Professor of Education and fellow Brit: Dr. Jo Boaler (who I had the pleasure of interviewing for the MathEd Out Podcast) of Stanford University. She argues that “tracking looks horrible when you look at the racial inequities, and we have to ask, ‘What do we want for this country? Do we want a country that has these racial divides in achievement?’ If we don’t, we need to work on a different model,”

Reaction has been emotional, political, swift, and strong, especially from those who associate the initiative with a leftist agenda. I have often heard the argument that this will water down the curriculum and hold back ‘more advanced students’ and will not help those in need of support. There are even studies that seem to back up both sides of the argument. A 2014 Fordham Institute paper argues that tracking helps disadvantaged students be more successful when it comes to preparing for AP math classes. On the other hand, Boaler has published multiple peer-reviewed papers on the negative consequences of tracking both academically and culturally.

However, de-tracking vs. tracking is a simplistic question which ignores a large and crucial part of the conversation that needs to be simultaneously considered if not reflected upon first: What is the nature of the mathematics that we are teaching and learning in our schools? One question simply cannot be considered without the other. When taken at face value, with the narrow and linear curriculum that students go through in many schools, it makes complete sense to track in ability groupings. Students need to be able to learn how to find the square root of a number before they can solve a quadratic equation. They need to know how to simplify fractions before they study before they can consider rates, ratio, and proportion. The way the curriculum is structured is why we track in mathematics and not in History or Science, for example, which are more multi-dimensional and complex. If a major aim of mathematics is preparing students for AP level math, then I can see the benefit of working with students at different levels, at least for more advanced students. My question is: Is that it?! Is that why we teach math? To prepare students to succeed in AP? If that is it, this is nothing short of a travesty.

Beyond mere preparation for AP level mathematics, there is a big problem with this model. For students who are struggling, placing them in a lower ability group can cement the fallacy that they are not capable of accessing even basic mathematical principles and skills. And, it can be challenging to move upward once students are set on a particular path. As the new framework puts it: “the subject and community of mathematics has a history of exclusion and filtering, rather than inclusion and welcoming. This is a much bigger issue to consider when compared with whether advanced students can understands imaginary numbers or not and this conversation should take up proportionally more time in our curriculum reviews.

Tracking even advanced students with current curricula is not without its drawbacks. I have had multiple parents of 6th-grade students in my conferences who are deeply concerned that their daughter/son is on track to get into good colleges or into a certain profession. Is this really what we want our 11-year-olds to be worrying about?

I enjoyed math at school, working through a linear narrow curriculum that I found straightforward, and I was able to demonstrate mastery in assessments. I also went to a school where there was an admissions exam so I should have been surrounded by similar minded peers. However, I was still in the minority in terms of those that enjoyed mathematics. Today, I tell friends that I teach math and I often see the shudder as they reply “oh I was never great at math”. I believe this to be a cultural legacy after decades of students working their way through a linear and monochromatic curriculum. This also makes me sad.

I only began to love math after I started to learn to teach and was blessed to be inspired to teach and learn great math by a phenomenal professor. He showed us that mathematics was so much richer and complex, so much more useful, so much more interesting than was often found in textbooks. After for teaching for over a decade, I have seen how such ‘Low-Floor-High-Ceiling’ activities can provide noticeably more engagement and a deeper level of learning. If we don’t start with the consideration of the nature of the mathematics we are teaching in our classrooms, there is no point in asking whether we should track or de-track. Thankfully, organizations such as YouCubed, Desmos, and Nrich are leading the charge to bring in a new kind of mathematics learning where de-tracking makes a lot more sense.

It is an uphill battle. Progressives are up against divisive politics and decades of established narrow curricula. This is much is for sure, however: Rather than simply just preparing students for AP level mathematics we should be asking:

  • Why do we teach mathematics?
  • How can we design a curriculum and learning space that is rich, engaging, and useful at every level?
  • How can we help students see how different parts of the curriculum are deeply connected?
  • How can we help students of every ability, culture and socio-economic background get excited about learning mathematics and their own capabilities to solve problems?

Math is so much richer, so much more useful and crucial than many curricula portray. In the real world, there is no neat and narrow linear progression of skills, there is are complex problems to be solved in whatever scrappy way possible. Let us at a minimum give every student, irrespective of academic prowess or culture, the chance to access the confidence to say ‘I can do this’.

*Opinions are my own and not associated with any school or organization with which I am associated

What makes a growth mindset math game? Hint: Avoid worksheets in disguise!

