Algebraic Reasoning For Teachers- Algebraic Reasoning

Today's resources on drive.

Growth mindset

What core belief about their learning are you conveying to students?

    1. Hard work changes your intelligence.

    2. Believing that will increase achievement!

    3. Mistakes are how students learn. They LITERALLY cause your brain to spark. That makes it grow! When we help our students have a growth mindset, they are more likely to fix their errors because of their increased attention to mistakes. My Favorite No.

    4. Offer strategies for solutions, not answers. Build off what they get right or wrong.

    5. Be authentically positive about future successes.


We are born to do complicated equations in our mind! Instead of memorizing math, why don't we build on what babies have?!

What IS algebra? Yes, it is patterns! No, it is not ONLY patterns!

Algebra is based on relationships. That is handy, because people make meaning through relations.

Algebra uses symbols to represent both numbers and relationships (=).

Structures need to be generalized and then can be represented with symbols (+1 -1).

Computation needs to move from counting to counting on to known facts and LIVE in derived facts- compose, decompose, recompose!! Develop flexibility with numbers. Move from counting down and "taking away" to composing, decomposing, recomposing! This is huge. Help students learn the facts through fluency with number. Jo Boaler's Youcubed.

Algebraic reasoning? Huh?

Representing, generalizing and formalizing (putting into numbers/equations).

Justifying solutions

Even in early years, students can understand the problem, make a plan, do it then check it (aka: look back!).

But how do we do that with a 6 year old?! Just like you do with a 32 year old!

    • realistic real-life applications and math tasks

    • everything start with hands-on conceptual learning and that is allowed until students decide to stop using it.

    • represent the hands-on and make sense of learning- make explicit connections!

    • THEN move into abstract because the symbols have no meaning on their own. We must connect them to real things.

What strategies work?

    • teach in sequence, starting with 3 or less items

      • count

      • subitize

      • count 1 to 1

      • count to compare

      • what number is next, after

      • comparative sizes

      • +1

    • teach and model that here is not just one way to get an answer!

    • place value fluency with representations

    • number lines - mental and physical, preferably open

    • number blocks/bar model- real, represented, abstract and digital.

    • visualizing- draw a picture

    • spiral/cumulative review

    • Rekenreks

    • terminology- compose and decompose

    • equality is a basic part of algebraic reasoning. Use balance. Equality is a relationship.

    • Number bonds help with equality and understanding part/whole.

    • model relationships (jack-o-lantern)

    • daily formative assessment to drive instruction

    • pre and post assessment to remediate and enrich

    • offer 4 chunks at a time

    • repetition over time to get information into long term memory

    • mix focused work on diffuse chance to learn

    • sleep

    • movement- jump the number line!

What about programs?

Your most important job is questioning!


How do you know?

Can you explain?

What is another way?

How can you check it?

Some research/articles

Singapore Math report by NC visitors

Mind For Numbers


Teaching Math to Young Children

5 Recommendations

Practice Guide

Book List