If you were to search for the term ‘math games’ on google, you would get instant access to many sites where you get the chance to practice skills in the guise of a ‘fun game’. For example, in the game Candy Stacker, you get to practice pretty much any skill in any grade and it stacks cake on top of an animal until it reaches a candy. As with many of these games, it is timed, and you ‘fail’ if you get on wrong. These types of games reinforce the fixed mindset that is so often observed in mathematics classrooms.

There are better ways of building fluency and understanding of number. Number talks and Formulator Tarsia are just two great examples. But, are there activities out there that both have that gaming element, and help build a deep understanding of number while promoting a growth mindset, that depth is more important than speed? Thankfully, yes, and here is a list of growth mindset games that I love my students to play, either in a dedicated lesson or in those moments where you have 10 minutes spare. None of these will have timers or ‘fail’ notices. (Note: There are many great card, board, and paper games out there, but I am going to focus on online interactive activities for this post). Click on the images to go to the game itself.

Factors and Multiples (nrich)

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This is probably my favorite of all, building sense of multiplication tables, but students also quickly come to realize that prime numbers are key in this game.

1 Player version: What is the longest chain you can make, clicking factors and multiples of the previous number (see the example in the picture). What is the longest chain that anyone can make? Are there numbers to be avoided at the beginning? Are there good numbers to start with at the beginning?

2 Player version: The winner is the person that can force the other player not to be able to build on the chain, cutting off all possible factors and multiples. Again, are there good numbers to start with? What are the key numbers to minimize the chances of losing the game?

Connect 4 Factors (Transum)

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Another fantastic game to build a sense of numbers, factors and multiples. No timer necessary!

One Player Game: Fill the game board with the counters from both boxes. Avoid lining up four numbers with a common factor (other than one).

Two Player Game: Each player has a box of counters to choose from. Take it in turns to drop a counter into the game board. The winner is the first to line up four numbers with a common factor (other than one).

Broken Calculator (author unknown)

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There are probably other versions out there but this one is the best I have found, even if is relatively basic. The idea is simple: Can you get the target amount using the keys on the calculator that are working. The levels get increasingly difficult, and it is a great way of building a sense of operations.

Double Take (Transum)

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Similar to the popular game on mobile devices, this game builds a sense of base-2 numbers and exponential growth. This one is a favorite with the students. Try to encourage your students to think of and discuss strategies to get further than you would by just trying different ways.

Got it! (nrich)

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A simple game with less than simple strategies. Once again, try to steer students to thinking strategically. Is there a better addition to start with? Is it better to go first or second? If you want a more challenging version of this game where you can’t choose the option that your opponent last chose, you could try transum’s 23 or bust.

Square it (nrich)

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Not a number game, but a great strategy game nonetheless. Students can play either another person or the computer. They must claim all 4 corners of one square to win. Can blue always win? Is there a way to force this every time? What is happening in the middle of the game? What does the end of the game look like?

I hope you enjoyed reading about these growth mindset math games. What are your favorites? Comment in the section below.

 

Ten Webites I Check Before Starting Every Unit

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There are many excellent resource sites out there with inspirational teachers making great resources. However, when it comes down to it, there are only a handful of websites I check on a very regular basis. To make it to this list, I am generally looking for tasks that:

  • Are ‘low floor/high ceiling‘ giving access and challenging everyone at their own level
  • Have multiple approaches, giving students room for creativity
  • Are challenging and hence interesting
  • Will give me permission to teach skills
  • Make skill practice, more interesting
  • Have a story for students to follow

There are many activities out there that fit these descriptions, but there is only so much time in the week. During breaks I have more time to explore but these are the sites that I can check quickly with high yield for your classroom (in no particular order):

1. Mathematics Assessment Project – A website produced from a partnership with the Shell Centre (University of Nottingham, UK) and the University of California at Berkley. Here you will find tasks that will help the common core standards come alive in your classroom. Full of rich activities that encourage discussion and investigation.

2. Standards Unit – This has been around for a while but is still one of the first publications I look at and is close to my picture of what mathematics education should like. A rich, dense set of activities that will give you great ways to work on various skills and topics.

3. YouCubed – A relatively new site that has come out of Prof. Jo Boaler’s (Stanford) efforts to encourage the ‘growth mindset’ in the classroom. I am interested in anything that comes from the idea that anyone can be good at math.

4. Open Middle – I came across this site relatively recently and am sad that I didn’t find out about it sooner. it’s tag-line is: ‘Challenging Math Problems Worth Solving’ and tips the proverbial hat to the school of thought that says that you don’t need to have tenuous links to real world problems in order to get buy-in from students. It appeals to the problem solver in all of us.

5. Emergent Math – With its routes in Problem Based Learning (PBL), emergent math is useful not only for those who want to tear up the textbook and start again with an integrated curriculum, but also for people who just want great projects that they can slot into their established curriculum. Lots of links to sites not mentioned here so worth a look.

6. Mr. Barton Maths – Lots of great resources and activities that will make your classroom a more interesting place. Enough said.

7. Tarsia – Sometimes students just need to practice. There I just said it. But using Tarsia you can avoid ‘death by worksheet’ and get students to practice without really realizing it. It also great for discussion and you can tell very quickly if students have answered everything correctly or not. For more, see Why I Love Tarsia.

8. NRich – Great site for ‘low floor/high ceiling’ problems that will challenge anyone in your classroom. Students may like to explore this outside the classroom, too. Sortable by topic and I believe they are coming out with a common core curriculum map, quite soon.

9. Dan Meyer 3 Acts Spreadsheet – Act 1 –  You show a video or picture prompting discussion, prediction and estimation as well as the all important step of coming up with the variables that are to be investigated. Act 2 – Students get the information they need to solve the problem. Act 3 – Once students have solved the problem in various ways and presented you show them the solution. There is a lot more to it than this and to pull these lessons off well is a true art form. But the only way to get better at these is to try them. So try them! More info here.

10. Mathalicious – Math lessons based on the real world problems. $185 for 12 months subscription (I think currently they are also doing pay-what you can) but well worth it. Excellent for creating the need for the math you teach.

It takes me around 90 minutes to trawl through these websites at the beginning of a unit but is well worth it when it comes to lesson planning and I know half of the activities I am going to do already. As with any of these activities they will need to be (and should be) adapted for your classes and situation but they provide an excellent starting point from which to plan.

What websites are a must-check when you are planning a unit? Leave a comment, below.

Walk the Line – Adding and Subtracting with Negative Numbers

I wanted an activity that would give students a deep sense of adding and subtraction involving negative numbers. Both. At the same time. I figured the best way would be for them to actually walk through what happens to a number when it goes through these operations; so began ‘walk the line’.

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The idea is simple, have students walk through various sums increasing from adding two positive numbers, going through to subtracting two negative numbers, reviewing each time.

They will first need to stand in a line, then using sidewalk chalk, draw a zero on the floor and draw out a number line to the right and left of where they are in the positive and negative direction.

Here is the general dialogue that happens:

  • I am going to call out a sum and I would like to see how quickly you can get to the result. You start with the first number I call out and are permitted to move before I have finished calling out the sum.
  • 3 + 2 (call out slowly to give students time to move)
    • Review Questions
      • Now, what did you do when I first said 3? – Run to that number
      • What did you do when I said “add”? – Got ready to run away from zero
      • What did you do when I said 2? – Ran two spaces away from zero
  • Next:  5 – 8
      • Review Questions
        • Now, what did you do when I first said 5? – Ran to 5
        • What did you do when I said “subtract”? – Turned to face the zero (Why? etc)
        • What did you do when I said 8? – Ran 8 spaces and ended up at -3
  • What about  5 – ( -8)    (call out slowly)
    • Review: What was different when I said subtract -8 instead of 8? – Had to go the other way (Why? etc.)
  • At this point I bring student thinking together and explicitly clarify the rules:
    • Whatever number I say first is where you start
    • If I then say “add” you face this way (pointing in the positive direction) and if I say “subtract” you face the other way (pointing in the negative direction)
    • If my second number is positive you walk forward by that amount, if my second number is negative you walk backward by that amount (for example: 4 would be four steps forward, -4 would be 4 steps backward)
    • Let’s try this out
  • 3 – 8 (call out slowly to give students time to think and move)
  • -4 + 10
  • 2 + (-5)
  • -9 + 12
  • 2 – (-5)
  • -4 + (-7)
  • -6 – (-10)
  • etc
  • Then return to classroom and do the same sort of thing but students write (just) their answers on their mini whiteboards. We review each time going through the 3 step process.

This activity really worked and students were doing this all in their head by the end of 50 minutes which is what I was aiming for. There was no separation of addition and subtraction or positive and negative numbers. They were just different points on the number line and different ways to move.

How do you introduce adding/subtracting negative numbers?